Using Geostatistical Methods to Improve Radar Rainfall Estimates

Accuracy

----Thesis Proposal

Geological Sciences

University of Texas at San Antonio

Student: Beibei Yu

Date: September 11th, 2008

Thesis Committee

Committee Chair: Dr. Hongjie Xie

Committee Member: Dr. Hatim Sharif

Committee Member: Dr. Minghe Sun

1

Contents

1. Introduction 2. Literature Review 3. Objectives 4. Study Area 5. Data Source 6. Methods 7. Preliminary Results 8. Future Work References

2

1. INTRODUCTION

Rainfall data are major input of many hydrological models, since it is the driving

force behind all the hydrologic processes. Generally, rain gauges are able to provide

accurate measurement of precipitation at a certain point, however, simulation of

hydrological processes requires not only the point measurements but also the spatial

distribution of the precipitation. Rainfall estimation using satellite data has been used in

many hydrological applications and analysis (Bedient et al, 2000), but several studies

show that radar are characteristically subject to both random and systematic errors

(Young et al., 1998). Therefore, to evaluate and improve the accuracy of radar products

becomes a prerequisite for hydrological modeling. Many validation studies have been

done regarding the radar products. For instance, Wang et al. (2008) validated the

NEXRAD MPE (2004) and Stage III (2001) rainfall products, using rain gauge as the

ground truth data. Their results showed that MPE products slightly underestimate the

total amount of rainfall but have higher probability of rain detection (POD). On the

contrast, Stage III detected fewer rainfall counts, but overestimated the total rainfall

amount. However, few works has been done to improve the accuracy of NEXRAD

products. In this study, we tried to use the geostatistical methods to merge two source of

information (gage and radar), thus improve the NEXRAD accuracy and create a more

applicable and spatially distributed rainfall map, which can be a better input into the

models.

2. LITERATURE REVIEW

3

Several geostatistical methods can be used in analyzing spatial variability and

spatial interpolation, including Thiessen, inverse distance weighting (IDW), moving

window regression (MWR), Nearest Neighbour (NN) and Kriging. The Kriging based

methods include univariate Kriging (e.g. Ordinary Kriging (OK) and Simple Kriging

(SK)) and multivariate Kriging (e.g. simple kriging with a locally varying mean (SKlm),

Kriging with External Drift (KED), Regression Kriging (RK),and more computationally

demanding Co-kriging (CK)). Generally, geostatistical approaches (i.e. OK, SKlm and

KED) provide better performances than IDW, MWR, Thiessen, areal-mean methods,

among which, multivariate geostatistical methods (e.g. KED, SKlm, CK and RK)

perform better than univariate methods (e.g. Nearest Neighbour, IDW, OK and indicator

Kriging) (Lloyd, 2005; Haberlandt et al., 2006; Goovaerts, 2000; Diodato and Ceccarelli,

2005; Severino 2005; Ciach et al. 2007). But different data sets and study area may lead

to different results. For example, Wilk et al. (2006) conducted a study of rainfall

interpolation in the Okavango region and their results showed that inverse distance

weighting provided lower errors than other methods like Kriging for the period before

1974. And the performances of these two methods are similar after 1974 because the

rainfall stations were becoming scarce. They found that merging the discontinuous and

non-coincident gauge and satellite datasets into a unified dataset is problematic with very

high systematic biases from satellite rainfall estimates. In terms of multivariate

interpolation, several kinds of information can be used as secondary source to

complement the primary attribute. First, digital elevation model (DEM) was generally

used to incorporate the topography information as complementary into the observed

precipitation in interpolation (Lloyd, 2004; Goovaerts, 2000; Diodato and Ceccarelli,

4

2005). Pardo-igu´ Zquiza (2005) compared four different geostatistical methods (i.e.

Thiessen, OK, CK, KED) used to estimate the areal average climatological rainfall mean

and concluded that inclusion of topographic information tends to improve the estimates.

In another study, Lloyd et al. (2005) compared the performance of five techniques (i.e.

IDW, OK, MWR, SKlm and KED) for mapping monthly precipitation in England. The

last three methods integrate elevation into the estimation. Since the relation between

elevation and precipitation varies locally, the benefits in using elevation data to inform

estimation vary locally. Goovaerts (2000) also used three multivariate geostatistical

algorithms (i.e. SKlm, KED and CK) to incorporate DEM into the spatial prediction of

rainfall. He concluded that the three multivariate geostatistical algorithms outperform

other interpolators, of which SKlm performs best. Besides DEM, satellite data is widely

used for interpolation. In the study of Grimes at al (1999), they used a new algorithm for

rainfall estimation by merging the Kriging estimates from raingauge data and satellite

estimates weighted by the inverse of its uncertainty. The merging procedure has provided

an improvement of the estimates of the decadal rainfall and their spatial distribution,

because satellite estimates provided a better estimation of the spatial pattern while the

Kriging estimates work with accurate values of point rainfall. However, there are some

drawbacks of the multivariate geostatistcal approaches. For instance, if too many random

variables (rain-gauges) exist, the computing complexities of the cross-semivariogram will

increase (Cheng et al., 2007). At this point of view, KED has the advantage of requiring a

less demanding variogram analysis than CK (Gimes et al, 1999). .

Compared to the straightforward univariate geostatistical techniques, multivariate

Kriging provides the sparsely sampled observations of the primary variable with a

5

complemented secondary variable, which is densely distributed in the domain.

