ORIGINAL PAPER

Multivariate geostatistical methods of mean annualand seasonal rainfall in southwest Saudi Arabia

Ali M. Subyani & Abdulrahman M. Al-Dakheel

Received: 21 July 2008 /Accepted: 30 September 2008 / Published online: 20 December 2008# Saudi Society for Geosciences 2008

Abstract A multivariate geostatistics (cokriging) is usedfor regional analysis and prediction of rainfall throughoutthe southwest region of Saudi Arabia. Elevation isintruded as a covariant factor in order to bring topograph-ic influences into the methodology. Sixty-three represen-tative weather stations are selected for a 21-year periodcovering different microclimate conditions. Results showthat on an annual basis, there is no significant changeusing elevation. On the seasonal basis, the cokrigingmethod gave more information about rainfall occurrencevalues, its accuracy related to the degree of correlationbetween elevation and rainfall by season. The predictionof spring and winter rainfall was improved owing to theimportance of orographic processes, while the summerseason was not affected within its monsoon climatology.In addition, fall season shows inverse and weak correla-tion of elevation with rainfall. Cross-validation andcokriging variances are also used for more accuracy ofrainfall regional estimation. Moreover, even though thecorrelation is not significant, the isohyet values ofcokriged estimates provided more information on rainfallchanges with elevation. Finally, adding more secondaryvariables in addition to elevation could improve theaccuracy of cokriging estimates.

Keywords Multivariate geostatistics . Cokriging . Rainfall .

Elevation . Saudi Arabia

Introduction

It has been stated in many studies that the high elevationreceives more rainfall than low elevation on the basis ofannual data (Chua and Bras 1982; Dingman et al. 1988;Hevesi et al. 1992; Johnson and Hanson 1995). However,the network of the rainfall measuring stations in thesouthwest is sparse and available data are insufficient tocharacterize the highly variable spatial distribution ofrainfall (Alehaideb 1985; Subyani 2004). The generalcharacteristics of rainfall and kriging estimates on the basisof annual data shows that the maximum amount of rainfalldoes not always occur at high elevations; in addition, itshows a little increase in its variance due to the complexityof terrain. Other factors such as the distance from thesource of moisture and seasonality are also important. Oneof the advantages of geostatistics is to use additionalinformation to improve rainfall estimations.

During recent years, cokriging has been used to bringtopographic influences into the calculations (Aboufirassiand Mariño 1984; Phillips et al. 1992; Hevesi et al. 1992)The second step constructs contour maps for the primaryvariable of this section. The cokriging system will bedeveloped with the basic two conditions of unbiasednessand minimum error variance (Myers 1982; Ahmed andMarsily 1987). Cokriging is a multivariate geostatisticalmethod that is used to estimate the spatial correlation oftwo variables that are interdependent in a physical sense.It represents more accurately the expected local oro-graphic influence on rainfall. It is also used to reduceestimation variances when one of these variables isundersampled.

Arab J Geosci (2009) 2:19–27DOI 10.1007/s12517-008-0015-z

A. M. Subyani (*)Hydrogeology Department, King Abdulaziz University,P. O. Box 80206, Jeddah 21589, Saudi Arabiae-mail: asubyani@hotmail.com

A. M. Al-DakheelGeology Department, King Saud University,P. O. Box 2455, Riyadh 11451, Saudi Arabiae-mail: amadakheel@yahoo.com

The southwest region lies within the subtropical climatezone of Saudi Arabia and receives the highest amount ofrainfall in comparison to other regions, because it ismountainous with elevations reaching to over 2,000 mabove mean sea level. It is selected as the study area, whichlies between latitudes 17°:00′ N and 22°:00′ N andlongitudes 40°:00′ E and 43°:00′ E (Fig. 1).

The main purpose of this paper is to describe thecharacterization and modeling for the distribution of annualand seasonal rainfall by using elevation as a covariate tobring topographic influence into the calculations usingmultivariate geostatistics (cokriging) technique. This tech-nique can be used to improve estimation accuracy byreducing estimation variances.

