Research ArticleSpatial and Temporal Variation of Annual Precipitation in aRiver of the Loess Plateau in China

Cui Shen and Huang Qiang

State Key Laboratory of Eco-Hydraulic Engineering in Shaanxi School of Water Resources and HydropowerXirsquoan University of Technology Xirsquoan 710048 China

Correspondence should be addressed to Huang Qiang wreshmailxauteducn

Received 9 October 2013 Accepted 20 December 2013 Published 19 February 2014

Academic Editor G H Huang

Copyright copy 2014 C Shen and H Qiang This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

Empirical orthogonal function (EOF) decomposition and geostatistical (semivariogram) analysis are used to analyze the spatialand temporal patterns of annual precipitation in theWeihe basin based on the data from 1960 to 2011 at 30 national meteorologicalstations within and surrounding the Weihe basin area of the Loess Plateau in China Then the Mann-Kendall method is used todiscriminate the variation points of precipitation series at each stationThe results show that during the study period there was anoverall reduction in annual precipitation across the basin and there were two spatial patterns increase in the northwest decrease insoutheast and increase in the west decrease in east Variation points in the annual precipitation series at eachmeteorological stationin the basin were detected and they occur earliest in the southeast and downstream then progressively later in the northwest andupstream areas These variations were most frequent during the 1970s and 1980s The findings have significant implications for thevariations research of runoff in the basin and formulate robust strategies to adapt to climate change and mitigate its effects

1 Introduction

Climate change is causing increasing concerns globally Pre-cipitation is a key climatic parameter that strongly affectsrunoff and diverse hydrological variablesThus knowledge ofprecipitation patterns is essential for characterizing past andpresent climate changes predicting future changes and eval-uating their effects Several studies have shown that annualprecipitation in China is declining [1 2] while the frequencyof extreme weather events is increasing [3] The changes inprecipitation are closely related to changes in runoff and thusare affecting ecosystems and restricting socioeconomic devel-opment [4] In the Weihe basin located in the eastern partof northwest China there are fragile ecosystems and frequentnatural disasters Thus analysis of the spatial and temporalvariation in precipitation in the basin is essential for under-standing changes in runoff and the associated socioeconomicand ecological impact in the region and for formulatingrobust cost-effective strategies to adapt to and mitigate theeffects of climate change

2 Study Area

Weihe River of the Loess Plateau in China originates in thenorth of Niaoshu mountain in Weiyuan county of Gansuprovince emptying into the Yellow River at Tongguan countyof Shaanxi province which has a drainage area of 135000 km2and flows 818 km eastwards through 84 counties and threeprovinces across the guanzhong basin [5] The Weihe basinspans longitude and latitude ranges of 1035∘ sim1105∘ E and5∘sim375∘N respectively and the two main tributaries are therivers Jinghe and Beiluohe [6] The southeast part of thebasin is located in the continental monsoon zone while thenorthwest part is in the transitional zone between arid andhumid regions controll by West Pacific subtropical high andMongolia High the basin is drought in spring rainy hot insummer but drought frequent dry cold in autumn andwinter[7]There are major annual variations in precipitation whichis concentrated in July to October when about 65 of annualprecipitation occurs [8] The average temperature is minus1∘ tominus3∘C in the coldest month (January) and 23∘sim26∘C in thewarmest month (July) [9]

Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2014 Article ID 827120 11 pageshttpdxdoiorg1011552014827120

2 Journal of Applied Mathematics

BeidaoHuaxian

Lintong

XianyangLinjiacun

Zhuangtou

Weijiabao

Zhangjiashan

Xiji

Wuqi

Baoji

Guyuan

Lintao

Foping

Wugong

Minxian Huashan

Changwu

Huanxian

Tianshui

Luochuan

Shangxian

Tongchuan

Pingliang

xifengzhen

Huajialing

0 68000 136000(KM)

Hydrologic stationMeteorological stationRiverJinghe basin

Beiluohe basinB-H section of Weihe basinWeihe basin above Beidao station

N

E

S

WYanrsquoan

Zhenrsquoan

Xirsquoan

Figure 1 Map of the Weihe basin showing the four subbasins and locations of hydrological and meteorological stations

Climate changes have had marked effects in the Weihebasin Thus we have examined changes in annual precipita-tion in the basin from 1960 to 2011 in order to elucidate theirspatial and temporal patterns thereby facilitating attempts toformulate effective mitigation strategies For the purposes ofthe study theWeihe basin is divided into four subbasins withdistinct hydrological characteristics and control stations thebasin above Beidao station Jinghe basin Beiluohe basin andthe Beidao to Huaxian section of the Weihe mainstream Amap showing the areas covered by these subbasins and loca-tions of hydrological andmeteorological stations is presentedin Figure 1

3 Data Sources and Methodology

31 Data Sources The annual precipitation data used inthis paper are recorded from 1960 to 2011 at 30 NationalStandard Stations in the Weihe basin and the surrounding

area obtained from the China Meteorological Data SharingService System (httpcdccmagovcnhomedo)

32 Methodology First we calculate means in annual precip-itation in the Weihe basin by statistical methods and deter-mine trends by regression analysis Then we use empiricalorthogonal function (EOF) decomposition and geostatisticalanalysis method to analyze spatial and temporal patternsassociated with precipitation Eigenvectors obtained fromEOF decomposition are then multiplied by trend vectors inthe matrix of the precipitation series to obtain eigenvectorfields thus describing the spatial trends in annual precipi-tation Finally we use the Mann-Kendall (M-K) method toidentify variation points in the precipitation series for eachstation thereby comprehensively describing the temporaland spatial patterns of annual precipitation in each of thesubbasins

Journal of Applied Mathematics 3

321 Regression Analysis Regression analysis is applied toanalyze the correlationship of two variables which deter-mines relative direction and intensity In this study regressionanalysis is used to get the linear regression of precipitationand time and the equation is as follows

119875 = 119886119879 + 119887 (1)

where 119875 is the precipitation 119879 is the time and 119886 and 119887

are parameters If the unit of 119879 is the year usually 119886 is thechanging trend of annual precipitation

322 Empirical Orthogonal Function Decomposition EOFdecomposition involves decomposition of a matrix X of datadescribing the distribution of a variable in both space andtime into a spatial function V and time function T as shownin

X = VT

X =

[

[

[

[

[

[

[

[

119909

11sdot sdot sdot 119909

1119895sdot sdot sdot 119909

1119899

119909

1198941sdot sdot sdot 119909

119894119895sdot sdot sdot 119909

119894119899

119909

1198981sdot sdot sdot 119909

119898119895sdot sdot sdot 119909

119898119899

]

]

]

]

]

]

]

]

(2)

where119898 and 119899 are points in space and time respectively and119909

119894119895is the value of the variable at point 119894 and time 119895 (here

annual precipitation at station 119894 in year 119895)If V and T are orthogonal

119898

sum

119894=1

V119894119896V119894119897= 1 when 119896 = 119897

119899

sum

119895=1

119905

119896119895119905

119897119895= 0 when 119896 = 119897

(3)

If X is an anomaly matrix (reference) multiplying theequation X = VT by XT gives

XXT= VTXT

= VTTTVT= VΛV (4)

where Λ is a diagonal matrix composed of eigenvalues 120582

119894of

matrix XXT and

TTT= Λ (5)

Due to the characteristics of the eigenvectors VTV isa unit matrix meeting the requirements of orthogonalityThus the spatial function matrix V can be calculated fromeigenvectors of the matrix XXT

Here we multiply the spatial functionmatrixV of annualprecipitation in the Weihe basin by the trend vector D =

1198891 1198892 119889119898 composed of trends in annual precipitationrecorded at each station (V

119889= DV) thus describing the

spatial trends of annual precipitation in the basinEOF decomposition was first established in 1902 [10]

It decomposes elements of a set of space-time data into

two functions one dependent only on time and anotherdependent only on space to analyze the spatial structure ofelement fields [11] Thus EOF decomposition is commonlyused to analyze spatial and temporal variations in meteoro-logical elements In contrast principal component analysis(PCA) is mainly used to find a few independent vectors ina dataset to reduce its dimensionality thereby simplifyingdata handling and interpretation However the operationalsteps are basically the same when the original variables areanomaly data or standardized anomaly data [12] Hence wecan use the PCA module of SPSS (statistical product andservice solution) software for EOF decomposition [13]

EOF decomposition and several variants thereof havebeen widely applied for analyzing spatial changes in mete-orological elements Notably EOF decomposition has beenused to analyze drought occurrence in both time and spacein Korea [14] It has also been applied to establish contribu-tions of annual evaporation annual precipitation and bothwarm- and cold-season precipitation to the annual runoffof rivers in an arid zone to obtain quantitative estimates ofthe effects of climatic factors on reconstructions of annualrunoff series in an ungauged area [15] A data-interpolatingempirical orthogonal function (DINEOF) has been used toanalyze distributions of sea surface temperatures in a studydemonstrating that DINEOF decomposition can be highlyeffective evenwhen very high proportions of data aremissing[16] In addition rotated empirical orthogonal function(REOF) decomposition has been applied to analyze space-time patterns of groundwater fluctuations in the ChoshuiRiver alluvial fan Taiwan based on monthly observations ofpiezometric heads from66wells during the period 1997ndash2002[17]

323 Mann-Kendall Variation Analysis The Mann-Kendalltest has been widely used for identifying trends in time-series data because it is nonparametric and thus suitable forhandling data that are not normally distributed and it isnot sensitive to interference from a few outliers The Mann-Kendall test can also be used for variation diagnosis and thecalculation method is as follows

For a time series 119883 = 119909

1 119909

2 119909

119899 the Mann-Kendall

test statistic 119863

120591is given by the following model

119863

120591=

120591

sum

119894=1

119877

119894(120591 = 2 3 119894) (6)

where

119877

119894=

+1 119909

119894gt 119909

119895

0 119909

119894le 119909

119895

(119895 = 1 2 119894) (7)

The sequential statistic UF120591(in which UF

1= 0) is

calculated by is

UF120591=

1003816

1003816

1003816

1003816

119863

120591minus 119864 (119863

120591)

1003816

1003816

1003816

1003816

radic119881 (119863

120591)

(8)

4 Journal of Applied Mathematics

520

500

540560

480

460

580

600

620

640 66

0

440

420

0

400

700

380

720

360

740

340

760780800

420

740

680

760

780

440

540

700

700

720

400

700

560

480

Xiji

Wuqi

SuideYanci

B

Jintai

Foping

Wugong

Guyuan

Lintao

Tongxin

Haiyuan

Yuzhong

HuashanMinxian

Changwu

Hengshan

Jingyuan

Tianshui

Luochuan

Huanxian

Zhongning

Shangzhou

Tongchuan

PingliangXifengzhen

Huajialing

662

612

2643

3725

4412

369228962036

3842

2311

7806

6888

8982

81855599568935

51735805

59515838

55461842

4136

53194332

4529

4609

48785286

18505

aoji

Yanrsquoan

Xirsquoan

Zhenrsquoan50 25 0 50 100 150 200

(km)

Basin boundaryWater systemPrecipitation isoline

Station

103∘09984000998400998400E 104

∘09984000998400998400E 105

∘09984000998400998400E 106

∘09984000998400998400E 107

∘09984000998400998400E 108

∘09984000998400998400E 109

∘09984000998400998400E 110

∘09984000998400998400E

103∘09984000998400998400E 104

∘09984000998400998400E 105

∘09984000998400998400E 106

∘09984000998400998400E 107

∘09984000998400998400E 108

∘09984000998400998400E 109

∘09984000998400998400E 110

∘09984000998400998400E

34∘09984000998400998400N

33∘09984000998400998400N

35∘09984000998400998400N

36∘09984000998400998400N

37∘09984000998400998400N

38∘09984000998400998400N

39∘09984000998400998400N

34∘09984000998400998400N

33∘09984000998400998400N

35∘09984000998400998400N

36∘09984000998400998400N

37∘09984000998400998400N

38∘09984000998400998400N

39∘09984000998400998400N

N

E

S

W

High 819619

Low 320163

Annual precipitation

Figure 2 Isohyets showing mean annual precipitation in the Weihe basin

where 119864(119863

120591) and 119881(119863

120591) are the mean and variance of the

series given respectively by

119864 (119863

120591) =

120591 (120591 minus 1)

4

119881 (119863

120591) =

120591 (120591 minus 1) (2120591 + 5)

72

(9)

Then repeat the calculation with the inversion series1198831015840 =119909

119899 119909

119899minus1 119909

1 and let UB

120591= minusUF

120591 (120591 = 119899 119899 minus 1 1)

Define the significant level 120572 and find out the critical valueplusmn119880

(1minus120572)2 If the curves of UF

120591and UB

120591intersect and the

intersections are between the critical line (in this study thesignificant level 120572 is 095 and the critical line is plusmn119880

0052=

plusmn196) the time of curve intersections is the starting time ofthe variation

Originally published in 1945 [18] it was initially onlyused to detect trends in sequences However subsequentmodifications have enabled its use for detecting variationpoints in various trends [19] and it has been applied inreversed sequence to detect variation points in trends inclimate data time series [20] Since then the method hasbeen widely used to diagnose variation points in hydrologicaland meteorological data sequences [21 22] Notably the M-K method has been used to determine whether there werepositive or negative trends in seven meteorological variables(and if so their significance) recorded at 12 weather stations in

Serbia during 1980ndash2010 [23] Amodified formof themethodhas also been used to test scaling effects [24] and four variantshave been applied to determine trends in selected streamflowstatistics from Indiana [25] In addition a procedure forcalculating exact distributions of the M-K trend test statisticfor persistent data with an arbitrary correlation structure hasbeen presented [26] and the test has been used to detectdirections and magnitudes of changes in monotonic trendsin annual and seasonal precipitation over time in annualprecipitation inMadhya Pradesh India [27] Here we appliedthe M-K test to detect trends variation points in the timeseries of the data obtained from the 30 stations in the Weihebasin (Figure 1)

4 Results

41 Spatial Patterns of Annual Precipitation in the WeiheBasin Means and trends in annual precipitation in theWeihebasin from 1960 to 2011 are shown in Figures 2 and 3 Themean annual precipitation at each station changed greatlySpatially it declined from the southeast to northwest Meanannual precipitation in the basin amounted to 580 mm witha maximum exceeding 800mm at Huashan and Zhenrsquoan areaand a minimum of 340mm in the northwest The annualprecipitation trend at each station was negative except atXirsquoan where it increased by 0123mma The mean trend

Journal of Applied Mathematics 5

Xiji

Wuqi

SuideYanci

Baoji

Jintai

Foping

Wugong

Guyuan

Lintao

Tongxin

Haiyuan

Yuzhong

HuashanMinxian

Changwu

Hengshan

Jingyuan

Tianshui

Luochuan

Huanxian

Zhongning

Shangzhou

Tongchuan

PingliangXifengzhen

Huajialing

minus20382

Xirsquoan

Zhenrsquoan

Yanrsquoan

50 25 0 50 100 150 200

(km)

N

E

S

W

Trend of annual precipitation High minus0884939

Low minus179305

103∘09984000998400998400E 104

∘09984000998400998400E 105

∘09984000998400998400E 106

∘09984000998400998400E 107

∘09984000998400998400E 108

∘09984000998400998400E 109

∘09984000998400998400E 110

∘09984000998400998400E

103∘09984000998400998400E 104

∘09984000998400998400E 105

∘09984000998400998400E 106

∘09984000998400998400E 107

∘09984000998400998400E 108

∘09984000998400998400E 109

∘09984000998400998400E 110

∘09984000998400998400E

34∘09984000998400998400N

33∘09984000998400998400N

35∘09984000998400998400N

36∘09984000998400998400N

37∘09984000998400998400N

38∘09984000998400998400N

39∘09984000998400998400N

34∘09984000998400998400N

33∘09984000998400998400N

35∘09984000998400998400N

36∘09984000998400998400N

37∘09984000998400998400N

38∘09984000998400998400N

39∘09984000998400998400N

01229

minus 0 785

minus 1366

minus 1 076minus 04934

minus 0 9699

minus 09262

minus 08778

minus 03846

minus 08691

minus 086 72minus 09866

minus 01212

minus 02984

minus 1821

minus 10304

minus 12346

0036300

minus 3 8975minus 12491

minus 20241

minus 14991

minus 21349

minus 21367minus 19091

minus 20868

minus 14559

minus 26897minus 26428

Basin boundaryWater systemStation

Figure 3 Annual precipitation trends in the Weihe basin

Table 1 Eigenvalues and their contributions to the variance of annual precipitation in the Weihe basin obtained from EOF decomposition

Component

Initial eigenvalues Extraction sums of squared loadings

Eigenvalues120582

119894

Proportion ofvariation ()

Cumulativeproportion ofvariation ()

Eigenvalues120582

119894

Proportion ofvariation ()

Cumulativeproportion ofvariation ()

1 13648 45494 45494 13648 45494 454942 5363 17875 63369 5363 17875 633693 1997 6658 70027 1997 6658 700274 1404 4681 747085 1196 3985 786936 0912 3039 817327 0685 2285 840178 0634 2113 861309 0506 1686 8781610 0422 1408 89224Extraction method principal component analysis

across the basin was minus1662mma clearly showing thatannual precipitation declined and that this trend was mostpronounced in upstream areas

42 EOF Decomposition of Annual Precipitation in the WeiheBasin The precipitation data analyzed herein originatedfrom the Weihe basin and the surrounding 30 nationalstandard stations during the period 1960ndash2011 From this

data we derived the normalized anomaly matrix X forannual precipitation Empirical orthogonal function (run inSPSS) was used to calculate the eigenvalues of the covariancematrix of X and the amount of variation that each principalcomponent explained as shown in Table 1

Eigenvalues for the covariance matrix of the Weihebasin annual precipitation matrix and their contributionsto variance (specific and cumulative) obtained by empirical

6 Journal of Applied Mathematics

orthogonal function (using SPSS software) are shown inTable 1 The precipitation field rapidly converged the firstprincipal component (PC) accounted for almost half (455)of the variation in annual precipitation and the first threecomponents (all of which had eigenvalues exceeding 1) for700 of the variationThus the first three PCs and eigenvec-tors describe the spatial distribution of annual precipitationin the Weihe basin quite well

Eigenvectors of the covariance matrix cannot be directlyobtained from the SPSS output but can be obtained by div-iding the 119894 vectors of the matrix by the square roots of thecorresponding eigenvalues The vector corresponding to thelargest eigenvalue is eigenvector 1 with progressively smalleigenvalues associated with subsequent eigenvectors (2 34 etc) The first three eigenvectors obtained from the EOFdecomposition of the annual precipitation in theWeihe basindataset are shown in Table 2

The first three eigenvectors obtained from the EOF anal-ysis provide an approximate description of the precipitationvalues If their values are consistently positive or negativethere is a consistent precipitation trend and the area withthe largest absolute value is the center of the variable field Incontrast if the eigenvector values fluctuate between positiveand negative the trend is nonuniform

We used the Geostatistical Analyst module of ArcGIS toundertake a semivariogram analysis of the first three eigen-vectors and thus to obtain the spatial variability parametersshown in Table 3We then usedKriging interpolation to drawa map of each eigenvector field

As can be seen from Table 3 the azimuth angles ofeigenvectors 1 2 and 3 are 47∘ 655∘ and 3096∘ approx-imately south to north east-northeast to west-southwestand northwest to southeast orientations respectively All theeigenvectors have a small Nugget value indicating that theprecipitationmeasurement error is small Indeed the Nuggetof eigenvector 2 is zero indicating that themeasurement errorfor precipitation in this eigenvector field can be ignored Thepartial sill of eigenvector 1 is significantly lower than that ofthe other eigenvectors indicating that it reflects smaller-scalevariations The Weihe basin spans about 7∘ longitude and 4∘latitude Thus since the major ranges in the eigenvectors are629∘ latitude and longitude the variability of precipitationchanges with distance across the basin

43 VariationDiagnosis of Annual Rainfall in theWeihe BasinThe Weihe basin is divided into four subbasins with differ-ing hydrological characteristics and controlled by differenthydrological stations The areas covered by the subbasinsand locations of hydrological and meteorological stations areshown in Figure 1TheM-Kmethodwas used to discriminatevariation points in the trends in the annual precipitationseries recorded at each station during 1960ndash2011 and theresults are shown in Table 4

5 Analysis and Discussion of Results

51 Analysis of Spatial Variation of Annual Precipitation inthe Weihe Basin As shown in Table 2 all components of

Table 2 Eigenvectors obtained byEOFdecomposition of the annualprecipitation in the Weihe basin dataset

EigenvectorsEigenvector 1

119881

1

Eigenvector 2119881

2

Eigenvector 3119881

3

Lintao 0190 minus0137 minus0291Huajialing 0190 minus0043 minus0383Wuqi 0196 minus0157 0112Guyuan 0224 minus0082 minus0086Huanxia 0210 minus0140 0086Yanrsquoan 0208 0011 0251Xiji 0192 minus0131 minus0308Pingliang

minus0070 0045 minus0019Xifengzhen 0222 0104 0067Changwu 0204 0201 0058Luochuan 0200 0139 0179Tongchua 0154 0248 0168Minxian 0202 0001 minus0271Tianshui 0204 0057 minus0241Baoji 0167 0287 minus0021Wugong 0180 0273 minus0010Xirsquoan 0158 0277 0051Huashan 0163 0215 minus0100Foping 0177 0218 0009Zhenrsquoan 0157 0245 0042Shangzhou 0167 0223 minus0021Jingtai 0119 minus0198 0272Jingyuan 0167 minus0250 minus0084Yuzhong 0178 minus0156 minus0281Zhongning 0182 minus0206 0127Yanci 0170 minus0200 0250Hengshan 0183 minus0120 0240Suide 0175 minus0108 0260Haiyuan 0199 minus0231 0004Tongxin 0192 minus0221 0072

Table 3 Spatial variability parameters of eigenvectors of annualprecipitation in the Weihe basin

EigenvectorsEigenvector 1

119881

1

Eigenvector 2119881

2

Eigenvector 3119881

3

Azimuth angle (∘) 47 655 3096Nugget (119862

0) 00012685 0 00013054

Partial sill (119862) 00014968 041084 0044187Major range(∘ latitude andlongitude)

628911 628911 628911

eigenvector 1 are positive except the data from Pingliang sta-tion indicating that precipitation in theWeihe basin had highspatial consistency during the study period The downward

Journal of Applied Mathematics 7

Table 4 Variation points in annual precipitation trends in the Weihe basin

Subbasins Meteorologicalstations

Variation points1960ndash1969 1970ndash1979 1980ndash1989 1990ndash1999 2000ndash2011

Weihe basin aboveBeidao station

Tianshui 1971 1983 1985 2002Minxian 1986 1988 2006Huajialing 1986Lintao 1986 1987 2003 2006 2008Xiji 2005

Jinghe basin

Huanxian 1970 2002Guyuan 1972 1986 1996Pingliang 1971 1977 1980ndash1985lowast 1990Xifengzhen 1972 1979Changwu 1978 1979 1983 1985

Beiluohe basinWuqi 1971ndash1974lowast 1981 1983 1986ndash1995lowast 1998 2000 2003 2007

Yanrsquoan 1972ndash1975lowast1978 1979 1982

Luochuan 1970 1976

Beidao to Huaxiansection of Weihemainstream

Baoji 1969 1971 19741976 1978 1980 1982

Wugong 19611964ndash1970lowast

Xirsquoan 1962 1965Tongchuan 1976 1983 1985Zhenrsquoan 1972 1975

Foping 1966 1968 1973 19771979 1986 1988

Shangxian 1960 1963 1971 1975 1984Huanshan 1986

Extraction method Mann-Kendall method lowastindicates that variation occurred frequently during the indicated period

trend of annual precipitation shows that annual precipitationin the basin declined in the region and this consistent trendaccounts for 455 of the variance The basin covers parts ofthe continental monsoon zone and transitional zone betweenarid and humid regions of northwest China The overallconsistency and decline represented by eigenvector 1 are themain spatial features of precipitation patterns in the Weihebasin during the study period

The eigenvector 2 values obtained from EOF decomposi-tion of the annual precipitation data are shown in Figure 4The values can be divided by a line passing roughly fromYanrsquoan through Pingliang to Tianshui into positive andnegative anomaly areas covering the southeastern parts of theWeihe basin (with themost-positive anomaly center in Baoji)and northwestern parts (with the most-negative anomalycenter in Wu Qi) respectively These results confirm thatthe annual precipitation in the basin has a clear northwest-southeast pattern

Combining the eigenvalues with the trends in annualprecipitation recorded at each of the stations gives the eigen-vector 2 trend values (Figure 5) which show that precipita-tion declined in the southeast and increased in the northwestThe decline was most pronounced at the downstream stationHuashan

The eigenvector 3 values have a clear eastwest pattern(Figure 6) with a negative anomaly area covering upstreamparts of the basin and a positive anomaly center located in theLuochuan region north of ShaanxiThese findings show thatin addition to the consistent overall pattern and northwest-southeast precipitation patterns there is a clear eastwestpattern In combination with the precipitation trends at the30 stations the results clearly show that annual precipitationdeclined in northern Shaanxi and increased in the upstreamareas (Figure 7)

To summarize there were three spatial patterns of annualprecipitation in the Weihe basin during the study period (1)consistent overall decline (2) declined in the southeast andincreased in the northwest (3) declined in northern Shaanxiand increased in the upstream areas

52 Analysis of Annual Precipitation Variation in the WeiheBasin The results summarized in Table 4 show that varia-tion points occurred in trends of the annual precipitationsequence recorded at every meteorological station in theWeihe basin but their timings differed in the followingmanner

(1) The timing of variation points has clear spatial char-acteristics with gradual progressions from southeast

8 Journal of Applied Mathematics

Xiji

Wuqi

SuideYanci

Baoji

Jintai

Foping

Wugong

Guyuan

Lintao

Tongxin

Haiyuan

Yuzhong

HuashanMinxian

Changwu

Hengshan

Jingyuan

Tianshui

Luochuan

Huanxian

Zhongning

Shangzhou

Tongchuan

PingliangXifengzhen

Huajialing

0223

0245

0218

0215027702730287

00570001

0248

0

0201

01040045

Yanrsquoan

Zhenrsquoan

Xirsquoan

139

minus02

minus0206

minus0198

minus025

minus0156

minus0137minus00428

minus0131minus0082

minus014

minus0157

minus012

minus0108

minus0221

minus0231

N

E

S

W

103∘09984000998400998400E 104

∘09984000998400998400E 105

∘09984000998400998400E 106

∘09984000998400998400E 107

∘09984000998400998400E 108

∘09984000998400998400E 109

∘09984000998400998400E 110

∘09984000998400998400E

103∘09984000998400998400E 104

∘09984000998400998400E 105

∘09984000998400998400E 106

∘09984000998400998400E 107

∘09984000998400998400E 108

∘09984000998400998400E 109

∘09984000998400998400E 110

∘09984000998400998400E

34∘09984000998400998400N

33∘09984000998400998400N

35∘09984000998400998400N

36∘09984000998400998400N

37∘09984000998400998400N

38∘09984000998400998400N

39∘09984000998400998400N

34∘09984000998400998400N

33∘09984000998400998400N

35∘09984000998400998400N

36∘09984000998400998400N

37∘09984000998400998400N

38∘09984000998400998400N

39∘09984000998400998400N

50 25 0 50 100 150 200

(km)

Basin boundaryWater systemStation

Eigenvector 2High 0286104

Low minus0196997

Figure 4 The eigenvector 2 values for annual precipitation in the Weihe basin

Xiji

Wuqi

SuideYanci

Baoji

Jintai

Foping

Wugong

Guyuan

Lintao

Tongxin

Haiyuan

Yuzhong

HuashanMinxian

Changwu

Hengshan

Jingyuan

Tianshui

Luochuan

Huanxian

Zhongning

Shangzhou

Tongchuan

PingliangXifengzhen

Huajialing

022

023

028

0227

0316

01290099

0199

0192

0034

0266

0172

0228

01150363

50 25 0 50 100 150 200

(km)

103∘09984000998400998400E 104

∘09984000998400998400E 105

∘09984000998400998400E 106

∘09984000998400998400E 107

∘09984000998400998400E 108

∘09984000998400998400E 109

∘09984000998400998400E 110

∘09984000998400998400E

103∘09984000998400998400E 104

∘09984000998400998400E 105

∘09984000998400998400E 106

∘09984000998400998400E 107

∘09984000998400998400E 108

∘09984000998400998400E 109

∘09984000998400998400E 110

∘09984000998400998400E

34∘09984000998400998400N

33∘09984000998400998400N

35∘09984000998400998400N

36∘09984000998400998400N

37∘09984000998400998400N

38∘09984000998400998400N

39∘09984000998400998400N

34∘09984000998400998400N

33∘09984000998400998400N

35∘09984000998400998400N

36∘09984000998400998400N

37∘09984000998400998400N

38∘09984000998400998400N

39∘09984000998400998400N

N

E

S

W

Basin boundaryWater systemStation

Trend of eigenvector 2value

High 0229457

Low minus0461604

minus0023

minus0109

minus003

minus0838

minus0094

minus0196minus019

minus0236

minus006

minus0156

minus0083

minus0283

minus0072minus00018

minus0007

Yanrsquoan

Xirsquoan

Zhenrsquoan

Figure 5 The eigenvector 2 trend field for annual precipitation in the Weihe basin

Journal of Applied Mathematics 9

Xiji

Wuqi

SuideYanci

Baoji

Jintai

Foping

Wugong

Guyuan

Lintao

Tongxin

Haiyuan

Yuzhong

HuashanMinxian

Changwu

Hengshan

Jingyuan

Tianshui

Luochuan

Huanxian

Zhongning

Shangzhou

Tongchuan

PingliangXifengzhen

Huajialing

026

024025

0072

0004

0127

0272

0042

0009

0051

0168

0179

0058

0067

025100860112

50 25 0 50 100 150 200

(km)

103∘09984000998400998400E 104

∘09984000998400998400E 105

∘09984000998400998400E 106

∘09984000998400998400E 107

∘09984000998400998400E 108

∘09984000998400998400E 109

∘09984000998400998400E 110

∘09984000998400998400E

103∘09984000998400998400E 104

∘09984000998400998400E 105

∘09984000998400998400E 106

∘09984000998400998400E 107

∘09984000998400998400E 108

∘09984000998400998400E 109

∘09984000998400998400E 110

∘09984000998400998400E

34∘09984000998400998400N

33∘09984000998400998400N

35∘09984000998400998400N

36∘09984000998400998400N

37∘09984000998400998400N

38∘09984000998400998400N

39∘09984000998400998400N

34∘09984000998400998400N

33∘09984000998400998400N

35∘09984000998400998400N

36∘09984000998400998400N

37∘09984000998400998400N

38∘09984000998400998400N

39∘09984000998400998400N

N

E

S

W

Basin boundaryWater systemStation

Eigenvector 3value

High 0223536

Low minus0365226

Yanrsquoan

Xirsquoan

Zhenrsquoan

minus01

minus0021

minus001minus0021

minus0019

minus0241

minus0383

minus0271

minus0291

minus0281

minus0084

minus0308minus0086

Figure 6 The eigenvector 3 values for annual precipitation in the Weihe basin

to northwest and downstream to upstream The firstvariation points occurred in the early 1960s at theShangxian Wugong and Xirsquoan stations in the Beidaoto Huaxian section of the Weihe mainstream andlater in the late 1960s and early 1970s other stationsvariation occurred in this section In the Jingheand Beiluohe basins the first variation occurred inthe early 1970s In the Weihe basin above Beidaostationand variation occurred most recently in themid- and late 1980s except at Tianshui station (1971)

(2) Variation points in the trends of annual precipitationin the basin above Beidao station were concentratedin the 1980s while they covered longer periods ofthe 1970s and 1980s in other subbasins These resultsindicate that precipitation variation points occurredfrequently during the 1970s to 1980s in each subbasin

(3) After the 1990s the frequency and scope of precipi-tation variation declined No variation points in the1990s were detected except at Guyuan Pingliangand Wuqi stations After the 1990s there were novariation points in the annual precipitation sequencesrecorded in the Beidao to Huaxian section of theWeihe mainstream Jinghe basin or Beiluohe basinThese results show that after an active period ofprecipitation variation in 1970s and 1980s the annual

precipitation sequence in the Weihe basin was rela-tively stableThus during the study period there wereshifts in precipitation from one stable state to anotherafter disturbance

(4) From 2000 to 2011 further changes in annual pre-cipitation occurred in the basin above the Beidaostation and at the most northern stations of thebasin (Huanxian and Wuqi) These variations wereclearly regional concentrated in the northwest of theWeihe River basin and the source areas of the riversJinghe and Beiluohe Due to the limited length of thetime series it is not possible to distinguish whetherthese variations were due to small-scale regionalvariations or represent initial stages of wide-rangingvariation across the whole basin or were the resultof lingering temporal and spatial effects of the basin-wide variation that occurred in the 1970s and 1980s

(5) The variation in annual precipitation in the Weihebasin propagated from the southeast to the northwestThis is consistent with the orientation of the eigen-vector 2 field of annual precipitation obtained fromEOF decomposition and the spatial pattern of meanannual precipitation in the basin The factors causingthese patterns are not clear from the data presentedhere but will be addressed in future studies