(Goovaerts, 2000). Thus, more precise information can be provided, and it is likely to get

more accurate rainfall map. Haberlandt et al. (2006) used KED and indicator kriging with

external drift (IKED) to interpolate the hourly rainfall from raingauges with secondary

information from radar, daily precipitation of a dense network and elevation. The best

performance is achieved when all additional information are used simultaneously with

KED (Haberlandt, 2006).

3. OBJECTIVES

The purpose of this study is to evaluate and improve the accuracy of Next Generation

Weather Radar (NEXRAD) data through incorporating rain gauge data. The corrected

NEXRAD is expected to provide more accurate and better spatially distributed rainfall

data sets as input into the hydrological models or used for other applications. In order to

achieve this goal, three objectives must be addressed as following:

(1) First, use three multivariate geostatistical methods (Sklm, KED and RK) to

interpolate the rainfall fields in a domain of Guadalupe River Basin.

(2) Second, spatially evaluate the radar estimates interpolated by the three methods

using the hourly rainfall measurements collected from fifty rain gauges scattered

in this area. Compare the hourly prediction performances of the three

geostatistical interpolation techniques.

(3) Third, time series evaluation of the three kinds of radar estimation in the scale of

day, year and season.

4. STUDY AREA

6

Figure 1. Study area: Guadalupe River Basin in Texas

The study area is Guadalupe River Basin. The reason of taking Guadalupe

watershed as our study area is that there are 50 gauges scattering in this area (green

circles), which can be used as ground truth data for validation. San Antonio and

Guadalupe watersheds provide an excellent study system for calibrating and testing many

hydrological and biological models, because their estuaries have strongly contrasting

physical and biological attributes despite their physical proximity. The greatest difference

is an extreme disparity in freshwater inflow resulting largely from differences in runoff

caused by soil differences in the watersheds (Montagna and Kalke, 1992). This disparity

in freshwater inflow leads to great differences in salinity and a strongly contrasting

system in which to study how ecological processes differ due to inflow differences over

7

broad spatial scales. Precipitation is an important input for many atmospheric or climate

models. Besides, the Edwards Aquifer is the major water source for over 1.7 million

people along this corridor, while precipitation is the only recharge water source. Thus, to

get better estimation of precipitation distribution and knowing the accuracy of radar

precipitation mapping is crucial.

5. DATA SOURCE

The NEXRAD system installed by the National Weather Services (NWS) consists

a network of WSR-88D (Weather Surveillance Radar -1988 Doppler) radars, providing

meterological data for hydrological models, weather forecasting and flood prediction

(Young et al, 1999; Young et al, 1998). As documented by Bradley et al. (2002), the

NEXRAD estimates are capable of providing valuable information for empirical

simulations of rainfall fields, which can be used in raingage network design. The current

NEXRAD precipitation product has a spatial resolution about 4 km by 4 km To replace

the Stage III rainfall products, the NWS Office of Hydrology developed a new product

called MPE (e multisensory Precipitation Estimator) in March 2000, which incorporated

the rainfall measurements from gages and rainfall estimates from Geostationary

Operational Environmental Satellite (GOES). The MPE precipitation products are hourly

accumulation with a spatial resolution of about 4 km by 4 km. As documented by Wang

et al (2008), in validating NEXRAD MPE and Stage III precipitation, MPE product fixed

the truncation error of Stage III product and has better agreement with gauge

observations than Stage III does. The NEXRAD MPE data used in this study was

provided by Greg Story, who is a hydrologist in the NWS West Gulf River Forecast

Center (WGRFC).

8

Rain gauge data was provided by Guadalupe-Blanco River Authority (GBRA)

There are 50 rain gages covering four counties: Kerr, Kendall, Comal, and Guadalupe in

the study area. All rain gauge data have a 6-min accumulation time-step. We used the

Visual Basic for Applications (VBA) scripts developed by Xianwei Wang to aggregate

the 6-min gauge data into hourly accumulations. Since the NEXRAD products use

Coordinated Universal Time (UTC), the local time (Central Standard Time) of rain gage

data were also converted to UTC time to match the time period of NEXRAD, using VBA

in excel (Wang et al, 2008).

6. METHOD

6.1 Semivariogram model

Before Kriging is performed, a valid semiviariogram model has to be selected. In the

study of Severino and Alpum (2005), kriging and cokriging were used to get the optimal

combination of weather radar and rain gauge measurement. They concluded that

exponential model led to a better adjustment to the radar semivariogram estimates for

distances smaller than 10km, and spherical model has the best performance for larger

distances (Severino and Alpum, 2005).

The semi-variance )(hγ is computed using the equation below,

∑ +−=)(

2))()(()(2

1)(hN

iii hzz

hNh xxγ (1)

where is the difference between two point locations, is the number of pairs of

points separated by ,

h )(hN

h )()( hzz ii +− xx is the value difference between point and

another point separated by distance . The Kriging methods require semivariogram

ix

h

9

models to be fitted to the experimental semivariogram values. In this study, two types of

semivariogram models (i.e. Spherical and Cubic models) were applied,

⎪⎩

⎪⎨⎧

>

≤−=

ac

aah

ahc

hhfor

hfor ])(5.05.1[)(

3

γ

⎪⎩

⎪⎨⎧

>

≤−+−=

ac

aah

ah

ah

ahc

hhfor

hfor ])(75.0)(5.3)(75.8)(7[)(

7532

γ

These two different types of semivariogram models were combined with a nugget-effect

model for the fitting of the experimental semivariogram of daily precipitation. Following

Goovaerts’ (2000) methods, the two models were fitted using regression and such that the

weighted sum of squares (WSS) of differences between experimental )(ˆ khγ and model

)( khγ semivariogram values is minimum:

∑=

−=K

kkkk hhhWSS

1

2)]()(ˆ)[( γγω

The weights )( khω were taken as in order to give more importance to the

first lags and the ones computed from more data pairs. For each day, the two

semivariogram models were trained to fit the empirical semivariogram values, and the

model with smaller WSS value was used in the Kriging interpolation.