Theory of cokriging

The first step in multivariate geostatistics is to establish asuitable model for cross continuity and dependencybetween two or more variables. This positive correlationbetween variables is called cross-regionalization or core-gionalization and it can be estimated by cross-covariance

and cross-variogram. These models are used to describe andinterpret the cross continuity and dependency between twoor more variables. As an example, let Zi xð Þ and Zj xð Þ betwo random variables. Hence, under the second-orderstationarity, the cross-variogram as:

gij hð Þ ¼ 12E Zi xþ hð Þ � Zi xð Þ½ � Zj xþ hð Þ � Zj xð Þ� �� �

:

ð1Þ

Of course, the direct and cross-variogram models shouldsatisfy the nonnegative-definite conditions (Christakos1992). The linear model of coregionalization, in terms ofvariogram, is defined as a linear combination as shown inEq. 1. However, cross-variogram model parameters wereselected with additional criteria of satisfying the Cauchy–Schwarz inequality as follows (Myers 1982) as:

gij hð Þ � gii hð Þgjj hð Þ� �1=2: ð2Þ

where γij (h) is the cross-variogram model, and γii (h) andγjj (h) are the direct-variogram models for primary andsecondary variables, respectively.

Ri ya dh

Jeddah

1 4

1 8

2 2

2 6

3 8 4 2 4 6 50 54 58

SAUDI ARAB I A

0 5 0 010 0 01 50 020 0 02 50 00 50 0 K m

STU D Y

A RE A

0 1 2 3 4

RE

DS

EA

0.0 100 Km

Taif

Bishah

Abha

YEMENJizan

Najran

Lith

Turabah

Kiyat

Qamah

Tathlith

Alaqiq

TI

HA

MA

H

Raingage Location

500 Elevation (m.a.s.l)

41 00 42 00 43 00 44 00

21 00

20 00

19 00

18 00

17 00

900

Riyadh

14

18

22

26

38 42 46 50 54 58

0 50010001500200025000 500 Km

STUDY

AREA

o '

o '

o '

o '

o '

o ' o ' o ' o '

100

500

900

1300

1700

2100

2500

Elevation (m.a.s.l)

Fig. 1 Sample location and to-pographic features of study area

20 Arab J Geosci (2009) 2:19–27

Consequently, Hevesi et al. (1992) proposed a graphicaltest of the positive-definite condition (PDC) with theCauchy–Schwarz inequality as:

PDC ¼ gii hð Þgjj hð Þ� �1=2: ð3ÞThe model is considered to be positive definite if the

absolute value of the cross-variogram for any distance issmaller than the corresponding absolute PDC value and theslope of cross-variogram model did not exceed the slope ofthe PDC curve.

Cokriging

The cokriging technique is a modification of the krigingtechnique. It is used to merge two or more randomvariables. Estimation of cokriging contains a primaryvariable of interest, which is undersampled, and one ormore secondary variables that are better sampled. Whenthe variable of interest is costly or undersampled, it isuseful to apply cokriging by using secondary variable(s),which is easily sampled at cheap cost. These secondaryvariables are usually cross-correlated with the primaryvariable. Cokriging is a useful technique used to improvethe interpolation of one important variable by usinganother variable; in addition, the cross-variogram modelsmay get smoother than variogram models and improve thepredictions (Journel and Huijbregts 1978).

Consider the coregionalization of the two stationaryrandom functions Z(x) and Y(x) that are correlated, and weare interested in estimating at a location x0 the value ofunknown Z(x0). From the cross-correlation structure, theestimation of Z(x0) is not only based on the primaryvariable Z:Z(xi),…,Z(xn) but also based on the secondaryvariable Y:Y(xk),…,Y(xm). In general, m≥n, so we can writethe following equations:

E Z xið Þ½ � ¼ mz

E Y xkð Þ½ � ¼ myð4Þ

where mz and my are the constant means of Z(xi) and Y(xk),respectively. That is Z(xi) and Y(xk) are (jointly) second-order stationary. The best linear unbiased estimate of Zvalue at any location x0 can be written as:

bZ x0ð Þ ¼Xni¼1

liZ xið Þ þXmk¼1

wkY xkð Þ ð5Þ

where Z(xi) are the measured values of the primary variableZ at Z(xi), i=1,…, n, and Y(xk) are the measured values ofthe secondary variable Y at Y(xk), k=1,…, m. λi and ωk arethe cokriging weights that should be determined. As in the

case of the kriging system, the cokriging estimator shouldsatisfy the two conditions of unbiasedness and theestimation variance minimization.