10 Journal of Applied Mathematics

Xiji

Wuqi

SuideYanci

Baoji

Jintai

Foping

Wugong

Guyuan

Lintao

Tongxin

Haiyuan

Yuzhong

HuashanMinxian

Changwu

Hengshan

Jingyuan

Tianshui

Luochuan

Huanxian

Shangzhou

Zhongning

Tongchuan

PingliangXifengzhen

Huajialing

001

039

103077

0347

0078

0019

0009002103010548

0034

0658 0179

00062

N

E

S

W

Trend of eigenvector 3value

High 0761376

Low minus0354567

Yanrsquoan

Zhenrsquoan

Xirsquoan

minus0258

minus0531

minus0536

minus0141

minus002

minus0016

minus0008

minus0017

minus0164

minus0123

minus0123

minus0005

minus0074

minus01

minus0163

50 25 0 50 100 150 200

(km)

103∘09984000998400998400E 104

∘09984000998400998400E 105

∘09984000998400998400E 106

∘09984000998400998400E 107

∘09984000998400998400E 108

∘09984000998400998400E 109

∘09984000998400998400E 110

∘09984000998400998400E

103∘09984000998400998400E 104

∘09984000998400998400E 105

∘09984000998400998400E 106

∘09984000998400998400E 107

∘09984000998400998400E 108

∘09984000998400998400E 109

∘09984000998400998400E 110

∘09984000998400998400E

34∘09984000998400998400N

33∘09984000998400998400N

35∘09984000998400998400N

36∘09984000998400998400N

37∘09984000998400998400N

38∘09984000998400998400N

39∘09984000998400998400N

34∘09984000998400998400N

33∘09984000998400998400N

35∘09984000998400998400N

36∘09984000998400998400N

37∘09984000998400998400N

38∘09984000998400998400N

39∘09984000998400998400N

Basin boundaryWater systemStation

Figure 7 The eigenvector 3 trend field for annual precipitation in the Weihe basin

6 Conclusions

We have analyzed spatial and temporal patterns of annualprecipitation in the Weihe basin using data recorded from1960 to 2011 at 30 National Standard Stations in the basinand surrounding area by applying empirical orthogonalfunction decomposition in combination with geostatistical(semivariogram) analysis then using the Mann-Kendallmethod to discriminate variation points in the precipitationseries recorded at each station

(1) The main spatial patterns of annual precipitation inthe Weihe Basin during the study period were aconsistent overall decline which was strongest in thenorthwest and progressively weaker from northwestto southwest and from west to east The consistentoverall decline accounted for 455 of the totalvariation

(2) Variation points were detected in the annual precip-itation sequences recorded at all the meteorologicalstations in the basin but their timing had clear spatialpatterns starting gradually later from the southeastto the northwest and downstream to upstream Inthe 1970s and 1980s they occurred frequently in eachsubbasin

(3) After the 1990s following the active period of precip-itation variation in the 1970s and 1980s the frequencyand scope of precipitation variations declined Thusduring the study period there were shifts in precipita-tion fromone stable state to another after disturbance

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was financially supported by the Natural ScienceFoundation of China (Grant nos 51190093 51179148 and51179149) the National Key Basic Research 973 of China(Grant no 2012CB417003) and the MWR Welfare SectorFunds (Grant nos 201101043 201101049) Constructive com-ments from reviewers of a previous version of the paper aregratefully acknowledged

References

[1] H Zhang Y Chen G Ren et al ldquoThe characteristics of pre-cipitation variation of Weihe River Basin in Shaanxi Province

Journal of Applied Mathematics 11

during recent 50 yearsrdquo Agricultural Research in the Arid Areasvol 26 no 04 pp 236ndash241 2008

[2] K Che and J Jian ldquoA study of climate variability in Zedangduring 1960ndash1999rdquo Meteorological Monthly vol 26 no 1 pp54ndash56 2000

[3] Y Ding G Ren G Shi et al ldquoNational assessment report ofclimate change (I) climate change inChina and its future trendrdquoAdvances in Climate Change Research vol 2 no 1 pp 3ndash8 2006

[4] Y He F Wang X Mu et al ldquoTemporal and spatial character-istics of temperature and precipitation in Weihe River basinrdquoBulletin of Soil and Water Conservation vol 32 no 4 pp 102ndash105 2012

[5] Y-X Zhang F Wang and X-M Mu ldquoResearch of quantitativerelations between Geographical environment factor and waterand sediment in the Weihe River basinrdquo Journal of Northwest Aamp FUniversity (Natural Science Edition) vol 37 no 1 pp 61ndash652009

[6] C Jiang F Wang and X Mu ldquoEffect of climate change on netprimary productivity (NPP) of natural vegetation in Wei riverbasin (II) NPP of natural vegetation inWei river basinrdquo Journalof Arid LandResources and Environment vol 27 no 5 pp 53ndash572013

[7] Y Zhang ldquoHydrologic feature analysis for Weihe basin inShaanxi provincerdquo Northwest Water Resources amp Water Engi-neering no 2 pp 62ndash64 2002

[8] G Wang and Y Wang ldquoAnalysis on causes of runoff andsediment variation in Weihe basin based on hydrologicalmodelrdquo Journal of Soil Water Conservation no 4 pp 22ndash252000

[9] W He and Z Xu ldquoSpatial and temporal characteristics of thelong-term trend for temperature and pan evaporation in theWei River Basinrdquo Journal of Beijing Normal University (NaturalScience) vol 42 no 1 pp 102ndash106 2006

[10] K Pearson ldquoOn lines and planes of closest fit to system of pointsin spacerdquo Philosophical Magazine vol 6 no 2 pp 559ndash5721902

[11] D-X Cui and W-Y Liu ldquoApplication of empirical orthogonalfunction resolution to analysis of crustal vertical deformationfieldrdquo Earthquake vol 20 no 3 pp 82ndash86 2000

[12] T Liu and H Zhang ldquoCompared with principal componentanalysis and empirical orthogonal function decompositionrdquoStatistics and Decision vol 340 no 16 pp 159ndash162 2011

[13] H Lin and W Zhang ldquoSimilarities and differences of principalcomponent analysis and factor analysis and SPSS softwarerdquoStatistics Research no 3 pp 65ndash69 2005

[14] D H Kim C Yoo and T-W Kim ldquoApplication of spatial EOFand multivariate time series model for evaluating agriculturaldrought vulnerability in Koreardquo Advances in Water Resourcesvol 34 no 3 pp 340ndash350 2011

[15] N S Loboda A V Glushkov and V N Khokhlov ldquoUsingmeteorological data for reconstruction of annual runoff seriesover an ungauged area empirical orthogonal function approachto Moldova-Southwest Ukraine regionrdquo Atmospheric Researchvol 77 no 1ndash4 pp 100ndash113 2005

[16] D Sirjacobs A Alvera-Azcarate A Barth et al ldquoCloud fillingof ocean colour and sea surface temperature remote sensingproducts over the SouthernNorth Sea by theData InterpolatingEmpirical Orthogonal Functions methodologyrdquo Journal of SeaResearch vol 65 no 1 pp 114ndash130 2011

[17] H-L Yu and H-J Chu ldquoUnderstanding space-time patternsof groundwater system by empirical orthogonal functions a

case study in the Choshui River alluvial fan Taiwanrdquo Journalof Hydrology vol 381 no 3-4 pp 239ndash247 2010

[18] H BMann ldquoNonparametric tests against trendrdquo Econometricavol 13 pp 245ndash259 1945

[19] R Snyers ldquoSur LaDetermination de la stabilite des series clima-tologiquesrdquo in Proceedings of the UNESCO-WMO Symposium ofChanges of Climate vol 20 ofUneseo Arid Zone Research Seriespp 37ndash44 Unesco Paris France 1963

[20] C Goossens and A Berger ldquoAnnual and seasonal climaticvariations over the northern hemisphere and Europe during thelast centuryrdquo Annales Geophysicae B vol 4 no 4 pp 385ndash4001986

[21] C Fu and QWang ldquoThe definition and detection of the abruptclimatic changerdquo Chinese Journal of Atmospheric Sciences vol16 no 4 pp 482ndash493 1992

[22] P Xie G Chen D Li et al ldquoHydrological variation inte-grated diagnosis method and application researchrdquo Hydroelec-tric Energy vol 23 no 2 pp 11ndash14 2005

[23] M Gocic and S Trajkovic ldquoAnalysis of changes in meteoro-logical variables using Mann-Kendall and Senrsquos slope estimatorstatistical tests in SerbiardquoGlobal and Planetary Change vol 100pp 172ndash182 2013

[24] K H Hamed ldquoTrend detection in hydrologic data the Mann-Kendall trend test under the scaling hypothesisrdquo Journal ofHydrology vol 349 no 3-4 pp 350ndash363 2008

[25] S Kumar V Merwade J Kam and K Thurner ldquoStreamflowtrends in Indiana effects of long term persistence precipitationand subsurface drainsrdquo Journal of Hydrology vol 374 no 1-2pp 171ndash183 2009

[26] K H Hamed ldquoExact distribution of the Mann-Kendall trendtest statistic for persistent datardquo Journal of Hydrology vol 365no 1-2 pp 86ndash94 2009

[27] D Duhan and A Pandey ldquoStatistical analysis of long termspatial and temporal trends of precipitation during 1901ndash2002at Madhya Pradesh Indiardquo Atmospheric Research vol 122 pp136ndash149 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Journal of Applied Mathematics

BeidaoHuaxian

Lintong

XianyangLinjiacun

Zhuangtou

Weijiabao

Zhangjiashan

Xiji

Wuqi

Baoji

Guyuan

Lintao

Foping

Wugong

Minxian Huashan

Changwu

Huanxian

Tianshui

Luochuan

Shangxian

Tongchuan

Pingliang

xifengzhen

Huajialing

0 68000 136000(KM)

Hydrologic stationMeteorological stationRiverJinghe basin

Beiluohe basinB-H section of Weihe basinWeihe basin above Beidao station

N

E

S

WYanrsquoan

Zhenrsquoan

Xirsquoan

Figure 1 Map of the Weihe basin showing the four subbasins and locations of hydrological and meteorological stations

Climate changes have had marked effects in the Weihebasin Thus we have examined changes in annual precipita-tion in the basin from 1960 to 2011 in order to elucidate theirspatial and temporal patterns thereby facilitating attempts toformulate effective mitigation strategies For the purposes ofthe study theWeihe basin is divided into four subbasins withdistinct hydrological characteristics and control stations thebasin above Beidao station Jinghe basin Beiluohe basin andthe Beidao to Huaxian section of the Weihe mainstream Amap showing the areas covered by these subbasins and loca-tions of hydrological andmeteorological stations is presentedin Figure 1

3 Data Sources and Methodology

31 Data Sources The annual precipitation data used inthis paper are recorded from 1960 to 2011 at 30 NationalStandard Stations in the Weihe basin and the surrounding

area obtained from the China Meteorological Data SharingService System (httpcdccmagovcnhomedo)

32 Methodology First we calculate means in annual precip-itation in the Weihe basin by statistical methods and deter-mine trends by regression analysis Then we use empiricalorthogonal function (EOF) decomposition and geostatisticalanalysis method to analyze spatial and temporal patternsassociated with precipitation Eigenvectors obtained fromEOF decomposition are then multiplied by trend vectors inthe matrix of the precipitation series to obtain eigenvectorfields thus describing the spatial trends in annual precipi-tation Finally we use the Mann-Kendall (M-K) method toidentify variation points in the precipitation series for eachstation thereby comprehensively describing the temporaland spatial patterns of annual precipitation in each of thesubbasins

Journal of Applied Mathematics 3

321 Regression Analysis Regression analysis is applied toanalyze the correlationship of two variables which deter-mines relative direction and intensity In this study regressionanalysis is used to get the linear regression of precipitationand time and the equation is as follows

119875 = 119886119879 + 119887 (1)

where 119875 is the precipitation 119879 is the time and 119886 and 119887

are parameters If the unit of 119879 is the year usually 119886 is thechanging trend of annual precipitation

322 Empirical Orthogonal Function Decomposition EOFdecomposition involves decomposition of a matrix X of datadescribing the distribution of a variable in both space andtime into a spatial function V and time function T as shownin

X = VT

X =

[

[

[

[

[

[

[

[

119909

11sdot sdot sdot 119909

1119895sdot sdot sdot 119909

1119899

119909

1198941sdot sdot sdot 119909

119894119895sdot sdot sdot 119909

119894119899

119909

1198981sdot sdot sdot 119909

119898119895sdot sdot sdot 119909

119898119899

]

]

]

]

]

]

]

]

(2)

where119898 and 119899 are points in space and time respectively and119909

119894119895is the value of the variable at point 119894 and time 119895 (here

annual precipitation at station 119894 in year 119895)If V and T are orthogonal

119898

sum

119894=1

V119894119896V119894119897= 1 when 119896 = 119897

119899

sum

119895=1

119905

119896119895119905

119897119895= 0 when 119896 = 119897

(3)

If X is an anomaly matrix (reference) multiplying theequation X = VT by XT gives

XXT= VTXT

= VTTTVT= VΛV (4)

where Λ is a diagonal matrix composed of eigenvalues 120582

119894of

matrix XXT and

TTT= Λ (5)

Due to the characteristics of the eigenvectors VTV isa unit matrix meeting the requirements of orthogonalityThus the spatial function matrix V can be calculated fromeigenvectors of the matrix XXT

Here we multiply the spatial functionmatrixV of annualprecipitation in the Weihe basin by the trend vector D =

1198891 1198892 119889119898 composed of trends in annual precipitationrecorded at each station (V

119889= DV) thus describing the

spatial trends of annual precipitation in the basinEOF decomposition was first established in 1902 [10]

It decomposes elements of a set of space-time data into

two functions one dependent only on time and anotherdependent only on space to analyze the spatial structure ofelement fields [11] Thus EOF decomposition is commonlyused to analyze spatial and temporal variations in meteoro-logical elements In contrast principal component analysis(PCA) is mainly used to find a few independent vectors ina dataset to reduce its dimensionality thereby simplifyingdata handling and interpretation However the operationalsteps are basically the same when the original variables areanomaly data or standardized anomaly data [12] Hence wecan use the PCA module of SPSS (statistical product andservice solution) software for EOF decomposition [13]

EOF decomposition and several variants thereof havebeen widely applied for analyzing spatial changes in mete-orological elements Notably EOF decomposition has beenused to analyze drought occurrence in both time and spacein Korea [14] It has also been applied to establish contribu-tions of annual evaporation annual precipitation and bothwarm- and cold-season precipitation to the annual runoffof rivers in an arid zone to obtain quantitative estimates ofthe effects of climatic factors on reconstructions of annualrunoff series in an ungauged area [15] A data-interpolatingempirical orthogonal function (DINEOF) has been used toanalyze distributions of sea surface temperatures in a studydemonstrating that DINEOF decomposition can be highlyeffective evenwhen very high proportions of data aremissing[16] In addition rotated empirical orthogonal function(REOF) decomposition has been applied to analyze space-time patterns of groundwater fluctuations in the ChoshuiRiver alluvial fan Taiwan based on monthly observations ofpiezometric heads from66wells during the period 1997ndash2002[17]

323 Mann-Kendall Variation Analysis The Mann-Kendalltest has been widely used for identifying trends in time-series data because it is nonparametric and thus suitable forhandling data that are not normally distributed and it isnot sensitive to interference from a few outliers The Mann-Kendall test can also be used for variation diagnosis and thecalculation method is as follows

For a time series 119883 = 119909

1 119909

2 119909

119899 the Mann-Kendall

test statistic 119863

120591is given by the following model

119863

120591=

120591

sum

119894=1

119877

119894(120591 = 2 3 119894) (6)

where

119877

119894=

+1 119909

119894gt 119909

119895

0 119909

119894le 119909

119895

(119895 = 1 2 119894) (7)

The sequential statistic UF120591(in which UF

1= 0) is

calculated by is

UF120591=

1003816

1003816

1003816

1003816

119863

120591minus 119864 (119863

120591)

1003816

1003816

1003816

1003816

radic119881 (119863

120591)

(8)

4 Journal of Applied Mathematics

520

500

540560

480

460

580

600

620

640 66

0

440

420

0

400

700

380

720

360

740

340

760780800

420

740

680

760

780

440

540

700

700

720

400

700

560

480

Xiji

Wuqi

SuideYanci

B

Jintai

Foping

Wugong

Guyuan

Lintao

Tongxin

Haiyuan

Yuzhong

HuashanMinxian

Changwu

Hengshan

Jingyuan

Tianshui

Luochuan

Huanxian

Zhongning

Shangzhou

Tongchuan

PingliangXifengzhen

Huajialing

662

612

2643

3725

4412

369228962036

3842

2311

7806

6888

8982

81855599568935

51735805

59515838

55461842

4136

53194332

4529

4609

48785286

18505

aoji

Yanrsquoan

Xirsquoan

Zhenrsquoan50 25 0 50 100 150 200

(km)

Basin boundaryWater systemPrecipitation isoline

Station

103∘09984000998400998400E 104

∘09984000998400998400E 105

∘09984000998400998400E 106

∘09984000998400998400E 107

∘09984000998400998400E 108

∘09984000998400998400E 109

∘09984000998400998400E 110

∘09984000998400998400E

103∘09984000998400998400E 104

∘09984000998400998400E 105

∘09984000998400998400E 106

∘09984000998400998400E 107

∘09984000998400998400E 108

∘09984000998400998400E 109

∘09984000998400998400E 110

∘09984000998400998400E

34∘09984000998400998400N

33∘09984000998400998400N

35∘09984000998400998400N

36∘09984000998400998400N

37∘09984000998400998400N

38∘09984000998400998400N

39∘09984000998400998400N

34∘09984000998400998400N

33∘09984000998400998400N

35∘09984000998400998400N

36∘09984000998400998400N

37∘09984000998400998400N

38∘09984000998400998400N

39∘09984000998400998400N

N

E

S

W

High 819619

Low 320163

Annual precipitation

Figure 2 Isohyets showing mean annual precipitation in the Weihe basin

where 119864(119863

120591) and 119881(119863

120591) are the mean and variance of the

series given respectively by

119864 (119863

120591) =

120591 (120591 minus 1)

4

119881 (119863

120591) =

120591 (120591 minus 1) (2120591 + 5)

72

(9)

Then repeat the calculation with the inversion series1198831015840 =119909

119899 119909

119899minus1 119909

1 and let UB

120591= minusUF

120591 (120591 = 119899 119899 minus 1 1)

Define the significant level 120572 and find out the critical valueplusmn119880

(1minus120572)2 If the curves of UF

120591and UB

120591intersect and the

intersections are between the critical line (in this study thesignificant level 120572 is 095 and the critical line is plusmn119880

0052=

plusmn196) the time of curve intersections is the starting time ofthe variation

Originally published in 1945 [18] it was initially onlyused to detect trends in sequences However subsequentmodifications have enabled its use for detecting variationpoints in various trends [19] and it has been applied inreversed sequence to detect variation points in trends inclimate data time series [20] Since then the method hasbeen widely used to diagnose variation points in hydrologicaland meteorological data sequences [21 22] Notably the M-K method has been used to determine whether there werepositive or negative trends in seven meteorological variables(and if so their significance) recorded at 12 weather stations in

Serbia during 1980ndash2010 [23] Amodified formof themethodhas also been used to test scaling effects [24] and four variantshave been applied to determine trends in selected streamflowstatistics from Indiana [25] In addition a procedure forcalculating exact distributions of the M-K trend test statisticfor persistent data with an arbitrary correlation structure hasbeen presented [26] and the test has been used to detectdirections and magnitudes of changes in monotonic trendsin annual and seasonal precipitation over time in annualprecipitation inMadhya Pradesh India [27] Here we appliedthe M-K test to detect trends variation points in the timeseries of the data obtained from the 30 stations in the Weihebasin (Figure 1)

4 Results

41 Spatial Patterns of Annual Precipitation in the WeiheBasin Means and trends in annual precipitation in theWeihebasin from 1960 to 2011 are shown in Figures 2 and 3 Themean annual precipitation at each station changed greatlySpatially it declined from the southeast to northwest Meanannual precipitation in the basin amounted to 580 mm witha maximum exceeding 800mm at Huashan and Zhenrsquoan areaand a minimum of 340mm in the northwest The annualprecipitation trend at each station was negative except atXirsquoan where it increased by 0123mma The mean trend

Journal of Applied Mathematics 5

Xiji

Wuqi

SuideYanci

Baoji

Jintai

Foping

Wugong

Guyuan

Lintao

Tongxin

Haiyuan

Yuzhong

HuashanMinxian

Changwu

Hengshan

Jingyuan

Tianshui

Luochuan

Huanxian

Zhongning

Shangzhou

Tongchuan

PingliangXifengzhen

Huajialing

minus20382

Xirsquoan

Zhenrsquoan

Yanrsquoan

50 25 0 50 100 150 200

(km)

N

E

S

W

Trend of annual precipitation High minus0884939

Low minus179305

103∘09984000998400998400E 104

∘09984000998400998400E 105

∘09984000998400998400E 106

∘09984000998400998400E 107

∘09984000998400998400E 108

∘09984000998400998400E 109

∘09984000998400998400E 110

∘09984000998400998400E

103∘09984000998400998400E 104

∘09984000998400998400E 105

∘09984000998400998400E 106

∘09984000998400998400E 107

∘09984000998400998400E 108

∘09984000998400998400E 109

∘09984000998400998400E 110

∘09984000998400998400E

34∘09984000998400998400N

33∘09984000998400998400N

35∘09984000998400998400N

36∘09984000998400998400N

37∘09984000998400998400N

38∘09984000998400998400N

39∘09984000998400998400N

34∘09984000998400998400N

33∘09984000998400998400N

35∘09984000998400998400N

36∘09984000998400998400N

37∘09984000998400998400N

38∘09984000998400998400N

39∘09984000998400998400N

01229

minus 0 785

minus 1366

minus 1 076minus 04934

minus 0 9699

minus 09262

minus 08778

minus 03846

minus 08691

minus 086 72minus 09866

minus 01212

minus 02984

minus 1821

minus 10304

minus 12346

0036300

minus 3 8975minus 12491

minus 20241

minus 14991

minus 21349

minus 21367minus 19091

minus 20868

minus 14559

minus 26897minus 26428

Basin boundaryWater systemStation

Figure 3 Annual precipitation trends in the Weihe basin

Table 1 Eigenvalues and their contributions to the variance of annual precipitation in the Weihe basin obtained from EOF decomposition

Component

Initial eigenvalues Extraction sums of squared loadings

Eigenvalues120582

119894

Proportion ofvariation ()

Cumulativeproportion ofvariation ()

Eigenvalues120582

119894

Proportion ofvariation ()

Cumulativeproportion ofvariation ()

1 13648 45494 45494 13648 45494 454942 5363 17875 63369 5363 17875 633693 1997 6658 70027 1997 6658 700274 1404 4681 747085 1196 3985 786936 0912 3039 817327 0685 2285 840178 0634 2113 861309 0506 1686 8781610 0422 1408 89224Extraction method principal component analysis

across the basin was minus1662mma clearly showing thatannual precipitation declined and that this trend was mostpronounced in upstream areas

42 EOF Decomposition of Annual Precipitation in the WeiheBasin The precipitation data analyzed herein originatedfrom the Weihe basin and the surrounding 30 nationalstandard stations during the period 1960ndash2011 From this

data we derived the normalized anomaly matrix X forannual precipitation Empirical orthogonal function (run inSPSS) was used to calculate the eigenvalues of the covariancematrix of X and the amount of variation that each principalcomponent explained as shown in Table 1

Eigenvalues for the covariance matrix of the Weihebasin annual precipitation matrix and their contributionsto variance (specific and cumulative) obtained by empirical

6 Journal of Applied Mathematics

orthogonal function (using SPSS software) are shown inTable 1 The precipitation field rapidly converged the firstprincipal component (PC) accounted for almost half (455)of the variation in annual precipitation and the first threecomponents (all of which had eigenvalues exceeding 1) for700 of the variationThus the first three PCs and eigenvec-tors describe the spatial distribution of annual precipitationin the Weihe basin quite well

Eigenvectors of the covariance matrix cannot be directlyobtained from the SPSS output but can be obtained by div-iding the 119894 vectors of the matrix by the square roots of thecorresponding eigenvalues The vector corresponding to thelargest eigenvalue is eigenvector 1 with progressively smalleigenvalues associated with subsequent eigenvectors (2 34 etc) The first three eigenvectors obtained from the EOFdecomposition of the annual precipitation in theWeihe basindataset are shown in Table 2

The first three eigenvectors obtained from the EOF anal-ysis provide an approximate description of the precipitationvalues If their values are consistently positive or negativethere is a consistent precipitation trend and the area withthe largest absolute value is the center of the variable field Incontrast if the eigenvector values fluctuate between positiveand negative the trend is nonuniform

We used the Geostatistical Analyst module of ArcGIS toundertake a semivariogram analysis of the first three eigen-vectors and thus to obtain the spatial variability parametersshown in Table 3We then usedKriging interpolation to drawa map of each eigenvector field

As can be seen from Table 3 the azimuth angles ofeigenvectors 1 2 and 3 are 47∘ 655∘ and 3096∘ approx-imately south to north east-northeast to west-southwestand northwest to southeast orientations respectively All theeigenvectors have a small Nugget value indicating that theprecipitationmeasurement error is small Indeed the Nuggetof eigenvector 2 is zero indicating that themeasurement errorfor precipitation in this eigenvector field can be ignored Thepartial sill of eigenvector 1 is significantly lower than that ofthe other eigenvectors indicating that it reflects smaller-scalevariations The Weihe basin spans about 7∘ longitude and 4∘latitude Thus since the major ranges in the eigenvectors are629∘ latitude and longitude the variability of precipitationchanges with distance across the basin

43 VariationDiagnosis of Annual Rainfall in theWeihe BasinThe Weihe basin is divided into four subbasins with differ-ing hydrological characteristics and controlled by differenthydrological stations The areas covered by the subbasinsand locations of hydrological and meteorological stations areshown in Figure 1TheM-Kmethodwas used to discriminatevariation points in the trends in the annual precipitationseries recorded at each station during 1960ndash2011 and theresults are shown in Table 4

5 Analysis and Discussion of Results

51 Analysis of Spatial Variation of Annual Precipitation inthe Weihe Basin As shown in Table 2 all components of

Table 2 Eigenvectors obtained byEOFdecomposition of the annualprecipitation in the Weihe basin dataset

EigenvectorsEigenvector 1

119881

1

Eigenvector 2119881

2

Eigenvector 3119881

3

Lintao 0190 minus0137 minus0291Huajialing 0190 minus0043 minus0383Wuqi 0196 minus0157 0112Guyuan 0224 minus0082 minus0086Huanxia 0210 minus0140 0086Yanrsquoan 0208 0011 0251Xiji 0192 minus0131 minus0308Pingliang

minus0070 0045 minus0019Xifengzhen 0222 0104 0067Changwu 0204 0201 0058Luochuan 0200 0139 0179Tongchua 0154 0248 0168Minxian 0202 0001 minus0271Tianshui 0204 0057 minus0241Baoji 0167 0287 minus0021Wugong 0180 0273 minus0010Xirsquoan 0158 0277 0051Huashan 0163 0215 minus0100Foping 0177 0218 0009Zhenrsquoan 0157 0245 0042Shangzhou 0167 0223 minus0021Jingtai 0119 minus0198 0272Jingyuan 0167 minus0250 minus0084Yuzhong 0178 minus0156 minus0281Zhongning 0182 minus0206 0127Yanci 0170 minus0200 0250Hengshan 0183 minus0120 0240Suide 0175 minus0108 0260Haiyuan 0199 minus0231 0004Tongxin 0192 minus0221 0072

Table 3 Spatial variability parameters of eigenvectors of annualprecipitation in the Weihe basin

EigenvectorsEigenvector 1

119881

1

Eigenvector 2119881

2

Eigenvector 3119881

3

Azimuth angle (∘) 47 655 3096Nugget (119862

0) 00012685 0 00013054

Partial sill (119862) 00014968 041084 0044187Major range(∘ latitude andlongitude)

628911 628911 628911

eigenvector 1 are positive except the data from Pingliang sta-tion indicating that precipitation in theWeihe basin had highspatial consistency during the study period The downward

Journal of Applied Mathematics 7

Table 4 Variation points in annual precipitation trends in the Weihe basin

Subbasins Meteorologicalstations

Variation points1960ndash1969 1970ndash1979 1980ndash1989 1990ndash1999 2000ndash2011

Weihe basin aboveBeidao station

Tianshui 1971 1983 1985 2002Minxian 1986 1988 2006Huajialing 1986Lintao 1986 1987 2003 2006 2008Xiji 2005

Jinghe basin

Huanxian 1970 2002Guyuan 1972 1986 1996Pingliang 1971 1977 1980ndash1985lowast 1990Xifengzhen 1972 1979Changwu 1978 1979 1983 1985

Beiluohe basinWuqi 1971ndash1974lowast 1981 1983 1986ndash1995lowast 1998 2000 2003 2007

Yanrsquoan 1972ndash1975lowast1978 1979 1982

Luochuan 1970 1976

Beidao to Huaxiansection of Weihemainstream

Baoji 1969 1971 19741976 1978 1980 1982

Wugong 19611964ndash1970lowast

Xirsquoan 1962 1965Tongchuan 1976 1983 1985Zhenrsquoan 1972 1975

Foping 1966 1968 1973 19771979 1986 1988

Shangxian 1960 1963 1971 1975 1984Huanshan 1986

Extraction method Mann-Kendall method lowastindicates that variation occurred frequently during the indicated period

trend of annual precipitation shows that annual precipitationin the basin declined in the region and this consistent trendaccounts for 455 of the variance The basin covers parts ofthe continental monsoon zone and transitional zone betweenarid and humid regions of northwest China The overallconsistency and decline represented by eigenvector 1 are themain spatial features of precipitation patterns in the Weihebasin during the study period

The eigenvector 2 values obtained from EOF decomposi-tion of the annual precipitation data are shown in Figure 4The values can be divided by a line passing roughly fromYanrsquoan through Pingliang to Tianshui into positive andnegative anomaly areas covering the southeastern parts of theWeihe basin (with themost-positive anomaly center in Baoji)and northwestern parts (with the most-negative anomalycenter in Wu Qi) respectively These results confirm thatthe annual precipitation in the basin has a clear northwest-southeast pattern

Combining the eigenvalues with the trends in annualprecipitation recorded at each of the stations gives the eigen-vector 2 trend values (Figure 5) which show that precipita-tion declined in the southeast and increased in the northwestThe decline was most pronounced at the downstream stationHuashan

The eigenvector 3 values have a clear eastwest pattern(Figure 6) with a negative anomaly area covering upstreamparts of the basin and a positive anomaly center located in theLuochuan region north of ShaanxiThese findings show thatin addition to the consistent overall pattern and northwest-southeast precipitation patterns there is a clear eastwestpattern In combination with the precipitation trends at the30 stations the results clearly show that annual precipitationdeclined in northern Shaanxi and increased in the upstreamareas (Figure 7)

To summarize there were three spatial patterns of annualprecipitation in the Weihe basin during the study period (1)consistent overall decline (2) declined in the southeast andincreased in the northwest (3) declined in northern Shaanxiand increased in the upstream areas

52 Analysis of Annual Precipitation Variation in the WeiheBasin The results summarized in Table 4 show that varia-tion points occurred in trends of the annual precipitationsequence recorded at every meteorological station in theWeihe basin but their timings differed in the followingmanner

(1) The timing of variation points has clear spatial char-acteristics with gradual progressions from southeast

8 Journal of Applied Mathematics

Xiji

Wuqi

SuideYanci

Baoji

Jintai

Foping

Wugong

Guyuan

Lintao

Tongxin

Haiyuan

Yuzhong

HuashanMinxian

Changwu

Hengshan

Jingyuan

Tianshui

Luochuan

Huanxian

Zhongning

Shangzhou

Tongchuan

PingliangXifengzhen

Huajialing

0223

0245

0218

0215027702730287

00570001

0248

0

0201

01040045

Yanrsquoan

Zhenrsquoan

Xirsquoan

139

minus02

minus0206

minus0198

minus025

minus0156

minus0137minus00428

minus0131minus0082

minus014

minus0157

minus012

minus0108

minus0221

minus0231

N

E

S

W

103∘09984000998400998400E 104

∘09984000998400998400E 105

∘09984000998400998400E 106

∘09984000998400998400E 107

∘09984000998400998400E 108

∘09984000998400998400E 109

∘09984000998400998400E 110

∘09984000998400998400E

103∘09984000998400998400E 104

∘09984000998400998400E 105

∘09984000998400998400E 106

∘09984000998400998400E 107

∘09984000998400998400E 108

∘09984000998400998400E 109

∘09984000998400998400E 110

∘09984000998400998400E

34∘09984000998400998400N

33∘09984000998400998400N

35∘09984000998400998400N

36∘09984000998400998400N

37∘09984000998400998400N

38∘09984000998400998400N

39∘09984000998400998400N

34∘09984000998400998400N

33∘09984000998400998400N

35∘09984000998400998400N

36∘09984000998400998400N

37∘09984000998400998400N

38∘09984000998400998400N

39∘09984000998400998400N

50 25 0 50 100 150 200

(km)

Basin boundaryWater systemStation

Eigenvector 2High 0286104

Low minus0196997

Figure 4 The eigenvector 2 values for annual precipitation in the Weihe basin

Xiji

Wuqi

SuideYanci

Baoji

Jintai

Foping

Wugong

Guyuan

Lintao

Tongxin

Haiyuan

Yuzhong

HuashanMinxian

Changwu

Hengshan

Jingyuan

Tianshui

Luochuan

Huanxian

Zhongning

Shangzhou

Tongchuan

PingliangXifengzhen

Huajialing

022

023

028

0227

0316

01290099

0199

0192

0034

0266

0172

0228

01150363

50 25 0 50 100 150 200

(km)