2)](/[)( kk hhN γ

A global optimization algorithm, Particle Swarm Optimizater (PSO), was used to

calibrate the nonlinear semivariogram models. PSO is a population based stochastic

optimization technique inspired by social behavior of bird flocking or fish schooling

(Kennedy and Eberhart, 2001). During the optimization process, in order to find global

optimum each particle in the population adjusts its “flying” according to its own flying

experience and its companions’ flying experience (Eberhart and Shi, 1998). The basic

10

PSO algorithm consists of three steps: 1) generating particles’ positions (coordinate in t

parameter space) and velocities (“flying” direction and speed), 2) update the velocity of

each particle using the information from the best solution it has achieved so far (personal

best) and another particle with the best fitness value that has been obtained so far by all

the particles in the population (global best), 3) finally, the new position of each particle i

calculated by adding the updated velocity to the current position. For further information

about PSO, please refer to Kennedy and Eberhart (2001).

6.2 Kriging

he

s

a group of advanced geostatistical techniques that that provides the best

line

Kriging is

ar unbiased estimation of the variable of interest at an unobserved location from

observations of the random field at nearby locations. In Kriging methods, the random

variable Z is decomposed into a trend ( m ) and a residual (ε ), where )()( xx )(xε+= mZ

The Kriging estimator is given by a linea combination of th surround

(Goovaerts, 1997). The weights of the points that surround the predicted points are

calculated based on the spatial dependence (i.e. semivariogram or covariance) of the

random field. Of the different Kriging techniques, the SK, OK, UK, KED, and SKlm are

introduced as follows.

Ordinary Kriging

.

r e ing observations

(OK). OK is a common type of Kriging in practice. In the OK, the

tren

iui

=f

linear function known as the “ordinary Kriging system” (Goovaerts, 2000)

d is considered as unknown and constant. OK estimates the unknown precipitation

depth at the unsampled location u as a linear combination of neighboring observations,

that is, )]([)(n

ZuZ x∑= λ . The optimal weights are obtained through solving a series o1

i

11

⎪⎪⎩

⎪⎪⎨

⎧

=∑

==−∑

=

=

1

,...,1 )()()(

1

1

n

juj

uiijn

juj nihuh

λ

γµγλ

where )(uµ is the Lagrange parameter accounting for the constraint on the weights,

denotes the separation distance between sampled location and .

ijh

ix jx )(hγ is the semi-

variance.

Simple Kriging (SK). The SK estimator is . SK

assumes the trend of the random variable is known and constant. The equation system

used to estimate the weights in equation (1) is

)]()([)()(1

imZumuZ in

iui −∑=−

=xλ

)()(1

uiij

n

juj hChC =∑

=

λ ni ,...,1=

where is the spatial covariance between two points separated by distance . )(hC h

Kriging with External Drift (KED). In OK and SK, the trend of the random variable

is constant. While in real world problems some spatial processes include varying trend or

‘drift’ (Webster and Oliver, 2007). In KED, the trend of the random variable is not

stationary, which can tank into account both the spatial dependence of the variable and its

linear relation to one or more additional variables (Ahmed and De Marsily, 1987). The

basic form of can be expressed as , where are

known and the

)(xm

)(xm ∑=

K

kkk y

0

)(xβ )( , ),( ),( 21 xxx Kyyy K

kβ are unknown coefficients to be determined (Webster and Oliver, 2007).

are ‘external’ variables. The usual expression for the KED estimate is the same as )(xky

12

that of OK, but the equation system used to obtain the optimal weights of KED is

different. These equations are expressed as

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

⎨

⎧

==

=

==++

∑

∑

∑∑

=

=

==

,,2 1for )()(

,1

,,...,2,1for )()()(

1

n

1i

10

1

,K, k yy

nihyh

n

iikikui

ui

ui

K

kikkij

n

juj

Kxx

x

λ

λ

γψψγλ εε

where )( ijhεγ is the semivariances of the residuals between the data at and ; ix jx kψ ,

, are Lagrange multipliers; The number of equations need to be solved

depends on the number of additional variables that are used to estimate the trend. In this

study, the NEXRAD value at point x is taken as the external variables to estimate the

trend of the primary variable (precipitation). The trend estimation is

,K, k K,2 1=

NEXRADm 21)( ββ +=x For a more thorough introduction of KED, please refer to

Webster and Oliver (2007). It is worth noting that the semivariance function )( ijhεγ is

estimated from the residuals but not the original observed data. It is a difficult task to

obtain such a task, because often we do not have direct observation of the residuals.

Different methods have been proposed to estimate )( ijhεγ . One way of dealing with drift

is to use trend surface analysis, and remove it from the data to obtain the residuals, then

variogram is computed and modeled (Webster and Oliver 2007).

