Correlation fields for solar radiation in northeast Brazil

INTERNATIONAL JOURNAL OF CLIMATOLOGY

Int. J. Climatol. 21: 887–902 (2001)

CORRELATION FIELDS FOR SOLAR RADIATION IN NORTHEASTBRAZIL

JUAN CARLOS CEBALLOSa,*, JAIDETE MONTEIRO DE SOUZAa and ADMIR CRESO TARGINO DANTASb

a Centro de Pre�isao de Tempo e Estudos Climaticos, Instituto Nacional de Pesquisas Espaciais, INPE, 12630 Cachoeira Paulista,SP, Brazil

b Facultad de Ingenierıa, Centro de Calculo, CeCal, Monte�ideo, Uruguay

Recei�ed 10 January 2000Re�ised 3 January 2001

Accepted 26 January 2001

ABSTRACT

Daily data of solar irradiation were used for analysing spatial decrease of correlation coefficients within northeastBrazil (a region extending over more than l.5 million km2). Available data of the regional climatological networkcorrespond to typical distances between stations of 150 km and more. Two 3-month periods were chosen:March–April–May (MAM) and August–September–October (ASO) 1975. It was found that spatial distribution ofcorrelation for daily irradiations does not allow linear interpolations based on network data. However, factor analysisof 5-day means (particularly useful for agricultural purposes) allowed identification of four regions with internalsimilarity of time series behaviour, which is induced by the influence of typical meteorological systems. For thistime-scale, correlation coefficients may be 0.7 and higher at rather long distances (in some cases, more than 600 km),so that the climatological network can provide data for a detailed description of the regional distribution of meanirradiation. For shorter time and space scales, a too dense solarimetric network would be necessary, and the use ofmodern techniques using satellite monitoring over the region may be more appropriate. Copyright © 2001 RoyalMeteorological Society.

KEY WORDS: correlation coefficients; factor analysis; northeast Brazil; solar irradiation

DOI: 10.1002/joc.644

1. INTRODUCTION

The importance of solar radiation as the main source of energy for meteorological phenomena, forevaporation processes in agriculture and forests, and last, but not least, as a non-conventional, ‘clean’energy resource is frequently mentioned. The description of spatial and temporal distribution of solarradiation is a necessary step for its proper management, but solarimetric measurements have traditionallybeen scarce as a consequence of the costs of the usual instruments, their maintenance, and/or dataprocessing. On the one hand, traditional instruments such as heliographs or actinographs have onlygraphical output, whose management implies a considerable effort of data processing (Azevedo et al.,1981; Drake and Mulugetta, 1996); on the other hand, electrical devices with automatic processing arerelatively expensive. Therefore, it would be desirable that a solarimetric network contains a minimal (notredundant) but meaningful set of stations, in order to save technical and economical efforts.

Extensive data sets have been organized in a Global Energy Balance Archive (GEBA) since 1985,including irradiation measured by pyranometers throughout the world; however, GEBA distribution doesnot include data over Brazil (Wild et al., 1995). Indeed, as a general rule the national climatologicalnetwork in this country (extended over 8.4 million km2) includes, at most, actinographs; pyranometershave not been used except during short periods. Only recently did the National Institute for Space

* Correspondence to: Centro de Previsao de Tempo e Estudos Climaticos (CPTEC), Instituto Nacional de Pesquisas Espaciais(INPE), 12630 Cachoeira Paulista, SP, Brazil; e-mail: [email protected]

Copyright © 2001 Royal Meteorological Society

J.C. CEBALLOS ET AL.888

Research (INPE) undertook the installation of a network of automatic data collecting platforms linked bysatellite to a control centre (see internet address http://www.cmcd.inpe.br).

Not all actinographic records for the Brazilian network have been processed. Nevertheless, the completeset of available records for northeast Brazil (extended over more than 1.5 million km2) for the period1974–1978 was integrated at the Federal University of Paraıba (Azevedo et al., 1981). The instrumentswere installed during 1974–1975. The error associated to graphic integration and calibration is not higherthan 4–5% for daily global irradiations (Fattori and Ceballos, 1988), a value much lower than thoseattributed to instruments manufactured before the 1950s (Robinson, 1966).

Concerning the accuracy of GEBA data, Gilgen et al., (1998) showed that daily mean irradiancemeasured by two neighbouring pyranometers with different manufacturers and maintenance schemes mayexhibit mean bias of about 3.4 W m−2 and standard deviation associated to instrumental errors not lowerthan 7.7 W m−2. At the monthly scale, daily means of two instruments at the same city may show asimilar bias, and a standard error of about 5–5.5 W m−2 (or 2–2.5 MJ m−2 for daily irradiation).Considering typical values of solar radiation over the northeast (Figure 2), and the errors attributed toactinographs with recent installation, this means that data from Azevedo et al. (1981) may be useful foran analysis of regional distribution of solar radiation, at least with the same order of reliability of somepyranometer networks across the world.