Unbiasedness Condition

From Eqs. 4 and 5, the unbiased condition for Z x0ð Þ can bewritten as:

E bZ x0ð Þh i

¼ EPni¼1

liZ xið Þ þ Pmk¼1

wkY xkð Þ� �

¼ Pni¼1

liE Z xið Þ½ � þ Pmk¼1

wkE Y xkð Þ½ �

¼ mzPni¼1

li þ myPmk¼1

wk ¼ mz :

ð6Þ

To guarantee the unbiased condition, the followingconstraints on the cokriging weights can be established as:

Pni¼1

li ¼ 1 andPmk¼1

wk ¼ 0: ð7Þ

Variance of Estimation

Minimizing the variance of the estimation error can bewritten as:

s2 ¼ E bZ x0ð Þ � Z x0ð Þh i 2

¼ Minimum: ð8Þ

That is, the variance minimization of the estimationerror, σ2, is subject to the two conditions in Eqs. 5 and 6and therefore can be achieved by the method of Lagrangemultipliers (μ1 and μ2). This leads to the followingcokriging system in the form of variogram and cross-variogram as:

Pnj¼1

ljgz xi � xj� þ Pm

k¼1wkgzy xi � xkð Þ þ m1 ¼ gz xi � x0ð Þ

Pni¼1

ligzy xi � xkð Þ þPml¼1

wlgy xl � xkð Þ þ m2 ¼ gzy xk � x0ð Þi ¼ 1; . . . ; nk ¼ 1; . . . ;m

ð9Þ

where:

γz (xi−xj) = variogram model between sample points ofthe primary ReV z(x) separated by a distance xi−xjγz (xj−x0) = variogram model between sample pointsof the primary ReV z(x) and unknown point x0separated by a distance xj−x0γy (xl−xk) = variogram model between sample points ofthe secondary ReV y(x) separated by a distance xl−xk

Arab J Geosci (2009) 2:19–27 21

γzy (xi−xk) = cross-variogram model between samplepoints of the primary ReV z(x) and secondary ReV y(x)separated by a distance xi−xkγzy (xk−x0) = cross-variogram model between samplepoints of the primary ReV z(x) and secondary ReV y(x)with unknown point x0 separated by a distance xk−x0

The system of Eq. 9 gives (n+m+2) linear equations in(n+m+2) unknowns (nλ, mω, μ1 and μ2). The variance ofthe estimation error is expressed by:

s2 ¼Xni¼1

ligz xi � x0ð Þ þXmk¼1

wkgzy xk � x0ð Þ þ m1 ð10Þ

In our case, rainfall characteristics in the southwest regionof Saudi Arabia are mainly attributed to the elevation factor.So the cokriging method will be taken into consideration todetect the rainfall–elevation relationship.

Preliminary statistical analysis

Sixty-three weather stations for average annual and season-al rainfall records, which were discussed and analyzed,have been selected in the southwest region of Saudi Arabia(Subyani 2004). These data are called the primary varia-bles. Elevation values, the secondary variables, wereprovided from sites at which the weather stations werelocated. These elevation values, given in meters above sealevel, are contoured in Fig. 1. The elevation data wereconsidered to be normally distributed.

Table 1 shows the results of the descriptive and cross-statistics of the annual and seasonal rainfall and elevationdata. It also indicates whether the respective correlationcoefficients (r) were statistically significant. For annualrainfall, the r value of 0.44 is significance. The correlationcoefficient in winter, r=0.34, was statistically significant,but it is low. However, the weather stations are located atsites far away from the source of the moisture (Mediterra-nean), and the Red Sea effect is weak in the south and

strong in the north at this time of year. In spring, thecorrelation coefficient was strong, r=0.77, and it was thehighest among all seasons. This suggests that the springrainfall increases positively with elevation (orographic),and it is the only season that showed a strong relationshipbetween rainfall and elevation. In summer, the correlationcoefficient was not statistically significant, r=0.15, and therainfall was mainly related to monsoons. In fall, thecorrelation coefficient was negative and not statisticallysignificant, r=−0.2, which may indicate an inverse corre-lation between these two variables. In addition, if not underthe monsoons, the region receives the least amount ofrainfall during fall season.