Simple Kriging with varying local means (SKlm). Goovaerts (2000) presents

another type of Kriging, which replaces the known stationary mean in the SK with known

varying means ( ) derived from secondary information, to improve spatial prediction

of precipitation. The basic procedures of implementing the SKlm are described as follows:

)(xm

13

first, the Ordinary Least Squares (OLS) is used to estimate the known varying means

using secondary information; second the residuals from subtracting the original data from

the varying means are taken as the random variable and to be estimated using the SK

method; finally, the estimated residuals are added back to the varying means to get the

SKlm estimates. The form of the equations used by the SKlm is similar to those used by

SK. The estimated precipitation is expressed as , where is

the weight of residual at point , which is obtained through solving the following

equations,

)()()(1

in

iuiumuZ ελε∑+=

=ui

ελ

ix

nihChC uijin

juj ,...,1 )()(

1==∑

=εε

ελ

where is the covariance function of the residual )(hCR ε . Other variables denote the

same meaning as stated above. In this study, the trend surface used by SKlm was

obtained using NEXRADm 21)( ββ +=x , which has the same form as KED’s trend

estimate. But it is worth noting that the coefficients 1β and 2β are estimated using

ordinary least squares, and a covariance function was fitted to the residuals. For more

detailed information on SKlm, please refer to Goovaerts (1997).

Regression Kriging (RK). RK has similar procedures to those of SKlm. The

estimated precipitation is also expressed as . The trend surface

used by RK is also calculated as

)()()(1

in

iuiumuZ ελε∑+=

=

NEXRADm 21)( ββ +=x . In stead of using SK to ,

RK uses the “Ordinary Kriging System” of OK.

uiελ

6.3 Ratio Corrected (RC)

14

In this method, the simple assumption is that there is systematic error in

NEXRAD rainfall estimates, and there is a particular difference between radar estimates

and rain gauge measurements. Thus, a ratio between the radar and gauge precipitation is

calculated based on the areal mean of the original radar rainfall estimates and gauge

precipitation. The ratio corrected method is to correct the radar rainfall by multiplying

this ratio to the NEXRAD estimates.

6.4 Evaluation of the performance of different interpolation methods

Several evaluation coefficients that are used to compare the characteristics of the

spatial precipitation estimated by different interpolation methods, which include areal

mean precipitation depth (AMPD), maximum precipitation depth (MaxP), minimum

precipitation depth (MinP), and coefficient of variation (CV). CV is calculated as the

ratio between the standard deviation and areal mean depth of the spatial precipitation.

Cross-validation is a popular method that has been used to compare the prediction

performances of spatial interpolation methods (Isaaks and Srivastava, 1989). In cross-

validation, each of the raingauge data is temporarily removed at a time and the remaining

data are used to estimate the value of the deleted datum. The difference between the

observed and estimated values is used to evaluate the accuracy of interpolation methods. )

For hourly spatial evalutation, the evaluation coefficients used in this study are

coefficient of determination (R2), Nash-Sutcliffe efficiency (Ens), and Relative mean

absolute error (RMAE) The formula for calculating coefficient R2 for spatial evaluation is:

( )( )( ) ( )

2

50

1

250

1

2

12R

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

⎥⎦⎤

⎢⎣⎡∑ −⎥⎦

⎤⎢⎣⎡∑ −

∑ −−=

==

=.N

i

ppi

.N

i

ooi

N

i

ppi

ooi

ZZZZ

ZZZZ

15

where is the estimated value, is the observed data, the over bar is the areal mean

of the spatial precipitation, and i = 1, 2, ..., N, where N is the total number of simulated

and observed data pairs. ρ is an indicator of the strength of the relationship between the

observed and simulated values. The formula to calculate E is (Nash and Sutcliffe, 1970),

piZ o

iZ

( )( )∑

=

∑=

−

−−=

N

i

ooi

N

i

pi

oi

nsZZ

ZZ.E

1

21

2

01

where the symbols are the same as described above. Ens indicates how well the plot of the

observed value versus the simulated value fits the 1:1 line, and ranges from to 1.

When the values for E and R

−∞

2 are equal to one, the model prediction is considered to be

"perfect". The formula for calculation of RMAE is,

o

n

l

pi

oi

Z

ZZabsnRMAE

∑−

−= 1

)(1

where is the estimated value, is the observed data. The smaller value of RMAE

means that the simulated value is close to the observed one.

piZ o

iZ

For time series evaluation, absolute difference (AD), R2, and Ens were used. The

form of the equations used to calculate the three coefficients are the same as those used to

evaluate spatial accuracy. The spatially distributed observed and estimated precipitation

will be replaced with the observed and estimated precipitation at a specific rain gauge for

a specified time period.

7. PRELIMINARY RESULTS

7.1 Interpolated rainfall map using different methods

16

One hour of the date October 2nd, 2004 is selected to output the grid file with a

cell size of 1000 km. The basic idea behind the weighting scheme is that observations

that are close to each other on the ground tend to be more alike than those further apart,

hence observations that are closer to the interpolated point should receive a larger weight.

Instead of the Euclidian distance, geostatistics uses the semivariogram as a measure of

the dissimilarity between observations. According to Figure 2, we can see that the rainfall

pattern mapped by SKlm estimation seems mostly like the original NEXRAD grid and

followed by RK estimation. The rainfall map estimated by KED is not as similar as the

other rainfall maps.

Comparing figure A and figure B, some white and red area in figure A turns blue and

orange in figure B, which means SKlm estimation increased the rainfall value at this

period of time. As well, RK estimation has a higher value than the original radar

estimation in most areas in this case.