A method for the design of a solarimetric network in Argentina was described by Grossi Gallegos andLopardo (1988). Daily values of irradiation obtained by pyranometers were available, and distancebetween stations was at least 90 km. The ‘neighbourhood’ between two stations was defined by thestandard deviation, S, of the difference between their irradiations; two stations were ‘closer’ for lowervalues of S. The region of analysis was the ‘Pampa Humeda’ (provinces of Buenos Aires, La Pampa,Santa Fe and Cordoba), a region where mean daily irradiation is remarkably constant throughout. The‘influence radius’, L, of a station was defined by the expected fluctuation of S at the 90% level (say, avalue S*). Their results showed that expected differences of daily irradiation are excessively large whencompared with their mean value. If monthly means are considered, S* is 8.5% of the regional mean forL=150 km, and 10% for L=225 km.

The present paper focuses on the ability to assess irradiation at a site by linear regression on data fromneighbouring stations. Therefore, the correlation coefficient, r, and its regional behaviour will be studiedin this paper. The field of correlation coefficients between stations is a most important source ofinformation for the assessment of missing data in the time series of a station, as well as for generatinghighly reliable time series of a ‘virtual’ site, where no station exists (Ceballos and Braga, 1995).

Northeast Brazil extends over an area of 1.5 million km2. Insolation (daily number of hours withsunshine) is influenced by several circulation systems, in such a manner that monthly means of dailyglobal irradiation, as well as the behaviour of time fluctuations, cannot be considered to be regionallyhomogeneous. Obviously, cloudiness associated to meteorological systems in the region must be directlyassociated to the spatial variability of r. Conversely, the functional decrease with distance shown by rmust provide information about the temporal coherence of those systems. It is expected that solarradiation exhibits mesoscale spatial variability. However, solarimetric instruments are usually included inclimatic stations whose distribution follows the requirements of synoptic scale dynamics, not alwaysadequate for the analysis of mesoscale phenomena.

This paper seeks to determine regions of northeast Brazil for which the solar regime may be consideredhomogeneous in a proper scale (as short as possible), in the sense that regional behaviour of parameterr may be described mainly as a function of distance between two sites in a spatial scale as large aspossible.

2. CLIMATIC ASPECTS

Figure 1 shows the northeast Brazil distribution of climatological stations for 1974–1978 (see also TableI). It is expected that the behaviour of solar irradiation follows fluctuations of cloudiness associated with

Copyright © 2001 Royal Meteorological Society Int. J. Climatol. 21: 887–902 (2001)

SOLAR RADIATION IN NORTHEAST BRAZIL 889

Figure 1. Climatological network of the northeast in 1974–1978. Only stations with available solarimetric data are included. Labelsfor stations are explained in Table I

regional meteorological systems. Three main systems act on this region (IBGE, 1985), as described below.Obviously, departures from climatic means may produce modifications to the geographical pattern in agiven year.

The Intertropical Convergence Zone (ITCZ) has an annual cycle oscillating between mean extremelatitudes of about 2°S (March) and 14°N (September). It induces the rainy season over the northern partof the northeast in the period February–March–April (FMA). Severe droughts ocurr in years when theITCZ is absent. The inner northeast has semi-arid characteristics.

The eastern part of the northeast is influenced by the easterlies, which penetrate over the continent bysome hundreds of kilometres. Parallel to the coast (at about 100 km of it) a low-height elevation(Borborema system) is the edge of a plateau with gentle slope from 500 m down to a typical 200 maltitude in the inner northeast. Easterlies blow mainly from east to southeast, oscillating during the yearaccording to the displacement of the permanent high pressure centre of the tropical Atlantic. Along theeastern coast, the rainy season occurs mainly between May and August.

Cold fronts come from Argentina into southern Brazil, and are progressively deflected towards the east,going into the Atlantic after crossing the southernmost part of northeast Brazil. Nevertheless, althoughweakened, they frequently penetrate into the northeast and increase cloudiness. The rainy season in thesouthern northeast is verified mainly in summer, from November to March.

In addition to this behaviour, the westernmost region has the influence of the Amazonian climateregime.