103∘09984000998400998400E 104

∘09984000998400998400E 105

∘09984000998400998400E 106

∘09984000998400998400E 107

∘09984000998400998400E 108

∘09984000998400998400E 109

∘09984000998400998400E 110

∘09984000998400998400E

103∘09984000998400998400E 104

∘09984000998400998400E 105

∘09984000998400998400E 106

∘09984000998400998400E 107

∘09984000998400998400E 108

∘09984000998400998400E 109

∘09984000998400998400E 110

∘09984000998400998400E

34∘09984000998400998400N

33∘09984000998400998400N

35∘09984000998400998400N

36∘09984000998400998400N

37∘09984000998400998400N

38∘09984000998400998400N

39∘09984000998400998400N

34∘09984000998400998400N

33∘09984000998400998400N

35∘09984000998400998400N

36∘09984000998400998400N

37∘09984000998400998400N

38∘09984000998400998400N

39∘09984000998400998400N

N

E

S

W

Basin boundaryWater systemStation

Trend of eigenvector 2value

High 0229457

Low minus0461604

minus0023

minus0109

minus003

minus0838

minus0094

minus0196minus019

minus0236

minus006

minus0156

minus0083

minus0283

minus0072minus00018

minus0007

Yanrsquoan

Xirsquoan

Zhenrsquoan

Figure 5 The eigenvector 2 trend field for annual precipitation in the Weihe basin

Journal of Applied Mathematics 9

Xiji

Wuqi

SuideYanci

Baoji

Jintai

Foping

Wugong

Guyuan

Lintao

Tongxin

Haiyuan

Yuzhong

HuashanMinxian

Changwu

Hengshan

Jingyuan

Tianshui

Luochuan

Huanxian

Zhongning

Shangzhou

Tongchuan

PingliangXifengzhen

Huajialing

026

024025

0072

0004

0127

0272

0042

0009

0051

0168

0179

0058

0067

025100860112

50 25 0 50 100 150 200

(km)

103∘09984000998400998400E 104

∘09984000998400998400E 105

∘09984000998400998400E 106

∘09984000998400998400E 107

∘09984000998400998400E 108

∘09984000998400998400E 109

∘09984000998400998400E 110

∘09984000998400998400E

103∘09984000998400998400E 104

∘09984000998400998400E 105

∘09984000998400998400E 106

∘09984000998400998400E 107

∘09984000998400998400E 108

∘09984000998400998400E 109

∘09984000998400998400E 110

∘09984000998400998400E

34∘09984000998400998400N

33∘09984000998400998400N

35∘09984000998400998400N

36∘09984000998400998400N

37∘09984000998400998400N

38∘09984000998400998400N

39∘09984000998400998400N

34∘09984000998400998400N

33∘09984000998400998400N

35∘09984000998400998400N

36∘09984000998400998400N

37∘09984000998400998400N

38∘09984000998400998400N

39∘09984000998400998400N

N

E

S

W

Basin boundaryWater systemStation

Eigenvector 3value

High 0223536

Low minus0365226

Yanrsquoan

Xirsquoan

Zhenrsquoan

minus01

minus0021

minus001minus0021

minus0019

minus0241

minus0383

minus0271

minus0291

minus0281

minus0084

minus0308minus0086

Figure 6 The eigenvector 3 values for annual precipitation in the Weihe basin

to northwest and downstream to upstream The firstvariation points occurred in the early 1960s at theShangxian Wugong and Xirsquoan stations in the Beidaoto Huaxian section of the Weihe mainstream andlater in the late 1960s and early 1970s other stationsvariation occurred in this section In the Jingheand Beiluohe basins the first variation occurred inthe early 1970s In the Weihe basin above Beidaostationand variation occurred most recently in themid- and late 1980s except at Tianshui station (1971)

(2) Variation points in the trends of annual precipitationin the basin above Beidao station were concentratedin the 1980s while they covered longer periods ofthe 1970s and 1980s in other subbasins These resultsindicate that precipitation variation points occurredfrequently during the 1970s to 1980s in each subbasin

(3) After the 1990s the frequency and scope of precipi-tation variation declined No variation points in the1990s were detected except at Guyuan Pingliangand Wuqi stations After the 1990s there were novariation points in the annual precipitation sequencesrecorded in the Beidao to Huaxian section of theWeihe mainstream Jinghe basin or Beiluohe basinThese results show that after an active period ofprecipitation variation in 1970s and 1980s the annual

precipitation sequence in the Weihe basin was rela-tively stableThus during the study period there wereshifts in precipitation from one stable state to anotherafter disturbance

(4) From 2000 to 2011 further changes in annual pre-cipitation occurred in the basin above the Beidaostation and at the most northern stations of thebasin (Huanxian and Wuqi) These variations wereclearly regional concentrated in the northwest of theWeihe River basin and the source areas of the riversJinghe and Beiluohe Due to the limited length of thetime series it is not possible to distinguish whetherthese variations were due to small-scale regionalvariations or represent initial stages of wide-rangingvariation across the whole basin or were the resultof lingering temporal and spatial effects of the basin-wide variation that occurred in the 1970s and 1980s

(5) The variation in annual precipitation in the Weihebasin propagated from the southeast to the northwestThis is consistent with the orientation of the eigen-vector 2 field of annual precipitation obtained fromEOF decomposition and the spatial pattern of meanannual precipitation in the basin The factors causingthese patterns are not clear from the data presentedhere but will be addressed in future studies

10 Journal of Applied Mathematics

Xiji

Wuqi

SuideYanci

Baoji

Jintai

Foping

Wugong

Guyuan

Lintao

Tongxin

Haiyuan

Yuzhong

HuashanMinxian

Changwu

Hengshan

Jingyuan

Tianshui

Luochuan

Huanxian

Shangzhou

Zhongning

Tongchuan

PingliangXifengzhen

Huajialing

001

039

103077

0347

0078

0019

0009002103010548

0034

0658 0179

00062

N

E

S

W

Trend of eigenvector 3value

High 0761376

Low minus0354567

Yanrsquoan

Zhenrsquoan

Xirsquoan

minus0258

minus0531

minus0536

minus0141

minus002

minus0016

minus0008

minus0017

minus0164

minus0123

minus0123

minus0005

minus0074

minus01

minus0163

50 25 0 50 100 150 200

(km)

103∘09984000998400998400E 104

∘09984000998400998400E 105

∘09984000998400998400E 106

∘09984000998400998400E 107

∘09984000998400998400E 108

∘09984000998400998400E 109

∘09984000998400998400E 110

∘09984000998400998400E

103∘09984000998400998400E 104

∘09984000998400998400E 105

∘09984000998400998400E 106

∘09984000998400998400E 107

∘09984000998400998400E 108

∘09984000998400998400E 109

∘09984000998400998400E 110

∘09984000998400998400E

34∘09984000998400998400N

33∘09984000998400998400N

35∘09984000998400998400N

36∘09984000998400998400N

37∘09984000998400998400N

38∘09984000998400998400N

39∘09984000998400998400N

34∘09984000998400998400N

33∘09984000998400998400N

35∘09984000998400998400N

36∘09984000998400998400N

37∘09984000998400998400N

38∘09984000998400998400N

39∘09984000998400998400N

Basin boundaryWater systemStation

Figure 7 The eigenvector 3 trend field for annual precipitation in the Weihe basin

6 Conclusions

We have analyzed spatial and temporal patterns of annualprecipitation in the Weihe basin using data recorded from1960 to 2011 at 30 National Standard Stations in the basinand surrounding area by applying empirical orthogonalfunction decomposition in combination with geostatistical(semivariogram) analysis then using the Mann-Kendallmethod to discriminate variation points in the precipitationseries recorded at each station

(1) The main spatial patterns of annual precipitation inthe Weihe Basin during the study period were aconsistent overall decline which was strongest in thenorthwest and progressively weaker from northwestto southwest and from west to east The consistentoverall decline accounted for 455 of the totalvariation

(2) Variation points were detected in the annual precip-itation sequences recorded at all the meteorologicalstations in the basin but their timing had clear spatialpatterns starting gradually later from the southeastto the northwest and downstream to upstream Inthe 1970s and 1980s they occurred frequently in eachsubbasin

(3) After the 1990s following the active period of precip-itation variation in the 1970s and 1980s the frequencyand scope of precipitation variations declined Thusduring the study period there were shifts in precipita-tion fromone stable state to another after disturbance

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was financially supported by the Natural ScienceFoundation of China (Grant nos 51190093 51179148 and51179149) the National Key Basic Research 973 of China(Grant no 2012CB417003) and the MWR Welfare SectorFunds (Grant nos 201101043 201101049) Constructive com-ments from reviewers of a previous version of the paper aregratefully acknowledged

References

[1] H Zhang Y Chen G Ren et al ldquoThe characteristics of pre-cipitation variation of Weihe River Basin in Shaanxi Province

Journal of Applied Mathematics 11

during recent 50 yearsrdquo Agricultural Research in the Arid Areasvol 26 no 04 pp 236ndash241 2008

[2] K Che and J Jian ldquoA study of climate variability in Zedangduring 1960ndash1999rdquo Meteorological Monthly vol 26 no 1 pp54ndash56 2000

[3] Y Ding G Ren G Shi et al ldquoNational assessment report ofclimate change (I) climate change inChina and its future trendrdquoAdvances in Climate Change Research vol 2 no 1 pp 3ndash8 2006

[4] Y He F Wang X Mu et al ldquoTemporal and spatial character-istics of temperature and precipitation in Weihe River basinrdquoBulletin of Soil and Water Conservation vol 32 no 4 pp 102ndash105 2012

[5] Y-X Zhang F Wang and X-M Mu ldquoResearch of quantitativerelations between Geographical environment factor and waterand sediment in the Weihe River basinrdquo Journal of Northwest Aamp FUniversity (Natural Science Edition) vol 37 no 1 pp 61ndash652009

[6] C Jiang F Wang and X Mu ldquoEffect of climate change on netprimary productivity (NPP) of natural vegetation in Wei riverbasin (II) NPP of natural vegetation inWei river basinrdquo Journalof Arid LandResources and Environment vol 27 no 5 pp 53ndash572013

[7] Y Zhang ldquoHydrologic feature analysis for Weihe basin inShaanxi provincerdquo Northwest Water Resources amp Water Engi-neering no 2 pp 62ndash64 2002

[8] G Wang and Y Wang ldquoAnalysis on causes of runoff andsediment variation in Weihe basin based on hydrologicalmodelrdquo Journal of Soil Water Conservation no 4 pp 22ndash252000

[9] W He and Z Xu ldquoSpatial and temporal characteristics of thelong-term trend for temperature and pan evaporation in theWei River Basinrdquo Journal of Beijing Normal University (NaturalScience) vol 42 no 1 pp 102ndash106 2006

[10] K Pearson ldquoOn lines and planes of closest fit to system of pointsin spacerdquo Philosophical Magazine vol 6 no 2 pp 559ndash5721902

[11] D-X Cui and W-Y Liu ldquoApplication of empirical orthogonalfunction resolution to analysis of crustal vertical deformationfieldrdquo Earthquake vol 20 no 3 pp 82ndash86 2000

[12] T Liu and H Zhang ldquoCompared with principal componentanalysis and empirical orthogonal function decompositionrdquoStatistics and Decision vol 340 no 16 pp 159ndash162 2011

[13] H Lin and W Zhang ldquoSimilarities and differences of principalcomponent analysis and factor analysis and SPSS softwarerdquoStatistics Research no 3 pp 65ndash69 2005

[14] D H Kim C Yoo and T-W Kim ldquoApplication of spatial EOFand multivariate time series model for evaluating agriculturaldrought vulnerability in Koreardquo Advances in Water Resourcesvol 34 no 3 pp 340ndash350 2011

[15] N S Loboda A V Glushkov and V N Khokhlov ldquoUsingmeteorological data for reconstruction of annual runoff seriesover an ungauged area empirical orthogonal function approachto Moldova-Southwest Ukraine regionrdquo Atmospheric Researchvol 77 no 1ndash4 pp 100ndash113 2005

[16] D Sirjacobs A Alvera-Azcarate A Barth et al ldquoCloud fillingof ocean colour and sea surface temperature remote sensingproducts over the SouthernNorth Sea by theData InterpolatingEmpirical Orthogonal Functions methodologyrdquo Journal of SeaResearch vol 65 no 1 pp 114ndash130 2011

[17] H-L Yu and H-J Chu ldquoUnderstanding space-time patternsof groundwater system by empirical orthogonal functions a

case study in the Choshui River alluvial fan Taiwanrdquo Journalof Hydrology vol 381 no 3-4 pp 239ndash247 2010

[18] H BMann ldquoNonparametric tests against trendrdquo Econometricavol 13 pp 245ndash259 1945

[19] R Snyers ldquoSur LaDetermination de la stabilite des series clima-tologiquesrdquo in Proceedings of the UNESCO-WMO Symposium ofChanges of Climate vol 20 ofUneseo Arid Zone Research Seriespp 37ndash44 Unesco Paris France 1963

[20] C Goossens and A Berger ldquoAnnual and seasonal climaticvariations over the northern hemisphere and Europe during thelast centuryrdquo Annales Geophysicae B vol 4 no 4 pp 385ndash4001986

[21] C Fu and QWang ldquoThe definition and detection of the abruptclimatic changerdquo Chinese Journal of Atmospheric Sciences vol16 no 4 pp 482ndash493 1992

[22] P Xie G Chen D Li et al ldquoHydrological variation inte-grated diagnosis method and application researchrdquo Hydroelec-tric Energy vol 23 no 2 pp 11ndash14 2005

[23] M Gocic and S Trajkovic ldquoAnalysis of changes in meteoro-logical variables using Mann-Kendall and Senrsquos slope estimatorstatistical tests in SerbiardquoGlobal and Planetary Change vol 100pp 172ndash182 2013

[24] K H Hamed ldquoTrend detection in hydrologic data the Mann-Kendall trend test under the scaling hypothesisrdquo Journal ofHydrology vol 349 no 3-4 pp 350ndash363 2008

[25] S Kumar V Merwade J Kam and K Thurner ldquoStreamflowtrends in Indiana effects of long term persistence precipitationand subsurface drainsrdquo Journal of Hydrology vol 374 no 1-2pp 171ndash183 2009

[26] K H Hamed ldquoExact distribution of the Mann-Kendall trendtest statistic for persistent datardquo Journal of Hydrology vol 365no 1-2 pp 86ndash94 2009

[27] D Duhan and A Pandey ldquoStatistical analysis of long termspatial and temporal trends of precipitation during 1901ndash2002at Madhya Pradesh Indiardquo Atmospheric Research vol 122 pp136ndash149 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Applied Mathematics 3

321 Regression Analysis Regression analysis is applied toanalyze the correlationship of two variables which deter-mines relative direction and intensity In this study regressionanalysis is used to get the linear regression of precipitationand time and the equation is as follows

119875 = 119886119879 + 119887 (1)

where 119875 is the precipitation 119879 is the time and 119886 and 119887

are parameters If the unit of 119879 is the year usually 119886 is thechanging trend of annual precipitation

322 Empirical Orthogonal Function Decomposition EOFdecomposition involves decomposition of a matrix X of datadescribing the distribution of a variable in both space andtime into a spatial function V and time function T as shownin

X = VT

X =

[

[

[

[

[

[

[

[

119909

11sdot sdot sdot 119909

1119895sdot sdot sdot 119909

1119899

119909

1198941sdot sdot sdot 119909

119894119895sdot sdot sdot 119909

119894119899

119909

1198981sdot sdot sdot 119909

119898119895sdot sdot sdot 119909

119898119899

]

]

]

]

]

]

]

]

(2)

where119898 and 119899 are points in space and time respectively and119909

119894119895is the value of the variable at point 119894 and time 119895 (here

annual precipitation at station 119894 in year 119895)If V and T are orthogonal

119898

sum

119894=1

V119894119896V119894119897= 1 when 119896 = 119897

119899

sum

119895=1

119905

119896119895119905

119897119895= 0 when 119896 = 119897

(3)

If X is an anomaly matrix (reference) multiplying theequation X = VT by XT gives

XXT= VTXT

= VTTTVT= VΛV (4)

where Λ is a diagonal matrix composed of eigenvalues 120582

119894of

matrix XXT and

TTT= Λ (5)

Due to the characteristics of the eigenvectors VTV isa unit matrix meeting the requirements of orthogonalityThus the spatial function matrix V can be calculated fromeigenvectors of the matrix XXT

Here we multiply the spatial functionmatrixV of annualprecipitation in the Weihe basin by the trend vector D =

1198891 1198892 119889119898 composed of trends in annual precipitationrecorded at each station (V

119889= DV) thus describing the

spatial trends of annual precipitation in the basinEOF decomposition was first established in 1902 [10]

It decomposes elements of a set of space-time data into

two functions one dependent only on time and anotherdependent only on space to analyze the spatial structure ofelement fields [11] Thus EOF decomposition is commonlyused to analyze spatial and temporal variations in meteoro-logical elements In contrast principal component analysis(PCA) is mainly used to find a few independent vectors ina dataset to reduce its dimensionality thereby simplifyingdata handling and interpretation However the operationalsteps are basically the same when the original variables areanomaly data or standardized anomaly data [12] Hence wecan use the PCA module of SPSS (statistical product andservice solution) software for EOF decomposition [13]

EOF decomposition and several variants thereof havebeen widely applied for analyzing spatial changes in mete-orological elements Notably EOF decomposition has beenused to analyze drought occurrence in both time and spacein Korea [14] It has also been applied to establish contribu-tions of annual evaporation annual precipitation and bothwarm- and cold-season precipitation to the annual runoffof rivers in an arid zone to obtain quantitative estimates ofthe effects of climatic factors on reconstructions of annualrunoff series in an ungauged area [15] A data-interpolatingempirical orthogonal function (DINEOF) has been used toanalyze distributions of sea surface temperatures in a studydemonstrating that DINEOF decomposition can be highlyeffective evenwhen very high proportions of data aremissing[16] In addition rotated empirical orthogonal function(REOF) decomposition has been applied to analyze space-time patterns of groundwater fluctuations in the ChoshuiRiver alluvial fan Taiwan based on monthly observations ofpiezometric heads from66wells during the period 1997ndash2002[17]

323 Mann-Kendall Variation Analysis The Mann-Kendalltest has been widely used for identifying trends in time-series data because it is nonparametric and thus suitable forhandling data that are not normally distributed and it isnot sensitive to interference from a few outliers The Mann-Kendall test can also be used for variation diagnosis and thecalculation method is as follows

For a time series 119883 = 119909

1 119909

2 119909

119899 the Mann-Kendall

test statistic 119863

120591is given by the following model

119863

120591=

120591

sum

119894=1

119877

119894(120591 = 2 3 119894) (6)

where

119877

119894=

+1 119909

119894gt 119909

119895

0 119909

119894le 119909

119895

(119895 = 1 2 119894) (7)

The sequential statistic UF120591(in which UF

1= 0) is

calculated by is

UF120591=

1003816

1003816

1003816

1003816

119863

120591minus 119864 (119863

120591)

1003816

1003816

1003816

1003816

radic119881 (119863

120591)

(8)

4 Journal of Applied Mathematics

520

500

540560

480

460

580

600

620

640 66

0

440

420

0

400

700

380

720

360

740

340

760780800

420

740

680

760

780

440

540

700

700

720

400

700

560

480

Xiji

Wuqi

SuideYanci

B

Jintai

Foping

Wugong

Guyuan

Lintao

Tongxin

Haiyuan

Yuzhong

HuashanMinxian

Changwu

Hengshan

Jingyuan

Tianshui

Luochuan

Huanxian

Zhongning

Shangzhou

Tongchuan

PingliangXifengzhen

Huajialing

662

612

2643

3725

4412

369228962036

3842

2311

7806

6888

8982

81855599568935

51735805

59515838

55461842

4136

53194332

4529

4609

48785286

18505

aoji

Yanrsquoan

Xirsquoan

Zhenrsquoan50 25 0 50 100 150 200

(km)

Basin boundaryWater systemPrecipitation isoline

Station

103∘09984000998400998400E 104

∘09984000998400998400E 105

∘09984000998400998400E 106

∘09984000998400998400E 107

∘09984000998400998400E 108

∘09984000998400998400E 109

∘09984000998400998400E 110

∘09984000998400998400E

103∘09984000998400998400E 104

∘09984000998400998400E 105

∘09984000998400998400E 106

∘09984000998400998400E 107

∘09984000998400998400E 108

∘09984000998400998400E 109

∘09984000998400998400E 110

∘09984000998400998400E

34∘09984000998400998400N

33∘09984000998400998400N

35∘09984000998400998400N

36∘09984000998400998400N

37∘09984000998400998400N

38∘09984000998400998400N

39∘09984000998400998400N

34∘09984000998400998400N

33∘09984000998400998400N

35∘09984000998400998400N

36∘09984000998400998400N

37∘09984000998400998400N

38∘09984000998400998400N

39∘09984000998400998400N

N

E

S

W

High 819619

Low 320163

Annual precipitation

Figure 2 Isohyets showing mean annual precipitation in the Weihe basin

where 119864(119863

120591) and 119881(119863

120591) are the mean and variance of the

series given respectively by

119864 (119863

120591) =

120591 (120591 minus 1)

4

119881 (119863

120591) =

120591 (120591 minus 1) (2120591 + 5)

72

(9)

Then repeat the calculation with the inversion series1198831015840 =119909

119899 119909

119899minus1 119909

1 and let UB

120591= minusUF

120591 (120591 = 119899 119899 minus 1 1)

Define the significant level 120572 and find out the critical valueplusmn119880

(1minus120572)2 If the curves of UF

120591and UB

120591intersect and the

intersections are between the critical line (in this study thesignificant level 120572 is 095 and the critical line is plusmn119880

0052=

plusmn196) the time of curve intersections is the starting time ofthe variation

Originally published in 1945 [18] it was initially onlyused to detect trends in sequences However subsequentmodifications have enabled its use for detecting variationpoints in various trends [19] and it has been applied inreversed sequence to detect variation points in trends inclimate data time series [20] Since then the method hasbeen widely used to diagnose variation points in hydrologicaland meteorological data sequences [21 22] Notably the M-K method has been used to determine whether there werepositive or negative trends in seven meteorological variables(and if so their significance) recorded at 12 weather stations in

Serbia during 1980ndash2010 [23] Amodified formof themethodhas also been used to test scaling effects [24] and four variantshave been applied to determine trends in selected streamflowstatistics from Indiana [25] In addition a procedure forcalculating exact distributions of the M-K trend test statisticfor persistent data with an arbitrary correlation structure hasbeen presented [26] and the test has been used to detectdirections and magnitudes of changes in monotonic trendsin annual and seasonal precipitation over time in annualprecipitation inMadhya Pradesh India [27] Here we appliedthe M-K test to detect trends variation points in the timeseries of the data obtained from the 30 stations in the Weihebasin (Figure 1)

4 Results

41 Spatial Patterns of Annual Precipitation in the WeiheBasin Means and trends in annual precipitation in theWeihebasin from 1960 to 2011 are shown in Figures 2 and 3 Themean annual precipitation at each station changed greatlySpatially it declined from the southeast to northwest Meanannual precipitation in the basin amounted to 580 mm witha maximum exceeding 800mm at Huashan and Zhenrsquoan areaand a minimum of 340mm in the northwest The annualprecipitation trend at each station was negative except atXirsquoan where it increased by 0123mma The mean trend

Journal of Applied Mathematics 5

Xiji

Wuqi

SuideYanci

Baoji

Jintai

Foping

Wugong

Guyuan

Lintao

Tongxin

Haiyuan

Yuzhong

HuashanMinxian

Changwu

Hengshan

Jingyuan

Tianshui

Luochuan

Huanxian

Zhongning

Shangzhou

Tongchuan

PingliangXifengzhen

Huajialing

minus20382

Xirsquoan

Zhenrsquoan

Yanrsquoan

50 25 0 50 100 150 200

(km)

N

E

S

W

Trend of annual precipitation High minus0884939

Low minus179305

103∘09984000998400998400E 104

∘09984000998400998400E 105

∘09984000998400998400E 106

∘09984000998400998400E 107

∘09984000998400998400E 108

∘09984000998400998400E 109

∘09984000998400998400E 110

∘09984000998400998400E

103∘09984000998400998400E 104

∘09984000998400998400E 105

∘09984000998400998400E 106

∘09984000998400998400E 107

∘09984000998400998400E 108

∘09984000998400998400E 109

∘09984000998400998400E 110

∘09984000998400998400E

34∘09984000998400998400N

33∘09984000998400998400N

35∘09984000998400998400N

36∘09984000998400998400N

37∘09984000998400998400N

38∘09984000998400998400N

39∘09984000998400998400N

34∘09984000998400998400N

33∘09984000998400998400N

35∘09984000998400998400N

36∘09984000998400998400N

37∘09984000998400998400N

38∘09984000998400998400N

39∘09984000998400998400N

01229

minus 0 785

minus 1366

minus 1 076minus 04934

minus 0 9699

minus 09262

minus 08778

minus 03846

minus 08691

minus 086 72minus 09866

minus 01212

minus 02984

minus 1821

minus 10304

minus 12346

0036300

minus 3 8975minus 12491

minus 20241

minus 14991

minus 21349

minus 21367minus 19091

minus 20868

minus 14559

minus 26897minus 26428

Basin boundaryWater systemStation

Figure 3 Annual precipitation trends in the Weihe basin

Table 1 Eigenvalues and their contributions to the variance of annual precipitation in the Weihe basin obtained from EOF decomposition

Component

Initial eigenvalues Extraction sums of squared loadings

Eigenvalues120582

119894

Proportion ofvariation ()

Cumulativeproportion ofvariation ()

Eigenvalues120582

119894

Proportion ofvariation ()

Cumulativeproportion ofvariation ()

1 13648 45494 45494 13648 45494 454942 5363 17875 63369 5363 17875 633693 1997 6658 70027 1997 6658 700274 1404 4681 747085 1196 3985 786936 0912 3039 817327 0685 2285 840178 0634 2113 861309 0506 1686 8781610 0422 1408 89224Extraction method principal component analysis

across the basin was minus1662mma clearly showing thatannual precipitation declined and that this trend was mostpronounced in upstream areas

42 EOF Decomposition of Annual Precipitation in the WeiheBasin The precipitation data analyzed herein originatedfrom the Weihe basin and the surrounding 30 nationalstandard stations during the period 1960ndash2011 From this

data we derived the normalized anomaly matrix X forannual precipitation Empirical orthogonal function (run inSPSS) was used to calculate the eigenvalues of the covariancematrix of X and the amount of variation that each principalcomponent explained as shown in Table 1

Eigenvalues for the covariance matrix of the Weihebasin annual precipitation matrix and their contributionsto variance (specific and cumulative) obtained by empirical

6 Journal of Applied Mathematics

orthogonal function (using SPSS software) are shown inTable 1 The precipitation field rapidly converged the firstprincipal component (PC) accounted for almost half (455)of the variation in annual precipitation and the first threecomponents (all of which had eigenvalues exceeding 1) for700 of the variationThus the first three PCs and eigenvec-tors describe the spatial distribution of annual precipitationin the Weihe basin quite well

Eigenvectors of the covariance matrix cannot be directlyobtained from the SPSS output but can be obtained by div-iding the 119894 vectors of the matrix by the square roots of thecorresponding eigenvalues The vector corresponding to thelargest eigenvalue is eigenvector 1 with progressively smalleigenvalues associated with subsequent eigenvectors (2 34 etc) The first three eigenvectors obtained from the EOFdecomposition of the annual precipitation in theWeihe basindataset are shown in Table 2

The first three eigenvectors obtained from the EOF anal-ysis provide an approximate description of the precipitationvalues If their values are consistently positive or negativethere is a consistent precipitation trend and the area withthe largest absolute value is the center of the variable field Incontrast if the eigenvector values fluctuate between positiveand negative the trend is nonuniform

We used the Geostatistical Analyst module of ArcGIS toundertake a semivariogram analysis of the first three eigen-vectors and thus to obtain the spatial variability parametersshown in Table 3We then usedKriging interpolation to drawa map of each eigenvector field

As can be seen from Table 3 the azimuth angles ofeigenvectors 1 2 and 3 are 47∘ 655∘ and 3096∘ approx-imately south to north east-northeast to west-southwestand northwest to southeast orientations respectively All theeigenvectors have a small Nugget value indicating that theprecipitationmeasurement error is small Indeed the Nuggetof eigenvector 2 is zero indicating that themeasurement errorfor precipitation in this eigenvector field can be ignored Thepartial sill of eigenvector 1 is significantly lower than that ofthe other eigenvectors indicating that it reflects smaller-scalevariations The Weihe basin spans about 7∘ longitude and 4∘latitude Thus since the major ranges in the eigenvectors are629∘ latitude and longitude the variability of precipitationchanges with distance across the basin

43 VariationDiagnosis of Annual Rainfall in theWeihe BasinThe Weihe basin is divided into four subbasins with differ-ing hydrological characteristics and controlled by differenthydrological stations The areas covered by the subbasinsand locations of hydrological and meteorological stations areshown in Figure 1TheM-Kmethodwas used to discriminatevariation points in the trends in the annual precipitationseries recorded at each station during 1960ndash2011 and theresults are shown in Table 4

5 Analysis and Discussion of Results

51 Analysis of Spatial Variation of Annual Precipitation inthe Weihe Basin As shown in Table 2 all components of

Table 2 Eigenvectors obtained byEOFdecomposition of the annualprecipitation in the Weihe basin dataset

EigenvectorsEigenvector 1

119881

1

Eigenvector 2119881

2

Eigenvector 3119881

3

Lintao 0190 minus0137 minus0291Huajialing 0190 minus0043 minus0383Wuqi 0196 minus0157 0112Guyuan 0224 minus0082 minus0086Huanxia 0210 minus0140 0086Yanrsquoan 0208 0011 0251Xiji 0192 minus0131 minus0308Pingliang

minus0070 0045 minus0019Xifengzhen 0222 0104 0067Changwu 0204 0201 0058Luochuan 0200 0139 0179Tongchua 0154 0248 0168Minxian 0202 0001 minus0271Tianshui 0204 0057 minus0241Baoji 0167 0287 minus0021Wugong 0180 0273 minus0010Xirsquoan 0158 0277 0051Huashan 0163 0215 minus0100Foping 0177 0218 0009Zhenrsquoan 0157 0245 0042Shangzhou 0167 0223 minus0021Jingtai 0119 minus0198 0272Jingyuan 0167 minus0250 minus0084Yuzhong 0178 minus0156 minus0281Zhongning 0182 minus0206 0127Yanci 0170 minus0200 0250Hengshan 0183 minus0120 0240Suide 0175 minus0108 0260Haiyuan 0199 minus0231 0004Tongxin 0192 minus0221 0072

Table 3 Spatial variability parameters of eigenvectors of annualprecipitation in the Weihe basin

EigenvectorsEigenvector 1

119881

1

Eigenvector 2119881

2

Eigenvector 3119881

3

Azimuth angle (∘) 47 655 3096Nugget (119862

0) 00012685 0 00013054

Partial sill (119862) 00014968 041084 0044187Major range(∘ latitude andlongitude)

628911 628911 628911

eigenvector 1 are positive except the data from Pingliang sta-tion indicating that precipitation in theWeihe basin had highspatial consistency during the study period The downward

Journal of Applied Mathematics 7

Table 4 Variation points in annual precipitation trends in the Weihe basin

Subbasins Meteorologicalstations

Variation points1960ndash1969 1970ndash1979 1980ndash1989 1990ndash1999 2000ndash2011

Weihe basin aboveBeidao station

Tianshui 1971 1983 1985 2002Minxian 1986 1988 2006Huajialing 1986Lintao 1986 1987 2003 2006 2008Xiji 2005

Jinghe basin

Huanxian 1970 2002Guyuan 1972 1986 1996Pingliang 1971 1977 1980ndash1985lowast 1990Xifengzhen 1972 1979Changwu 1978 1979 1983 1985

Beiluohe basinWuqi 1971ndash1974lowast 1981 1983 1986ndash1995lowast 1998 2000 2003 2007

Yanrsquoan 1972ndash1975lowast1978 1979 1982

Luochuan 1970 1976

Beidao to Huaxiansection of Weihemainstream

Baoji 1969 1971 19741976 1978 1980 1982

Wugong 19611964ndash1970lowast

Xirsquoan 1962 1965Tongchuan 1976 1983 1985Zhenrsquoan 1972 1975

Foping 1966 1968 1973 19771979 1986 1988

Shangxian 1960 1963 1971 1975 1984Huanshan 1986

Extraction method Mann-Kendall method lowastindicates that variation occurred frequently during the indicated period

trend of annual precipitation shows that annual precipitationin the basin declined in the region and this consistent trendaccounts for 455 of the variance The basin covers parts ofthe continental monsoon zone and transitional zone betweenarid and humid regions of northwest China The overallconsistency and decline represented by eigenvector 1 are themain spatial features of precipitation patterns in the Weihebasin during the study period

The eigenvector 2 values obtained from EOF decomposi-tion of the annual precipitation data are shown in Figure 4The values can be divided by a line passing roughly fromYanrsquoan through Pingliang to Tianshui into positive andnegative anomaly areas covering the southeastern parts of theWeihe basin (with themost-positive anomaly center in Baoji)and northwestern parts (with the most-negative anomalycenter in Wu Qi) respectively These results confirm thatthe annual precipitation in the basin has a clear northwest-southeast pattern

Combining the eigenvalues with the trends in annualprecipitation recorded at each of the stations gives the eigen-vector 2 trend values (Figure 5) which show that precipita-tion declined in the southeast and increased in the northwestThe decline was most pronounced at the downstream stationHuashan