Table 1 shows the corresponding statistics of the cross-validation of this particular

hour. Overall, SKlm has the best performance in interpolating rainfall distribution at this

hour, since both Ens and ρ between SKlm rainfall estimates and gauge precipitation are

the highest among all the methods. Besides, the areal mean of SKlm estimates (10.456

mm) is the closest to the gauge areal mean (10.317 mm). Consistent to what is showing in

the figures, SKlm and RK increased the average rainfall value compared to the NEXRAD

which made it closer to the average gauge value. According to the Mean Square Error

(MSE), the MSE of SKlm estimation is the smallest, and then followed by RK and KED

estimations. According to the statistical results in table 1, all the 3 methods we used to

correct the NEXRAD estimates improved the ρ and Ens and decreased the MSE, which

17

means all the interpolation methods have improved the radar estimation to some extent.

However, KED and RK techniques seemed to change the entire pattern of the rainfall

distribution when we look at the interpolated rainfall map (Figure 2).

(A) Radar

(B) SKlm

(C) KED

(D) RK

50

Figure 2. Original NEXRAD rainfall map and interpolated rainfall map for October 2nd, 13 (UTC), 2004: Original NEXRAD precipitation map(A); Precipitation map interpolated using SKlm (B); Precipitation map interpolated using KED (C); Precipitation map interpolated using RK (D).

Table 1. General Statistics of October 2nd, 13 (UTC), 2004

Radar SKlm KED RK R2 0.651 0.705 0.667 0.694

52.656 44.292 50.365 46.73 Ens 0.638 0.696 0.654 0.679 areal mean 8.95 10.456 10.758 10.557 cv 1.098 1.08 1.04 1.089 maxpcp 40.98 44.422 38.15 44.578 minpcp 0 0 0 0

0

18

gauge_mean 10.317 gauge_cv 1.17

7.2 Spatial Evaluation

7.2.1 Results of average hourly Cross-Validation

The performance of three multivariate geostatistical interpolation methods is

showing in table 2. The original radar estimation has the highest RMAE, where Sklm

estimation shows the smallest RMAE. Thus, the mean coefficient of correlation (ρ) of 50

gages has the consistent results that Sklm rainfall estimation has the highest ρ with the

observed precipitation. But it is unexpected to see that the mean ρ of rainfall estimated by

KED technique with gage precipitation is lower than the original radar to gage ρ.

However, Nash-Sutcliffe efficiency seems a more precise method to evaluated the

performance of each method. As we can see, the Ens of Sklm is much higher than the rest

methods, except RK, which has similar Ens with SKlm technique. The original radar

estimation has the smallest Ens, which is negative 14.2154. Therefore, the geostatistical

methods we used in this study seems to be applicable in precipitation interpolation.

You did not cite the Table 2

Table 2. Overall Performance of Sklm, KED, RK and Radar Corrected Methods for the year 2004.

ρ ENSMethod

mean Improved (%) mean Improved

(%) Original Radar 0.519163 -14.2154

RC 0.519163 0.258328 43.51% Sklm 0.53631 62.94% 0.492494 54.81% KED 0.421253 54.19% -0.95589 40.82% RK 0.527491 62.53% 0.482021 53.30%

7.2.2 The improvements of Ens using different interpolation methods.

19

There are 2194 counts of hours having the gauge mean larger than 0 mm/hour,

among which 1777 are lower than 0.5 mm/hour. The RC method increased the Ens

between radar estimates and gauge mostly when gage areal mean is around 3 and 5

mm/hour. The other methods (SKlm, KED and RK) improved the Ens most when the

gauge areal mean is relatively large. Generally, the interpolation methods do not perform

as well when the gauge areal mean is small as the areal mean is large. However, there is

overall improvement of Ens, where SKlm have the largest improvement (54.81%) and

followed by RK (53.3%) (Figure 3).

0.00%10.00%20.00%30.00%40.00%50.00%60.00%70.00%80.00%90.00%

100.00%

0.01-0

.50

0.51-1

.00

1.01-2

.00

2.01-3

.00

3.01-5

.00

5.01-8

.00>8.0

0

Gage Areal Mean (mm/h)

Ens

Impr

oved

/Tot

al C

ount

s

RCSKlmKEDRK

Figure 3. Ens improvement at different gage areal mean intervals

7.2.3 Spatial correlation between fifty gauges’ precipitation and radar estimates

Figure 4 shows the spatial correlation of fifty gauges’ precipitation and rainfall

estimates interpolated by three different geostatistical methods, which are SKlm, KED

and RK respectively for the hour October 2nd, 13 (UTC). The SKlm estimates and gauge

precipitation has the highest R2 (0.7052), following by RK estimates and KED estimates.

20

Generally, all the interpolation methods improved correlation between radar estimates

and gauge precipitation.