Table I. Northeast Climatological Stations with actinographic data used in this work

Number Station Label Longitude Latitude

1 Areia ARE −35.7 −6.972 Barbalha BLH −39.32 −7.323 Barra do Corda BRC −45.27 −5.504 Barreiras BAR −44.98 −12.155 Bom Jesus da Lapa BJL −43.42 −13.276 Cabrobo CAB −39.32 −8.507 Caetite CTE −42.48 −14.078 Carolina CAR −47.47 −7.339 Crateus CRA −40.67 −5.18

10 Cruzeta CRZ −36.78 −6.4211 Floriano FLO −43.02 −6.7512 Fortaleza FOR −38.53 −3.7313 Guaratinga GUA −39.55 −16.5714 Ilheus ILH −39.03 −14.8015 Irece IRE −41.87 −11.3016 Jaguarana JAG −37.80 −4.8317 Joao Pessoa JPS −34.88 −7.1218 Lencois LEN −41.38 −12.5219 Monte Santo MST −39.50 −10.3520 Morada Nova MNV −38.38 −5.1021 Morro do Cavalo MCV −42.42 −7.9222 Patos PAT −37.28 −7.0223 Paulo Afonso PAF −38.25 −9.3524 Propria PRP −36.83 −10.2225 Quixeramobim QUX −39.30 −5.2026 Recife REC −34.88 −8.0327 Salvador SSA −38.62 −13.0028 Sao Goncalo SGC −38.32 −6.8329 Sao Luiz SLZ −44.28 −2.5330 Sobral SOB −40.35 −3.7031 Surubim SUR −35.75 −7.82

3. SOME STATISTICAL ASPECTS OF CORRELATION COEFFICIENTS

A minimal set of conditions for network design (and regional interpolation) may be: (1) the stations’ensemble should provide information about the whole region; (2) in order to save costs, informationprovided by two neighbouring stations should be only weakly redundant; (3) nevertheless, someredundancy is needed in order to assess radiation values by proper interpolation within the network.

The use of standard deviation S helps to assess mean absolute (numerical) neighbourhood betweenstations, while correlation coefficient r describes the statistical distance between standardized time seriesdata, and estimates a ‘first order’ likelihood between fluctuations of two stations. It is worthwhile to notethat variance S2 and correlation coefficient r are related parameters: if D=y−x is generic deviationbetween two station series X={x1, x2 . . . , xn} and Y={y1, y2, . . . , yn}, with means �x�, �y� then

x=�x�+u, y=�y�+�, �D�=�y�−�x�,

S2=�[D−�D�]2�=�[�−u ]2�=Sx2 +Sy

2−2rSxSy, (3.1)

r=�u��SxSy

��=Cov(X, Y)

�x�y

.

The coefficient r is calculated from actual data, while � is a mathematical expectance. D allows theelimination of differences between climatic means of both stations. It is important to note that deviationsu and � are composed of local fluctuations U, V, owing to mesoscale or microscale characteristics, plusdeviations u �, ��, owing to local estimation errors which are weakly correlated with U or V, such that



�U�=�V�=�u ��=��=0,

Sx2 �U2+sx

2, Sy2�V2+sy

2, (3.2)

S2=�[(V−U)− (��−u �)]2��(V−U)2�+sx2 +sy

2.

From Equations (3.1), quadratic distance S2 will be zero if r= �1, but requires Sx=Sy. However, itbecomes evident in Equation (3.2) that S2 would not vanish, even if U, V local meteorological fluctuationswere low or almost identical, because of instrumental errors sx, sy ; at the limit, for two identicalneighbouring instruments located at the same station, we should obtain S2=2s2. Therefore, it is alwaysexpected that r�1.

3.1. Correlation le�els ha�ing ‘physical meaning’

Let us consider the solarimetric time series of two stations during a climatically well defined period(which avoids filtering of long term variations). Physical quantities may be mean irradiances (daily orweekly). Given two series, X and Y, a linear regression between them may be written as

Y�=a+bX, (3.3)

where Y� is a linear estimate of Y assessed from X, obtained by applying the condition of minimalquadratic distance between the series Y and Y�. If both series are standardized related to their meanvalues, �x�, �y�, and to standard deviations Sx, Sy, then the straight line (3.3) is described by

Y=rX, r=�xy�−�x��y�

SxSy

��, (3.4)

where r is an estimate of the ‘true’ correlation coefficient � between X and Y.It is easily shown that the errors e associated to the estimates Y’ related to Y have a standard deviation

sy such that

sy2= (1−�2)Sy

2, �2=1− (sy/Sy)2. (3.5)

Daily irradiation in the northeast has typical values of Q=500 langley (ly) or 20.9 MJ m−2 with standarddeviations Sy=75–100 ly or 3.6 MJ m−2 (Azevedo et al., 1981), while typical errors of actinographic dataare 4% for daily irradiation, or sac�0.84 MJ m−2 (Fattori and Ceballos, 1988). Assuming that deviationse=Y�−Y are mainly associated with instruments, then Equation (3.5) tells us that, for 2sac�s�3sac, itis expected to be about 0.87��0.67. A correlation value �=0.7 can be chosen as a good level oflikelihood between two stations, because: (1) 50% of information is explained by the linear regression; (2)for normally distributed errors, this correlation means that more than 90% of experimental points in (x, y)lie within distances e�2sac from the regression line.