Variogram for elevation

The experimental direct variogram was computed forweather station elevations for the spherical model usingGSLIB program (Deutsch and Journel 1992). Maintainingthe assumptions of stationarity and isotropism weredesirable to simplify model fitting during cross-validation.The root mean square error (RMSE) value is close to one,and mean estimation error (MEE) value is close to zero,which indicates an excellent model in terms of estimationaccuracy (Cressie 1993; Clark and Harper 2000). Thesensitivity of the model cross-validation results for thevariogram model parameters indicated that the model fittingwas important for distances between 0 and approximately115 km. For distances greater than 115 km, the results werenot sensitive (Fig. 2). However, an isotropic sphericalmodel was selected based on the cross-validation results asthe best representation of the spatial structure of theelevation as shown in Fig. 1. This model was defined withno nugget, as sill was equal to the sample variance of922,600 m2 and the range was 115 km.

Table 1 Descriptive and cross-statistics of rainfall and elevation

Variable Mean Standarddeviation

Correlationcoefficient

Significance

α=0.05

Elevation 1,228 910Annual 5.18 0.78 0.44 YesWinter 3.7 0.62 0.34 YesSpring 4.2 0.92 0.77 YesSummer 3.4 1.06 0.15 NoFall 3.1 1.02 −0.2 No

Distance (Km)

0

200

400

600

800

1000

1200

0 100 200 300 400

γ (h

) fo

r E

leva

tion

(X10

00)

m2

Sample SVSpher. Model

Range = 115 KmSill = 922600 m2

Fig. 2 Experimental and fitted variogram model for elevation

22 Arab J Geosci (2009) 2:19–27

Cross-variogram models for rainfall and elevation

Figures 3, 4, 5, 6, and 7 show the experimental and cross-variogram models for natural logs of annual and seasonalrainfall with elevation. The number of sample pairs in eachcase was the same as in the direct variogram. These figuresalso show the PDC curve computed to check the positivedefinite condition. In addition, cross-validation statisticswere used to find the suitable cross-variogram model.

For the annual case, the increase in the values of samplecross-variogram for distances from 0 to 130 km indicated apositive spatial cross-correlation between annual rainfalland elevation. For distances greater than 130 km, the cross-variogram sill is approximately as the same as the samplecovariance (Fig. 3). A negative cross-correlation wasobserved for distances between 130 and 220 km. Thismay be due to the weakness of the correlation, r=0.44, andthe fact that the annual rainfall contains different temporalrainfall mechanisms.

In winter (Fig. 4), the positive sample cross-variogrambetween winter rainfall and elevation increased for dis-

tances from 0 to approximately 50 km. The cross-correlation beyond 50 km fluctuated with distance. Thislow range of cross-variogram is due to the weak correlationbetween winter rainfall and elevation (r=0.34).

In spring, the positive sample cross-variogram betweenspring rainfall and elevation increased for distances from 0to approximately 140 km (Fig. 5). This wide range is due tothe strong correlation between spring rainfall and elevation(r=0.77). In other words, such a strong correlationconfirmed that the spring rainfall was mainly due to theorographic factors.

In summer, the low value of correlation between summerrainfall and elevation, r=0.15, made the sample cross-variogram start with negative slope for distances from 0 toapproximately 40 km and increase to the distance of 132 km(Fig. 6). The cross-variogram may take negative values;whereas a direct variogram is always positive (Journel andHuijbregts 1978). This negative value indicates that thesummer rainfall increase corresponds to a decrease inelevation. This inconsistency in relation between summerrainfall and elevation is mainly due to monsoonal rainfall.

Distance (Km)

0

100

200

300

400

500

600

700

800

0 100 200 300 400

Range = 130 KmSill = 380

PDC Curve

Sample Cross-SV

Cross-Sph. Model

γij

(h)

Fig. 3 Sample cross-variogram and fitted model for rainfall andelevation for annual data

Distance (Km)

0

200

400

600

800

0 100 200 300 400

Range = 50 KmSill = 218

Sample Cross-SV

PDC Curve

Cross-Sph. Model

γij

(h)

Fig. 4 Sample cross-variogram and fitted model for rainfall andelevation for winter data

Distance (Km)

0

200

400

600

800

1000

1200

0 100 200 300 400

Range = 140 KmSill = 786

Sample Cross-SV

PDC Curve

Cross Sph. Model

γij

(h)

Fig. 5 Sample cross-variogram and fitted model for rainfall andelevation for spring data

Distance (Km)

0

200

400

600

800

1000

1200

0 100 200 300 400

Range = 30 KmSill = 310

Sample Cross-SV

Cross Exp. ModelPDC Curveγ

ij (h

)

Fig. 6 Sample cross-variogram and fitted model for rainfall andelevation for summer data

Arab J Geosci (2009) 2:19–27 23

In fall, the negative correlation, r=−0.2, between fallrainfall and elevation (Fig. 7), showed a high amount ofscatter sample cross-variogram, and no obvious theoreticalcross-variogram models were evident. However, thesenegative values indicated that a positive increase in fallrainfall corresponded, on the average, to a decrease inelevation, and this was due to the fact that the fall season isa transitional period between the monsoon rainfall insummer and the Mediterranean rainfall in winter.