Radar_Gauge

y = 0.6573x + 2.1688R2 = 0.6512

0

10

20

30

40

50

0 10 20 30 40 50

SKlm_Gauge

y = 0.7863x + 2.3438R2 = 0.7052

0

10

20

30

40

50

0 10 20 30 40 50Observed Precipitation Depth (mm/hour)

Estim

ated

Pre

cipi

tatio

n D

epth

(mm

/hou

rObserved Precipitation Depth (mm/hour)

Estim

ated

Pre

cipi

tatio

n D

epth

(mm

/hou

r

KED_Gauge

y = 0.7574x + 2.9438R2 = 0.6675

0

10

20

30

40

50

0 10 20 30 40 50

RK_Gauge

y = 0.7936x + 2.3701R2 = 0.6938

0

10

20

30

40

50

0 10 20 30 40Observed Precipitation Depth (mm/hour)

Estim

ated

Pre

cipi

tatio

n D

epth

(mm

/hou

50

r

Observed Precipitation Depth (mm/hour)

Estim

ated

Pre

cipi

tatio

n D

epth

(mm

/hou

r

Figure 4. Scatter plot of fifty gauges’ precipitation and rainfall estimates interpolated by

three different geostatistical methods

7.3 Time Series Evaluation

Of the total fifty gauges, twelve gages in Comal county, which are ck9 to ck16,

gp4, gp7 , gp8 and gp9; eight in Guadalupe county, which are gp1, gp2, gp3 gp5 gp6

21

gp10 gp11 and gp12, ; twenty two in Kerr county, which are kr1 to kr22; and eight in

Kendell county, which are ck1 to ck8. We randomly picked five gauges (ck1, ck6, gp6,

gp11 and kr4) from these fifty gauges to do the preliminary analysis. The scatter plots

with R2 between the radar estimated interpolated by the 3 geostatistical methods and the

observed rain gauge precipitation were displayed in the following figures (Figures 5, 6, 7,

8 and 9). The R2 of the original radar at gauge ck1 is 0.7853. SKlm and RK estimation

improved the R2 to 0.8653 and 0.8604, respectively. However, KED did not perform well

in correcting NEXRAD estimates at this gauge. At gauge ck6, all the methods improved

the R2. Again, SKlm has the best performance, followed by RK and KED. At gage gp6,

each method has similar performance. But Sklm slightly beat the other methods. As can

be seen from figure 6, the trend line of the interpolated estimates and gauge scatter plots

are much closer to the 1:1 line comparing to the original radar estimates. The trend line of

gage gp11 scatter plot between original radar estimates and gauge shows the largest

deviation. However, the estimated precipitation shows the greatest improvement at this

gauge. At gauge kr4, the R2 between radar estimates and gauge measurements improved

from 0.8114 to 0.9465 after corrected by SKlm interpolation. RK has the similar

performance as SKlm, and KED has slight improvement on the R2.

22

Radar_ck1_Original

y = 0.8926x + 0.0196R2 = 0.7853

0

10

20

30

40

50

0 10 20 30 40 50

Radar_ck1_SKlm

y = 0.9772x + 0.0128R2 = 0.8653

0

10

20

30

40

50

0 10 20 30 40 50Observed Precipitation Depth (mm/hour)

Estim

ated

Pre

cipi

tatio

n D

epth

(mm

/hou

r

Observed Precipitation Depth (mm/hour)

Estim

ated

Pre

cipi

tatio

n D

epth

(mm

/hou

r

Radar_ck1_KED

y = 0.9806x + 0.0128R2 = 0.7767

0

10

20

30

40

50

0 10 20 30 40 50

Radar_ck1_RK

y = 0.9818x + 0.0129R2 = 0.8604

0

10

20

30

40

50

0 10 20 30 40 50Observed Precipitation Depth (mm/hour)

Estim

ated

Pre

cipi

tatio

n D

epth

(mm

/hou

r

Observed Precipitation Depth (mm/hour)

Estim

ated

Pre

cipi

tatio

n D

epth

(mm

/hou

r

Figure 5. Scatter plots of hourly precipitation estimated using different methods and observed rain gauge data at ck1.

23

Radar_ck6_Original Radar

y = 0.9749x + 0.0106R2 = 0.8733

0

10

20

30

40

50

0 10 20 30 40 50

Radar_ck6_SKlm

y = 0.8864x + 0.0122R2 = 0.9382

0

10

20

30

40

50

0 10 20 30 40 5Observed Precipitation Depth (mm/hour)

Estim

ated

Pre

cipi

tatio

n D

epth

(mm

/hou

0

r

Observed Precipitation Depth (mm/hour)

Estim

ated

Pre

cipi

tatio

n D

epth

(mm

/hou

r

Radar_ck6_KED

y = 0.8523x + 0.0108R2 = 0.9148

0

10

20

30

40

50

0 10 20 30 40 50Observed Precipitation Depth (mm/hour)

Estim

ated

Pre

cipi

tatio

n D

epth

(mm

/hou

r

Radar_ck6_RK

y = 0.8846x + 0.0127R2 = 0.9378

0

10

20

30

40

50

0 10 20 30 40 50Observed Precipitation Depth (mm/hour)

Estim

ated

Pre

cipi

tatio

n D

epth

(mm

/hou

r

Figure 6. Scatter plots of hourly precipitation estimated using different methods and observed rain gauge data at ck6.

24

Radar_gp6_Original Radar

y = 0.7017x + 0.0445

70

R2 = 0.744

0

10

20

30

40

50

60

0 10 20 30 40 50 60 7Observed Precipitation Depth (mm/hour)

Estim

ated

Pre

cipi

tatio

n D

epth

(mm

/hou

0

r

Radar_gp6_SKlm

y = 0.8032x + 0.0383R2 = 0.8195

0

10

20

30

40

50

60

70

0 10 20 30 40 50 60 70

Observed Precipitation Depth (mm/hour)

Estim

ated

Pre

cipi

tatio

n D

epth

(mm

/hou

r

Radar_gp6_KED

y = 0.8348x + 0.0352R2 = 0.8121

0

10

20

30

40

50

60

70

0 10 20 30 40 50 60 70Observed Precipitation Depth (mm/hour)

Estim

ated

Pre

cipi

tatio

n D

epth

(mm

/hou

r

Radar_gp6_RK

y = 0.8028x + 0.0386R2 = 0.8172

0

10

20

30

40

50

60

70

0 10 20 30 40 50 60 70Observed Precipitation Depth (mm/hour)

Estim

ated

Pre

cipi

tatio

n D

epth

(mm

/hou

r

Figure 7. Scatter plots of hourly precipitation estimated using different methods and observed rain gauge data at gp6.