The eigenvector 3 values have a clear eastwest pattern(Figure 6) with a negative anomaly area covering upstreamparts of the basin and a positive anomaly center located in theLuochuan region north of ShaanxiThese findings show thatin addition to the consistent overall pattern and northwest-southeast precipitation patterns there is a clear eastwestpattern In combination with the precipitation trends at the30 stations the results clearly show that annual precipitationdeclined in northern Shaanxi and increased in the upstreamareas (Figure 7)

To summarize there were three spatial patterns of annualprecipitation in the Weihe basin during the study period (1)consistent overall decline (2) declined in the southeast andincreased in the northwest (3) declined in northern Shaanxiand increased in the upstream areas

52 Analysis of Annual Precipitation Variation in the WeiheBasin The results summarized in Table 4 show that varia-tion points occurred in trends of the annual precipitationsequence recorded at every meteorological station in theWeihe basin but their timings differed in the followingmanner

(1) The timing of variation points has clear spatial char-acteristics with gradual progressions from southeast

8 Journal of Applied Mathematics

Xiji

Wuqi

SuideYanci

Baoji

Jintai

Foping

Wugong

Guyuan

Lintao

Tongxin

Haiyuan

Yuzhong

HuashanMinxian

Changwu

Hengshan

Jingyuan

Tianshui

Luochuan

Huanxian

Zhongning

Shangzhou

Tongchuan

PingliangXifengzhen

Huajialing

0223

0245

0218

0215027702730287

00570001

0248

0

0201

01040045

Yanrsquoan

Zhenrsquoan

Xirsquoan

139

minus02

minus0206

minus0198

minus025

minus0156

minus0137minus00428

minus0131minus0082

minus014

minus0157

minus012

minus0108

minus0221

minus0231

N

E

S

W

103∘09984000998400998400E 104

∘09984000998400998400E 105

∘09984000998400998400E 106

∘09984000998400998400E 107

∘09984000998400998400E 108

∘09984000998400998400E 109

∘09984000998400998400E 110

∘09984000998400998400E

103∘09984000998400998400E 104

∘09984000998400998400E 105

∘09984000998400998400E 106

∘09984000998400998400E 107

∘09984000998400998400E 108

∘09984000998400998400E 109

∘09984000998400998400E 110

∘09984000998400998400E

34∘09984000998400998400N

33∘09984000998400998400N

35∘09984000998400998400N

36∘09984000998400998400N

37∘09984000998400998400N

38∘09984000998400998400N

39∘09984000998400998400N

34∘09984000998400998400N

33∘09984000998400998400N

35∘09984000998400998400N

36∘09984000998400998400N

37∘09984000998400998400N

38∘09984000998400998400N

39∘09984000998400998400N

50 25 0 50 100 150 200

(km)

Basin boundaryWater systemStation

Eigenvector 2High 0286104

Low minus0196997

Figure 4 The eigenvector 2 values for annual precipitation in the Weihe basin

Xiji

Wuqi

SuideYanci

Baoji

Jintai

Foping

Wugong

Guyuan

Lintao

Tongxin

Haiyuan

Yuzhong

HuashanMinxian

Changwu

Hengshan

Jingyuan

Tianshui

Luochuan

Huanxian

Zhongning

Shangzhou

Tongchuan

PingliangXifengzhen

Huajialing

022

023

028

0227

0316

01290099

0199

0192

0034

0266

0172

0228

01150363

50 25 0 50 100 150 200

(km)

103∘09984000998400998400E 104

∘09984000998400998400E 105

∘09984000998400998400E 106

∘09984000998400998400E 107

∘09984000998400998400E 108

∘09984000998400998400E 109

∘09984000998400998400E 110

∘09984000998400998400E

103∘09984000998400998400E 104

∘09984000998400998400E 105

∘09984000998400998400E 106

∘09984000998400998400E 107

∘09984000998400998400E 108

∘09984000998400998400E 109

∘09984000998400998400E 110

∘09984000998400998400E

34∘09984000998400998400N

33∘09984000998400998400N

35∘09984000998400998400N

36∘09984000998400998400N

37∘09984000998400998400N

38∘09984000998400998400N

39∘09984000998400998400N

34∘09984000998400998400N

33∘09984000998400998400N

35∘09984000998400998400N

36∘09984000998400998400N

37∘09984000998400998400N

38∘09984000998400998400N

39∘09984000998400998400N

N

E

S

W

Basin boundaryWater systemStation

Trend of eigenvector 2value

High 0229457

Low minus0461604

minus0023

minus0109

minus003

minus0838

minus0094

minus0196minus019

minus0236

minus006

minus0156

minus0083

minus0283

minus0072minus00018

minus0007

Yanrsquoan

Xirsquoan

Zhenrsquoan

Figure 5 The eigenvector 2 trend field for annual precipitation in the Weihe basin

Journal of Applied Mathematics 9

Xiji

Wuqi

SuideYanci

Baoji

Jintai

Foping

Wugong

Guyuan

Lintao

Tongxin

Haiyuan

Yuzhong

HuashanMinxian

Changwu

Hengshan

Jingyuan

Tianshui

Luochuan

Huanxian

Zhongning

Shangzhou

Tongchuan

PingliangXifengzhen

Huajialing

026

024025

0072

0004

0127

0272

0042

0009

0051

0168

0179

0058

0067

025100860112

50 25 0 50 100 150 200

(km)

103∘09984000998400998400E 104

∘09984000998400998400E 105

∘09984000998400998400E 106

∘09984000998400998400E 107

∘09984000998400998400E 108

∘09984000998400998400E 109

∘09984000998400998400E 110

∘09984000998400998400E

103∘09984000998400998400E 104

∘09984000998400998400E 105

∘09984000998400998400E 106

∘09984000998400998400E 107

∘09984000998400998400E 108

∘09984000998400998400E 109

∘09984000998400998400E 110

∘09984000998400998400E

34∘09984000998400998400N

33∘09984000998400998400N

35∘09984000998400998400N

36∘09984000998400998400N

37∘09984000998400998400N

38∘09984000998400998400N

39∘09984000998400998400N

34∘09984000998400998400N

33∘09984000998400998400N

35∘09984000998400998400N

36∘09984000998400998400N

37∘09984000998400998400N

38∘09984000998400998400N

39∘09984000998400998400N

N

E

S

W

Basin boundaryWater systemStation

Eigenvector 3value

High 0223536

Low minus0365226

Yanrsquoan

Xirsquoan

Zhenrsquoan

minus01

minus0021

minus001minus0021

minus0019

minus0241

minus0383

minus0271

minus0291

minus0281

minus0084

minus0308minus0086

Figure 6 The eigenvector 3 values for annual precipitation in the Weihe basin

to northwest and downstream to upstream The firstvariation points occurred in the early 1960s at theShangxian Wugong and Xirsquoan stations in the Beidaoto Huaxian section of the Weihe mainstream andlater in the late 1960s and early 1970s other stationsvariation occurred in this section In the Jingheand Beiluohe basins the first variation occurred inthe early 1970s In the Weihe basin above Beidaostationand variation occurred most recently in themid- and late 1980s except at Tianshui station (1971)

(2) Variation points in the trends of annual precipitationin the basin above Beidao station were concentratedin the 1980s while they covered longer periods ofthe 1970s and 1980s in other subbasins These resultsindicate that precipitation variation points occurredfrequently during the 1970s to 1980s in each subbasin

(3) After the 1990s the frequency and scope of precipi-tation variation declined No variation points in the1990s were detected except at Guyuan Pingliangand Wuqi stations After the 1990s there were novariation points in the annual precipitation sequencesrecorded in the Beidao to Huaxian section of theWeihe mainstream Jinghe basin or Beiluohe basinThese results show that after an active period ofprecipitation variation in 1970s and 1980s the annual

precipitation sequence in the Weihe basin was rela-tively stableThus during the study period there wereshifts in precipitation from one stable state to anotherafter disturbance

(4) From 2000 to 2011 further changes in annual pre-cipitation occurred in the basin above the Beidaostation and at the most northern stations of thebasin (Huanxian and Wuqi) These variations wereclearly regional concentrated in the northwest of theWeihe River basin and the source areas of the riversJinghe and Beiluohe Due to the limited length of thetime series it is not possible to distinguish whetherthese variations were due to small-scale regionalvariations or represent initial stages of wide-rangingvariation across the whole basin or were the resultof lingering temporal and spatial effects of the basin-wide variation that occurred in the 1970s and 1980s

(5) The variation in annual precipitation in the Weihebasin propagated from the southeast to the northwestThis is consistent with the orientation of the eigen-vector 2 field of annual precipitation obtained fromEOF decomposition and the spatial pattern of meanannual precipitation in the basin The factors causingthese patterns are not clear from the data presentedhere but will be addressed in future studies

10 Journal of Applied Mathematics

Xiji

Wuqi

SuideYanci

Baoji

Jintai

Foping

Wugong

Guyuan

Lintao

Tongxin

Haiyuan

Yuzhong

HuashanMinxian

Changwu

Hengshan

Jingyuan

Tianshui

Luochuan

Huanxian

Shangzhou

Zhongning

Tongchuan

PingliangXifengzhen

Huajialing

001

039

103077

0347

0078

0019

0009002103010548

0034

0658 0179

00062

N

E

S

W

Trend of eigenvector 3value

High 0761376

Low minus0354567

Yanrsquoan

Zhenrsquoan

Xirsquoan

minus0258

minus0531

minus0536

minus0141

minus002

minus0016

minus0008

minus0017

minus0164

minus0123

minus0123

minus0005

minus0074

minus01

minus0163

50 25 0 50 100 150 200

(km)

103∘09984000998400998400E 104

∘09984000998400998400E 105

∘09984000998400998400E 106

∘09984000998400998400E 107

∘09984000998400998400E 108

∘09984000998400998400E 109

∘09984000998400998400E 110

∘09984000998400998400E

103∘09984000998400998400E 104

∘09984000998400998400E 105

∘09984000998400998400E 106

∘09984000998400998400E 107

∘09984000998400998400E 108

∘09984000998400998400E 109

∘09984000998400998400E 110

∘09984000998400998400E

34∘09984000998400998400N

33∘09984000998400998400N

35∘09984000998400998400N

36∘09984000998400998400N

37∘09984000998400998400N

38∘09984000998400998400N

39∘09984000998400998400N

34∘09984000998400998400N

33∘09984000998400998400N

35∘09984000998400998400N

36∘09984000998400998400N

37∘09984000998400998400N

38∘09984000998400998400N

39∘09984000998400998400N

Basin boundaryWater systemStation

Figure 7 The eigenvector 3 trend field for annual precipitation in the Weihe basin

6 Conclusions

We have analyzed spatial and temporal patterns of annualprecipitation in the Weihe basin using data recorded from1960 to 2011 at 30 National Standard Stations in the basinand surrounding area by applying empirical orthogonalfunction decomposition in combination with geostatistical(semivariogram) analysis then using the Mann-Kendallmethod to discriminate variation points in the precipitationseries recorded at each station

(1) The main spatial patterns of annual precipitation inthe Weihe Basin during the study period were aconsistent overall decline which was strongest in thenorthwest and progressively weaker from northwestto southwest and from west to east The consistentoverall decline accounted for 455 of the totalvariation

(2) Variation points were detected in the annual precip-itation sequences recorded at all the meteorologicalstations in the basin but their timing had clear spatialpatterns starting gradually later from the southeastto the northwest and downstream to upstream Inthe 1970s and 1980s they occurred frequently in eachsubbasin

(3) After the 1990s following the active period of precip-itation variation in the 1970s and 1980s the frequencyand scope of precipitation variations declined Thusduring the study period there were shifts in precipita-tion fromone stable state to another after disturbance

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was financially supported by the Natural ScienceFoundation of China (Grant nos 51190093 51179148 and51179149) the National Key Basic Research 973 of China(Grant no 2012CB417003) and the MWR Welfare SectorFunds (Grant nos 201101043 201101049) Constructive com-ments from reviewers of a previous version of the paper aregratefully acknowledged

References

[1] H Zhang Y Chen G Ren et al ldquoThe characteristics of pre-cipitation variation of Weihe River Basin in Shaanxi Province

Journal of Applied Mathematics 11

during recent 50 yearsrdquo Agricultural Research in the Arid Areasvol 26 no 04 pp 236ndash241 2008

[2] K Che and J Jian ldquoA study of climate variability in Zedangduring 1960ndash1999rdquo Meteorological Monthly vol 26 no 1 pp54ndash56 2000

[3] Y Ding G Ren G Shi et al ldquoNational assessment report ofclimate change (I) climate change inChina and its future trendrdquoAdvances in Climate Change Research vol 2 no 1 pp 3ndash8 2006

[4] Y He F Wang X Mu et al ldquoTemporal and spatial character-istics of temperature and precipitation in Weihe River basinrdquoBulletin of Soil and Water Conservation vol 32 no 4 pp 102ndash105 2012

[5] Y-X Zhang F Wang and X-M Mu ldquoResearch of quantitativerelations between Geographical environment factor and waterand sediment in the Weihe River basinrdquo Journal of Northwest Aamp FUniversity (Natural Science Edition) vol 37 no 1 pp 61ndash652009

[6] C Jiang F Wang and X Mu ldquoEffect of climate change on netprimary productivity (NPP) of natural vegetation in Wei riverbasin (II) NPP of natural vegetation inWei river basinrdquo Journalof Arid LandResources and Environment vol 27 no 5 pp 53ndash572013

[7] Y Zhang ldquoHydrologic feature analysis for Weihe basin inShaanxi provincerdquo Northwest Water Resources amp Water Engi-neering no 2 pp 62ndash64 2002

[8] G Wang and Y Wang ldquoAnalysis on causes of runoff andsediment variation in Weihe basin based on hydrologicalmodelrdquo Journal of Soil Water Conservation no 4 pp 22ndash252000

[9] W He and Z Xu ldquoSpatial and temporal characteristics of thelong-term trend for temperature and pan evaporation in theWei River Basinrdquo Journal of Beijing Normal University (NaturalScience) vol 42 no 1 pp 102ndash106 2006

[10] K Pearson ldquoOn lines and planes of closest fit to system of pointsin spacerdquo Philosophical Magazine vol 6 no 2 pp 559ndash5721902

[11] D-X Cui and W-Y Liu ldquoApplication of empirical orthogonalfunction resolution to analysis of crustal vertical deformationfieldrdquo Earthquake vol 20 no 3 pp 82ndash86 2000

[12] T Liu and H Zhang ldquoCompared with principal componentanalysis and empirical orthogonal function decompositionrdquoStatistics and Decision vol 340 no 16 pp 159ndash162 2011

[13] H Lin and W Zhang ldquoSimilarities and differences of principalcomponent analysis and factor analysis and SPSS softwarerdquoStatistics Research no 3 pp 65ndash69 2005

[14] D H Kim C Yoo and T-W Kim ldquoApplication of spatial EOFand multivariate time series model for evaluating agriculturaldrought vulnerability in Koreardquo Advances in Water Resourcesvol 34 no 3 pp 340ndash350 2011

[15] N S Loboda A V Glushkov and V N Khokhlov ldquoUsingmeteorological data for reconstruction of annual runoff seriesover an ungauged area empirical orthogonal function approachto Moldova-Southwest Ukraine regionrdquo Atmospheric Researchvol 77 no 1ndash4 pp 100ndash113 2005

[16] D Sirjacobs A Alvera-Azcarate A Barth et al ldquoCloud fillingof ocean colour and sea surface temperature remote sensingproducts over the SouthernNorth Sea by theData InterpolatingEmpirical Orthogonal Functions methodologyrdquo Journal of SeaResearch vol 65 no 1 pp 114ndash130 2011

[17] H-L Yu and H-J Chu ldquoUnderstanding space-time patternsof groundwater system by empirical orthogonal functions a

case study in the Choshui River alluvial fan Taiwanrdquo Journalof Hydrology vol 381 no 3-4 pp 239ndash247 2010

[18] H BMann ldquoNonparametric tests against trendrdquo Econometricavol 13 pp 245ndash259 1945

[19] R Snyers ldquoSur LaDetermination de la stabilite des series clima-tologiquesrdquo in Proceedings of the UNESCO-WMO Symposium ofChanges of Climate vol 20 ofUneseo Arid Zone Research Seriespp 37ndash44 Unesco Paris France 1963

[20] C Goossens and A Berger ldquoAnnual and seasonal climaticvariations over the northern hemisphere and Europe during thelast centuryrdquo Annales Geophysicae B vol 4 no 4 pp 385ndash4001986

[21] C Fu and QWang ldquoThe definition and detection of the abruptclimatic changerdquo Chinese Journal of Atmospheric Sciences vol16 no 4 pp 482ndash493 1992

[22] P Xie G Chen D Li et al ldquoHydrological variation inte-grated diagnosis method and application researchrdquo Hydroelec-tric Energy vol 23 no 2 pp 11ndash14 2005

[23] M Gocic and S Trajkovic ldquoAnalysis of changes in meteoro-logical variables using Mann-Kendall and Senrsquos slope estimatorstatistical tests in SerbiardquoGlobal and Planetary Change vol 100pp 172ndash182 2013

[24] K H Hamed ldquoTrend detection in hydrologic data the Mann-Kendall trend test under the scaling hypothesisrdquo Journal ofHydrology vol 349 no 3-4 pp 350ndash363 2008

[25] S Kumar V Merwade J Kam and K Thurner ldquoStreamflowtrends in Indiana effects of long term persistence precipitationand subsurface drainsrdquo Journal of Hydrology vol 374 no 1-2pp 171ndash183 2009

[26] K H Hamed ldquoExact distribution of the Mann-Kendall trendtest statistic for persistent datardquo Journal of Hydrology vol 365no 1-2 pp 86ndash94 2009

[27] D Duhan and A Pandey ldquoStatistical analysis of long termspatial and temporal trends of precipitation during 1901ndash2002at Madhya Pradesh Indiardquo Atmospheric Research vol 122 pp136ndash149 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Journal of Applied Mathematics

520

500

540560

480

460

580

600

620

640 66

0

440

420

0

400

700

380

720

360

740

340

760780800

420

740

680

760

780

440

540

700

700

720

400

700

560

480

Xiji

Wuqi

SuideYanci

B

Jintai

Foping

Wugong

Guyuan

Lintao

Tongxin

Haiyuan

Yuzhong

HuashanMinxian

Changwu

Hengshan

Jingyuan

Tianshui

Luochuan

Huanxian

Zhongning

Shangzhou

Tongchuan

PingliangXifengzhen

Huajialing

662

612

2643

3725

4412

369228962036

3842

2311

7806

6888

8982

81855599568935

51735805

59515838

55461842

4136

53194332

4529

4609

48785286

18505

aoji

Yanrsquoan

Xirsquoan

Zhenrsquoan50 25 0 50 100 150 200

(km)

Basin boundaryWater systemPrecipitation isoline

Station

103∘09984000998400998400E 104

∘09984000998400998400E 105

∘09984000998400998400E 106

∘09984000998400998400E 107

∘09984000998400998400E 108

∘09984000998400998400E 109

∘09984000998400998400E 110

∘09984000998400998400E

103∘09984000998400998400E 104

∘09984000998400998400E 105

∘09984000998400998400E 106

∘09984000998400998400E 107

∘09984000998400998400E 108

∘09984000998400998400E 109

∘09984000998400998400E 110

∘09984000998400998400E

34∘09984000998400998400N

33∘09984000998400998400N

35∘09984000998400998400N

36∘09984000998400998400N

37∘09984000998400998400N

38∘09984000998400998400N

39∘09984000998400998400N

34∘09984000998400998400N

33∘09984000998400998400N

35∘09984000998400998400N

36∘09984000998400998400N

37∘09984000998400998400N

38∘09984000998400998400N

39∘09984000998400998400N

N

E

S

W

High 819619

Low 320163

Annual precipitation

Figure 2 Isohyets showing mean annual precipitation in the Weihe basin

where 119864(119863

120591) and 119881(119863

120591) are the mean and variance of the

series given respectively by

119864 (119863

120591) =

120591 (120591 minus 1)

4

119881 (119863

120591) =

120591 (120591 minus 1) (2120591 + 5)

72

(9)

Then repeat the calculation with the inversion series1198831015840 =119909

119899 119909

119899minus1 119909

1 and let UB

120591= minusUF

120591 (120591 = 119899 119899 minus 1 1)

Define the significant level 120572 and find out the critical valueplusmn119880

(1minus120572)2 If the curves of UF

120591and UB

120591intersect and the

intersections are between the critical line (in this study thesignificant level 120572 is 095 and the critical line is plusmn119880

0052=

plusmn196) the time of curve intersections is the starting time ofthe variation

Originally published in 1945 [18] it was initially onlyused to detect trends in sequences However subsequentmodifications have enabled its use for detecting variationpoints in various trends [19] and it has been applied inreversed sequence to detect variation points in trends inclimate data time series [20] Since then the method hasbeen widely used to diagnose variation points in hydrologicaland meteorological data sequences [21 22] Notably the M-K method has been used to determine whether there werepositive or negative trends in seven meteorological variables(and if so their significance) recorded at 12 weather stations in

Serbia during 1980ndash2010 [23] Amodified formof themethodhas also been used to test scaling effects [24] and four variantshave been applied to determine trends in selected streamflowstatistics from Indiana [25] In addition a procedure forcalculating exact distributions of the M-K trend test statisticfor persistent data with an arbitrary correlation structure hasbeen presented [26] and the test has been used to detectdirections and magnitudes of changes in monotonic trendsin annual and seasonal precipitation over time in annualprecipitation inMadhya Pradesh India [27] Here we appliedthe M-K test to detect trends variation points in the timeseries of the data obtained from the 30 stations in the Weihebasin (Figure 1)

4 Results

41 Spatial Patterns of Annual Precipitation in the WeiheBasin Means and trends in annual precipitation in theWeihebasin from 1960 to 2011 are shown in Figures 2 and 3 Themean annual precipitation at each station changed greatlySpatially it declined from the southeast to northwest Meanannual precipitation in the basin amounted to 580 mm witha maximum exceeding 800mm at Huashan and Zhenrsquoan areaand a minimum of 340mm in the northwest The annualprecipitation trend at each station was negative except atXirsquoan where it increased by 0123mma The mean trend

Journal of Applied Mathematics 5

Xiji

Wuqi

SuideYanci

Baoji

Jintai

Foping

Wugong

Guyuan

Lintao

Tongxin

Haiyuan

Yuzhong

HuashanMinxian

Changwu

Hengshan

Jingyuan

Tianshui

Luochuan

Huanxian

Zhongning

Shangzhou

Tongchuan

PingliangXifengzhen

Huajialing

minus20382

Xirsquoan

Zhenrsquoan

Yanrsquoan

50 25 0 50 100 150 200

(km)

N

E

S

W

Trend of annual precipitation High minus0884939

Low minus179305

103∘09984000998400998400E 104

∘09984000998400998400E 105

∘09984000998400998400E 106

∘09984000998400998400E 107

∘09984000998400998400E 108

∘09984000998400998400E 109

∘09984000998400998400E 110

∘09984000998400998400E

103∘09984000998400998400E 104

∘09984000998400998400E 105

∘09984000998400998400E 106

∘09984000998400998400E 107

∘09984000998400998400E 108

∘09984000998400998400E 109

∘09984000998400998400E 110

∘09984000998400998400E

34∘09984000998400998400N

33∘09984000998400998400N

35∘09984000998400998400N

36∘09984000998400998400N

37∘09984000998400998400N

38∘09984000998400998400N

39∘09984000998400998400N

34∘09984000998400998400N

33∘09984000998400998400N

35∘09984000998400998400N

36∘09984000998400998400N

37∘09984000998400998400N

38∘09984000998400998400N

39∘09984000998400998400N

01229

minus 0 785

minus 1366

minus 1 076minus 04934

minus 0 9699

minus 09262

minus 08778

minus 03846

minus 08691

minus 086 72minus 09866

minus 01212

minus 02984

minus 1821

minus 10304

minus 12346

0036300

minus 3 8975minus 12491

minus 20241

minus 14991

minus 21349

minus 21367minus 19091

minus 20868

minus 14559

minus 26897minus 26428

Basin boundaryWater systemStation

Figure 3 Annual precipitation trends in the Weihe basin

Table 1 Eigenvalues and their contributions to the variance of annual precipitation in the Weihe basin obtained from EOF decomposition

Component

Initial eigenvalues Extraction sums of squared loadings

Eigenvalues120582

119894

Proportion ofvariation ()

Cumulativeproportion ofvariation ()

Eigenvalues120582

119894

Proportion ofvariation ()

Cumulativeproportion ofvariation ()

1 13648 45494 45494 13648 45494 454942 5363 17875 63369 5363 17875 633693 1997 6658 70027 1997 6658 700274 1404 4681 747085 1196 3985 786936 0912 3039 817327 0685 2285 840178 0634 2113 861309 0506 1686 8781610 0422 1408 89224Extraction method principal component analysis

across the basin was minus1662mma clearly showing thatannual precipitation declined and that this trend was mostpronounced in upstream areas

42 EOF Decomposition of Annual Precipitation in the WeiheBasin The precipitation data analyzed herein originatedfrom the Weihe basin and the surrounding 30 nationalstandard stations during the period 1960ndash2011 From this

data we derived the normalized anomaly matrix X forannual precipitation Empirical orthogonal function (run inSPSS) was used to calculate the eigenvalues of the covariancematrix of X and the amount of variation that each principalcomponent explained as shown in Table 1

Eigenvalues for the covariance matrix of the Weihebasin annual precipitation matrix and their contributionsto variance (specific and cumulative) obtained by empirical

6 Journal of Applied Mathematics

orthogonal function (using SPSS software) are shown inTable 1 The precipitation field rapidly converged the firstprincipal component (PC) accounted for almost half (455)of the variation in annual precipitation and the first threecomponents (all of which had eigenvalues exceeding 1) for700 of the variationThus the first three PCs and eigenvec-tors describe the spatial distribution of annual precipitationin the Weihe basin quite well

Eigenvectors of the covariance matrix cannot be directlyobtained from the SPSS output but can be obtained by div-iding the 119894 vectors of the matrix by the square roots of thecorresponding eigenvalues The vector corresponding to thelargest eigenvalue is eigenvector 1 with progressively smalleigenvalues associated with subsequent eigenvectors (2 34 etc) The first three eigenvectors obtained from the EOFdecomposition of the annual precipitation in theWeihe basindataset are shown in Table 2

The first three eigenvectors obtained from the EOF anal-ysis provide an approximate description of the precipitationvalues If their values are consistently positive or negativethere is a consistent precipitation trend and the area withthe largest absolute value is the center of the variable field Incontrast if the eigenvector values fluctuate between positiveand negative the trend is nonuniform

We used the Geostatistical Analyst module of ArcGIS toundertake a semivariogram analysis of the first three eigen-vectors and thus to obtain the spatial variability parametersshown in Table 3We then usedKriging interpolation to drawa map of each eigenvector field

As can be seen from Table 3 the azimuth angles ofeigenvectors 1 2 and 3 are 47∘ 655∘ and 3096∘ approx-imately south to north east-northeast to west-southwestand northwest to southeast orientations respectively All theeigenvectors have a small Nugget value indicating that theprecipitationmeasurement error is small Indeed the Nuggetof eigenvector 2 is zero indicating that themeasurement errorfor precipitation in this eigenvector field can be ignored Thepartial sill of eigenvector 1 is significantly lower than that ofthe other eigenvectors indicating that it reflects smaller-scalevariations The Weihe basin spans about 7∘ longitude and 4∘latitude Thus since the major ranges in the eigenvectors are629∘ latitude and longitude the variability of precipitationchanges with distance across the basin

43 VariationDiagnosis of Annual Rainfall in theWeihe BasinThe Weihe basin is divided into four subbasins with differ-ing hydrological characteristics and controlled by differenthydrological stations The areas covered by the subbasinsand locations of hydrological and meteorological stations areshown in Figure 1TheM-Kmethodwas used to discriminatevariation points in the trends in the annual precipitationseries recorded at each station during 1960ndash2011 and theresults are shown in Table 4

5 Analysis and Discussion of Results

51 Analysis of Spatial Variation of Annual Precipitation inthe Weihe Basin As shown in Table 2 all components of

Table 2 Eigenvectors obtained byEOFdecomposition of the annualprecipitation in the Weihe basin dataset

EigenvectorsEigenvector 1

119881

1

Eigenvector 2119881

2

Eigenvector 3119881

3

Lintao 0190 minus0137 minus0291Huajialing 0190 minus0043 minus0383Wuqi 0196 minus0157 0112Guyuan 0224 minus0082 minus0086Huanxia 0210 minus0140 0086Yanrsquoan 0208 0011 0251Xiji 0192 minus0131 minus0308Pingliang

minus0070 0045 minus0019Xifengzhen 0222 0104 0067Changwu 0204 0201 0058Luochuan 0200 0139 0179Tongchua 0154 0248 0168Minxian 0202 0001 minus0271Tianshui 0204 0057 minus0241Baoji 0167 0287 minus0021Wugong 0180 0273 minus0010Xirsquoan 0158 0277 0051Huashan 0163 0215 minus0100Foping 0177 0218 0009Zhenrsquoan 0157 0245 0042Shangzhou 0167 0223 minus0021Jingtai 0119 minus0198 0272Jingyuan 0167 minus0250 minus0084Yuzhong 0178 minus0156 minus0281Zhongning 0182 minus0206 0127Yanci 0170 minus0200 0250Hengshan 0183 minus0120 0240Suide 0175 minus0108 0260Haiyuan 0199 minus0231 0004Tongxin 0192 minus0221 0072

Table 3 Spatial variability parameters of eigenvectors of annualprecipitation in the Weihe basin

EigenvectorsEigenvector 1

119881

1

Eigenvector 2119881

2

Eigenvector 3119881

3

Azimuth angle (∘) 47 655 3096Nugget (119862

0) 00012685 0 00013054

Partial sill (119862) 00014968 041084 0044187Major range(∘ latitude andlongitude)

628911 628911 628911

eigenvector 1 are positive except the data from Pingliang sta-tion indicating that precipitation in theWeihe basin had highspatial consistency during the study period The downward

Journal of Applied Mathematics 7

Table 4 Variation points in annual precipitation trends in the Weihe basin

Subbasins Meteorologicalstations

Variation points1960ndash1969 1970ndash1979 1980ndash1989 1990ndash1999 2000ndash2011

Weihe basin aboveBeidao station

Tianshui 1971 1983 1985 2002Minxian 1986 1988 2006Huajialing 1986Lintao 1986 1987 2003 2006 2008Xiji 2005

Jinghe basin

Huanxian 1970 2002Guyuan 1972 1986 1996Pingliang 1971 1977 1980ndash1985lowast 1990Xifengzhen 1972 1979Changwu 1978 1979 1983 1985

Beiluohe basinWuqi 1971ndash1974lowast 1981 1983 1986ndash1995lowast 1998 2000 2003 2007

Yanrsquoan 1972ndash1975lowast1978 1979 1982

Luochuan 1970 1976

Beidao to Huaxiansection of Weihemainstream

Baoji 1969 1971 19741976 1978 1980 1982

Wugong 19611964ndash1970lowast

Xirsquoan 1962 1965Tongchuan 1976 1983 1985Zhenrsquoan 1972 1975

Foping 1966 1968 1973 19771979 1986 1988

Shangxian 1960 1963 1971 1975 1984Huanshan 1986

Extraction method Mann-Kendall method lowastindicates that variation occurred frequently during the indicated period

trend of annual precipitation shows that annual precipitationin the basin declined in the region and this consistent trendaccounts for 455 of the variance The basin covers parts ofthe continental monsoon zone and transitional zone betweenarid and humid regions of northwest China The overallconsistency and decline represented by eigenvector 1 are themain spatial features of precipitation patterns in the Weihebasin during the study period

The eigenvector 2 values obtained from EOF decomposi-tion of the annual precipitation data are shown in Figure 4The values can be divided by a line passing roughly fromYanrsquoan through Pingliang to Tianshui into positive andnegative anomaly areas covering the southeastern parts of theWeihe basin (with themost-positive anomaly center in Baoji)and northwestern parts (with the most-negative anomalycenter in Wu Qi) respectively These results confirm thatthe annual precipitation in the basin has a clear northwest-southeast pattern

Combining the eigenvalues with the trends in annualprecipitation recorded at each of the stations gives the eigen-vector 2 trend values (Figure 5) which show that precipita-tion declined in the southeast and increased in the northwestThe decline was most pronounced at the downstream stationHuashan

The eigenvector 3 values have a clear eastwest pattern(Figure 6) with a negative anomaly area covering upstreamparts of the basin and a positive anomaly center located in theLuochuan region north of ShaanxiThese findings show thatin addition to the consistent overall pattern and northwest-southeast precipitation patterns there is a clear eastwestpattern In combination with the precipitation trends at the30 stations the results clearly show that annual precipitationdeclined in northern Shaanxi and increased in the upstreamareas (Figure 7)

To summarize there were three spatial patterns of annualprecipitation in the Weihe basin during the study period (1)consistent overall decline (2) declined in the southeast andincreased in the northwest (3) declined in northern Shaanxiand increased in the upstream areas

52 Analysis of Annual Precipitation Variation in the WeiheBasin The results summarized in Table 4 show that varia-tion points occurred in trends of the annual precipitationsequence recorded at every meteorological station in theWeihe basin but their timings differed in the followingmanner

(1) The timing of variation points has clear spatial char-acteristics with gradual progressions from southeast