Correlation must decrease with distance L between stations. Defining a ‘critical distance’ L* when�=0.7 is attained, it should be longer if irradiations are considered in 5-day means rather than dailyvalues. Standard deviations Sy would be reduced by a factor of �5, as well as those sy of errors e for thespecified correlation �.

It should be noted that sampling values r are only estimators of ��0. Assuming a normal distributionof errors, the probability distribution of sampling (stochastic) estimators r admits the definition of arandom variable Z first introduced by Fischer (Paradine and Rivett, 1991) with normal distributionaround Z� =Z(�) and properties

Z=12

ln1+r1−r

, Var{Z}=�Z2 =

1n−3

. (3.6)

Considering samples of n=30 days and a rather strict interval Z� ��Z for accepting Z values, it is evidentthat a minimal sampling correlation r=0.6 leads to an interval 0.693�Z�1.077 corresponding to�=0.709. Therefore, experimental values r�0.6 may be assumed representative of ‘true’ correlation��0.7.



4. DATA PROCESSING

Two solarimetric data series are available in the Atmospheric Sciences Department of the FederalUniversity of Paraiba (UFPb) corresponding to Fuess actinographic records (heliographic records are notconsidered in this study). One corresponds to daily global irradiation in the climatological network of thenortheast, held by the National Meteorological Institute (INEMET). Azevedo et al. (1981) processed allthe available actinograms of the period 1974–1978. The second data series was provided by thesolarimetric network of the UFPb, with stations located mainly in Paraiba State, partially covering theperiod 1975–1996.

Two periods of 3 months were selected for daily irradiation: FMA (corresponding to the rainy seasonover the most part of inner semiarid region), and August–September–October (ASO) (dry season overmost parts of the northeast). Not all years have complete data series for the whole set of stations, whichis a necessary condition for the derivation of a regional correlation matrix. The year 1975 is the best dataset, including 33 stations. Table I presents the set of stations used in this study, including theirgeographical location. The stations of the UFPb network were not included in the first step. Dailyirradiation, as well as mean values over periods of 5 days (pentads), were considered, the latter definingdata for a 3-month time interval. Only pentads with not more than 1 missing day were accepted. Figure2(a) and (b) illustrate the mean irradiance for 3-month periods in 1975.

Correlation matrices for the network stations were calculated, considering daily values for March andfor September, as well as 5-day means for FMA and ASO. The resulting correlation matrices haddimensions 22×22 and 24×24, respectively.

Figure 2. Distribution of mean daily irradiance over periods FMA 1975 (a) and ASO 1975 (b). Labels for stations are detailed inTable I



Figure 2 (Continued)

The distance L between two stations 1 and 2 was assessed as a geodesic (minimum arc length betweentwo points lying on a sphere), given by

L=RT�, cos �=cos �1 cos �2 cos(�1−�2)+sin �1 sin �2 (4.1)

where (�, �) are geographical co-ordinates (latitude, longitude) of a station; RT=earth radius.

5. RESULTS

As shown in Section 2, if X1 and X2 are time series of two neighbouring stations, and the linear expressionX�=a+bX1 assesses the time series X2 based on linear regression X2�X1, then values r�0.6 are acceptableas a sampling correlation coefficient. These r values allow us to expect that the estimate X� includes morethan 50% of the variance; in addition, more than 90% of experimental values differ from estimate X� inless than twice a typical instrument error.

Figure 3(a) and (b) illustrate the regional distribution of correlation coefficient for several stations.Correlation fields for daily irradiation in March 1975 are shown for six sites, namely BRC, SLZ, CAB,REC, SSA and BJL (Figure 3(a)) as well as for 5-day means in FMA (Figure 3(b)). Levels of correlationbetween 0.6 and 0.9 are presented in 0.1 steps. The kriging method was applied for interpolation,supposing that resulting bull’s eyes (tendency to show closed lines around a given site) are less markedcompared with those of other numerical methods.



Daily irradiance correlation shows a steep decrease with distance, especially on the eastern coast. Itsuggests that local or mesoscale fluctuations of convective cloudiness are prevailing, except over thenorthernmost region, where the ITCZ induces larger scale cloudiness (thus a spatially homogeneousbehaviour) during March. It is evident that the coefficient r is usually lower than the critical value of 0.6over less than 200–300 km. We conclude that a network design with typical synoptic criteria (distancesof about 200 km between stations) is not appropriate for interpolation of daily irradiance, at least in thisregion.