25

Radar_gp11_Original Radar

y = 0.6597x + 0.0425R2 = 0.7651

0

10

20

30

40

50

60

0 10 20 30 40 50 60

Radar_gp11_SKlm

y = 0.9004x + 0.0087R2 = 0.8912

0

10

20

30

40

50

60

0 10 20 30 40 50 6Observed Precipitation Depth (mm/hour)

Estim

ated

Pre

cipi

tatio

n D

epth

(mm

/hou

0

r

Observed Precipitation Depth (mm/hour)

Estim

ated

Pre

cipi

tatio

n D

epth

(mm

/hou

r

Radar_gp11_KED

y = 0.8284x + 0.0097R2 = 0.8106

0

10

20

30

40

50

60

0 10 20 30 40 50 60

Radar_gp11_RK

y = 0.9023x + 0.0084R2 = 0.8876

0

10

20

30

40

50

60

0 10 20 30 40 50 6Observed Precipitation Depth (mm/hour)

Estim

ated

Pre

cipi

tatio

n D

epth

(mm

/hou

0

r

Observed Precipitation Depth (mm/hour)

Estim

ated

Pre

cipi

tatio

n D

epth

(mm

/hou

r

Figure 8. Scatter plots of hourly precipitation estimated using different methods and observed rain gauge data at gp11.

26

Radar_kr4_Original Radar

y = 0.8835x + 0.0225R2 = 0.8114

0

10

20

30

40

50

60

0 10 20 30 40 50 60

Radar_kr4_SKlmy = 0.9228x + 0.0104

R2 = 0.9465

0

10

20

30

40

50

60

0 10 20 30 40 50

Observed Precipitation Depth (mm/hour)

Estim

ated

Pre

cipi

tatio

n D

epth

(mm

/hou

60

r

Observed Precipitation Depth (mm/hour)

Estim

ated

Pre

cipi

tatio

n D

epth

(mm

/hou

r

Radar_kr4_KEDy = 0.9644x + 0.0095

R2 = 0.905

0

10

20

30

40

50

60

0 10 20 30 40 50 60

Radar_kr4_RKy = 0.9228x + 0.0104

R2 = 0.9464

0

10

20

30

40

50

60

0 10 20 30 40 50 60

Observed Precipitation Depth (mm/hour)

Estim

ated

Pre

cipi

tatio

n D

epth

(mm

/hou

r

Observed Precipitation Depth (mm/hour)

Estim

ated

Pre

cipi

tatio

n D

epth

(mm

/hou

r

Figure 9. Scatter plots of hourly precipitation estimated using different methods and observed rain gauge data at kr4.

Table 3. Performance of Sklm, KED and RK comparing to original radar rainfall using rain gauge precipitation as a constraint at selected gauges

Ens Absolute Difference R2

Gage Radar Sklm RK Radar SKlm RK Radar SKlm RK KED ck1 0.7723 0.8518 0.8443 554 474 480 0.7844 0.8653 0.8604 0.7767ck6 0.8614 0.9353 0.9348 563 419 423 0.8730 0.9382 0.9378 0.9148gp6 0.7234 0.8216 0.7958 754 640 665 0.7440 0.8195 0.8172 0.8121gp11 0.5843 0.7963 0.7673 731 503 515 0.7651 0.8912 0.8876 0.8106kr4 0.6803 0.8245 0.8249 579 480 479 0.8094 0.9465 0.9464 0.9050

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According to the results we’ve achieved till now, SKlm is recognized as the best

method in correcting NEXRAD rainfall estimates. RK has similar performance. The

corrected Radar increased the Ens of the hourly estimates and gauge to a large extent,

which means there is a certain difference between radar rainfall and observed gauge

precipitation. Ens is an important parameter in evaluating the performance of the

interpolation methods, since it takes into account the difference besides the trend. KED

has been recognized as one of the most preferred stochastic surface interpolation

techniques. However, it does not perform well in this case. Some of the previous studies

also stated that Kriging is a fragile method, since most of the uncertainty involved in the

interpolation is related to the determination of external trends. (Several limitations

inherited in the KED interpolation method may cause its ineffectiveness, such as

selection of a semivariogram model, assignment of arbitrary values to sill and nugget

parameters, and distance intervals, and the computational burden involved in

interpolation of surfaces. Unless we can properly solve all these difficulties associated

with this method, this method is not recommended in correcting the NEXRAD rainfall.

8. FUTURE STUDIES 1. Further analysis of the data: previous study led by Xianwei Wang indicated that

CV value has some influence on the rainfall estimation. Uniformed rainfall turned

to be has a higher correlation with the observed gage precipitation. I will try to

find the pattern and threshold of the influence of CV value on rainfall estimation.

2. Finish the whole year’s spatial and time series analysis.

3. Evaluate the effectiveness of the interpolation methods in improving the radar

estimates accuracy by day and season.

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4. Use RK with General Least Squares function instead of OLS to interpolation the

precipitation to see if there is improvements on the rainfall pattern distribution.