8 Journal of Applied Mathematics

Xiji

Wuqi

SuideYanci

Baoji

Jintai

Foping

Wugong

Guyuan

Lintao

Tongxin

Haiyuan

Yuzhong

HuashanMinxian

Changwu

Hengshan

Jingyuan

Tianshui

Luochuan

Huanxian

Zhongning

Shangzhou

Tongchuan

PingliangXifengzhen

Huajialing

0223

0245

0218

0215027702730287

00570001

0248

0

0201

01040045

Yanrsquoan

Zhenrsquoan

Xirsquoan

139

minus02

minus0206

minus0198

minus025

minus0156

minus0137minus00428

minus0131minus0082

minus014

minus0157

minus012

minus0108

minus0221

minus0231

N

E

S

W

103∘09984000998400998400E 104

∘09984000998400998400E 105

∘09984000998400998400E 106

∘09984000998400998400E 107

∘09984000998400998400E 108

∘09984000998400998400E 109

∘09984000998400998400E 110

∘09984000998400998400E

103∘09984000998400998400E 104

∘09984000998400998400E 105

∘09984000998400998400E 106

∘09984000998400998400E 107

∘09984000998400998400E 108

∘09984000998400998400E 109

∘09984000998400998400E 110

∘09984000998400998400E

34∘09984000998400998400N

33∘09984000998400998400N

35∘09984000998400998400N

36∘09984000998400998400N

37∘09984000998400998400N

38∘09984000998400998400N

39∘09984000998400998400N

34∘09984000998400998400N

33∘09984000998400998400N

35∘09984000998400998400N

36∘09984000998400998400N

37∘09984000998400998400N

38∘09984000998400998400N

39∘09984000998400998400N

50 25 0 50 100 150 200

(km)

Basin boundaryWater systemStation

Eigenvector 2High 0286104

Low minus0196997

Figure 4 The eigenvector 2 values for annual precipitation in the Weihe basin

Xiji

Wuqi

SuideYanci

Baoji

Jintai

Foping

Wugong

Guyuan

Lintao

Tongxin

Haiyuan

Yuzhong

HuashanMinxian

Changwu

Hengshan

Jingyuan

Tianshui

Luochuan

Huanxian

Zhongning

Shangzhou

Tongchuan

PingliangXifengzhen

Huajialing

022

023

028

0227

0316

01290099

0199

0192

0034

0266

0172

0228

01150363

50 25 0 50 100 150 200

(km)

103∘09984000998400998400E 104

∘09984000998400998400E 105

∘09984000998400998400E 106

∘09984000998400998400E 107

∘09984000998400998400E 108

∘09984000998400998400E 109

∘09984000998400998400E 110

∘09984000998400998400E

103∘09984000998400998400E 104

∘09984000998400998400E 105

∘09984000998400998400E 106

∘09984000998400998400E 107

∘09984000998400998400E 108

∘09984000998400998400E 109

∘09984000998400998400E 110

∘09984000998400998400E

34∘09984000998400998400N

33∘09984000998400998400N

35∘09984000998400998400N

36∘09984000998400998400N

37∘09984000998400998400N

38∘09984000998400998400N

39∘09984000998400998400N

34∘09984000998400998400N

33∘09984000998400998400N

35∘09984000998400998400N

36∘09984000998400998400N

37∘09984000998400998400N

38∘09984000998400998400N

39∘09984000998400998400N

N

E

S

W

Basin boundaryWater systemStation

Trend of eigenvector 2value

High 0229457

Low minus0461604

minus0023

minus0109

minus003

minus0838

minus0094

minus0196minus019

minus0236

minus006

minus0156

minus0083

minus0283

minus0072minus00018

minus0007

Yanrsquoan

Xirsquoan

Zhenrsquoan

Figure 5 The eigenvector 2 trend field for annual precipitation in the Weihe basin

Journal of Applied Mathematics 9

Xiji

Wuqi

SuideYanci

Baoji

Jintai

Foping

Wugong

Guyuan

Lintao

Tongxin

Haiyuan

Yuzhong

HuashanMinxian

Changwu

Hengshan

Jingyuan

Tianshui

Luochuan

Huanxian

Zhongning

Shangzhou

Tongchuan

PingliangXifengzhen

Huajialing

026

024025

0072

0004

0127

0272

0042

0009

0051

0168

0179

0058

0067

025100860112

50 25 0 50 100 150 200

(km)

103∘09984000998400998400E 104

∘09984000998400998400E 105

∘09984000998400998400E 106

∘09984000998400998400E 107

∘09984000998400998400E 108

∘09984000998400998400E 109

∘09984000998400998400E 110

∘09984000998400998400E

103∘09984000998400998400E 104

∘09984000998400998400E 105

∘09984000998400998400E 106

∘09984000998400998400E 107

∘09984000998400998400E 108

∘09984000998400998400E 109

∘09984000998400998400E 110

∘09984000998400998400E

34∘09984000998400998400N

33∘09984000998400998400N

35∘09984000998400998400N

36∘09984000998400998400N

37∘09984000998400998400N

38∘09984000998400998400N

39∘09984000998400998400N

34∘09984000998400998400N

33∘09984000998400998400N

35∘09984000998400998400N

36∘09984000998400998400N

37∘09984000998400998400N

38∘09984000998400998400N

39∘09984000998400998400N

N

E

S

W

Basin boundaryWater systemStation

Eigenvector 3value

High 0223536

Low minus0365226

Yanrsquoan

Xirsquoan

Zhenrsquoan

minus01

minus0021

minus001minus0021

minus0019

minus0241

minus0383

minus0271

minus0291

minus0281

minus0084

minus0308minus0086

Figure 6 The eigenvector 3 values for annual precipitation in the Weihe basin

to northwest and downstream to upstream The firstvariation points occurred in the early 1960s at theShangxian Wugong and Xirsquoan stations in the Beidaoto Huaxian section of the Weihe mainstream andlater in the late 1960s and early 1970s other stationsvariation occurred in this section In the Jingheand Beiluohe basins the first variation occurred inthe early 1970s In the Weihe basin above Beidaostationand variation occurred most recently in themid- and late 1980s except at Tianshui station (1971)

(2) Variation points in the trends of annual precipitationin the basin above Beidao station were concentratedin the 1980s while they covered longer periods ofthe 1970s and 1980s in other subbasins These resultsindicate that precipitation variation points occurredfrequently during the 1970s to 1980s in each subbasin

(3) After the 1990s the frequency and scope of precipi-tation variation declined No variation points in the1990s were detected except at Guyuan Pingliangand Wuqi stations After the 1990s there were novariation points in the annual precipitation sequencesrecorded in the Beidao to Huaxian section of theWeihe mainstream Jinghe basin or Beiluohe basinThese results show that after an active period ofprecipitation variation in 1970s and 1980s the annual

precipitation sequence in the Weihe basin was rela-tively stableThus during the study period there wereshifts in precipitation from one stable state to anotherafter disturbance

(4) From 2000 to 2011 further changes in annual pre-cipitation occurred in the basin above the Beidaostation and at the most northern stations of thebasin (Huanxian and Wuqi) These variations wereclearly regional concentrated in the northwest of theWeihe River basin and the source areas of the riversJinghe and Beiluohe Due to the limited length of thetime series it is not possible to distinguish whetherthese variations were due to small-scale regionalvariations or represent initial stages of wide-rangingvariation across the whole basin or were the resultof lingering temporal and spatial effects of the basin-wide variation that occurred in the 1970s and 1980s

(5) The variation in annual precipitation in the Weihebasin propagated from the southeast to the northwestThis is consistent with the orientation of the eigen-vector 2 field of annual precipitation obtained fromEOF decomposition and the spatial pattern of meanannual precipitation in the basin The factors causingthese patterns are not clear from the data presentedhere but will be addressed in future studies

10 Journal of Applied Mathematics

Xiji

Wuqi

SuideYanci

Baoji

Jintai

Foping

Wugong

Guyuan

Lintao

Tongxin

Haiyuan

Yuzhong

HuashanMinxian

Changwu

Hengshan

Jingyuan

Tianshui

Luochuan

Huanxian

Shangzhou

Zhongning

Tongchuan

PingliangXifengzhen

Huajialing

001

039

103077

0347

0078

0019

0009002103010548

0034

0658 0179

00062

N

E

S

W

Trend of eigenvector 3value

High 0761376

Low minus0354567

Yanrsquoan

Zhenrsquoan

Xirsquoan

minus0258

minus0531

minus0536

minus0141

minus002

minus0016

minus0008

minus0017

minus0164

minus0123

minus0123

minus0005

minus0074

minus01

minus0163

50 25 0 50 100 150 200

(km)

103∘09984000998400998400E 104

∘09984000998400998400E 105

∘09984000998400998400E 106

∘09984000998400998400E 107

∘09984000998400998400E 108

∘09984000998400998400E 109

∘09984000998400998400E 110

∘09984000998400998400E

103∘09984000998400998400E 104

∘09984000998400998400E 105

∘09984000998400998400E 106

∘09984000998400998400E 107

∘09984000998400998400E 108

∘09984000998400998400E 109

∘09984000998400998400E 110

∘09984000998400998400E

34∘09984000998400998400N

33∘09984000998400998400N

35∘09984000998400998400N

36∘09984000998400998400N

37∘09984000998400998400N

38∘09984000998400998400N

39∘09984000998400998400N

34∘09984000998400998400N

33∘09984000998400998400N

35∘09984000998400998400N

36∘09984000998400998400N

37∘09984000998400998400N

38∘09984000998400998400N

39∘09984000998400998400N

Basin boundaryWater systemStation

Figure 7 The eigenvector 3 trend field for annual precipitation in the Weihe basin

6 Conclusions

We have analyzed spatial and temporal patterns of annualprecipitation in the Weihe basin using data recorded from1960 to 2011 at 30 National Standard Stations in the basinand surrounding area by applying empirical orthogonalfunction decomposition in combination with geostatistical(semivariogram) analysis then using the Mann-Kendallmethod to discriminate variation points in the precipitationseries recorded at each station

(1) The main spatial patterns of annual precipitation inthe Weihe Basin during the study period were aconsistent overall decline which was strongest in thenorthwest and progressively weaker from northwestto southwest and from west to east The consistentoverall decline accounted for 455 of the totalvariation

(2) Variation points were detected in the annual precip-itation sequences recorded at all the meteorologicalstations in the basin but their timing had clear spatialpatterns starting gradually later from the southeastto the northwest and downstream to upstream Inthe 1970s and 1980s they occurred frequently in eachsubbasin

(3) After the 1990s following the active period of precip-itation variation in the 1970s and 1980s the frequencyand scope of precipitation variations declined Thusduring the study period there were shifts in precipita-tion fromone stable state to another after disturbance

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was financially supported by the Natural ScienceFoundation of China (Grant nos 51190093 51179148 and51179149) the National Key Basic Research 973 of China(Grant no 2012CB417003) and the MWR Welfare SectorFunds (Grant nos 201101043 201101049) Constructive com-ments from reviewers of a previous version of the paper aregratefully acknowledged

References

[1] H Zhang Y Chen G Ren et al ldquoThe characteristics of pre-cipitation variation of Weihe River Basin in Shaanxi Province

Journal of Applied Mathematics 11

during recent 50 yearsrdquo Agricultural Research in the Arid Areasvol 26 no 04 pp 236ndash241 2008

[2] K Che and J Jian ldquoA study of climate variability in Zedangduring 1960ndash1999rdquo Meteorological Monthly vol 26 no 1 pp54ndash56 2000

[3] Y Ding G Ren G Shi et al ldquoNational assessment report ofclimate change (I) climate change inChina and its future trendrdquoAdvances in Climate Change Research vol 2 no 1 pp 3ndash8 2006

[4] Y He F Wang X Mu et al ldquoTemporal and spatial character-istics of temperature and precipitation in Weihe River basinrdquoBulletin of Soil and Water Conservation vol 32 no 4 pp 102ndash105 2012

[5] Y-X Zhang F Wang and X-M Mu ldquoResearch of quantitativerelations between Geographical environment factor and waterand sediment in the Weihe River basinrdquo Journal of Northwest Aamp FUniversity (Natural Science Edition) vol 37 no 1 pp 61ndash652009

[6] C Jiang F Wang and X Mu ldquoEffect of climate change on netprimary productivity (NPP) of natural vegetation in Wei riverbasin (II) NPP of natural vegetation inWei river basinrdquo Journalof Arid LandResources and Environment vol 27 no 5 pp 53ndash572013

[7] Y Zhang ldquoHydrologic feature analysis for Weihe basin inShaanxi provincerdquo Northwest Water Resources amp Water Engi-neering no 2 pp 62ndash64 2002

[8] G Wang and Y Wang ldquoAnalysis on causes of runoff andsediment variation in Weihe basin based on hydrologicalmodelrdquo Journal of Soil Water Conservation no 4 pp 22ndash252000

[9] W He and Z Xu ldquoSpatial and temporal characteristics of thelong-term trend for temperature and pan evaporation in theWei River Basinrdquo Journal of Beijing Normal University (NaturalScience) vol 42 no 1 pp 102ndash106 2006

[10] K Pearson ldquoOn lines and planes of closest fit to system of pointsin spacerdquo Philosophical Magazine vol 6 no 2 pp 559ndash5721902

[11] D-X Cui and W-Y Liu ldquoApplication of empirical orthogonalfunction resolution to analysis of crustal vertical deformationfieldrdquo Earthquake vol 20 no 3 pp 82ndash86 2000

[12] T Liu and H Zhang ldquoCompared with principal componentanalysis and empirical orthogonal function decompositionrdquoStatistics and Decision vol 340 no 16 pp 159ndash162 2011

[13] H Lin and W Zhang ldquoSimilarities and differences of principalcomponent analysis and factor analysis and SPSS softwarerdquoStatistics Research no 3 pp 65ndash69 2005

[14] D H Kim C Yoo and T-W Kim ldquoApplication of spatial EOFand multivariate time series model for evaluating agriculturaldrought vulnerability in Koreardquo Advances in Water Resourcesvol 34 no 3 pp 340ndash350 2011

[15] N S Loboda A V Glushkov and V N Khokhlov ldquoUsingmeteorological data for reconstruction of annual runoff seriesover an ungauged area empirical orthogonal function approachto Moldova-Southwest Ukraine regionrdquo Atmospheric Researchvol 77 no 1ndash4 pp 100ndash113 2005

[16] D Sirjacobs A Alvera-Azcarate A Barth et al ldquoCloud fillingof ocean colour and sea surface temperature remote sensingproducts over the SouthernNorth Sea by theData InterpolatingEmpirical Orthogonal Functions methodologyrdquo Journal of SeaResearch vol 65 no 1 pp 114ndash130 2011

[17] H-L Yu and H-J Chu ldquoUnderstanding space-time patternsof groundwater system by empirical orthogonal functions a

case study in the Choshui River alluvial fan Taiwanrdquo Journalof Hydrology vol 381 no 3-4 pp 239ndash247 2010

[18] H BMann ldquoNonparametric tests against trendrdquo Econometricavol 13 pp 245ndash259 1945

[19] R Snyers ldquoSur LaDetermination de la stabilite des series clima-tologiquesrdquo in Proceedings of the UNESCO-WMO Symposium ofChanges of Climate vol 20 ofUneseo Arid Zone Research Seriespp 37ndash44 Unesco Paris France 1963

[20] C Goossens and A Berger ldquoAnnual and seasonal climaticvariations over the northern hemisphere and Europe during thelast centuryrdquo Annales Geophysicae B vol 4 no 4 pp 385ndash4001986

[21] C Fu and QWang ldquoThe definition and detection of the abruptclimatic changerdquo Chinese Journal of Atmospheric Sciences vol16 no 4 pp 482ndash493 1992

[22] P Xie G Chen D Li et al ldquoHydrological variation inte-grated diagnosis method and application researchrdquo Hydroelec-tric Energy vol 23 no 2 pp 11ndash14 2005

[23] M Gocic and S Trajkovic ldquoAnalysis of changes in meteoro-logical variables using Mann-Kendall and Senrsquos slope estimatorstatistical tests in SerbiardquoGlobal and Planetary Change vol 100pp 172ndash182 2013

[24] K H Hamed ldquoTrend detection in hydrologic data the Mann-Kendall trend test under the scaling hypothesisrdquo Journal ofHydrology vol 349 no 3-4 pp 350ndash363 2008

[25] S Kumar V Merwade J Kam and K Thurner ldquoStreamflowtrends in Indiana effects of long term persistence precipitationand subsurface drainsrdquo Journal of Hydrology vol 374 no 1-2pp 171ndash183 2009

[26] K H Hamed ldquoExact distribution of the Mann-Kendall trendtest statistic for persistent datardquo Journal of Hydrology vol 365no 1-2 pp 86ndash94 2009

[27] D Duhan and A Pandey ldquoStatistical analysis of long termspatial and temporal trends of precipitation during 1901ndash2002at Madhya Pradesh Indiardquo Atmospheric Research vol 122 pp136ndash149 2013

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Applied Mathematics 5

Xiji

Wuqi

SuideYanci

Baoji

Jintai

Foping

Wugong

Guyuan

Lintao

Tongxin

Haiyuan

Yuzhong

HuashanMinxian

Changwu

Hengshan

Jingyuan

Tianshui

Luochuan

Huanxian

Zhongning

Shangzhou

Tongchuan

PingliangXifengzhen

Huajialing

minus20382

Xirsquoan

Zhenrsquoan

Yanrsquoan

50 25 0 50 100 150 200

(km)

N

E

S

W

Trend of annual precipitation High minus0884939

Low minus179305

103∘09984000998400998400E 104

∘09984000998400998400E 105

∘09984000998400998400E 106

∘09984000998400998400E 107

∘09984000998400998400E 108

∘09984000998400998400E 109

∘09984000998400998400E 110

∘09984000998400998400E

103∘09984000998400998400E 104

∘09984000998400998400E 105

∘09984000998400998400E 106

∘09984000998400998400E 107

∘09984000998400998400E 108

∘09984000998400998400E 109

∘09984000998400998400E 110

∘09984000998400998400E

34∘09984000998400998400N

33∘09984000998400998400N

35∘09984000998400998400N

36∘09984000998400998400N

37∘09984000998400998400N

38∘09984000998400998400N

39∘09984000998400998400N

34∘09984000998400998400N

33∘09984000998400998400N

35∘09984000998400998400N

36∘09984000998400998400N

37∘09984000998400998400N

38∘09984000998400998400N

39∘09984000998400998400N

01229

minus 0 785

minus 1366

minus 1 076minus 04934

minus 0 9699

minus 09262

minus 08778

minus 03846

minus 08691

minus 086 72minus 09866

minus 01212

minus 02984

minus 1821

minus 10304

minus 12346

0036300

minus 3 8975minus 12491

minus 20241

minus 14991

minus 21349

minus 21367minus 19091

minus 20868

minus 14559

minus 26897minus 26428

Basin boundaryWater systemStation

Figure 3 Annual precipitation trends in the Weihe basin

Table 1 Eigenvalues and their contributions to the variance of annual precipitation in the Weihe basin obtained from EOF decomposition

Component

Initial eigenvalues Extraction sums of squared loadings

Eigenvalues120582

119894

Proportion ofvariation ()

Cumulativeproportion ofvariation ()

Eigenvalues120582

119894

Proportion ofvariation ()

Cumulativeproportion ofvariation ()

1 13648 45494 45494 13648 45494 454942 5363 17875 63369 5363 17875 633693 1997 6658 70027 1997 6658 700274 1404 4681 747085 1196 3985 786936 0912 3039 817327 0685 2285 840178 0634 2113 861309 0506 1686 8781610 0422 1408 89224Extraction method principal component analysis

across the basin was minus1662mma clearly showing thatannual precipitation declined and that this trend was mostpronounced in upstream areas

42 EOF Decomposition of Annual Precipitation in the WeiheBasin The precipitation data analyzed herein originatedfrom the Weihe basin and the surrounding 30 nationalstandard stations during the period 1960ndash2011 From this

data we derived the normalized anomaly matrix X forannual precipitation Empirical orthogonal function (run inSPSS) was used to calculate the eigenvalues of the covariancematrix of X and the amount of variation that each principalcomponent explained as shown in Table 1

Eigenvalues for the covariance matrix of the Weihebasin annual precipitation matrix and their contributionsto variance (specific and cumulative) obtained by empirical

6 Journal of Applied Mathematics

orthogonal function (using SPSS software) are shown inTable 1 The precipitation field rapidly converged the firstprincipal component (PC) accounted for almost half (455)of the variation in annual precipitation and the first threecomponents (all of which had eigenvalues exceeding 1) for700 of the variationThus the first three PCs and eigenvec-tors describe the spatial distribution of annual precipitationin the Weihe basin quite well

Eigenvectors of the covariance matrix cannot be directlyobtained from the SPSS output but can be obtained by div-iding the 119894 vectors of the matrix by the square roots of thecorresponding eigenvalues The vector corresponding to thelargest eigenvalue is eigenvector 1 with progressively smalleigenvalues associated with subsequent eigenvectors (2 34 etc) The first three eigenvectors obtained from the EOFdecomposition of the annual precipitation in theWeihe basindataset are shown in Table 2

The first three eigenvectors obtained from the EOF anal-ysis provide an approximate description of the precipitationvalues If their values are consistently positive or negativethere is a consistent precipitation trend and the area withthe largest absolute value is the center of the variable field Incontrast if the eigenvector values fluctuate between positiveand negative the trend is nonuniform

We used the Geostatistical Analyst module of ArcGIS toundertake a semivariogram analysis of the first three eigen-vectors and thus to obtain the spatial variability parametersshown in Table 3We then usedKriging interpolation to drawa map of each eigenvector field

As can be seen from Table 3 the azimuth angles ofeigenvectors 1 2 and 3 are 47∘ 655∘ and 3096∘ approx-imately south to north east-northeast to west-southwestand northwest to southeast orientations respectively All theeigenvectors have a small Nugget value indicating that theprecipitationmeasurement error is small Indeed the Nuggetof eigenvector 2 is zero indicating that themeasurement errorfor precipitation in this eigenvector field can be ignored Thepartial sill of eigenvector 1 is significantly lower than that ofthe other eigenvectors indicating that it reflects smaller-scalevariations The Weihe basin spans about 7∘ longitude and 4∘latitude Thus since the major ranges in the eigenvectors are629∘ latitude and longitude the variability of precipitationchanges with distance across the basin

43 VariationDiagnosis of Annual Rainfall in theWeihe BasinThe Weihe basin is divided into four subbasins with differ-ing hydrological characteristics and controlled by differenthydrological stations The areas covered by the subbasinsand locations of hydrological and meteorological stations areshown in Figure 1TheM-Kmethodwas used to discriminatevariation points in the trends in the annual precipitationseries recorded at each station during 1960ndash2011 and theresults are shown in Table 4

5 Analysis and Discussion of Results

51 Analysis of Spatial Variation of Annual Precipitation inthe Weihe Basin As shown in Table 2 all components of

Table 2 Eigenvectors obtained byEOFdecomposition of the annualprecipitation in the Weihe basin dataset

EigenvectorsEigenvector 1

119881

1

Eigenvector 2119881

2

Eigenvector 3119881

3

Lintao 0190 minus0137 minus0291Huajialing 0190 minus0043 minus0383Wuqi 0196 minus0157 0112Guyuan 0224 minus0082 minus0086Huanxia 0210 minus0140 0086Yanrsquoan 0208 0011 0251Xiji 0192 minus0131 minus0308Pingliang

minus0070 0045 minus0019Xifengzhen 0222 0104 0067Changwu 0204 0201 0058Luochuan 0200 0139 0179Tongchua 0154 0248 0168Minxian 0202 0001 minus0271Tianshui 0204 0057 minus0241Baoji 0167 0287 minus0021Wugong 0180 0273 minus0010Xirsquoan 0158 0277 0051Huashan 0163 0215 minus0100Foping 0177 0218 0009Zhenrsquoan 0157 0245 0042Shangzhou 0167 0223 minus0021Jingtai 0119 minus0198 0272Jingyuan 0167 minus0250 minus0084Yuzhong 0178 minus0156 minus0281Zhongning 0182 minus0206 0127Yanci 0170 minus0200 0250Hengshan 0183 minus0120 0240Suide 0175 minus0108 0260Haiyuan 0199 minus0231 0004Tongxin 0192 minus0221 0072

Table 3 Spatial variability parameters of eigenvectors of annualprecipitation in the Weihe basin

EigenvectorsEigenvector 1

119881

1

Eigenvector 2119881

2

Eigenvector 3119881

3

Azimuth angle (∘) 47 655 3096Nugget (119862

0) 00012685 0 00013054

Partial sill (119862) 00014968 041084 0044187Major range(∘ latitude andlongitude)

628911 628911 628911

eigenvector 1 are positive except the data from Pingliang sta-tion indicating that precipitation in theWeihe basin had highspatial consistency during the study period The downward

Journal of Applied Mathematics 7

Table 4 Variation points in annual precipitation trends in the Weihe basin

Subbasins Meteorologicalstations

Variation points1960ndash1969 1970ndash1979 1980ndash1989 1990ndash1999 2000ndash2011

Weihe basin aboveBeidao station

Tianshui 1971 1983 1985 2002Minxian 1986 1988 2006Huajialing 1986Lintao 1986 1987 2003 2006 2008Xiji 2005

Jinghe basin

Huanxian 1970 2002Guyuan 1972 1986 1996Pingliang 1971 1977 1980ndash1985lowast 1990Xifengzhen 1972 1979Changwu 1978 1979 1983 1985

Beiluohe basinWuqi 1971ndash1974lowast 1981 1983 1986ndash1995lowast 1998 2000 2003 2007

Yanrsquoan 1972ndash1975lowast1978 1979 1982

Luochuan 1970 1976

Beidao to Huaxiansection of Weihemainstream

Baoji 1969 1971 19741976 1978 1980 1982

Wugong 19611964ndash1970lowast

Xirsquoan 1962 1965Tongchuan 1976 1983 1985Zhenrsquoan 1972 1975

Foping 1966 1968 1973 19771979 1986 1988

Shangxian 1960 1963 1971 1975 1984Huanshan 1986

Extraction method Mann-Kendall method lowastindicates that variation occurred frequently during the indicated period

trend of annual precipitation shows that annual precipitationin the basin declined in the region and this consistent trendaccounts for 455 of the variance The basin covers parts ofthe continental monsoon zone and transitional zone betweenarid and humid regions of northwest China The overallconsistency and decline represented by eigenvector 1 are themain spatial features of precipitation patterns in the Weihebasin during the study period

The eigenvector 2 values obtained from EOF decomposi-tion of the annual precipitation data are shown in Figure 4The values can be divided by a line passing roughly fromYanrsquoan through Pingliang to Tianshui into positive andnegative anomaly areas covering the southeastern parts of theWeihe basin (with themost-positive anomaly center in Baoji)and northwestern parts (with the most-negative anomalycenter in Wu Qi) respectively These results confirm thatthe annual precipitation in the basin has a clear northwest-southeast pattern

Combining the eigenvalues with the trends in annualprecipitation recorded at each of the stations gives the eigen-vector 2 trend values (Figure 5) which show that precipita-tion declined in the southeast and increased in the northwestThe decline was most pronounced at the downstream stationHuashan

The eigenvector 3 values have a clear eastwest pattern(Figure 6) with a negative anomaly area covering upstreamparts of the basin and a positive anomaly center located in theLuochuan region north of ShaanxiThese findings show thatin addition to the consistent overall pattern and northwest-southeast precipitation patterns there is a clear eastwestpattern In combination with the precipitation trends at the30 stations the results clearly show that annual precipitationdeclined in northern Shaanxi and increased in the upstreamareas (Figure 7)

To summarize there were three spatial patterns of annualprecipitation in the Weihe basin during the study period (1)consistent overall decline (2) declined in the southeast andincreased in the northwest (3) declined in northern Shaanxiand increased in the upstream areas

52 Analysis of Annual Precipitation Variation in the WeiheBasin The results summarized in Table 4 show that varia-tion points occurred in trends of the annual precipitationsequence recorded at every meteorological station in theWeihe basin but their timings differed in the followingmanner

(1) The timing of variation points has clear spatial char-acteristics with gradual progressions from southeast

8 Journal of Applied Mathematics

Xiji

Wuqi

SuideYanci

Baoji

Jintai

Foping

Wugong

Guyuan

Lintao

Tongxin

Haiyuan

Yuzhong

HuashanMinxian

Changwu

Hengshan

Jingyuan

Tianshui

Luochuan

Huanxian

Zhongning

Shangzhou

Tongchuan

PingliangXifengzhen

Huajialing

0223

0245

0218

0215027702730287

00570001

0248

0

0201

01040045

Yanrsquoan

Zhenrsquoan

Xirsquoan

139

minus02

minus0206

minus0198

minus025

minus0156

minus0137minus00428

minus0131minus0082

minus014

minus0157

minus012

minus0108

minus0221

minus0231

N

E

S

W

103∘09984000998400998400E 104

∘09984000998400998400E 105

∘09984000998400998400E 106

∘09984000998400998400E 107

∘09984000998400998400E 108

∘09984000998400998400E 109

∘09984000998400998400E 110

∘09984000998400998400E

103∘09984000998400998400E 104

∘09984000998400998400E 105

∘09984000998400998400E 106

∘09984000998400998400E 107

∘09984000998400998400E 108

∘09984000998400998400E 109

∘09984000998400998400E 110

∘09984000998400998400E

34∘09984000998400998400N

33∘09984000998400998400N

35∘09984000998400998400N

36∘09984000998400998400N

37∘09984000998400998400N

38∘09984000998400998400N

39∘09984000998400998400N

34∘09984000998400998400N

33∘09984000998400998400N

35∘09984000998400998400N

36∘09984000998400998400N

37∘09984000998400998400N

38∘09984000998400998400N

39∘09984000998400998400N

50 25 0 50 100 150 200

(km)

Basin boundaryWater systemStation

Eigenvector 2High 0286104

Low minus0196997

Figure 4 The eigenvector 2 values for annual precipitation in the Weihe basin

Xiji

Wuqi

SuideYanci

Baoji

Jintai

Foping

Wugong

Guyuan

Lintao

Tongxin

Haiyuan

Yuzhong

HuashanMinxian

Changwu

Hengshan

Jingyuan

Tianshui

Luochuan

Huanxian

Zhongning

Shangzhou

Tongchuan

PingliangXifengzhen

Huajialing

022

023

028

0227

0316

01290099

0199

0192

0034

0266

0172

0228

01150363

50 25 0 50 100 150 200

(km)

103∘09984000998400998400E 104

∘09984000998400998400E 105

∘09984000998400998400E 106

∘09984000998400998400E 107

∘09984000998400998400E 108

∘09984000998400998400E 109

∘09984000998400998400E 110

∘09984000998400998400E

103∘09984000998400998400E 104

∘09984000998400998400E 105

∘09984000998400998400E 106

∘09984000998400998400E 107

∘09984000998400998400E 108

∘09984000998400998400E 109

∘09984000998400998400E 110

∘09984000998400998400E

34∘09984000998400998400N

33∘09984000998400998400N

35∘09984000998400998400N

36∘09984000998400998400N

37∘09984000998400998400N

38∘09984000998400998400N

39∘09984000998400998400N

34∘09984000998400998400N

33∘09984000998400998400N

35∘09984000998400998400N

36∘09984000998400998400N

37∘09984000998400998400N

38∘09984000998400998400N

39∘09984000998400998400N

N

E

S

W

Basin boundaryWater systemStation

Trend of eigenvector 2value

High 0229457

Low minus0461604

minus0023

minus0109

minus003

minus0838

minus0094

minus0196minus019

minus0236

minus006

minus0156

minus0083

minus0283

minus0072minus00018

minus0007

Yanrsquoan

Xirsquoan

Zhenrsquoan

Figure 5 The eigenvector 2 trend field for annual precipitation in the Weihe basin

Journal of Applied Mathematics 9

Xiji

Wuqi

SuideYanci

Baoji

Jintai

Foping

Wugong

Guyuan

Lintao

Tongxin

Haiyuan

Yuzhong

HuashanMinxian

Changwu

Hengshan

Jingyuan

Tianshui

Luochuan

Huanxian

Zhongning

Shangzhou

Tongchuan

PingliangXifengzhen

Huajialing

026

024025

0072

0004

0127

0272

0042

0009

0051

0168

0179

0058

0067

025100860112

50 25 0 50 100 150 200

(km)

103∘09984000998400998400E 104

∘09984000998400998400E 105

∘09984000998400998400E 106

∘09984000998400998400E 107

∘09984000998400998400E 108

∘09984000998400998400E 109

∘09984000998400998400E 110

∘09984000998400998400E

103∘09984000998400998400E 104

∘09984000998400998400E 105

∘09984000998400998400E 106

∘09984000998400998400E 107

∘09984000998400998400E 108

∘09984000998400998400E 109

∘09984000998400998400E 110

∘09984000998400998400E

34∘09984000998400998400N

33∘09984000998400998400N

35∘09984000998400998400N

36∘09984000998400998400N

37∘09984000998400998400N

38∘09984000998400998400N

39∘09984000998400998400N

34∘09984000998400998400N

33∘09984000998400998400N

35∘09984000998400998400N

36∘09984000998400998400N

37∘09984000998400998400N

38∘09984000998400998400N

39∘09984000998400998400N

N

E

S

W

Basin boundaryWater systemStation

Eigenvector 3value

High 0223536

Low minus0365226

Yanrsquoan

Xirsquoan

Zhenrsquoan

minus01

minus0021

minus001minus0021

minus0019

minus0241

minus0383

minus0271

minus0291

minus0281

minus0084

minus0308minus0086

Figure 6 The eigenvector 3 values for annual precipitation in the Weihe basin

to northwest and downstream to upstream The firstvariation points occurred in the early 1960s at theShangxian Wugong and Xirsquoan stations in the Beidaoto Huaxian section of the Weihe mainstream andlater in the late 1960s and early 1970s other stationsvariation occurred in this section In the Jingheand Beiluohe basins the first variation occurred inthe early 1970s In the Weihe basin above Beidaostationand variation occurred most recently in themid- and late 1980s except at Tianshui station (1971)