Table 2 lists the parameters of the cross-variogrammodels and the cross-validation statistics for all annualand seasonal cases. The range of influence from cross-variogram of annual data is increased more slightly than therange of annual variogram, but the other model character-istics are still the same. In winter and spring, the cross-variogram models show more consistency in their structuralparameters than their variogram models due to the effect ofelevation. For example, exponential variogram models withnugget become more consistent in their shapes withspherical cross-variogram models and absence of thenugget effect and that the range of influence is increased.Also, the MEE for the seasonal cross-variogram models iscloser to zero compared to the MEE of the seasonal direct-variogram models, and the RMSE for the seasonal cross-variogram models is closer to 1.0 compared to RMSE forthe seasonal direct-variogram models. However, summercross-variogram becomes less consistent with lower range

and presence of the nugget effect due to the insignificanteffect of elevation in this season.

The PDC curves were computed for the annual andseasonal cross-variogram models to check the positivedefinite conditions. With the exception of the fall season,most of the cases that showed the plotted PDC curveprovided a close fit to the sample cross-variogram for thesmall distances, and the absolute experimental values forthe cross-variogram were smaller than the absolute valuesof the PDC curve for most of the distances computed.However, the PDC curve may give a relative indication forthe degree of correlation. For example, in spring, the PDCcurve is closer to the cross-variogram model curve, but inother seasons, it moves farther from the cross-variogrammodel curve depending on the degree of correlation asshown in Figs. 3, 4, 5, 6, and 7.

Cokriging for rainfall and elevation

The cokriging interpolation technique was applied to bothannual and seasonal data estimate rainfall and its variances.With the exception of the fall season, the strength of theinfluence of the elevation in the estimation accuracydepends on the degree of correlation between rainfall andelevation.

For annual rainfall (Fig. 8), cokriging estimates showthat the isohyets values increased gradually with elevation.Cokriging estimation variances indicated similar trends inestimation accuracy of the kriging throughout the studyarea. Moreover, estimation variances (Fig. 9) were reducedin average in the east.

In winter, cokriging estimates show very little effect of thetopographic factor in the mountain area, and the cokrigingcontours follow these topographic changes as shown inFig. 10. This may be due to existence of no significanteffect on elevation factor (i.e., r=0.34). The cokrigingestimation variances (Fig. 11) show high variances in thenorth part of mountain due to the Mediterranean effect inwinter and low variance in the east part of the study area.

In spring, cokriging estimates gave more detailedinformation as shown in Fig. 12, and the elevation factor

Table 2 Structures and cross-validation statistics for fitted cross-variogram models

Variable Model Nugget Sill Number Range (km) MEE RMSE

Annual Spherical 0.0 380 62 130 −0.02 0.85Winter – 0.0 218 62 50 −0.035 1.1Spring – 0.0 786 62 140 −0.07 0.81Summer Exponential 0.0 310 62 30 −0.02 1.03Fall NA NA NA 62 NA NA NA

NA not applicable

Distance (Km)

-250

-150

-50

50

150

250

350

450

0 50 100 150 200 250 300 350

γij

(h)

Sample Cross-SV

Fig. 7 Sample cross-variogram for rainfall and elevation for fall data

24 Arab J Geosci (2009) 2:19–27

0 100000 200000 300000 4000000

100000

200000

300000

400000

500000

600000

0 1 2 3 4

RE

DS

EA

0.0 100 Km

NTaif

Bishah

Abha

YEMENJizan

Najran

Lith

5000

Fig. 9 Cokriging estimation variances for annual rainfall (mm2)

0 100000 200000 300000 4000000

100000

200000

300000

400000

500000

600000

0 1 2 3 4

RE

DS

EA

0.0 100 Km

NTaif

Bishah

Abha

YEMENJizan

Najran

Lith

500

Fig. 11 Cokriging estimation variances of winter rainfall (mm2)