5. Compare the performance of each method in hourly rainfall interpolation using

other parameters, such as RMSE.

REFERENCES:

Ahmed, S. and G. de Marsily. 1987. Comparison of geostatistical methods for estimating transmissivity using data on transmissivity and specific capacity. Water Resources Research 23(9): 1727-1737.

Bedient PB, Hoblit BC, Gladwell DC, Vieux BE, 2000. NEXRAD Radar for Flood

Prediction in Houston. Journal of Hydrologic Engineering 5:269-277.

Bradley AA, Peters-Lidard C, Nelson BR, Smith JA, Yong CB, 2002. Raingauge

Network Design Using NEXRAD Precipitation Estimate Journal of the American

Water Resources Association. 38:1393-1407.Buytaert W, Celleri R, Willems P,

Bievre B, Wyseur G, 2006. Spatial and Temporal Rainfall Variability in Mountainous

Areas: A case study from the south ecuadorian Andes. Journal of Hydrology 329:

413-421.

Cheng S, Hsiedh H, Wang Y, 2007. Geostatistical interpolation of space-time rainfall on

tamshui river basin, taiwan. Hydrological Processes 21: 3136-3145.

Christel Prudhomme DW, 1999. Mapping extreme rainfall in a mountainous region

using geostatistical techniques: a case study in Scotland. International Journal of

Climatology 19(12): 1337-1356.

Ciach G J, Krajewski WF, Villarini G, 2007. Product-Error-Driven uncertainty Model for

probabilistic quantitative precipitation estimation with NEXRAD Data. Journal of

Hydrometeorology 8(6): 1325-1347.

29

Diodato N, Ceccarelli M, 2005. Interpolation Processes using multivariate geostatistics

for mapping of climatological precipitation mean in the sannio mountains (southern

Italy). Earth Surface Processes and Landforms 30: 259-268.

Dirks KN, Hay JE, Stow CD, Harris D, 1998. High-resolution studies of rainfall on

Norfolk Island Part II: Interpolation of rainfall data. Journal of Hydrology 208:187-

193.

Eberhart, R.C., and Y. Shi, 1998. Comparison between genetic algorithms and particle swarm optimization. In Proceedings of the Seventh Annual Conference on Evolutionary Programming: Evolutionary Programming VII, 611-616. Springer: New York.

Eduardo Severino TA, 2005. Spatiotemporal models in the estimation of area

precipitation. Environmetrics 16(8): 773-802.

Goovaerts P, 2000. Geostatistical approaches for incorporating elevation into the spatial

interpolation of rainfall. Journal of Hydrology 228: 113-129.

Goovaerts, P., 1997. Geostatistics for Natural Resources Evaluation. Oxford University Press: New York.

Grimes DIF, Pardo-igu´ Zquiza E, Bonifacio R, 1999. Optimal areal rainfall estimation

using raingauges and satellite data. Journal of Hydrology 222: 93-108.

Haberlandt U,2007. Geostatistical interpolation of hourly precipitation from rain gauges

and radar for a large-scale extreme rainfall event. Journal of Hydrology 332:144-157.

Kennedy, J., and R.C. Eberhart, 2001. Swarm Intelligence. Morgan Kaufmann: San

Mateo, CA.

Lloyd CD, 2005. Assessing the effect of integrating elevation data into the estimation of

monthly precipitation in Great Britain. Journal of Hydrology 308: 128-150.

Montagna PA, Kalke RD, 1992. The effect of freshwater inflow on meiofaunal and

macrofaunal populations in the Guadalupe and Nueces estuaries, Texas. Estuaries 15

(3): 307–326.

30

Nash, J.E., and J.V. Sutcliffe, 1970. River flow forecasting through conceptual models: Part I. A discussion of principles. Journal of Hydrology 10(3): 282-290.Isaaks EH, Srivastava R M, 1989. Applied Geostatistics. New York: Oxford University Press.

Pardo-igu´ Zquiza E, 1998. Comparison of geostatistical methods for estimating the areal

average cclimatological rainfall mean using data on precipitation and topography.

International Journal of Climatology 18: 1031-1047.

Wang X, Xie H, Sharif H, Zeitler J, 2008. Validating NEXRDA MPE and Stage III

precipitation products for uniform rainfall on the Upper Guadalupe River Basin of the

Texas Hill Country. Journal of Hydrology 348(1-2): 73-86.

Webster, R. and M.A. Oliver, 2007. Geostatistics for Environmental Scientists. John Wiley & Sons: Chichester.

Wilk J, Kniveton D, Andersson L, Layberry R, Todd MC, Hughes D, Ringrose S,

Vanderpost C, 2006. Estimating rainfall and water balance over Okavango River

Basin for hydrological applications. Journal of Hydrology 331: 18-29.

Xie H, Zhou J, HendrickxE, Vivoni H, Guan H, Tian YQ Small EE, 2006. comparison of

NEXRAD Stage III and gauge precipitation estimates in central New Mexico.

Journal of the American Water Resources Association 42(1): 237-256

Young CB, Nelson BR, Bradley AA, Krajewski WF, Kruger A, 1999. Evaluating

NEXRAD Multisensor Precipitation Estimates for operational hydrologic forecasting.

Journal of Hydrometeorology

Young CB, Nelson BR, Bradley AA, Smith JA, Perters-Lidard CD, Kruger A, Baeck

ML,1998. An evaluation of NEXRAD precipitation estimates in complex terrain.

Journal of Geophysical Research 104 (D16), 19691-19703.

 

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