(2) Variation points in the trends of annual precipitationin the basin above Beidao station were concentratedin the 1980s while they covered longer periods ofthe 1970s and 1980s in other subbasins These resultsindicate that precipitation variation points occurredfrequently during the 1970s to 1980s in each subbasin

(3) After the 1990s the frequency and scope of precipi-tation variation declined No variation points in the1990s were detected except at Guyuan Pingliangand Wuqi stations After the 1990s there were novariation points in the annual precipitation sequencesrecorded in the Beidao to Huaxian section of theWeihe mainstream Jinghe basin or Beiluohe basinThese results show that after an active period ofprecipitation variation in 1970s and 1980s the annual

precipitation sequence in the Weihe basin was rela-tively stableThus during the study period there wereshifts in precipitation from one stable state to anotherafter disturbance

(4) From 2000 to 2011 further changes in annual pre-cipitation occurred in the basin above the Beidaostation and at the most northern stations of thebasin (Huanxian and Wuqi) These variations wereclearly regional concentrated in the northwest of theWeihe River basin and the source areas of the riversJinghe and Beiluohe Due to the limited length of thetime series it is not possible to distinguish whetherthese variations were due to small-scale regionalvariations or represent initial stages of wide-rangingvariation across the whole basin or were the resultof lingering temporal and spatial effects of the basin-wide variation that occurred in the 1970s and 1980s

(5) The variation in annual precipitation in the Weihebasin propagated from the southeast to the northwestThis is consistent with the orientation of the eigen-vector 2 field of annual precipitation obtained fromEOF decomposition and the spatial pattern of meanannual precipitation in the basin The factors causingthese patterns are not clear from the data presentedhere but will be addressed in future studies

10 Journal of Applied Mathematics

Xiji

Wuqi

SuideYanci

Baoji

Jintai

Foping

Wugong

Guyuan

Lintao

Tongxin

Haiyuan

Yuzhong

HuashanMinxian

Changwu

Hengshan

Jingyuan

Tianshui

Luochuan

Huanxian

Shangzhou

Zhongning

Tongchuan

PingliangXifengzhen

Huajialing

001

039

103077

0347

0078

0019

0009002103010548

0034

0658 0179

00062

N

E

S

W

Trend of eigenvector 3value

High 0761376

Low minus0354567

Yanrsquoan

Zhenrsquoan

Xirsquoan

minus0258

minus0531

minus0536

minus0141

minus002

minus0016

minus0008

minus0017

minus0164

minus0123

minus0123

minus0005

minus0074

minus01

minus0163

50 25 0 50 100 150 200

(km)

103∘09984000998400998400E 104

∘09984000998400998400E 105

∘09984000998400998400E 106

∘09984000998400998400E 107

∘09984000998400998400E 108

∘09984000998400998400E 109

∘09984000998400998400E 110

∘09984000998400998400E

103∘09984000998400998400E 104

∘09984000998400998400E 105

∘09984000998400998400E 106

∘09984000998400998400E 107

∘09984000998400998400E 108

∘09984000998400998400E 109

∘09984000998400998400E 110

∘09984000998400998400E

34∘09984000998400998400N

33∘09984000998400998400N

35∘09984000998400998400N

36∘09984000998400998400N

37∘09984000998400998400N

38∘09984000998400998400N

39∘09984000998400998400N

34∘09984000998400998400N

33∘09984000998400998400N

35∘09984000998400998400N

36∘09984000998400998400N

37∘09984000998400998400N

38∘09984000998400998400N

39∘09984000998400998400N

Basin boundaryWater systemStation

Figure 7 The eigenvector 3 trend field for annual precipitation in the Weihe basin

6 Conclusions

We have analyzed spatial and temporal patterns of annualprecipitation in the Weihe basin using data recorded from1960 to 2011 at 30 National Standard Stations in the basinand surrounding area by applying empirical orthogonalfunction decomposition in combination with geostatistical(semivariogram) analysis then using the Mann-Kendallmethod to discriminate variation points in the precipitationseries recorded at each station

(1) The main spatial patterns of annual precipitation inthe Weihe Basin during the study period were aconsistent overall decline which was strongest in thenorthwest and progressively weaker from northwestto southwest and from west to east The consistentoverall decline accounted for 455 of the totalvariation

(2) Variation points were detected in the annual precip-itation sequences recorded at all the meteorologicalstations in the basin but their timing had clear spatialpatterns starting gradually later from the southeastto the northwest and downstream to upstream Inthe 1970s and 1980s they occurred frequently in eachsubbasin

(3) After the 1990s following the active period of precip-itation variation in the 1970s and 1980s the frequencyand scope of precipitation variations declined Thusduring the study period there were shifts in precipita-tion fromone stable state to another after disturbance

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was financially supported by the Natural ScienceFoundation of China (Grant nos 51190093 51179148 and51179149) the National Key Basic Research 973 of China(Grant no 2012CB417003) and the MWR Welfare SectorFunds (Grant nos 201101043 201101049) Constructive com-ments from reviewers of a previous version of the paper aregratefully acknowledged

References

[1] H Zhang Y Chen G Ren et al ldquoThe characteristics of pre-cipitation variation of Weihe River Basin in Shaanxi Province

Journal of Applied Mathematics 11

during recent 50 yearsrdquo Agricultural Research in the Arid Areasvol 26 no 04 pp 236ndash241 2008

[2] K Che and J Jian ldquoA study of climate variability in Zedangduring 1960ndash1999rdquo Meteorological Monthly vol 26 no 1 pp54ndash56 2000

[3] Y Ding G Ren G Shi et al ldquoNational assessment report ofclimate change (I) climate change inChina and its future trendrdquoAdvances in Climate Change Research vol 2 no 1 pp 3ndash8 2006

[4] Y He F Wang X Mu et al ldquoTemporal and spatial character-istics of temperature and precipitation in Weihe River basinrdquoBulletin of Soil and Water Conservation vol 32 no 4 pp 102ndash105 2012

[5] Y-X Zhang F Wang and X-M Mu ldquoResearch of quantitativerelations between Geographical environment factor and waterand sediment in the Weihe River basinrdquo Journal of Northwest Aamp FUniversity (Natural Science Edition) vol 37 no 1 pp 61ndash652009

[6] C Jiang F Wang and X Mu ldquoEffect of climate change on netprimary productivity (NPP) of natural vegetation in Wei riverbasin (II) NPP of natural vegetation inWei river basinrdquo Journalof Arid LandResources and Environment vol 27 no 5 pp 53ndash572013

[7] Y Zhang ldquoHydrologic feature analysis for Weihe basin inShaanxi provincerdquo Northwest Water Resources amp Water Engi-neering no 2 pp 62ndash64 2002

[8] G Wang and Y Wang ldquoAnalysis on causes of runoff andsediment variation in Weihe basin based on hydrologicalmodelrdquo Journal of Soil Water Conservation no 4 pp 22ndash252000

[9] W He and Z Xu ldquoSpatial and temporal characteristics of thelong-term trend for temperature and pan evaporation in theWei River Basinrdquo Journal of Beijing Normal University (NaturalScience) vol 42 no 1 pp 102ndash106 2006

[10] K Pearson ldquoOn lines and planes of closest fit to system of pointsin spacerdquo Philosophical Magazine vol 6 no 2 pp 559ndash5721902

[11] D-X Cui and W-Y Liu ldquoApplication of empirical orthogonalfunction resolution to analysis of crustal vertical deformationfieldrdquo Earthquake vol 20 no 3 pp 82ndash86 2000

[12] T Liu and H Zhang ldquoCompared with principal componentanalysis and empirical orthogonal function decompositionrdquoStatistics and Decision vol 340 no 16 pp 159ndash162 2011

[13] H Lin and W Zhang ldquoSimilarities and differences of principalcomponent analysis and factor analysis and SPSS softwarerdquoStatistics Research no 3 pp 65ndash69 2005

[14] D H Kim C Yoo and T-W Kim ldquoApplication of spatial EOFand multivariate time series model for evaluating agriculturaldrought vulnerability in Koreardquo Advances in Water Resourcesvol 34 no 3 pp 340ndash350 2011

[15] N S Loboda A V Glushkov and V N Khokhlov ldquoUsingmeteorological data for reconstruction of annual runoff seriesover an ungauged area empirical orthogonal function approachto Moldova-Southwest Ukraine regionrdquo Atmospheric Researchvol 77 no 1ndash4 pp 100ndash113 2005

[16] D Sirjacobs A Alvera-Azcarate A Barth et al ldquoCloud fillingof ocean colour and sea surface temperature remote sensingproducts over the SouthernNorth Sea by theData InterpolatingEmpirical Orthogonal Functions methodologyrdquo Journal of SeaResearch vol 65 no 1 pp 114ndash130 2011

[17] H-L Yu and H-J Chu ldquoUnderstanding space-time patternsof groundwater system by empirical orthogonal functions a

case study in the Choshui River alluvial fan Taiwanrdquo Journalof Hydrology vol 381 no 3-4 pp 239ndash247 2010

[18] H BMann ldquoNonparametric tests against trendrdquo Econometricavol 13 pp 245ndash259 1945

[19] R Snyers ldquoSur LaDetermination de la stabilite des series clima-tologiquesrdquo in Proceedings of the UNESCO-WMO Symposium ofChanges of Climate vol 20 ofUneseo Arid Zone Research Seriespp 37ndash44 Unesco Paris France 1963

[20] C Goossens and A Berger ldquoAnnual and seasonal climaticvariations over the northern hemisphere and Europe during thelast centuryrdquo Annales Geophysicae B vol 4 no 4 pp 385ndash4001986

[21] C Fu and QWang ldquoThe definition and detection of the abruptclimatic changerdquo Chinese Journal of Atmospheric Sciences vol16 no 4 pp 482ndash493 1992

[22] P Xie G Chen D Li et al ldquoHydrological variation inte-grated diagnosis method and application researchrdquo Hydroelec-tric Energy vol 23 no 2 pp 11ndash14 2005

[23] M Gocic and S Trajkovic ldquoAnalysis of changes in meteoro-logical variables using Mann-Kendall and Senrsquos slope estimatorstatistical tests in SerbiardquoGlobal and Planetary Change vol 100pp 172ndash182 2013

[24] K H Hamed ldquoTrend detection in hydrologic data the Mann-Kendall trend test under the scaling hypothesisrdquo Journal ofHydrology vol 349 no 3-4 pp 350ndash363 2008

[25] S Kumar V Merwade J Kam and K Thurner ldquoStreamflowtrends in Indiana effects of long term persistence precipitationand subsurface drainsrdquo Journal of Hydrology vol 374 no 1-2pp 171ndash183 2009

[26] K H Hamed ldquoExact distribution of the Mann-Kendall trendtest statistic for persistent datardquo Journal of Hydrology vol 365no 1-2 pp 86ndash94 2009

[27] D Duhan and A Pandey ldquoStatistical analysis of long termspatial and temporal trends of precipitation during 1901ndash2002at Madhya Pradesh Indiardquo Atmospheric Research vol 122 pp136ndash149 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Discrete Dynamics in Nature and Society

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Journal of Applied Mathematics

orthogonal function (using SPSS software) are shown inTable 1 The precipitation field rapidly converged the firstprincipal component (PC) accounted for almost half (455)of the variation in annual precipitation and the first threecomponents (all of which had eigenvalues exceeding 1) for700 of the variationThus the first three PCs and eigenvec-tors describe the spatial distribution of annual precipitationin the Weihe basin quite well

Eigenvectors of the covariance matrix cannot be directlyobtained from the SPSS output but can be obtained by div-iding the 119894 vectors of the matrix by the square roots of thecorresponding eigenvalues The vector corresponding to thelargest eigenvalue is eigenvector 1 with progressively smalleigenvalues associated with subsequent eigenvectors (2 34 etc) The first three eigenvectors obtained from the EOFdecomposition of the annual precipitation in theWeihe basindataset are shown in Table 2

The first three eigenvectors obtained from the EOF anal-ysis provide an approximate description of the precipitationvalues If their values are consistently positive or negativethere is a consistent precipitation trend and the area withthe largest absolute value is the center of the variable field Incontrast if the eigenvector values fluctuate between positiveand negative the trend is nonuniform

We used the Geostatistical Analyst module of ArcGIS toundertake a semivariogram analysis of the first three eigen-vectors and thus to obtain the spatial variability parametersshown in Table 3We then usedKriging interpolation to drawa map of each eigenvector field

As can be seen from Table 3 the azimuth angles ofeigenvectors 1 2 and 3 are 47∘ 655∘ and 3096∘ approx-imately south to north east-northeast to west-southwestand northwest to southeast orientations respectively All theeigenvectors have a small Nugget value indicating that theprecipitationmeasurement error is small Indeed the Nuggetof eigenvector 2 is zero indicating that themeasurement errorfor precipitation in this eigenvector field can be ignored Thepartial sill of eigenvector 1 is significantly lower than that ofthe other eigenvectors indicating that it reflects smaller-scalevariations The Weihe basin spans about 7∘ longitude and 4∘latitude Thus since the major ranges in the eigenvectors are629∘ latitude and longitude the variability of precipitationchanges with distance across the basin

43 VariationDiagnosis of Annual Rainfall in theWeihe BasinThe Weihe basin is divided into four subbasins with differ-ing hydrological characteristics and controlled by differenthydrological stations The areas covered by the subbasinsand locations of hydrological and meteorological stations areshown in Figure 1TheM-Kmethodwas used to discriminatevariation points in the trends in the annual precipitationseries recorded at each station during 1960ndash2011 and theresults are shown in Table 4

5 Analysis and Discussion of Results

51 Analysis of Spatial Variation of Annual Precipitation inthe Weihe Basin As shown in Table 2 all components of

Table 2 Eigenvectors obtained byEOFdecomposition of the annualprecipitation in the Weihe basin dataset

EigenvectorsEigenvector 1

119881

1

Eigenvector 2119881

2

Eigenvector 3119881

3

Lintao 0190 minus0137 minus0291Huajialing 0190 minus0043 minus0383Wuqi 0196 minus0157 0112Guyuan 0224 minus0082 minus0086Huanxia 0210 minus0140 0086Yanrsquoan 0208 0011 0251Xiji 0192 minus0131 minus0308Pingliang

minus0070 0045 minus0019Xifengzhen 0222 0104 0067Changwu 0204 0201 0058Luochuan 0200 0139 0179Tongchua 0154 0248 0168Minxian 0202 0001 minus0271Tianshui 0204 0057 minus0241Baoji 0167 0287 minus0021Wugong 0180 0273 minus0010Xirsquoan 0158 0277 0051Huashan 0163 0215 minus0100Foping 0177 0218 0009Zhenrsquoan 0157 0245 0042Shangzhou 0167 0223 minus0021Jingtai 0119 minus0198 0272Jingyuan 0167 minus0250 minus0084Yuzhong 0178 minus0156 minus0281Zhongning 0182 minus0206 0127Yanci 0170 minus0200 0250Hengshan 0183 minus0120 0240Suide 0175 minus0108 0260Haiyuan 0199 minus0231 0004Tongxin 0192 minus0221 0072

Table 3 Spatial variability parameters of eigenvectors of annualprecipitation in the Weihe basin

EigenvectorsEigenvector 1

119881

1

Eigenvector 2119881

2

Eigenvector 3119881

3

Azimuth angle (∘) 47 655 3096Nugget (119862

0) 00012685 0 00013054

Partial sill (119862) 00014968 041084 0044187Major range(∘ latitude andlongitude)

628911 628911 628911

eigenvector 1 are positive except the data from Pingliang sta-tion indicating that precipitation in theWeihe basin had highspatial consistency during the study period The downward

Journal of Applied Mathematics 7

Table 4 Variation points in annual precipitation trends in the Weihe basin

Subbasins Meteorologicalstations

Variation points1960ndash1969 1970ndash1979 1980ndash1989 1990ndash1999 2000ndash2011

Weihe basin aboveBeidao station

Tianshui 1971 1983 1985 2002Minxian 1986 1988 2006Huajialing 1986Lintao 1986 1987 2003 2006 2008Xiji 2005

Jinghe basin

Huanxian 1970 2002Guyuan 1972 1986 1996Pingliang 1971 1977 1980ndash1985lowast 1990Xifengzhen 1972 1979Changwu 1978 1979 1983 1985

Beiluohe basinWuqi 1971ndash1974lowast 1981 1983 1986ndash1995lowast 1998 2000 2003 2007

Yanrsquoan 1972ndash1975lowast1978 1979 1982

Luochuan 1970 1976

Beidao to Huaxiansection of Weihemainstream

Baoji 1969 1971 19741976 1978 1980 1982

Wugong 19611964ndash1970lowast

Xirsquoan 1962 1965Tongchuan 1976 1983 1985Zhenrsquoan 1972 1975

Foping 1966 1968 1973 19771979 1986 1988

Shangxian 1960 1963 1971 1975 1984Huanshan 1986

Extraction method Mann-Kendall method lowastindicates that variation occurred frequently during the indicated period

trend of annual precipitation shows that annual precipitationin the basin declined in the region and this consistent trendaccounts for 455 of the variance The basin covers parts ofthe continental monsoon zone and transitional zone betweenarid and humid regions of northwest China The overallconsistency and decline represented by eigenvector 1 are themain spatial features of precipitation patterns in the Weihebasin during the study period

The eigenvector 2 values obtained from EOF decomposi-tion of the annual precipitation data are shown in Figure 4The values can be divided by a line passing roughly fromYanrsquoan through Pingliang to Tianshui into positive andnegative anomaly areas covering the southeastern parts of theWeihe basin (with themost-positive anomaly center in Baoji)and northwestern parts (with the most-negative anomalycenter in Wu Qi) respectively These results confirm thatthe annual precipitation in the basin has a clear northwest-southeast pattern

Combining the eigenvalues with the trends in annualprecipitation recorded at each of the stations gives the eigen-vector 2 trend values (Figure 5) which show that precipita-tion declined in the southeast and increased in the northwestThe decline was most pronounced at the downstream stationHuashan

The eigenvector 3 values have a clear eastwest pattern(Figure 6) with a negative anomaly area covering upstreamparts of the basin and a positive anomaly center located in theLuochuan region north of ShaanxiThese findings show thatin addition to the consistent overall pattern and northwest-southeast precipitation patterns there is a clear eastwestpattern In combination with the precipitation trends at the30 stations the results clearly show that annual precipitationdeclined in northern Shaanxi and increased in the upstreamareas (Figure 7)

To summarize there were three spatial patterns of annualprecipitation in the Weihe basin during the study period (1)consistent overall decline (2) declined in the southeast andincreased in the northwest (3) declined in northern Shaanxiand increased in the upstream areas

52 Analysis of Annual Precipitation Variation in the WeiheBasin The results summarized in Table 4 show that varia-tion points occurred in trends of the annual precipitationsequence recorded at every meteorological station in theWeihe basin but their timings differed in the followingmanner

(1) The timing of variation points has clear spatial char-acteristics with gradual progressions from southeast

8 Journal of Applied Mathematics

Xiji

Wuqi

SuideYanci

Baoji

Jintai

Foping

Wugong

Guyuan

Lintao

Tongxin

Haiyuan

Yuzhong

HuashanMinxian

Changwu

Hengshan

Jingyuan

Tianshui

Luochuan

Huanxian

Zhongning

Shangzhou

Tongchuan

PingliangXifengzhen

Huajialing

0223

0245

0218

0215027702730287

00570001

0248

0

0201

01040045

Yanrsquoan

Zhenrsquoan

Xirsquoan

139

minus02

minus0206

minus0198

minus025

minus0156

minus0137minus00428

minus0131minus0082

minus014

minus0157

minus012

minus0108

minus0221

minus0231

N

E

S

W

103∘09984000998400998400E 104

∘09984000998400998400E 105

∘09984000998400998400E 106

∘09984000998400998400E 107

∘09984000998400998400E 108

∘09984000998400998400E 109

∘09984000998400998400E 110

∘09984000998400998400E

103∘09984000998400998400E 104

∘09984000998400998400E 105

∘09984000998400998400E 106

∘09984000998400998400E 107

∘09984000998400998400E 108

∘09984000998400998400E 109

∘09984000998400998400E 110

∘09984000998400998400E

34∘09984000998400998400N

33∘09984000998400998400N

35∘09984000998400998400N

36∘09984000998400998400N

37∘09984000998400998400N

38∘09984000998400998400N

39∘09984000998400998400N

34∘09984000998400998400N

33∘09984000998400998400N

35∘09984000998400998400N

36∘09984000998400998400N

37∘09984000998400998400N

38∘09984000998400998400N

39∘09984000998400998400N

50 25 0 50 100 150 200

(km)

Basin boundaryWater systemStation

Eigenvector 2High 0286104

Low minus0196997

Figure 4 The eigenvector 2 values for annual precipitation in the Weihe basin

Xiji

Wuqi

SuideYanci

Baoji

Jintai

Foping

Wugong

Guyuan

Lintao

Tongxin

Haiyuan

Yuzhong

HuashanMinxian

Changwu

Hengshan

Jingyuan

Tianshui

Luochuan

Huanxian

Zhongning

Shangzhou

Tongchuan

PingliangXifengzhen

Huajialing

022

023

028

0227

0316

01290099

0199

0192

0034

0266

0172

0228

01150363

50 25 0 50 100 150 200

(km)

103∘09984000998400998400E 104

∘09984000998400998400E 105

∘09984000998400998400E 106

∘09984000998400998400E 107

∘09984000998400998400E 108

∘09984000998400998400E 109

∘09984000998400998400E 110

∘09984000998400998400E

103∘09984000998400998400E 104

∘09984000998400998400E 105

∘09984000998400998400E 106

∘09984000998400998400E 107

∘09984000998400998400E 108

∘09984000998400998400E 109

∘09984000998400998400E 110

∘09984000998400998400E

34∘09984000998400998400N

33∘09984000998400998400N

35∘09984000998400998400N

36∘09984000998400998400N

37∘09984000998400998400N

38∘09984000998400998400N

39∘09984000998400998400N

34∘09984000998400998400N

33∘09984000998400998400N

35∘09984000998400998400N

36∘09984000998400998400N

37∘09984000998400998400N

38∘09984000998400998400N

39∘09984000998400998400N

N

E

S

W

Basin boundaryWater systemStation

Trend of eigenvector 2value

High 0229457

Low minus0461604

minus0023

minus0109

minus003

minus0838

minus0094

minus0196minus019

minus0236

minus006

minus0156

minus0083

minus0283

minus0072minus00018

minus0007

Yanrsquoan

Xirsquoan

Zhenrsquoan

Figure 5 The eigenvector 2 trend field for annual precipitation in the Weihe basin

Journal of Applied Mathematics 9

Xiji

Wuqi

SuideYanci

Baoji

Jintai

Foping

Wugong

Guyuan

Lintao

Tongxin

Haiyuan

Yuzhong

HuashanMinxian

Changwu

Hengshan

Jingyuan

Tianshui

Luochuan

Huanxian

Zhongning

Shangzhou

Tongchuan

PingliangXifengzhen

Huajialing

026

024025

0072

0004

0127

0272

0042

0009

0051

0168

0179

0058

0067

025100860112

50 25 0 50 100 150 200

(km)

103∘09984000998400998400E 104

∘09984000998400998400E 105

∘09984000998400998400E 106

∘09984000998400998400E 107

∘09984000998400998400E 108

∘09984000998400998400E 109

∘09984000998400998400E 110

∘09984000998400998400E

103∘09984000998400998400E 104

∘09984000998400998400E 105

∘09984000998400998400E 106

∘09984000998400998400E 107

∘09984000998400998400E 108

∘09984000998400998400E 109

∘09984000998400998400E 110

∘09984000998400998400E

34∘09984000998400998400N

33∘09984000998400998400N

35∘09984000998400998400N

36∘09984000998400998400N

37∘09984000998400998400N

38∘09984000998400998400N

39∘09984000998400998400N

34∘09984000998400998400N

33∘09984000998400998400N

35∘09984000998400998400N

36∘09984000998400998400N

37∘09984000998400998400N

38∘09984000998400998400N

39∘09984000998400998400N

N

E

S

W

Basin boundaryWater systemStation

Eigenvector 3value

High 0223536

Low minus0365226

Yanrsquoan

Xirsquoan

Zhenrsquoan

minus01

minus0021

minus001minus0021

minus0019

minus0241

minus0383

minus0271

minus0291

minus0281

minus0084

minus0308minus0086

Figure 6 The eigenvector 3 values for annual precipitation in the Weihe basin

to northwest and downstream to upstream The firstvariation points occurred in the early 1960s at theShangxian Wugong and Xirsquoan stations in the Beidaoto Huaxian section of the Weihe mainstream andlater in the late 1960s and early 1970s other stationsvariation occurred in this section In the Jingheand Beiluohe basins the first variation occurred inthe early 1970s In the Weihe basin above Beidaostationand variation occurred most recently in themid- and late 1980s except at Tianshui station (1971)

(2) Variation points in the trends of annual precipitationin the basin above Beidao station were concentratedin the 1980s while they covered longer periods ofthe 1970s and 1980s in other subbasins These resultsindicate that precipitation variation points occurredfrequently during the 1970s to 1980s in each subbasin

(3) After the 1990s the frequency and scope of precipi-tation variation declined No variation points in the1990s were detected except at Guyuan Pingliangand Wuqi stations After the 1990s there were novariation points in the annual precipitation sequencesrecorded in the Beidao to Huaxian section of theWeihe mainstream Jinghe basin or Beiluohe basinThese results show that after an active period ofprecipitation variation in 1970s and 1980s the annual

precipitation sequence in the Weihe basin was rela-tively stableThus during the study period there wereshifts in precipitation from one stable state to anotherafter disturbance

(4) From 2000 to 2011 further changes in annual pre-cipitation occurred in the basin above the Beidaostation and at the most northern stations of thebasin (Huanxian and Wuqi) These variations wereclearly regional concentrated in the northwest of theWeihe River basin and the source areas of the riversJinghe and Beiluohe Due to the limited length of thetime series it is not possible to distinguish whetherthese variations were due to small-scale regionalvariations or represent initial stages of wide-rangingvariation across the whole basin or were the resultof lingering temporal and spatial effects of the basin-wide variation that occurred in the 1970s and 1980s

(5) The variation in annual precipitation in the Weihebasin propagated from the southeast to the northwestThis is consistent with the orientation of the eigen-vector 2 field of annual precipitation obtained fromEOF decomposition and the spatial pattern of meanannual precipitation in the basin The factors causingthese patterns are not clear from the data presentedhere but will be addressed in future studies

10 Journal of Applied Mathematics

Xiji

Wuqi

SuideYanci

Baoji

Jintai

Foping

Wugong

Guyuan

Lintao

Tongxin

Haiyuan

Yuzhong

HuashanMinxian

Changwu

Hengshan

Jingyuan

Tianshui

Luochuan

Huanxian

Shangzhou

Zhongning

Tongchuan

PingliangXifengzhen

Huajialing

001

039

103077

0347

0078

0019

0009002103010548

0034

0658 0179

00062

N

E

S

W

Trend of eigenvector 3value

High 0761376

Low minus0354567

Yanrsquoan

Zhenrsquoan

Xirsquoan

minus0258

minus0531

minus0536

minus0141

minus002

minus0016

minus0008

minus0017

minus0164

minus0123

minus0123

minus0005

minus0074

minus01

minus0163

50 25 0 50 100 150 200

(km)

103∘09984000998400998400E 104

∘09984000998400998400E 105

∘09984000998400998400E 106

∘09984000998400998400E 107

∘09984000998400998400E 108

∘09984000998400998400E 109

∘09984000998400998400E 110

∘09984000998400998400E

103∘09984000998400998400E 104

∘09984000998400998400E 105

∘09984000998400998400E 106

∘09984000998400998400E 107

∘09984000998400998400E 108

∘09984000998400998400E 109

∘09984000998400998400E 110

∘09984000998400998400E

34∘09984000998400998400N

33∘09984000998400998400N

35∘09984000998400998400N

36∘09984000998400998400N

37∘09984000998400998400N

38∘09984000998400998400N

39∘09984000998400998400N

34∘09984000998400998400N

33∘09984000998400998400N

35∘09984000998400998400N

36∘09984000998400998400N

37∘09984000998400998400N

38∘09984000998400998400N

39∘09984000998400998400N

Basin boundaryWater systemStation

Figure 7 The eigenvector 3 trend field for annual precipitation in the Weihe basin

6 Conclusions

We have analyzed spatial and temporal patterns of annualprecipitation in the Weihe basin using data recorded from1960 to 2011 at 30 National Standard Stations in the basinand surrounding area by applying empirical orthogonalfunction decomposition in combination with geostatistical(semivariogram) analysis then using the Mann-Kendallmethod to discriminate variation points in the precipitationseries recorded at each station

(1) The main spatial patterns of annual precipitation inthe Weihe Basin during the study period were aconsistent overall decline which was strongest in thenorthwest and progressively weaker from northwestto southwest and from west to east The consistentoverall decline accounted for 455 of the totalvariation

(2) Variation points were detected in the annual precip-itation sequences recorded at all the meteorologicalstations in the basin but their timing had clear spatialpatterns starting gradually later from the southeastto the northwest and downstream to upstream Inthe 1970s and 1980s they occurred frequently in eachsubbasin

(3) After the 1990s following the active period of precip-itation variation in the 1970s and 1980s the frequencyand scope of precipitation variations declined Thusduring the study period there were shifts in precipita-tion fromone stable state to another after disturbance

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was financially supported by the Natural ScienceFoundation of China (Grant nos 51190093 51179148 and51179149) the National Key Basic Research 973 of China(Grant no 2012CB417003) and the MWR Welfare SectorFunds (Grant nos 201101043 201101049) Constructive com-ments from reviewers of a previous version of the paper aregratefully acknowledged

References

[1] H Zhang Y Chen G Ren et al ldquoThe characteristics of pre-cipitation variation of Weihe River Basin in Shaanxi Province

Journal of Applied Mathematics 11

during recent 50 yearsrdquo Agricultural Research in the Arid Areasvol 26 no 04 pp 236ndash241 2008

[2] K Che and J Jian ldquoA study of climate variability in Zedangduring 1960ndash1999rdquo Meteorological Monthly vol 26 no 1 pp54ndash56 2000

[3] Y Ding G Ren G Shi et al ldquoNational assessment report ofclimate change (I) climate change inChina and its future trendrdquoAdvances in Climate Change Research vol 2 no 1 pp 3ndash8 2006

[4] Y He F Wang X Mu et al ldquoTemporal and spatial character-istics of temperature and precipitation in Weihe River basinrdquoBulletin of Soil and Water Conservation vol 32 no 4 pp 102ndash105 2012

[5] Y-X Zhang F Wang and X-M Mu ldquoResearch of quantitativerelations between Geographical environment factor and waterand sediment in the Weihe River basinrdquo Journal of Northwest Aamp FUniversity (Natural Science Edition) vol 37 no 1 pp 61ndash652009

[6] C Jiang F Wang and X Mu ldquoEffect of climate change on netprimary productivity (NPP) of natural vegetation in Wei riverbasin (II) NPP of natural vegetation inWei river basinrdquo Journalof Arid LandResources and Environment vol 27 no 5 pp 53ndash572013

[7] Y Zhang ldquoHydrologic feature analysis for Weihe basin inShaanxi provincerdquo Northwest Water Resources amp Water Engi-neering no 2 pp 62ndash64 2002

[8] G Wang and Y Wang ldquoAnalysis on causes of runoff andsediment variation in Weihe basin based on hydrologicalmodelrdquo Journal of Soil Water Conservation no 4 pp 22ndash252000

[9] W He and Z Xu ldquoSpatial and temporal characteristics of thelong-term trend for temperature and pan evaporation in theWei River Basinrdquo Journal of Beijing Normal University (NaturalScience) vol 42 no 1 pp 102ndash106 2006

[10] K Pearson ldquoOn lines and planes of closest fit to system of pointsin spacerdquo Philosophical Magazine vol 6 no 2 pp 559ndash5721902

[11] D-X Cui and W-Y Liu ldquoApplication of empirical orthogonalfunction resolution to analysis of crustal vertical deformationfieldrdquo Earthquake vol 20 no 3 pp 82ndash86 2000

[12] T Liu and H Zhang ldquoCompared with principal componentanalysis and empirical orthogonal function decompositionrdquoStatistics and Decision vol 340 no 16 pp 159ndash162 2011

[13] H Lin and W Zhang ldquoSimilarities and differences of principalcomponent analysis and factor analysis and SPSS softwarerdquoStatistics Research no 3 pp 65ndash69 2005

[14] D H Kim C Yoo and T-W Kim ldquoApplication of spatial EOFand multivariate time series model for evaluating agriculturaldrought vulnerability in Koreardquo Advances in Water Resourcesvol 34 no 3 pp 340ndash350 2011

[15] N S Loboda A V Glushkov and V N Khokhlov ldquoUsingmeteorological data for reconstruction of annual runoff seriesover an ungauged area empirical orthogonal function approachto Moldova-Southwest Ukraine regionrdquo Atmospheric Researchvol 77 no 1ndash4 pp 100ndash113 2005

[16] D Sirjacobs A Alvera-Azcarate A Barth et al ldquoCloud fillingof ocean colour and sea surface temperature remote sensingproducts over the SouthernNorth Sea by theData InterpolatingEmpirical Orthogonal Functions methodologyrdquo Journal of SeaResearch vol 65 no 1 pp 114ndash130 2011

[17] H-L Yu and H-J Chu ldquoUnderstanding space-time patternsof groundwater system by empirical orthogonal functions a

case study in the Choshui River alluvial fan Taiwanrdquo Journalof Hydrology vol 381 no 3-4 pp 239ndash247 2010