Uncorrelated short-scale fluctuations over a region tend to be filtered when 5-day means are considered,so that the presence of larger scale systems responsible for regional cloudiness become evident. Indeed, theITCZ still dominates the northernmost part of the region; in addition, the southern part of the northeast(BJL case) appears dominated by sampling correlation higher than 0.6, thus suggesting association withmeteorological systems coming from southern Brazil. Correlation with REC (eastern coast) are higherthan 0.6 because of the influence of easterlies. Finally, a large cell of high correlation around CAB isobserved over the inner northeast.

September shows similar features for daily irradiation, but the distribution of 5-day means in ASO hasa complex structure. As before, southern stations appear highly correlated. A new pattern consists of sitesin westernmost northeast with high-valued, but negative, correlation with large areas on the eastern andnorthern coast.

These results justify the consideration of the behaviour of 5-day means only. In order to detect regionswith distinctive behaviour, principal components analysis (PCA) was performed. It must be observed thatthe time series of a given variable is the result of the sum of influences of the time series of those factors

Figure 3. (a) Regional distribution of correlation coefficients for some selected stations considering daily irradiation in March 1975.(b) As part (a) but considering 5-day means for period FMA



Figure 3 (Continued)

(therefore, during limited periods only, one of them could be dominant, even being poorly correlated duringthe whole period in consideration).

5.1. Principal components analysis

Twenty-two stations were included for the period FMA and 24 for the period ASO. Each station wasconsidered as a vector with 19 components. Standardized variables were considered, in order to enhancethe importance of relative, rather than absolute, fluctuations of original variables. Therefore, PC loadingswere deduced from the correlation matrix. Tables II and III illustrate the results. Generally speaking, fiveprincipal components accounted for, more than 84% of the individual station variances. Varimax rotationof the first five components was applied to the matrix of components loadings. This procedure conservescommonality (Johnson and Wichern, 1982), and contributes to separate distinct groups of stations whichare highly correlated with independent (orthogonal) principal components. At first glance, not more thantwo rotated PCs concentrated the most part of the variance of the system.

5.2. Period Feb-Mar-Apr (FMA)

Although two PCs accounted for 57% of total variance (Table II), eigenvalues decrease rapidly only afterthe fifth one. The sixth component had the eigenvalue 0.934, and would not be more significant than onesingle variable (‘Kaiser criterion’). Therefore, five principal components were considered (accounting for84.6% of total variance). It is seen that cumulated communalities for individual stations were not lowerthan 0.7 (except for MCV Morro do Cavalo). Therefore, five temporal behaviours (or factors) seem to be



Table II. Principal components analysis for period FMA 1975, showing eigenvalues (eval),communality (comm) and component loadings for correlation matrix of 5-day means

Order Eval Label Comm PC1 PC2 PC3 PC4 PC5

SLZ 0.869 −0.1201 −0.011 (0.571) 0.153 0.7117.893SOB 0.880 0.027 0.0224.631 −0.0102 −0.091 0.933

2.6813 JAG 0.822 0.326 −0.080 0.740 −0.058 0.398QUX 0.789 (0.522) −0.022 (0.567) 0.367 0.2454 1.883MNV 0.930 0.169 −0.1681.537 0.909 0.188 −0.1115FLO 0.701 0.453 −0.341 0.3126 −0.464 −0.2580.934MCV 0.680 −0.067 0.1200.626 0.807 0.101 0.0097

0.5058 CRZ 0.897 (0.500) −0.197 0.348 0.694 0.071ARE 0.824 0.787 −0.1270.403 0.0559 0.392 0.182

0.34310 PAT 0.942 0.213 0.088 0.174 0.924 −0.0730.30511 JPS 0.814 0.874 0.039 0.062 0.185 −0.101

CAB 0.935 0.380 (0.531)0.228 0.33612 0.609 −0.1580.16613 REC 0.836 0.879 0.152 0.172 0.079 0.065

PET 0.770 0.409 0.609 0.01714 0.480 −0.0240.113PAF 0.990 0.971 0.1900.075 0.06115 0.081 0.012

0.04616 PRP 0.934 0.916 0.207 0.151 0.038 −0.167SSA 0.717 0.600 (0.506) −0.219 0.196 0.12217 0.039BAR 0.786 −0.071 0.8830.016 −0.011 0.042 0.01018

19 LEN 0.882 0.471 0.784 −0.058 −0.186 0.091BJL 0.893 −0.004 0.920 −0.060 −0.189 −0.08920CTE 0.855 −0.005 0.881 0.08821 0.212 −0.161ILH 0.881 0.20922 0.860 −0.091 0.158 0.254

Five main components (PC1, . . . , PC5) have been considered. Varimax rotation was applied.Values higher than 0.6 are underlined.