0 100000 200000 300000 4000000

100000

200000

300000

400000

500000

600000

0 1 2 3 4

RE

DS

EA

0.0 100 Km

NTaif

Bishah

Abha

YEMENJizan

Najran

Lith

60

Fig. 10 Isohyetal map of cokriging estimates of winter (mm)

0 100000 200000 300000 4000000

100000

200000

300000

400000

500000

600000

0 1 2 3 4

R E D

S E A

0.0 100 Km

NTaif

Bishah

Abha

YEMENJizan

Najran

Lith

Fig. 8 Isohyetal map of cokriging estimates for annual rainfall (mm)

Arab J Geosci (2009) 2:19–27 25

0 100000 200000 300000 4000000

100000

200000

300000

400000

500000

600000

0 1 2 3 4

RE

DS

EA

0.0 100 Km

NTaif

Bishah

Abha

YEMENJizan

Najran

Lith

1000

0

Fig. 15 Cokriging estimation variances of summer (mm2)

0 100000 200000 300000 4000000

100000

200000

300000

400000

500000

600000

0 1 2 3 4

RE

DS

EA

0.0 100 Km

NTaif

Bishah

Abha

YEMENJizan

Najran

Lith

10

Fig. 14 Isohyetal map of cokriging estimates of summer (mm)

0 100000 200000 300000 4000000

100000

200000

300000

400000

500000

600000

0 1 2 3 4

RE

DS

EA

0.0 100 Km

NTaif

Bishah

Abha

YEMENJizan

Najran

Lith

15000

Fig. 13 Cokriging estimation variances of spring (mm2)

0 100000 200000 300000 4000000

100000

200000

300000

400000

500000

600000

0 1 2 3 4

RE

DS

EA

0.0 100 Km

NTaif

Bishah

Abha

YEMENJizan

Najran

Lith

80

Fig. 12 Isohyetal map of cokriging estimates of spring (mm)

26 Arab J Geosci (2009) 2:19–27

was clear in the mountain area. Isohyetal lines in thecokriging map followed roughly the main features ofelevation. However, the estimation variances (Fig. 13) showthat there is no change in their values, which indicates thatthe spring rainfall is of orographic type.

In summer, there is no elevation influence which is notsignificant at this time of the year (Fig. 14). In addition, thecokriging variance estimation (Fig. 15) shows no change.

Generally, in all time cases, the cokriging method gave alittle more information about rainfall values than did theother methods, and its accuracy is related to the degree ofcorrelation between rainfall and elevation. Moreover, eventhough the correlation is not significant (winter andsummer), the isohyetal values of cokriged estimates providemore detail concerning rainfall values with elevationchanges. However, in the east and northeast areas, at lowelevations lacking information, there appears little reduc-tion in estimation variances.

Conclusions

A multivariate geostatistics method was developed to detecteffect of the elevation factor as a covariant variable addingto rainfall as a primary variable on the basis of annual andseasonal data. The major conclusions of this study can besummarized as follows:

& The rainfall–elevation cross-correlation revealed a spher-ical cross-variogrammodel for annual, winter, and springseasons, whereas the exponential cross-variogram modelwas fitted to summer season. For the fall season, noobvious theoretical variogram model was evident. Thecloser the PDC curve fit the cross-variogram modelcurve, the higher correlation between rainfall andelevation, which is in the case of spring season

& The cokriging method gave little more informationabout rainfall values and its accuracy was related to thedegree of correlation between rainfall and elevation.Moreover, even though the correlation is not significant,the isohyet values of cokriged estimates provided moreinformation on rainfall changes with elevation

& This study was global application of geostatistics.However, we proposed to study an area like southwestregion of Saudi Arabia separately according to homo-geneity in geographic features (i.e., Tihamah, Mountain,

and Plateau) and stationary in time scale (monthly orseasonal) within each defined region. This fact canreduce the uncertainty of the results

& Due to the complexity and type of rainfall formation ingeneral, adding more secondary variables such as temper-ature, distance from the moisture source, wind speed anddirection, and pressure, in addition to elevation, couldimprove the accuracy of cokriging estimates

Acknowledgment The authors express their appreciation to KingAbdulaziz University and Ministry of Water and Electricity in SaudiArabia for providing necessary facilities during the course of thisstudy. The comments of the reviewers are gratefully acknowledged.

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