[18] H BMann ldquoNonparametric tests against trendrdquo Econometricavol 13 pp 245ndash259 1945

[19] R Snyers ldquoSur LaDetermination de la stabilite des series clima-tologiquesrdquo in Proceedings of the UNESCO-WMO Symposium ofChanges of Climate vol 20 ofUneseo Arid Zone Research Seriespp 37ndash44 Unesco Paris France 1963

[20] C Goossens and A Berger ldquoAnnual and seasonal climaticvariations over the northern hemisphere and Europe during thelast centuryrdquo Annales Geophysicae B vol 4 no 4 pp 385ndash4001986

[21] C Fu and QWang ldquoThe definition and detection of the abruptclimatic changerdquo Chinese Journal of Atmospheric Sciences vol16 no 4 pp 482ndash493 1992

[22] P Xie G Chen D Li et al ldquoHydrological variation inte-grated diagnosis method and application researchrdquo Hydroelec-tric Energy vol 23 no 2 pp 11ndash14 2005

[23] M Gocic and S Trajkovic ldquoAnalysis of changes in meteoro-logical variables using Mann-Kendall and Senrsquos slope estimatorstatistical tests in SerbiardquoGlobal and Planetary Change vol 100pp 172ndash182 2013

[24] K H Hamed ldquoTrend detection in hydrologic data the Mann-Kendall trend test under the scaling hypothesisrdquo Journal ofHydrology vol 349 no 3-4 pp 350ndash363 2008

[25] S Kumar V Merwade J Kam and K Thurner ldquoStreamflowtrends in Indiana effects of long term persistence precipitationand subsurface drainsrdquo Journal of Hydrology vol 374 no 1-2pp 171ndash183 2009

[26] K H Hamed ldquoExact distribution of the Mann-Kendall trendtest statistic for persistent datardquo Journal of Hydrology vol 365no 1-2 pp 86ndash94 2009

[27] D Duhan and A Pandey ldquoStatistical analysis of long termspatial and temporal trends of precipitation during 1901ndash2002at Madhya Pradesh Indiardquo Atmospheric Research vol 122 pp136ndash149 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Applied Mathematics 7

Table 4 Variation points in annual precipitation trends in the Weihe basin

Subbasins Meteorologicalstations

Variation points1960ndash1969 1970ndash1979 1980ndash1989 1990ndash1999 2000ndash2011

Weihe basin aboveBeidao station

Tianshui 1971 1983 1985 2002Minxian 1986 1988 2006Huajialing 1986Lintao 1986 1987 2003 2006 2008Xiji 2005

Jinghe basin

Huanxian 1970 2002Guyuan 1972 1986 1996Pingliang 1971 1977 1980ndash1985lowast 1990Xifengzhen 1972 1979Changwu 1978 1979 1983 1985

Beiluohe basinWuqi 1971ndash1974lowast 1981 1983 1986ndash1995lowast 1998 2000 2003 2007

Yanrsquoan 1972ndash1975lowast1978 1979 1982

Luochuan 1970 1976

Beidao to Huaxiansection of Weihemainstream

Baoji 1969 1971 19741976 1978 1980 1982

Wugong 19611964ndash1970lowast

Xirsquoan 1962 1965Tongchuan 1976 1983 1985Zhenrsquoan 1972 1975

Foping 1966 1968 1973 19771979 1986 1988

Shangxian 1960 1963 1971 1975 1984Huanshan 1986

Extraction method Mann-Kendall method lowastindicates that variation occurred frequently during the indicated period

trend of annual precipitation shows that annual precipitationin the basin declined in the region and this consistent trendaccounts for 455 of the variance The basin covers parts ofthe continental monsoon zone and transitional zone betweenarid and humid regions of northwest China The overallconsistency and decline represented by eigenvector 1 are themain spatial features of precipitation patterns in the Weihebasin during the study period

The eigenvector 2 values obtained from EOF decomposi-tion of the annual precipitation data are shown in Figure 4The values can be divided by a line passing roughly fromYanrsquoan through Pingliang to Tianshui into positive andnegative anomaly areas covering the southeastern parts of theWeihe basin (with themost-positive anomaly center in Baoji)and northwestern parts (with the most-negative anomalycenter in Wu Qi) respectively These results confirm thatthe annual precipitation in the basin has a clear northwest-southeast pattern

Combining the eigenvalues with the trends in annualprecipitation recorded at each of the stations gives the eigen-vector 2 trend values (Figure 5) which show that precipita-tion declined in the southeast and increased in the northwestThe decline was most pronounced at the downstream stationHuashan

The eigenvector 3 values have a clear eastwest pattern(Figure 6) with a negative anomaly area covering upstreamparts of the basin and a positive anomaly center located in theLuochuan region north of ShaanxiThese findings show thatin addition to the consistent overall pattern and northwest-southeast precipitation patterns there is a clear eastwestpattern In combination with the precipitation trends at the30 stations the results clearly show that annual precipitationdeclined in northern Shaanxi and increased in the upstreamareas (Figure 7)

To summarize there were three spatial patterns of annualprecipitation in the Weihe basin during the study period (1)consistent overall decline (2) declined in the southeast andincreased in the northwest (3) declined in northern Shaanxiand increased in the upstream areas

52 Analysis of Annual Precipitation Variation in the WeiheBasin The results summarized in Table 4 show that varia-tion points occurred in trends of the annual precipitationsequence recorded at every meteorological station in theWeihe basin but their timings differed in the followingmanner

(1) The timing of variation points has clear spatial char-acteristics with gradual progressions from southeast

8 Journal of Applied Mathematics

Xiji

Wuqi

SuideYanci

Baoji

Jintai

Foping

Wugong

Guyuan

Lintao

Tongxin

Haiyuan

Yuzhong

HuashanMinxian

Changwu

Hengshan

Jingyuan

Tianshui

Luochuan

Huanxian

Zhongning

Shangzhou

Tongchuan

PingliangXifengzhen

Huajialing

0223

0245

0218

0215027702730287

00570001

0248

0

0201

01040045

Yanrsquoan

Zhenrsquoan

Xirsquoan

139

minus02

minus0206

minus0198

minus025

minus0156

minus0137minus00428

minus0131minus0082

minus014

minus0157

minus012

minus0108

minus0221

minus0231

N

E

S

W

103∘09984000998400998400E 104

∘09984000998400998400E 105

∘09984000998400998400E 106

∘09984000998400998400E 107

∘09984000998400998400E 108

∘09984000998400998400E 109

∘09984000998400998400E 110

∘09984000998400998400E

103∘09984000998400998400E 104

∘09984000998400998400E 105

∘09984000998400998400E 106

∘09984000998400998400E 107

∘09984000998400998400E 108

∘09984000998400998400E 109

∘09984000998400998400E 110

∘09984000998400998400E

34∘09984000998400998400N

33∘09984000998400998400N

35∘09984000998400998400N

36∘09984000998400998400N

37∘09984000998400998400N

38∘09984000998400998400N

39∘09984000998400998400N

34∘09984000998400998400N

33∘09984000998400998400N

35∘09984000998400998400N

36∘09984000998400998400N

37∘09984000998400998400N

38∘09984000998400998400N

39∘09984000998400998400N

50 25 0 50 100 150 200

(km)

Basin boundaryWater systemStation

Eigenvector 2High 0286104

Low minus0196997

Figure 4 The eigenvector 2 values for annual precipitation in the Weihe basin

Xiji

Wuqi

SuideYanci

Baoji

Jintai

Foping

Wugong

Guyuan

Lintao

Tongxin

Haiyuan

Yuzhong

HuashanMinxian

Changwu

Hengshan

Jingyuan

Tianshui

Luochuan

Huanxian

Zhongning

Shangzhou

Tongchuan

PingliangXifengzhen

Huajialing

022

023

028

0227

0316

01290099

0199

0192

0034

0266

0172

0228

01150363

50 25 0 50 100 150 200

(km)

103∘09984000998400998400E 104

∘09984000998400998400E 105

∘09984000998400998400E 106

∘09984000998400998400E 107

∘09984000998400998400E 108

∘09984000998400998400E 109

∘09984000998400998400E 110

∘09984000998400998400E

103∘09984000998400998400E 104

∘09984000998400998400E 105

∘09984000998400998400E 106

∘09984000998400998400E 107

∘09984000998400998400E 108

∘09984000998400998400E 109

∘09984000998400998400E 110

∘09984000998400998400E

34∘09984000998400998400N

33∘09984000998400998400N

35∘09984000998400998400N

36∘09984000998400998400N

37∘09984000998400998400N

38∘09984000998400998400N

39∘09984000998400998400N

34∘09984000998400998400N

33∘09984000998400998400N

35∘09984000998400998400N

36∘09984000998400998400N

37∘09984000998400998400N

38∘09984000998400998400N

39∘09984000998400998400N

N

E

S

W

Basin boundaryWater systemStation

Trend of eigenvector 2value

High 0229457

Low minus0461604

minus0023

minus0109

minus003

minus0838

minus0094

minus0196minus019

minus0236

minus006

minus0156

minus0083

minus0283

minus0072minus00018

minus0007

Yanrsquoan

Xirsquoan

Zhenrsquoan

Figure 5 The eigenvector 2 trend field for annual precipitation in the Weihe basin

Journal of Applied Mathematics 9

Xiji

Wuqi

SuideYanci

Baoji

Jintai

Foping

Wugong

Guyuan

Lintao

Tongxin

Haiyuan

Yuzhong

HuashanMinxian

Changwu

Hengshan

Jingyuan

Tianshui

Luochuan

Huanxian

Zhongning

Shangzhou

Tongchuan

PingliangXifengzhen

Huajialing

026

024025

0072

0004

0127

0272

0042

0009

0051

0168

0179

0058

0067

025100860112

50 25 0 50 100 150 200

(km)

103∘09984000998400998400E 104

∘09984000998400998400E 105

∘09984000998400998400E 106

∘09984000998400998400E 107

∘09984000998400998400E 108

∘09984000998400998400E 109

∘09984000998400998400E 110

∘09984000998400998400E

103∘09984000998400998400E 104

∘09984000998400998400E 105

∘09984000998400998400E 106

∘09984000998400998400E 107

∘09984000998400998400E 108

∘09984000998400998400E 109

∘09984000998400998400E 110

∘09984000998400998400E

34∘09984000998400998400N

33∘09984000998400998400N

35∘09984000998400998400N

36∘09984000998400998400N

37∘09984000998400998400N

38∘09984000998400998400N

39∘09984000998400998400N

34∘09984000998400998400N

33∘09984000998400998400N

35∘09984000998400998400N

36∘09984000998400998400N

37∘09984000998400998400N

38∘09984000998400998400N

39∘09984000998400998400N

N

E

S

W

Basin boundaryWater systemStation

Eigenvector 3value

High 0223536

Low minus0365226

Yanrsquoan

Xirsquoan

Zhenrsquoan

minus01

minus0021

minus001minus0021

minus0019

minus0241

minus0383

minus0271

minus0291

minus0281

minus0084

minus0308minus0086

Figure 6 The eigenvector 3 values for annual precipitation in the Weihe basin

to northwest and downstream to upstream The firstvariation points occurred in the early 1960s at theShangxian Wugong and Xirsquoan stations in the Beidaoto Huaxian section of the Weihe mainstream andlater in the late 1960s and early 1970s other stationsvariation occurred in this section In the Jingheand Beiluohe basins the first variation occurred inthe early 1970s In the Weihe basin above Beidaostationand variation occurred most recently in themid- and late 1980s except at Tianshui station (1971)

(2) Variation points in the trends of annual precipitationin the basin above Beidao station were concentratedin the 1980s while they covered longer periods ofthe 1970s and 1980s in other subbasins These resultsindicate that precipitation variation points occurredfrequently during the 1970s to 1980s in each subbasin

(3) After the 1990s the frequency and scope of precipi-tation variation declined No variation points in the1990s were detected except at Guyuan Pingliangand Wuqi stations After the 1990s there were novariation points in the annual precipitation sequencesrecorded in the Beidao to Huaxian section of theWeihe mainstream Jinghe basin or Beiluohe basinThese results show that after an active period ofprecipitation variation in 1970s and 1980s the annual

precipitation sequence in the Weihe basin was rela-tively stableThus during the study period there wereshifts in precipitation from one stable state to anotherafter disturbance

(4) From 2000 to 2011 further changes in annual pre-cipitation occurred in the basin above the Beidaostation and at the most northern stations of thebasin (Huanxian and Wuqi) These variations wereclearly regional concentrated in the northwest of theWeihe River basin and the source areas of the riversJinghe and Beiluohe Due to the limited length of thetime series it is not possible to distinguish whetherthese variations were due to small-scale regionalvariations or represent initial stages of wide-rangingvariation across the whole basin or were the resultof lingering temporal and spatial effects of the basin-wide variation that occurred in the 1970s and 1980s

(5) The variation in annual precipitation in the Weihebasin propagated from the southeast to the northwestThis is consistent with the orientation of the eigen-vector 2 field of annual precipitation obtained fromEOF decomposition and the spatial pattern of meanannual precipitation in the basin The factors causingthese patterns are not clear from the data presentedhere but will be addressed in future studies

10 Journal of Applied Mathematics

Xiji

Wuqi

SuideYanci

Baoji

Jintai

Foping

Wugong

Guyuan

Lintao

Tongxin

Haiyuan

Yuzhong

HuashanMinxian

Changwu

Hengshan

Jingyuan

Tianshui

Luochuan

Huanxian

Shangzhou

Zhongning

Tongchuan

PingliangXifengzhen

Huajialing

001

039

103077

0347

0078

0019

0009002103010548

0034

0658 0179

00062

N

E

S

W

Trend of eigenvector 3value

High 0761376

Low minus0354567

Yanrsquoan

Zhenrsquoan

Xirsquoan

minus0258

minus0531

minus0536

minus0141

minus002

minus0016

minus0008

minus0017

minus0164

minus0123

minus0123

minus0005

minus0074

minus01

minus0163

50 25 0 50 100 150 200

(km)

103∘09984000998400998400E 104

∘09984000998400998400E 105

∘09984000998400998400E 106

∘09984000998400998400E 107

∘09984000998400998400E 108

∘09984000998400998400E 109

∘09984000998400998400E 110

∘09984000998400998400E

103∘09984000998400998400E 104

∘09984000998400998400E 105

∘09984000998400998400E 106

∘09984000998400998400E 107

∘09984000998400998400E 108

∘09984000998400998400E 109

∘09984000998400998400E 110

∘09984000998400998400E

34∘09984000998400998400N

33∘09984000998400998400N

35∘09984000998400998400N

36∘09984000998400998400N

37∘09984000998400998400N

38∘09984000998400998400N

39∘09984000998400998400N

34∘09984000998400998400N

33∘09984000998400998400N

35∘09984000998400998400N

36∘09984000998400998400N

37∘09984000998400998400N

38∘09984000998400998400N

39∘09984000998400998400N

Basin boundaryWater systemStation

Figure 7 The eigenvector 3 trend field for annual precipitation in the Weihe basin

6 Conclusions

We have analyzed spatial and temporal patterns of annualprecipitation in the Weihe basin using data recorded from1960 to 2011 at 30 National Standard Stations in the basinand surrounding area by applying empirical orthogonalfunction decomposition in combination with geostatistical(semivariogram) analysis then using the Mann-Kendallmethod to discriminate variation points in the precipitationseries recorded at each station

(1) The main spatial patterns of annual precipitation inthe Weihe Basin during the study period were aconsistent overall decline which was strongest in thenorthwest and progressively weaker from northwestto southwest and from west to east The consistentoverall decline accounted for 455 of the totalvariation

(2) Variation points were detected in the annual precip-itation sequences recorded at all the meteorologicalstations in the basin but their timing had clear spatialpatterns starting gradually later from the southeastto the northwest and downstream to upstream Inthe 1970s and 1980s they occurred frequently in eachsubbasin

(3) After the 1990s following the active period of precip-itation variation in the 1970s and 1980s the frequencyand scope of precipitation variations declined Thusduring the study period there were shifts in precipita-tion fromone stable state to another after disturbance

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was financially supported by the Natural ScienceFoundation of China (Grant nos 51190093 51179148 and51179149) the National Key Basic Research 973 of China(Grant no 2012CB417003) and the MWR Welfare SectorFunds (Grant nos 201101043 201101049) Constructive com-ments from reviewers of a previous version of the paper aregratefully acknowledged

References

[1] H Zhang Y Chen G Ren et al ldquoThe characteristics of pre-cipitation variation of Weihe River Basin in Shaanxi Province

Journal of Applied Mathematics 11

during recent 50 yearsrdquo Agricultural Research in the Arid Areasvol 26 no 04 pp 236ndash241 2008

[2] K Che and J Jian ldquoA study of climate variability in Zedangduring 1960ndash1999rdquo Meteorological Monthly vol 26 no 1 pp54ndash56 2000

[3] Y Ding G Ren G Shi et al ldquoNational assessment report ofclimate change (I) climate change inChina and its future trendrdquoAdvances in Climate Change Research vol 2 no 1 pp 3ndash8 2006

[4] Y He F Wang X Mu et al ldquoTemporal and spatial character-istics of temperature and precipitation in Weihe River basinrdquoBulletin of Soil and Water Conservation vol 32 no 4 pp 102ndash105 2012

[5] Y-X Zhang F Wang and X-M Mu ldquoResearch of quantitativerelations between Geographical environment factor and waterand sediment in the Weihe River basinrdquo Journal of Northwest Aamp FUniversity (Natural Science Edition) vol 37 no 1 pp 61ndash652009

[6] C Jiang F Wang and X Mu ldquoEffect of climate change on netprimary productivity (NPP) of natural vegetation in Wei riverbasin (II) NPP of natural vegetation inWei river basinrdquo Journalof Arid LandResources and Environment vol 27 no 5 pp 53ndash572013

[7] Y Zhang ldquoHydrologic feature analysis for Weihe basin inShaanxi provincerdquo Northwest Water Resources amp Water Engi-neering no 2 pp 62ndash64 2002

[8] G Wang and Y Wang ldquoAnalysis on causes of runoff andsediment variation in Weihe basin based on hydrologicalmodelrdquo Journal of Soil Water Conservation no 4 pp 22ndash252000

[9] W He and Z Xu ldquoSpatial and temporal characteristics of thelong-term trend for temperature and pan evaporation in theWei River Basinrdquo Journal of Beijing Normal University (NaturalScience) vol 42 no 1 pp 102ndash106 2006

[10] K Pearson ldquoOn lines and planes of closest fit to system of pointsin spacerdquo Philosophical Magazine vol 6 no 2 pp 559ndash5721902

[11] D-X Cui and W-Y Liu ldquoApplication of empirical orthogonalfunction resolution to analysis of crustal vertical deformationfieldrdquo Earthquake vol 20 no 3 pp 82ndash86 2000

[12] T Liu and H Zhang ldquoCompared with principal componentanalysis and empirical orthogonal function decompositionrdquoStatistics and Decision vol 340 no 16 pp 159ndash162 2011

[13] H Lin and W Zhang ldquoSimilarities and differences of principalcomponent analysis and factor analysis and SPSS softwarerdquoStatistics Research no 3 pp 65ndash69 2005

[14] D H Kim C Yoo and T-W Kim ldquoApplication of spatial EOFand multivariate time series model for evaluating agriculturaldrought vulnerability in Koreardquo Advances in Water Resourcesvol 34 no 3 pp 340ndash350 2011

[15] N S Loboda A V Glushkov and V N Khokhlov ldquoUsingmeteorological data for reconstruction of annual runoff seriesover an ungauged area empirical orthogonal function approachto Moldova-Southwest Ukraine regionrdquo Atmospheric Researchvol 77 no 1ndash4 pp 100ndash113 2005

[16] D Sirjacobs A Alvera-Azcarate A Barth et al ldquoCloud fillingof ocean colour and sea surface temperature remote sensingproducts over the SouthernNorth Sea by theData InterpolatingEmpirical Orthogonal Functions methodologyrdquo Journal of SeaResearch vol 65 no 1 pp 114ndash130 2011

[17] H-L Yu and H-J Chu ldquoUnderstanding space-time patternsof groundwater system by empirical orthogonal functions a

case study in the Choshui River alluvial fan Taiwanrdquo Journalof Hydrology vol 381 no 3-4 pp 239ndash247 2010

[18] H BMann ldquoNonparametric tests against trendrdquo Econometricavol 13 pp 245ndash259 1945

[19] R Snyers ldquoSur LaDetermination de la stabilite des series clima-tologiquesrdquo in Proceedings of the UNESCO-WMO Symposium ofChanges of Climate vol 20 ofUneseo Arid Zone Research Seriespp 37ndash44 Unesco Paris France 1963

[20] C Goossens and A Berger ldquoAnnual and seasonal climaticvariations over the northern hemisphere and Europe during thelast centuryrdquo Annales Geophysicae B vol 4 no 4 pp 385ndash4001986

[21] C Fu and QWang ldquoThe definition and detection of the abruptclimatic changerdquo Chinese Journal of Atmospheric Sciences vol16 no 4 pp 482ndash493 1992

[22] P Xie G Chen D Li et al ldquoHydrological variation inte-grated diagnosis method and application researchrdquo Hydroelec-tric Energy vol 23 no 2 pp 11ndash14 2005

[23] M Gocic and S Trajkovic ldquoAnalysis of changes in meteoro-logical variables using Mann-Kendall and Senrsquos slope estimatorstatistical tests in SerbiardquoGlobal and Planetary Change vol 100pp 172ndash182 2013

[24] K H Hamed ldquoTrend detection in hydrologic data the Mann-Kendall trend test under the scaling hypothesisrdquo Journal ofHydrology vol 349 no 3-4 pp 350ndash363 2008

[25] S Kumar V Merwade J Kam and K Thurner ldquoStreamflowtrends in Indiana effects of long term persistence precipitationand subsurface drainsrdquo Journal of Hydrology vol 374 no 1-2pp 171ndash183 2009

[26] K H Hamed ldquoExact distribution of the Mann-Kendall trendtest statistic for persistent datardquo Journal of Hydrology vol 365no 1-2 pp 86ndash94 2009

[27] D Duhan and A Pandey ldquoStatistical analysis of long termspatial and temporal trends of precipitation during 1901ndash2002at Madhya Pradesh Indiardquo Atmospheric Research vol 122 pp136ndash149 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Journal of Applied Mathematics

Xiji

Wuqi

SuideYanci

Baoji

Jintai

Foping

Wugong

Guyuan

Lintao

Tongxin

Haiyuan

Yuzhong

HuashanMinxian

Changwu

Hengshan

Jingyuan

Tianshui

Luochuan

Huanxian

Zhongning

Shangzhou

Tongchuan

PingliangXifengzhen

Huajialing

0223

0245

0218

0215027702730287

00570001

0248

0

0201

01040045

Yanrsquoan

Zhenrsquoan

Xirsquoan

139

minus02

minus0206

minus0198

minus025

minus0156

minus0137minus00428

minus0131minus0082

minus014

minus0157

minus012

minus0108

minus0221

minus0231

N

E

S

W

103∘09984000998400998400E 104

∘09984000998400998400E 105

∘09984000998400998400E 106

∘09984000998400998400E 107

∘09984000998400998400E 108

∘09984000998400998400E 109

∘09984000998400998400E 110

∘09984000998400998400E

103∘09984000998400998400E 104

∘09984000998400998400E 105

∘09984000998400998400E 106

∘09984000998400998400E 107

∘09984000998400998400E 108

∘09984000998400998400E 109

∘09984000998400998400E 110

∘09984000998400998400E

34∘09984000998400998400N

33∘09984000998400998400N

35∘09984000998400998400N

36∘09984000998400998400N

37∘09984000998400998400N

38∘09984000998400998400N

39∘09984000998400998400N

34∘09984000998400998400N

33∘09984000998400998400N

35∘09984000998400998400N

36∘09984000998400998400N

37∘09984000998400998400N

38∘09984000998400998400N

39∘09984000998400998400N

50 25 0 50 100 150 200

(km)

Basin boundaryWater systemStation

Eigenvector 2High 0286104

Low minus0196997

Figure 4 The eigenvector 2 values for annual precipitation in the Weihe basin

Xiji

Wuqi

SuideYanci

Baoji

Jintai

Foping

Wugong

Guyuan

Lintao

Tongxin

Haiyuan

Yuzhong

HuashanMinxian

Changwu

Hengshan

Jingyuan

Tianshui

Luochuan

Huanxian

Zhongning

Shangzhou

Tongchuan

PingliangXifengzhen

Huajialing

022

023

028

0227

0316

01290099

0199

0192

0034

0266

0172

0228

01150363

50 25 0 50 100 150 200

(km)

103∘09984000998400998400E 104

∘09984000998400998400E 105

∘09984000998400998400E 106

∘09984000998400998400E 107

∘09984000998400998400E 108

∘09984000998400998400E 109

∘09984000998400998400E 110

∘09984000998400998400E

103∘09984000998400998400E 104

∘09984000998400998400E 105

∘09984000998400998400E 106

∘09984000998400998400E 107

∘09984000998400998400E 108

∘09984000998400998400E 109

∘09984000998400998400E 110

∘09984000998400998400E

34∘09984000998400998400N

33∘09984000998400998400N

35∘09984000998400998400N

36∘09984000998400998400N

37∘09984000998400998400N

38∘09984000998400998400N

39∘09984000998400998400N

34∘09984000998400998400N

33∘09984000998400998400N

35∘09984000998400998400N

36∘09984000998400998400N

37∘09984000998400998400N

38∘09984000998400998400N

39∘09984000998400998400N

N

E

S

W

Basin boundaryWater systemStation

Trend of eigenvector 2value

High 0229457

Low minus0461604

minus0023

minus0109

minus003

minus0838

minus0094

minus0196minus019

minus0236

minus006

minus0156

minus0083

minus0283

minus0072minus00018

minus0007

Yanrsquoan

Xirsquoan

Zhenrsquoan

Figure 5 The eigenvector 2 trend field for annual precipitation in the Weihe basin

Journal of Applied Mathematics 9

Xiji

Wuqi

SuideYanci

Baoji

Jintai

Foping

Wugong

Guyuan

Lintao

Tongxin

Haiyuan

Yuzhong

HuashanMinxian

Changwu

Hengshan

Jingyuan

Tianshui

Luochuan

Huanxian

Zhongning

Shangzhou

Tongchuan

PingliangXifengzhen

Huajialing

026

024025

0072

0004

0127

0272

0042

0009

0051

0168

0179

0058

0067

025100860112

50 25 0 50 100 150 200

(km)

103∘09984000998400998400E 104

∘09984000998400998400E 105

∘09984000998400998400E 106

∘09984000998400998400E 107

∘09984000998400998400E 108

∘09984000998400998400E 109

∘09984000998400998400E 110

∘09984000998400998400E

103∘09984000998400998400E 104

∘09984000998400998400E 105

∘09984000998400998400E 106

∘09984000998400998400E 107

∘09984000998400998400E 108

∘09984000998400998400E 109

∘09984000998400998400E 110

∘09984000998400998400E

34∘09984000998400998400N

33∘09984000998400998400N

35∘09984000998400998400N

36∘09984000998400998400N

37∘09984000998400998400N

38∘09984000998400998400N

39∘09984000998400998400N

34∘09984000998400998400N

33∘09984000998400998400N

35∘09984000998400998400N

36∘09984000998400998400N

37∘09984000998400998400N

38∘09984000998400998400N

39∘09984000998400998400N

N

E

S

W

Basin boundaryWater systemStation

Eigenvector 3value

High 0223536

Low minus0365226

Yanrsquoan

Xirsquoan

Zhenrsquoan

minus01

minus0021

minus001minus0021

minus0019

minus0241

minus0383

minus0271

minus0291

minus0281

minus0084

minus0308minus0086

Figure 6 The eigenvector 3 values for annual precipitation in the Weihe basin

to northwest and downstream to upstream The firstvariation points occurred in the early 1960s at theShangxian Wugong and Xirsquoan stations in the Beidaoto Huaxian section of the Weihe mainstream andlater in the late 1960s and early 1970s other stationsvariation occurred in this section In the Jingheand Beiluohe basins the first variation occurred inthe early 1970s In the Weihe basin above Beidaostationand variation occurred most recently in themid- and late 1980s except at Tianshui station (1971)

(2) Variation points in the trends of annual precipitationin the basin above Beidao station were concentratedin the 1980s while they covered longer periods ofthe 1970s and 1980s in other subbasins These resultsindicate that precipitation variation points occurredfrequently during the 1970s to 1980s in each subbasin

(3) After the 1990s the frequency and scope of precipi-tation variation declined No variation points in the1990s were detected except at Guyuan Pingliangand Wuqi stations After the 1990s there were novariation points in the annual precipitation sequencesrecorded in the Beidao to Huaxian section of theWeihe mainstream Jinghe basin or Beiluohe basinThese results show that after an active period ofprecipitation variation in 1970s and 1980s the annual

precipitation sequence in the Weihe basin was rela-tively stableThus during the study period there wereshifts in precipitation from one stable state to anotherafter disturbance

(4) From 2000 to 2011 further changes in annual pre-cipitation occurred in the basin above the Beidaostation and at the most northern stations of thebasin (Huanxian and Wuqi) These variations wereclearly regional concentrated in the northwest of theWeihe River basin and the source areas of the riversJinghe and Beiluohe Due to the limited length of thetime series it is not possible to distinguish whetherthese variations were due to small-scale regionalvariations or represent initial stages of wide-rangingvariation across the whole basin or were the resultof lingering temporal and spatial effects of the basin-wide variation that occurred in the 1970s and 1980s

(5) The variation in annual precipitation in the Weihebasin propagated from the southeast to the northwestThis is consistent with the orientation of the eigen-vector 2 field of annual precipitation obtained fromEOF decomposition and the spatial pattern of meanannual precipitation in the basin The factors causingthese patterns are not clear from the data presentedhere but will be addressed in future studies

10 Journal of Applied Mathematics

Xiji

Wuqi

SuideYanci

Baoji

Jintai

Foping

Wugong

Guyuan

Lintao

Tongxin

Haiyuan

Yuzhong

HuashanMinxian

Changwu

Hengshan

Jingyuan

Tianshui

Luochuan

Huanxian

Shangzhou

Zhongning

Tongchuan

PingliangXifengzhen

Huajialing

001

039

103077

0347

0078

0019

0009002103010548

0034

0658 0179

00062

N

E

S

W

Trend of eigenvector 3value

High 0761376

Low minus0354567

Yanrsquoan

Zhenrsquoan

Xirsquoan

minus0258

minus0531

minus0536

minus0141

minus002

minus0016

minus0008

minus0017

minus0164

minus0123

minus0123

minus0005

minus0074

minus01

minus0163

50 25 0 50 100 150 200

(km)

103∘09984000998400998400E 104

∘09984000998400998400E 105

∘09984000998400998400E 106

∘09984000998400998400E 107

∘09984000998400998400E 108

∘09984000998400998400E 109

∘09984000998400998400E 110

∘09984000998400998400E

103∘09984000998400998400E 104

∘09984000998400998400E 105

∘09984000998400998400E 106

∘09984000998400998400E 107

∘09984000998400998400E 108

∘09984000998400998400E 109

∘09984000998400998400E 110

∘09984000998400998400E

34∘09984000998400998400N

33∘09984000998400998400N

35∘09984000998400998400N

36∘09984000998400998400N

37∘09984000998400998400N

38∘09984000998400998400N

39∘09984000998400998400N

34∘09984000998400998400N

33∘09984000998400998400N

35∘09984000998400998400N

36∘09984000998400998400N

37∘09984000998400998400N

38∘09984000998400998400N

39∘09984000998400998400N

Basin boundaryWater systemStation

Figure 7 The eigenvector 3 trend field for annual precipitation in the Weihe basin

6 Conclusions

We have analyzed spatial and temporal patterns of annualprecipitation in the Weihe basin using data recorded from1960 to 2011 at 30 National Standard Stations in the basinand surrounding area by applying empirical orthogonalfunction decomposition in combination with geostatistical(semivariogram) analysis then using the Mann-Kendallmethod to discriminate variation points in the precipitationseries recorded at each station

(1) The main spatial patterns of annual precipitation inthe Weihe Basin during the study period were aconsistent overall decline which was strongest in thenorthwest and progressively weaker from northwestto southwest and from west to east The consistentoverall decline accounted for 455 of the totalvariation

(2) Variation points were detected in the annual precip-itation sequences recorded at all the meteorologicalstations in the basin but their timing had clear spatialpatterns starting gradually later from the southeastto the northwest and downstream to upstream Inthe 1970s and 1980s they occurred frequently in eachsubbasin

(3) After the 1990s following the active period of precip-itation variation in the 1970s and 1980s the frequencyand scope of precipitation variations declined Thusduring the study period there were shifts in precipita-tion fromone stable state to another after disturbance

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was financially supported by the Natural ScienceFoundation of China (Grant nos 51190093 51179148 and51179149) the National Key Basic Research 973 of China(Grant no 2012CB417003) and the MWR Welfare SectorFunds (Grant nos 201101043 201101049) Constructive com-ments from reviewers of a previous version of the paper aregratefully acknowledged

References

[1] H Zhang Y Chen G Ren et al ldquoThe characteristics of pre-cipitation variation of Weihe River Basin in Shaanxi Province

Journal of Applied Mathematics 11

during recent 50 yearsrdquo Agricultural Research in the Arid Areasvol 26 no 04 pp 236ndash241 2008

[2] K Che and J Jian ldquoA study of climate variability in Zedangduring 1960ndash1999rdquo Meteorological Monthly vol 26 no 1 pp54ndash56 2000