Table III. Principal components analysis for period ASO 1975

PC2 PC3 PC4Order Eval Station Comm PC1 PC5

0.0980.375 −0.069 −0.0380.7491 8.856 SLZ 0.7175−0.819 0.003 0.118 −0.073 0.0532 5.629 BRO 0.6921

0.844 −0.088 0.2083 2.469 CRA 0.7938 0.123 0.1260.156 0.0040.850 −0.0310.1654 1.919 QUX 0.7756

0.678 0.245 −0.4195 1.243 MNV 0.8312 −0.2700.252(0.507) 0.423 0.2496 0.795 MCV 0.7458 −0.462 0.187

0.0850.020(0.595) 0.308(0.577)7 0.765 SGC 0.79040.469 0.206 −0.0658 0.579 CRZ 0.8569 0.766 0.056

0.187−0.1430.0230.3260.8489 0.450 ARE 0.8823−0.757 0.241 0.197 0.499 0.04710 0.322 CAR 0.9222

0.228 0.242 0.3030.7770.17311 0.261 BLH 0.83490.356 0.340 0.064 −0.10812 0.206 PAT 0.8080.9117

0.158 −0.084 −0.20813 0.180 SUR 0.8325 0.870 −0.003−0.339−0.076−0.024 (0.548)0.69414 0.131 JPS 0.9027

0.819 0.334 −0.043 0.201 0.09915 0.100 CAB 0.8346−0.023 −0.144 0.00416 0.052 PAF 0.9378 0.932 0.220

0.373 0.2400.094 −0.2960.76717 0.039 PRP 0.88210.742 (0.507) −0.00618 0.025 IRE 0.8746 0.194 0.172

0.220 0.191 0.87019 0.017 SSA 0.8566 −0.053 −0.1100.1320.7210.125 0.126−0.47220 0.003 BAR 0.7915

0.759 −0.046 −0.40124 0.000 LEN 0.8907 0.373 −0.1160.0890.0310.8400.111−0.24722 BJL 0.7873

−0.106 −0.033 0.893 0.28623 CTE 0.0000.8910−0.053 0.141 0.353 0.85624 ILH −0.0410.8814

Same parameters as in Table II.



present in the region. Table II makes evident how Varimax procedure separates distinct clusters ofstations. Rotated component loadings higher than 0.6 (underlined in Table II) define the followinggroups:

� G1—ARE JPS REC PAF PRP SSA (QUX) (CRZ)� G2—PET BAR LEN BJL CTE ILH (CAB) (SSA)� G3—JAG MNV MCV (QUX) (SLZ)� G4—CRZ PAT CAB� G5—SLZ SOB

Stations in parentheses have r values, such that 0.6�r�0.5. Figure 4 illustrates the interpolation of rwithin the region. Loadings are analogous to a correlation coefficient with the temporal behaviour of thecorresponding component loading. A given component loading contributes to about 50% of local varianceover areas inside the closed line r=0.7, while 0.6 may be considered an acceptable critical value. Areasoutside r=0.6 are only weakly correlated with the component loading (contribution to individualcommunality is r2�0.36). QUX and FLO did not show high correlation with any component loading.Indeed, lines outside the northeast region (including lines over the ocean) are only illustrative as theyextrapolate fields based on inner regional data. Four distinct regions are sketched which suggest theexistence of the same number of factors:

� PC1 (group G1): associated with atmospheric flow of Atlantic trades, extended over the eastern coastalregion north from 13°S.

� PC2 (groups G3 and G4): limited to the inner northeast. This is a dry region frequently subject todrought. Figure 4 suggests the existence of two parallel southwest–northeast oriented regions;however, more station data would be desirable to clarify this.

� PC3 (group G2): associated with the influence of cold fronts coming from southern Brazil, havingtrajectories which typically tend towards the Atlantic ocean.

Figure 4. Spatial distribution of components loadings for the first five (varimax rotated) PCs; FMA 1975



� PC4 (group G5): associated with the ITCZ at the March equinox epoch, influencing mainly thenorthern coastal region, west from 38°W. The structure of lines suggests that an additional station onthis coast might show that this component extends its influence throughout longitudes up to 36°W.

5.3. Period ASO

Five principal components, with an eigenvalue higher than unity, accounted for 84% of total variance(equal to 24), as shown in Table III. It is seen that some r values are negative. Considering higher absolutevalues (�r ��0.6), five groups can be clearly drawn:

� G1—SLZ CRZ ARE SUR JPS CAB PAF PRP (SGC) BRC* CAR*� G2—CRA QUX MNV BLH (SGC) (MCV)� G3—IRE BAR LEN BJL CTE� G4—SSA ILH� GS—PAT (JPS)

Stations in parentheses exhibit values 0.5�r�0.6, meaning weaker association with a given component.Asterisks denote stations with negative component loadings. PAT (Patos) is associated with the lowesteigenvalue (�=1.24), which suggests a distinct behaviour; it will not be considered hereafter. Figure 5illustrates the spatial distribution of correlation with the main components, suggesting four regionalbehaviours:

� PC1 (group G1): as before, it is associated with Atlantic trades influencing the coastal environmentnorth from 12°S. The configuration of correlation fields suggests that data from an additional stationon the coast would provide evidence of spatial continuity for loadings component influence all alongthe northern coast. The ITCZ is located in the Northern Hemisphere during this 3-month period. Thedeep continental penetration of trade winds influence over an area between 8°S and 10°S is alsoevident. An outstanding feature is the high-valued negati�e correlation over the western northeast, at

Figure 5. The same as Figure 4 but for ASO 1975



latitudes similar to those of the high positi�e correlation field over the coast. This ‘bipolar pattern’ seemsunfortuitous, as BRC and CAR are in the eastern limit of Amazonia. Convective bursts in this regionmay originate circulation cells with subsidence (downstreaming branch) over the eastern northeast, suchthat solar radiation would be oppositely affected by resultant regional high and low cloudiness,respectively.

� PC2 (group G2): it is present in the inner northeast, extending over the driest region.� PC3 (group G3): it is associated to fronts arriving from southern Brazil.� PC5: suggests the influence of Atlantic Ocean at latitudes south from 12°S.

These results are coherent with climate analyses found in the literature (IBGE, 1985). Further detailed aspectsof the atmospheric dynamics are beyond the scope of this work.

5.4. Homogeneous regions

The precedent results suggest the presence of four main regimes of correlation when 5-day means areconsidered during

1. FMA and ASO.2. A northern coastal region is dominated by the presence of the ITCZ in March (group G5 during FMA).3. A southern region shows the remaining influence of fronts coming from southern Brazil (groups G2

during FMA and G3 during ASO).4. An eastern area is dominated by Atlantic wind circulation, and eventually, shows two typical axes for

the correlation field: along the coast and transversally to it, including the northern coast during ASO(G1 during FMA and G1 during ASO).

5. An inner region extends over the semiarid central northeast (groups G3 and G4 during FMA, G2 duringASO).

It seems sound that sites inside these regions have similar temporal regimes, even if they show someoverlapping of different components. Some areas in the inner northeast do not exhibit predominance ofa given component. It may be expected that time series alternate the influence of different components.

It might be asked if a lower number of components (for instance only two) could be enough to describethe characteristics of correlation fields. In order to highlight this question, it is worthwhile to make a criticalcomment about purely mathematical results. Principal components analysis defines the total variance ofthe system as the sum of the variances of all M original variables X. An eigenvalue assesses the contributionof a given principal component Z to the total variance. If variables X are standardized, then each singlevariance equals unity and total variance is equal to M. Let us suppose that the variables are grouped inK�M sets (clusters) with sizes M1, . . . , Mk, . . . , MK. Variables X are highly correlated within theircorresponding group only. Thus, (a) K main principal components will be found, characteristic of thoseclusters, and accumulating variances which account for the majority of the total variance; (b) on the otherhand, high internal correlation inside groups and weak correlation outside them means that the eigenvalue�k assigned to the kth group must be of the order of the size of that group, say Mk. This reasoning makesevident the fact that uncorrelated regions will be defined (in the sense that their corresponding variableZk is orthogonal to others), but also, that higher eigenvalues will usually be assigned to regions served bya higher number of solarimetric stations. Physically, this is not enough to say that the corresponding principalcomponent is ‘more important’ than others; at the limit, some groups may be constituted by a single stationdescribing well defined regional behaviours (or factors) of solar irradiation. Given that eigenvalues maybe mainly associated to the spatial density of stations, rather than to the geographical extension of theinfluence of each component, the Kaiser’s criterion becomes a useful tool for assessing the number ofdifferent behaviours present in a region.

5.5. Correlation inside homogeneous region

Figure 6 shows correlation coefficients in the southern northeast as a function of distance (factor 2, orgroups G2 in FMA, and G3 in ASO).



Figure 6. Correlation coefficients in the southern northeast as a function of distance (groups G2 in FMA and G3 in ASO)

It is evident that correlation with Ilheus (ILH) during FMA is higher than 0.6 for distances as long as700 km. Ilheus is located on the Atlantic coast, suggesting an association with a typical position of coldfronts when they leave the continent. On the other hand, Irece (IRE) and Lencois (LEN) show a steepdecay of correlation, probably because of lower correlation with PC2 and local topographic conditions(they are at the northernmost and southernmost extremes of a mountainous region, the ChapadaDiamantina, in central Bahia State). For the other stations, correlation decays to r=0.6 withinL=350–400 km. A functional relationship is suggested r(L), as follows:

� General (except ILH, LEN, IRE): r(L)= (L/60)−0.264, 100�L�600 km.� Ilheus (ILH): r(L)= (L/l62)−0.326, 350�L�700 km.