[3] Y Ding G Ren G Shi et al ldquoNational assessment report ofclimate change (I) climate change inChina and its future trendrdquoAdvances in Climate Change Research vol 2 no 1 pp 3ndash8 2006

[4] Y He F Wang X Mu et al ldquoTemporal and spatial character-istics of temperature and precipitation in Weihe River basinrdquoBulletin of Soil and Water Conservation vol 32 no 4 pp 102ndash105 2012

[5] Y-X Zhang F Wang and X-M Mu ldquoResearch of quantitativerelations between Geographical environment factor and waterand sediment in the Weihe River basinrdquo Journal of Northwest Aamp FUniversity (Natural Science Edition) vol 37 no 1 pp 61ndash652009

[6] C Jiang F Wang and X Mu ldquoEffect of climate change on netprimary productivity (NPP) of natural vegetation in Wei riverbasin (II) NPP of natural vegetation inWei river basinrdquo Journalof Arid LandResources and Environment vol 27 no 5 pp 53ndash572013

[7] Y Zhang ldquoHydrologic feature analysis for Weihe basin inShaanxi provincerdquo Northwest Water Resources amp Water Engi-neering no 2 pp 62ndash64 2002

[8] G Wang and Y Wang ldquoAnalysis on causes of runoff andsediment variation in Weihe basin based on hydrologicalmodelrdquo Journal of Soil Water Conservation no 4 pp 22ndash252000

[9] W He and Z Xu ldquoSpatial and temporal characteristics of thelong-term trend for temperature and pan evaporation in theWei River Basinrdquo Journal of Beijing Normal University (NaturalScience) vol 42 no 1 pp 102ndash106 2006

[10] K Pearson ldquoOn lines and planes of closest fit to system of pointsin spacerdquo Philosophical Magazine vol 6 no 2 pp 559ndash5721902

[11] D-X Cui and W-Y Liu ldquoApplication of empirical orthogonalfunction resolution to analysis of crustal vertical deformationfieldrdquo Earthquake vol 20 no 3 pp 82ndash86 2000

[12] T Liu and H Zhang ldquoCompared with principal componentanalysis and empirical orthogonal function decompositionrdquoStatistics and Decision vol 340 no 16 pp 159ndash162 2011

[13] H Lin and W Zhang ldquoSimilarities and differences of principalcomponent analysis and factor analysis and SPSS softwarerdquoStatistics Research no 3 pp 65ndash69 2005

[14] D H Kim C Yoo and T-W Kim ldquoApplication of spatial EOFand multivariate time series model for evaluating agriculturaldrought vulnerability in Koreardquo Advances in Water Resourcesvol 34 no 3 pp 340ndash350 2011

[15] N S Loboda A V Glushkov and V N Khokhlov ldquoUsingmeteorological data for reconstruction of annual runoff seriesover an ungauged area empirical orthogonal function approachto Moldova-Southwest Ukraine regionrdquo Atmospheric Researchvol 77 no 1ndash4 pp 100ndash113 2005

[16] D Sirjacobs A Alvera-Azcarate A Barth et al ldquoCloud fillingof ocean colour and sea surface temperature remote sensingproducts over the SouthernNorth Sea by theData InterpolatingEmpirical Orthogonal Functions methodologyrdquo Journal of SeaResearch vol 65 no 1 pp 114ndash130 2011

[17] H-L Yu and H-J Chu ldquoUnderstanding space-time patternsof groundwater system by empirical orthogonal functions a

case study in the Choshui River alluvial fan Taiwanrdquo Journalof Hydrology vol 381 no 3-4 pp 239ndash247 2010

[18] H BMann ldquoNonparametric tests against trendrdquo Econometricavol 13 pp 245ndash259 1945

[19] R Snyers ldquoSur LaDetermination de la stabilite des series clima-tologiquesrdquo in Proceedings of the UNESCO-WMO Symposium ofChanges of Climate vol 20 ofUneseo Arid Zone Research Seriespp 37ndash44 Unesco Paris France 1963

[20] C Goossens and A Berger ldquoAnnual and seasonal climaticvariations over the northern hemisphere and Europe during thelast centuryrdquo Annales Geophysicae B vol 4 no 4 pp 385ndash4001986

[21] C Fu and QWang ldquoThe definition and detection of the abruptclimatic changerdquo Chinese Journal of Atmospheric Sciences vol16 no 4 pp 482ndash493 1992

[22] P Xie G Chen D Li et al ldquoHydrological variation inte-grated diagnosis method and application researchrdquo Hydroelec-tric Energy vol 23 no 2 pp 11ndash14 2005

[23] M Gocic and S Trajkovic ldquoAnalysis of changes in meteoro-logical variables using Mann-Kendall and Senrsquos slope estimatorstatistical tests in SerbiardquoGlobal and Planetary Change vol 100pp 172ndash182 2013

[24] K H Hamed ldquoTrend detection in hydrologic data the Mann-Kendall trend test under the scaling hypothesisrdquo Journal ofHydrology vol 349 no 3-4 pp 350ndash363 2008

[25] S Kumar V Merwade J Kam and K Thurner ldquoStreamflowtrends in Indiana effects of long term persistence precipitationand subsurface drainsrdquo Journal of Hydrology vol 374 no 1-2pp 171ndash183 2009

[26] K H Hamed ldquoExact distribution of the Mann-Kendall trendtest statistic for persistent datardquo Journal of Hydrology vol 365no 1-2 pp 86ndash94 2009

[27] D Duhan and A Pandey ldquoStatistical analysis of long termspatial and temporal trends of precipitation during 1901ndash2002at Madhya Pradesh Indiardquo Atmospheric Research vol 122 pp136ndash149 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Applied Mathematics 9

Xiji

Wuqi

SuideYanci

Baoji

Jintai

Foping

Wugong

Guyuan

Lintao

Tongxin

Haiyuan

Yuzhong

HuashanMinxian

Changwu

Hengshan

Jingyuan

Tianshui

Luochuan

Huanxian

Zhongning

Shangzhou

Tongchuan

PingliangXifengzhen

Huajialing

026

024025

0072

0004

0127

0272

0042

0009

0051

0168

0179

0058

0067

025100860112

50 25 0 50 100 150 200

(km)

103∘09984000998400998400E 104

∘09984000998400998400E 105

∘09984000998400998400E 106

∘09984000998400998400E 107

∘09984000998400998400E 108

∘09984000998400998400E 109

∘09984000998400998400E 110

∘09984000998400998400E

103∘09984000998400998400E 104

∘09984000998400998400E 105

∘09984000998400998400E 106

∘09984000998400998400E 107

∘09984000998400998400E 108

∘09984000998400998400E 109

∘09984000998400998400E 110

∘09984000998400998400E

34∘09984000998400998400N

33∘09984000998400998400N

35∘09984000998400998400N

36∘09984000998400998400N

37∘09984000998400998400N

38∘09984000998400998400N

39∘09984000998400998400N

34∘09984000998400998400N

33∘09984000998400998400N

35∘09984000998400998400N

36∘09984000998400998400N

37∘09984000998400998400N

38∘09984000998400998400N

39∘09984000998400998400N

N

E

S

W

Basin boundaryWater systemStation

Eigenvector 3value

High 0223536

Low minus0365226

Yanrsquoan

Xirsquoan

Zhenrsquoan

minus01

minus0021

minus001minus0021

minus0019

minus0241

minus0383

minus0271

minus0291

minus0281

minus0084

minus0308minus0086

Figure 6 The eigenvector 3 values for annual precipitation in the Weihe basin

to northwest and downstream to upstream The firstvariation points occurred in the early 1960s at theShangxian Wugong and Xirsquoan stations in the Beidaoto Huaxian section of the Weihe mainstream andlater in the late 1960s and early 1970s other stationsvariation occurred in this section In the Jingheand Beiluohe basins the first variation occurred inthe early 1970s In the Weihe basin above Beidaostationand variation occurred most recently in themid- and late 1980s except at Tianshui station (1971)

(2) Variation points in the trends of annual precipitationin the basin above Beidao station were concentratedin the 1980s while they covered longer periods ofthe 1970s and 1980s in other subbasins These resultsindicate that precipitation variation points occurredfrequently during the 1970s to 1980s in each subbasin

(3) After the 1990s the frequency and scope of precipi-tation variation declined No variation points in the1990s were detected except at Guyuan Pingliangand Wuqi stations After the 1990s there were novariation points in the annual precipitation sequencesrecorded in the Beidao to Huaxian section of theWeihe mainstream Jinghe basin or Beiluohe basinThese results show that after an active period ofprecipitation variation in 1970s and 1980s the annual

precipitation sequence in the Weihe basin was rela-tively stableThus during the study period there wereshifts in precipitation from one stable state to anotherafter disturbance

(4) From 2000 to 2011 further changes in annual pre-cipitation occurred in the basin above the Beidaostation and at the most northern stations of thebasin (Huanxian and Wuqi) These variations wereclearly regional concentrated in the northwest of theWeihe River basin and the source areas of the riversJinghe and Beiluohe Due to the limited length of thetime series it is not possible to distinguish whetherthese variations were due to small-scale regionalvariations or represent initial stages of wide-rangingvariation across the whole basin or were the resultof lingering temporal and spatial effects of the basin-wide variation that occurred in the 1970s and 1980s

(5) The variation in annual precipitation in the Weihebasin propagated from the southeast to the northwestThis is consistent with the orientation of the eigen-vector 2 field of annual precipitation obtained fromEOF decomposition and the spatial pattern of meanannual precipitation in the basin The factors causingthese patterns are not clear from the data presentedhere but will be addressed in future studies

10 Journal of Applied Mathematics

Xiji

Wuqi

SuideYanci

Baoji

Jintai

Foping

Wugong

Guyuan

Lintao

Tongxin

Haiyuan

Yuzhong

HuashanMinxian

Changwu

Hengshan

Jingyuan

Tianshui

Luochuan

Huanxian

Shangzhou

Zhongning

Tongchuan

PingliangXifengzhen

Huajialing

001

039

103077

0347

0078

0019

0009002103010548

0034

0658 0179

00062

N

E

S

W

Trend of eigenvector 3value

High 0761376

Low minus0354567

Yanrsquoan

Zhenrsquoan

Xirsquoan

minus0258

minus0531

minus0536

minus0141

minus002

minus0016

minus0008

minus0017

minus0164

minus0123

minus0123

minus0005

minus0074

minus01

minus0163

50 25 0 50 100 150 200

(km)

103∘09984000998400998400E 104

∘09984000998400998400E 105

∘09984000998400998400E 106

∘09984000998400998400E 107

∘09984000998400998400E 108

∘09984000998400998400E 109

∘09984000998400998400E 110

∘09984000998400998400E

103∘09984000998400998400E 104

∘09984000998400998400E 105

∘09984000998400998400E 106

∘09984000998400998400E 107

∘09984000998400998400E 108

∘09984000998400998400E 109

∘09984000998400998400E 110

∘09984000998400998400E

34∘09984000998400998400N

33∘09984000998400998400N

35∘09984000998400998400N

36∘09984000998400998400N

37∘09984000998400998400N

38∘09984000998400998400N

39∘09984000998400998400N

34∘09984000998400998400N

33∘09984000998400998400N

35∘09984000998400998400N

36∘09984000998400998400N

37∘09984000998400998400N

38∘09984000998400998400N

39∘09984000998400998400N

Basin boundaryWater systemStation

Figure 7 The eigenvector 3 trend field for annual precipitation in the Weihe basin

6 Conclusions

We have analyzed spatial and temporal patterns of annualprecipitation in the Weihe basin using data recorded from1960 to 2011 at 30 National Standard Stations in the basinand surrounding area by applying empirical orthogonalfunction decomposition in combination with geostatistical(semivariogram) analysis then using the Mann-Kendallmethod to discriminate variation points in the precipitationseries recorded at each station

(1) The main spatial patterns of annual precipitation inthe Weihe Basin during the study period were aconsistent overall decline which was strongest in thenorthwest and progressively weaker from northwestto southwest and from west to east The consistentoverall decline accounted for 455 of the totalvariation

(2) Variation points were detected in the annual precip-itation sequences recorded at all the meteorologicalstations in the basin but their timing had clear spatialpatterns starting gradually later from the southeastto the northwest and downstream to upstream Inthe 1970s and 1980s they occurred frequently in eachsubbasin

(3) After the 1990s following the active period of precip-itation variation in the 1970s and 1980s the frequencyand scope of precipitation variations declined Thusduring the study period there were shifts in precipita-tion fromone stable state to another after disturbance

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was financially supported by the Natural ScienceFoundation of China (Grant nos 51190093 51179148 and51179149) the National Key Basic Research 973 of China(Grant no 2012CB417003) and the MWR Welfare SectorFunds (Grant nos 201101043 201101049) Constructive com-ments from reviewers of a previous version of the paper aregratefully acknowledged

References

[1] H Zhang Y Chen G Ren et al ldquoThe characteristics of pre-cipitation variation of Weihe River Basin in Shaanxi Province

Journal of Applied Mathematics 11

during recent 50 yearsrdquo Agricultural Research in the Arid Areasvol 26 no 04 pp 236ndash241 2008

[2] K Che and J Jian ldquoA study of climate variability in Zedangduring 1960ndash1999rdquo Meteorological Monthly vol 26 no 1 pp54ndash56 2000

[3] Y Ding G Ren G Shi et al ldquoNational assessment report ofclimate change (I) climate change inChina and its future trendrdquoAdvances in Climate Change Research vol 2 no 1 pp 3ndash8 2006

[4] Y He F Wang X Mu et al ldquoTemporal and spatial character-istics of temperature and precipitation in Weihe River basinrdquoBulletin of Soil and Water Conservation vol 32 no 4 pp 102ndash105 2012

[5] Y-X Zhang F Wang and X-M Mu ldquoResearch of quantitativerelations between Geographical environment factor and waterand sediment in the Weihe River basinrdquo Journal of Northwest Aamp FUniversity (Natural Science Edition) vol 37 no 1 pp 61ndash652009

[6] C Jiang F Wang and X Mu ldquoEffect of climate change on netprimary productivity (NPP) of natural vegetation in Wei riverbasin (II) NPP of natural vegetation inWei river basinrdquo Journalof Arid LandResources and Environment vol 27 no 5 pp 53ndash572013

[7] Y Zhang ldquoHydrologic feature analysis for Weihe basin inShaanxi provincerdquo Northwest Water Resources amp Water Engi-neering no 2 pp 62ndash64 2002

[8] G Wang and Y Wang ldquoAnalysis on causes of runoff andsediment variation in Weihe basin based on hydrologicalmodelrdquo Journal of Soil Water Conservation no 4 pp 22ndash252000

[9] W He and Z Xu ldquoSpatial and temporal characteristics of thelong-term trend for temperature and pan evaporation in theWei River Basinrdquo Journal of Beijing Normal University (NaturalScience) vol 42 no 1 pp 102ndash106 2006

[10] K Pearson ldquoOn lines and planes of closest fit to system of pointsin spacerdquo Philosophical Magazine vol 6 no 2 pp 559ndash5721902

[11] D-X Cui and W-Y Liu ldquoApplication of empirical orthogonalfunction resolution to analysis of crustal vertical deformationfieldrdquo Earthquake vol 20 no 3 pp 82ndash86 2000

[12] T Liu and H Zhang ldquoCompared with principal componentanalysis and empirical orthogonal function decompositionrdquoStatistics and Decision vol 340 no 16 pp 159ndash162 2011

[13] H Lin and W Zhang ldquoSimilarities and differences of principalcomponent analysis and factor analysis and SPSS softwarerdquoStatistics Research no 3 pp 65ndash69 2005

[14] D H Kim C Yoo and T-W Kim ldquoApplication of spatial EOFand multivariate time series model for evaluating agriculturaldrought vulnerability in Koreardquo Advances in Water Resourcesvol 34 no 3 pp 340ndash350 2011

[15] N S Loboda A V Glushkov and V N Khokhlov ldquoUsingmeteorological data for reconstruction of annual runoff seriesover an ungauged area empirical orthogonal function approachto Moldova-Southwest Ukraine regionrdquo Atmospheric Researchvol 77 no 1ndash4 pp 100ndash113 2005

[16] D Sirjacobs A Alvera-Azcarate A Barth et al ldquoCloud fillingof ocean colour and sea surface temperature remote sensingproducts over the SouthernNorth Sea by theData InterpolatingEmpirical Orthogonal Functions methodologyrdquo Journal of SeaResearch vol 65 no 1 pp 114ndash130 2011

[17] H-L Yu and H-J Chu ldquoUnderstanding space-time patternsof groundwater system by empirical orthogonal functions a

case study in the Choshui River alluvial fan Taiwanrdquo Journalof Hydrology vol 381 no 3-4 pp 239ndash247 2010

[18] H BMann ldquoNonparametric tests against trendrdquo Econometricavol 13 pp 245ndash259 1945

[19] R Snyers ldquoSur LaDetermination de la stabilite des series clima-tologiquesrdquo in Proceedings of the UNESCO-WMO Symposium ofChanges of Climate vol 20 ofUneseo Arid Zone Research Seriespp 37ndash44 Unesco Paris France 1963

[20] C Goossens and A Berger ldquoAnnual and seasonal climaticvariations over the northern hemisphere and Europe during thelast centuryrdquo Annales Geophysicae B vol 4 no 4 pp 385ndash4001986

[21] C Fu and QWang ldquoThe definition and detection of the abruptclimatic changerdquo Chinese Journal of Atmospheric Sciences vol16 no 4 pp 482ndash493 1992

[22] P Xie G Chen D Li et al ldquoHydrological variation inte-grated diagnosis method and application researchrdquo Hydroelec-tric Energy vol 23 no 2 pp 11ndash14 2005

[23] M Gocic and S Trajkovic ldquoAnalysis of changes in meteoro-logical variables using Mann-Kendall and Senrsquos slope estimatorstatistical tests in SerbiardquoGlobal and Planetary Change vol 100pp 172ndash182 2013

[24] K H Hamed ldquoTrend detection in hydrologic data the Mann-Kendall trend test under the scaling hypothesisrdquo Journal ofHydrology vol 349 no 3-4 pp 350ndash363 2008

[25] S Kumar V Merwade J Kam and K Thurner ldquoStreamflowtrends in Indiana effects of long term persistence precipitationand subsurface drainsrdquo Journal of Hydrology vol 374 no 1-2pp 171ndash183 2009

[26] K H Hamed ldquoExact distribution of the Mann-Kendall trendtest statistic for persistent datardquo Journal of Hydrology vol 365no 1-2 pp 86ndash94 2009

[27] D Duhan and A Pandey ldquoStatistical analysis of long termspatial and temporal trends of precipitation during 1901ndash2002at Madhya Pradesh Indiardquo Atmospheric Research vol 122 pp136ndash149 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Journal of Applied Mathematics

Xiji

Wuqi

SuideYanci

Baoji

Jintai

Foping

Wugong

Guyuan

Lintao

Tongxin

Haiyuan

Yuzhong

HuashanMinxian

Changwu

Hengshan

Jingyuan

Tianshui

Luochuan

Huanxian

Shangzhou

Zhongning

Tongchuan

PingliangXifengzhen

Huajialing

001

039

103077

0347

0078

0019

0009002103010548

0034

0658 0179

00062

N

E

S

W

Trend of eigenvector 3value

High 0761376

Low minus0354567

Yanrsquoan

Zhenrsquoan

Xirsquoan

minus0258

minus0531

minus0536

minus0141

minus002

minus0016

minus0008

minus0017

minus0164

minus0123

minus0123

minus0005

minus0074

minus01

minus0163

50 25 0 50 100 150 200

(km)

103∘09984000998400998400E 104

∘09984000998400998400E 105

∘09984000998400998400E 106

∘09984000998400998400E 107

∘09984000998400998400E 108

∘09984000998400998400E 109

∘09984000998400998400E 110

∘09984000998400998400E

103∘09984000998400998400E 104

∘09984000998400998400E 105

∘09984000998400998400E 106

∘09984000998400998400E 107

∘09984000998400998400E 108

∘09984000998400998400E 109

∘09984000998400998400E 110

∘09984000998400998400E

34∘09984000998400998400N

33∘09984000998400998400N

35∘09984000998400998400N

36∘09984000998400998400N

37∘09984000998400998400N

38∘09984000998400998400N

39∘09984000998400998400N

34∘09984000998400998400N

33∘09984000998400998400N

35∘09984000998400998400N

36∘09984000998400998400N

37∘09984000998400998400N

38∘09984000998400998400N

39∘09984000998400998400N

Basin boundaryWater systemStation

Figure 7 The eigenvector 3 trend field for annual precipitation in the Weihe basin

6 Conclusions

We have analyzed spatial and temporal patterns of annualprecipitation in the Weihe basin using data recorded from1960 to 2011 at 30 National Standard Stations in the basinand surrounding area by applying empirical orthogonalfunction decomposition in combination with geostatistical(semivariogram) analysis then using the Mann-Kendallmethod to discriminate variation points in the precipitationseries recorded at each station

(1) The main spatial patterns of annual precipitation inthe Weihe Basin during the study period were aconsistent overall decline which was strongest in thenorthwest and progressively weaker from northwestto southwest and from west to east The consistentoverall decline accounted for 455 of the totalvariation

(2) Variation points were detected in the annual precip-itation sequences recorded at all the meteorologicalstations in the basin but their timing had clear spatialpatterns starting gradually later from the southeastto the northwest and downstream to upstream Inthe 1970s and 1980s they occurred frequently in eachsubbasin

(3) After the 1990s following the active period of precip-itation variation in the 1970s and 1980s the frequencyand scope of precipitation variations declined Thusduring the study period there were shifts in precipita-tion fromone stable state to another after disturbance

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was financially supported by the Natural ScienceFoundation of China (Grant nos 51190093 51179148 and51179149) the National Key Basic Research 973 of China(Grant no 2012CB417003) and the MWR Welfare SectorFunds (Grant nos 201101043 201101049) Constructive com-ments from reviewers of a previous version of the paper aregratefully acknowledged

References

[1] H Zhang Y Chen G Ren et al ldquoThe characteristics of pre-cipitation variation of Weihe River Basin in Shaanxi Province

Journal of Applied Mathematics 11

during recent 50 yearsrdquo Agricultural Research in the Arid Areasvol 26 no 04 pp 236ndash241 2008

[2] K Che and J Jian ldquoA study of climate variability in Zedangduring 1960ndash1999rdquo Meteorological Monthly vol 26 no 1 pp54ndash56 2000

[3] Y Ding G Ren G Shi et al ldquoNational assessment report ofclimate change (I) climate change inChina and its future trendrdquoAdvances in Climate Change Research vol 2 no 1 pp 3ndash8 2006

[4] Y He F Wang X Mu et al ldquoTemporal and spatial character-istics of temperature and precipitation in Weihe River basinrdquoBulletin of Soil and Water Conservation vol 32 no 4 pp 102ndash105 2012

[5] Y-X Zhang F Wang and X-M Mu ldquoResearch of quantitativerelations between Geographical environment factor and waterand sediment in the Weihe River basinrdquo Journal of Northwest Aamp FUniversity (Natural Science Edition) vol 37 no 1 pp 61ndash652009

[6] C Jiang F Wang and X Mu ldquoEffect of climate change on netprimary productivity (NPP) of natural vegetation in Wei riverbasin (II) NPP of natural vegetation inWei river basinrdquo Journalof Arid LandResources and Environment vol 27 no 5 pp 53ndash572013

[7] Y Zhang ldquoHydrologic feature analysis for Weihe basin inShaanxi provincerdquo Northwest Water Resources amp Water Engi-neering no 2 pp 62ndash64 2002

[8] G Wang and Y Wang ldquoAnalysis on causes of runoff andsediment variation in Weihe basin based on hydrologicalmodelrdquo Journal of Soil Water Conservation no 4 pp 22ndash252000

[9] W He and Z Xu ldquoSpatial and temporal characteristics of thelong-term trend for temperature and pan evaporation in theWei River Basinrdquo Journal of Beijing Normal University (NaturalScience) vol 42 no 1 pp 102ndash106 2006

[10] K Pearson ldquoOn lines and planes of closest fit to system of pointsin spacerdquo Philosophical Magazine vol 6 no 2 pp 559ndash5721902

[11] D-X Cui and W-Y Liu ldquoApplication of empirical orthogonalfunction resolution to analysis of crustal vertical deformationfieldrdquo Earthquake vol 20 no 3 pp 82ndash86 2000

[12] T Liu and H Zhang ldquoCompared with principal componentanalysis and empirical orthogonal function decompositionrdquoStatistics and Decision vol 340 no 16 pp 159ndash162 2011

[13] H Lin and W Zhang ldquoSimilarities and differences of principalcomponent analysis and factor analysis and SPSS softwarerdquoStatistics Research no 3 pp 65ndash69 2005

[14] D H Kim C Yoo and T-W Kim ldquoApplication of spatial EOFand multivariate time series model for evaluating agriculturaldrought vulnerability in Koreardquo Advances in Water Resourcesvol 34 no 3 pp 340ndash350 2011

[15] N S Loboda A V Glushkov and V N Khokhlov ldquoUsingmeteorological data for reconstruction of annual runoff seriesover an ungauged area empirical orthogonal function approachto Moldova-Southwest Ukraine regionrdquo Atmospheric Researchvol 77 no 1ndash4 pp 100ndash113 2005

[16] D Sirjacobs A Alvera-Azcarate A Barth et al ldquoCloud fillingof ocean colour and sea surface temperature remote sensingproducts over the SouthernNorth Sea by theData InterpolatingEmpirical Orthogonal Functions methodologyrdquo Journal of SeaResearch vol 65 no 1 pp 114ndash130 2011

[17] H-L Yu and H-J Chu ldquoUnderstanding space-time patternsof groundwater system by empirical orthogonal functions a

case study in the Choshui River alluvial fan Taiwanrdquo Journalof Hydrology vol 381 no 3-4 pp 239ndash247 2010

[18] H BMann ldquoNonparametric tests against trendrdquo Econometricavol 13 pp 245ndash259 1945

[19] R Snyers ldquoSur LaDetermination de la stabilite des series clima-tologiquesrdquo in Proceedings of the UNESCO-WMO Symposium ofChanges of Climate vol 20 ofUneseo Arid Zone Research Seriespp 37ndash44 Unesco Paris France 1963

[20] C Goossens and A Berger ldquoAnnual and seasonal climaticvariations over the northern hemisphere and Europe during thelast centuryrdquo Annales Geophysicae B vol 4 no 4 pp 385ndash4001986

[21] C Fu and QWang ldquoThe definition and detection of the abruptclimatic changerdquo Chinese Journal of Atmospheric Sciences vol16 no 4 pp 482ndash493 1992

[22] P Xie G Chen D Li et al ldquoHydrological variation inte-grated diagnosis method and application researchrdquo Hydroelec-tric Energy vol 23 no 2 pp 11ndash14 2005

[23] M Gocic and S Trajkovic ldquoAnalysis of changes in meteoro-logical variables using Mann-Kendall and Senrsquos slope estimatorstatistical tests in SerbiardquoGlobal and Planetary Change vol 100pp 172ndash182 2013

[24] K H Hamed ldquoTrend detection in hydrologic data the Mann-Kendall trend test under the scaling hypothesisrdquo Journal ofHydrology vol 349 no 3-4 pp 350ndash363 2008

[25] S Kumar V Merwade J Kam and K Thurner ldquoStreamflowtrends in Indiana effects of long term persistence precipitationand subsurface drainsrdquo Journal of Hydrology vol 374 no 1-2pp 171ndash183 2009

[26] K H Hamed ldquoExact distribution of the Mann-Kendall trendtest statistic for persistent datardquo Journal of Hydrology vol 365no 1-2 pp 86ndash94 2009

[27] D Duhan and A Pandey ldquoStatistical analysis of long termspatial and temporal trends of precipitation during 1901ndash2002at Madhya Pradesh Indiardquo Atmospheric Research vol 122 pp136ndash149 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Applied Mathematics 11

during recent 50 yearsrdquo Agricultural Research in the Arid Areasvol 26 no 04 pp 236ndash241 2008

[2] K Che and J Jian ldquoA study of climate variability in Zedangduring 1960ndash1999rdquo Meteorological Monthly vol 26 no 1 pp54ndash56 2000

[3] Y Ding G Ren G Shi et al ldquoNational assessment report ofclimate change (I) climate change inChina and its future trendrdquoAdvances in Climate Change Research vol 2 no 1 pp 3ndash8 2006

[4] Y He F Wang X Mu et al ldquoTemporal and spatial character-istics of temperature and precipitation in Weihe River basinrdquoBulletin of Soil and Water Conservation vol 32 no 4 pp 102ndash105 2012

[5] Y-X Zhang F Wang and X-M Mu ldquoResearch of quantitativerelations between Geographical environment factor and waterand sediment in the Weihe River basinrdquo Journal of Northwest Aamp FUniversity (Natural Science Edition) vol 37 no 1 pp 61ndash652009

[6] C Jiang F Wang and X Mu ldquoEffect of climate change on netprimary productivity (NPP) of natural vegetation in Wei riverbasin (II) NPP of natural vegetation inWei river basinrdquo Journalof Arid LandResources and Environment vol 27 no 5 pp 53ndash572013

[7] Y Zhang ldquoHydrologic feature analysis for Weihe basin inShaanxi provincerdquo Northwest Water Resources amp Water Engi-neering no 2 pp 62ndash64 2002

[8] G Wang and Y Wang ldquoAnalysis on causes of runoff andsediment variation in Weihe basin based on hydrologicalmodelrdquo Journal of Soil Water Conservation no 4 pp 22ndash252000

[9] W He and Z Xu ldquoSpatial and temporal characteristics of thelong-term trend for temperature and pan evaporation in theWei River Basinrdquo Journal of Beijing Normal University (NaturalScience) vol 42 no 1 pp 102ndash106 2006

[10] K Pearson ldquoOn lines and planes of closest fit to system of pointsin spacerdquo Philosophical Magazine vol 6 no 2 pp 559ndash5721902

[11] D-X Cui and W-Y Liu ldquoApplication of empirical orthogonalfunction resolution to analysis of crustal vertical deformationfieldrdquo Earthquake vol 20 no 3 pp 82ndash86 2000

[12] T Liu and H Zhang ldquoCompared with principal componentanalysis and empirical orthogonal function decompositionrdquoStatistics and Decision vol 340 no 16 pp 159ndash162 2011

[13] H Lin and W Zhang ldquoSimilarities and differences of principalcomponent analysis and factor analysis and SPSS softwarerdquoStatistics Research no 3 pp 65ndash69 2005

[14] D H Kim C Yoo and T-W Kim ldquoApplication of spatial EOFand multivariate time series model for evaluating agriculturaldrought vulnerability in Koreardquo Advances in Water Resourcesvol 34 no 3 pp 340ndash350 2011

[15] N S Loboda A V Glushkov and V N Khokhlov ldquoUsingmeteorological data for reconstruction of annual runoff seriesover an ungauged area empirical orthogonal function approachto Moldova-Southwest Ukraine regionrdquo Atmospheric Researchvol 77 no 1ndash4 pp 100ndash113 2005

[16] D Sirjacobs A Alvera-Azcarate A Barth et al ldquoCloud fillingof ocean colour and sea surface temperature remote sensingproducts over the SouthernNorth Sea by theData InterpolatingEmpirical Orthogonal Functions methodologyrdquo Journal of SeaResearch vol 65 no 1 pp 114ndash130 2011

[17] H-L Yu and H-J Chu ldquoUnderstanding space-time patternsof groundwater system by empirical orthogonal functions a

case study in the Choshui River alluvial fan Taiwanrdquo Journalof Hydrology vol 381 no 3-4 pp 239ndash247 2010

[18] H BMann ldquoNonparametric tests against trendrdquo Econometricavol 13 pp 245ndash259 1945

[19] R Snyers ldquoSur LaDetermination de la stabilite des series clima-tologiquesrdquo in Proceedings of the UNESCO-WMO Symposium ofChanges of Climate vol 20 ofUneseo Arid Zone Research Seriespp 37ndash44 Unesco Paris France 1963

[20] C Goossens and A Berger ldquoAnnual and seasonal climaticvariations over the northern hemisphere and Europe during thelast centuryrdquo Annales Geophysicae B vol 4 no 4 pp 385ndash4001986

[21] C Fu and QWang ldquoThe definition and detection of the abruptclimatic changerdquo Chinese Journal of Atmospheric Sciences vol16 no 4 pp 482ndash493 1992

[22] P Xie G Chen D Li et al ldquoHydrological variation inte-grated diagnosis method and application researchrdquo Hydroelec-tric Energy vol 23 no 2 pp 11ndash14 2005

[23] M Gocic and S Trajkovic ldquoAnalysis of changes in meteoro-logical variables using Mann-Kendall and Senrsquos slope estimatorstatistical tests in SerbiardquoGlobal and Planetary Change vol 100pp 172ndash182 2013

[24] K H Hamed ldquoTrend detection in hydrologic data the Mann-Kendall trend test under the scaling hypothesisrdquo Journal ofHydrology vol 349 no 3-4 pp 350ndash363 2008

[25] S Kumar V Merwade J Kam and K Thurner ldquoStreamflowtrends in Indiana effects of long term persistence precipitationand subsurface drainsrdquo Journal of Hydrology vol 374 no 1-2pp 171ndash183 2009

[26] K H Hamed ldquoExact distribution of the Mann-Kendall trendtest statistic for persistent datardquo Journal of Hydrology vol 365no 1-2 pp 86ndash94 2009

[27] D Duhan and A Pandey ldquoStatistical analysis of long termspatial and temporal trends of precipitation during 1901ndash2002at Madhya Pradesh Indiardquo Atmospheric Research vol 122 pp136ndash149 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of