Figure 7 illustrates a correlation for stations associated with the eastern region during FMA and ASOperiods (group G1). It is found that the correlation between coastal and near-coastal stations decayssomewhat steadily to r=0.6 within 400 km, although one pair (REC/SSA) shows r=0.63 for L=700km. Within the inland region, and along transects orientated transversally to the coast, correlation isusually higher than 0.8, but falls steeply to r=0.6 for L=400 km. The first behaviour might beassociated with latitudinal variation of typical meteorological conditions along the coastal region, whilethe last case suggests penetration of the trades’ influence on cloudiness up to circa 400 km from the coast.

Concerning correlation with western region, Sao Luiz (SLZ) shows high values of r up to L=1100 kmduring the ASO period, denoting a widespread homogeneous influence of the trades circulation along thenortheast coast. However, a high valued negative correlation with the inland western region does notshow evidence of a general rule.

The group G3 influenced by the ITCZ during FMA is not analysed because of the reduced number ofstations. Nevertheless, Figure 3 strongly suggests high-valued correlation along the northern coastalregion west from 38°W.

Paraıba State data (at about 7°S on eastern region, oriented west–east along 500 km) providedinformation for distances lower than 150 km. The state is classified as group G1 or G2 during FMA andASO, depending on distance to the coast. An inspection of correlation matrices for 5-day means showstypical values of 0.85 and higher for distances of 100 km, being even higher than 0.90 for about 70 km.At the daily scale, correlation falls to 0.8 within 70 km, and 0.6–0.7 within 100 km.



Figure 7. Correlation coefficients as a function of distance. Trade winds influence over the eastern northeast (G1 in FMA) extendsto the northern coast in ASO

6. CONCLUSIONS

The possibility of linear interpolation of solar irradiation in northeast Brazil using the standardclimatological network data was analysed. The criterion for acceptable linear regression was that itaccounted for at least 50% of the local variance (correlation coefficient r�0.7). As expected, it was foundthat fluctuations of daily values do not allow the use of network data at this time scale, as correlationcoefficients decrease at scales shorter than the distance between stations (typically 150–200 km, followinga synoptic criterion). Only a few situations generate extended homogeneity suitable for interpolations atthe scale of 1 day (for instance, when ITCZ is located along northern coastal region during FMA).

Means of 5 days provide a filter for small-scale, short-term fluctuations. Principal component analysismakes evident the presence and extent of the influence of larger scale systems which are responsible forcloudiness fluctuations, suggesting an interesting application of solar irradiation time series as acomplement for monitoring and diagnosis of climate. An example is the evidence of connections betweenwestern (pre-Amazonian) and eastern (trades-influenced) regions. Basically, three factors have aninfluence on cloudiness, and hence, on solar irradiation: atmospheric flow of Atlantic trades on theeastern region, and cold fronts extending their influence over the southern region (both acting duringFMA, as well as during ASO) and ITCZ over the northern region (during FMA). A central, semiaridregion may be influenced by these factors during FMA (thus, varying its extension year-to-year), and haveits own behaviour during ASO. The period chosen for analysis was 1975. It might be argued thatconclusions are limited to that year, but they clearly agree with the structure of regional climate.

It has been found that spatial linear interpolation using sequences of 5-day means may beaccomplished, usually for distances of up to 400 km (even 600 km, and more in some cases) withinhomogeneous regions. These time series are highly important for monitoring potential evapotranspirationfor agricultural purposes, and certainly provide better information than monthly means. Standardnetworks contain solarimeters, which do not perform automatic integration of daily irradiation.Considering the cost of such an infrastructure, it is seen that a solarimetric network could have ageographical density lower than that of the synoptic one.



Daily irradiation at scales between 10 and 100 km are important for other uses, like solar energyengineering. Results from the solarimetric network in Paraiba State show that a high density of stationswould be required. Given cost limitations, this time and scale behaviour might be better monitored bysatellite methods with acceptable errors (Pereira et al., 1996; Ceballos and Moura, 1997; Ceballos et al.,1998). At the same time, daily data produced in this way can be the source for assessing 5-day or weeklymeans with greater spatial detail. This effort has recently been undertaken. For instance, satelliteassessments for Brazil are produced by CPTEC/INPE since 2000, and released through the Internet (seehttp://www.cptec.inpe.br, ‘Satellite Products’).

ACKNOWLEDGEMENTS

This work was partially supported by CNPq/Brazil. Solar irradiation data were furnished by theDepartment of Atmospheric Sciences/UFPB. The authors were formerly part of this Department.

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Correlation fields for solar radiation in northeast Brazil