COMPARISON OF GEOSTATISTICAL METHODS FOR ESTIMATING THE AREAL AVERAGE CLIMATOLOGICAL RAINFALL MEAN USING DATA ON PRECIPITATION AND TOPOGRAPHY

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1 INTERNATIONAL JOURNAL OF CLIMATOLOGY It. J. Climatol. 18: (1998) COMPARISON OF GEOSTATISTICAL METHODS FOR ESTIMATING THE AREAL AVERAGE CLIMATOLOGICAL RAINFALL MEAN USING DATA ON PRECIPITATION AND TOPOGRAPHY EULOGIO PARDO-IGÚZQUIZA* TAMSAT, Departmet of Meteorology, Uiersity of Readig, Readig RG6 6BB, Berkshire, Eglad Receied 7 August 1997 Reised 22 February 1998 Accepted 28 February 1998 ABSTRACT The results of estimatig the areal average climatological raifall mea i the Guadalhorce river basi i souther Spai are preseted i this paper. The classical Thiesse method ad three differet geostatistical approaches (ordiary krigig, cokrigig ad krigig with a exteral drift) have bee used as estimators ad their results are compared ad discussed. The first two methods use oly raifall iformatio, while cokrigig ad krigig with a exteral drift use both precipitatio data ad orographic iformatio (easily accessible from topographic maps). I the case study preseted, krigig with a exteral drift seems to give the most coheret results i accordace with cross-validatio statistics. If there is a correlatio betwee climatological raifall mea ad altitude, it seems logical that the iclusio of topographic iformatio should improve the estimates. Krigig with a exteral drift has the advatage of requirig a less demadig variogram aalysis tha cokrigig Royal Meteorological Society. KEY WORDS: techiques; Thiesse method; ordiary krigig; cokrigig; krigig with a exteral drift; raifall; areal average; Guadalhorce river basi; Spai; geostatistical methods 1. INTRODUCTION Most of the water received by a river basi occurs as raifall evets over the basi. Differet quatities are of iterest, from total raifall i a sigle storm to log-term aggregates over time. Amog the latter, mea aual raifall is used widely i differet applicatios, ad varies from year-to-year. The mea of the aual raifall meas for all the years available i the historical record is kow as the climatological mea ad gives the expected aual raifall. However the climatological mea is usually kow oly at the locatios of raigauge statios; to estimate it at usampled locatios over the whole basi (ad the expected total water iput by raifall), a procedure is required to calculate the average spatial climatological raifall mea. To calculate average spatial statistics i climatology, the Thiesse method (Thiesse, 1911) was the stadard for about half a cetury util the developmet of geostatistics i the middle 1960s (Mathero, 1963). The geostatistical method of spatial estimatio, kow geerically as krigig, takes ito accout the spatial correlatio betwee experimetal data through the variogram fuctio. Krigig improves o the Thiesse method i two ways: (i) More precise estimates due to usig the iformatio from surroudig samples. (ii) It miimises the estimatio variace which i tur provide a measure of the ucertaity of the estimates. * Correspodece to. Departmet of Meteorology, Uiversity of Readig, Readig RG6 6BB, UK; tel.: ; fax: ; E.Pardo@readig.ac.uk Cotract grat sposor: Europea Commuity Traiig Fellowship (part of the Traiig ad Mobility of Researches (TMR) programme) CCC /98/ $ Royal Meteorological Society

2 1032 E. PARDO-IGÚZQUIZA Thus krigig has curretly become the stadard method of estimatig average raifall values at sites with o records. Early applicatios of krigig i raifall estimatio were described by Delhomme ad Delfier (1973). Sice the there have bee may applicatios ad differet kids of krigig have also bee developed. For example, multivariate techiques to deal with at least two variables that are coregioalized. This is the case where the raifall patter i a area is iflueced by physiography, i.e. orographic raifall. The umber of raigauges is usually small ad their spatial distributio sparse while, topographical iformatio is widely available. The, if the raifall is coregioalized with altitude, the topographical data may be used to improve the precisio of the raifall estimates. The aim of this paper is to show the results of the applicatio of differet geostatistical methods to estimate the areal average climatological mea usig precipitatio data ad topography. A review of the geostatistical techiques ad the results of their applicatio to the Guadalhorce river basi i the south of Spai are preseted ad discussed i the followig sectios. 2. THEORY REVIEW ON GEOSTATISTICAL METHODS Give a river basi of area A (Figure 1(a)), the expected aual iput Q by raifall is give by: Q= A p(x)dx (1) Figure 1. River basi (a) ad discretized river basi (b)

3 GEOSTATISTICAL METHODS AND CLIMATOLOGICAL RAINFALL 1033 where p(x) is the value of the climatological raifall mea at spatial locatio x (x deotig the coordiates of a poit i the plae). p(x) is oly kow at those poits where raigauge statios are located, ad the it must be estimated at the o- sampled poits. I practice, the itegral (1) is evaluated usig the discrete sum approximatio: N Q= p B (x i ) (2) where the area A of the river basi has bee discretized i N cells of equal area B (Figure 1(b)); ad p B (x i ) is the areal average climatological raifall mea over cell B with cetroid x i. It is clear that p B (x i ) is a areal average estimate (also referred to as a block estimate): p B (x i )= 1 B B(x i ) p(x) dx where B is called the support of the estimate, beig: N A= B(x i ) (4) the whole regio, ad B(x i ) deotes the support B at the locatio x i. Block krigig provides estimates of Equatio (3) from the experimetal data with puctual support (for example data measured i raigauge statios): pˆb(x i )= j p(x j ) (5) with p(x j ) beig the climatological mea at locatio x j. I equatio (5) oly raifall data (the climatological mea) are used. There are differet possibilities for performig such a estimatio usig geostatistics; there are also differet ways of icorporatig the topographical iformatio for the purpose of improvig estimates pˆb(x i ), ad the Q. The methods of estimatio are briefly explaied below Thiesse method The Thiesse polygos method is equivalet to the earest eighbor method; all the poits iside the polygo have the same value which is equal to the closest sample value; p(x) i Equatio (3) is the estimated as the earest experimetal poit. I practice, this is solved by the well-kow method of drawig Vorooï polygos aroud each experimetal data. However, a easier procedure has bee adopted here as it is more suitable for computatio. The support B is subdivided ito smaller square uits ad each small uit is estimated by the earest eighbor method. If each subuit is sufficietly small this procedure gives a good approximatio to the classical triagulatio method Ordiary krigig (OK) I ordiary krigig the estimate pˆb(x o ) is give by equatio (5) where the weights i are obtaied by solvig to the ordiary krigig system (Jourel ad Huijbregts, 1978): j p (x i, x j )+= p(x i, B(x o )) j =1 Ad the estimatio variace is give by: ;,..., (6) (3)

4 1034 E. PARDO-IGÚZQUIZA ˆ 2= i p(x i, B(x o ))+ (7) where, Lagrage multiplier; p (x i, x j ), variogram fuctio betwee poits x i ad x j ; p(x i, B(x o )), mea variogram fuctio betwee poit x i ad support B with cetroid x o ; ad p, deotes variogram of the climatological raifall mea. The two previous methods use oly raifall data (the climatological mea). Iformatio o altitude may be icluded i the estimatio procedure i differet ways, two of them beig cokrigig ad krigig with a exteral drift Cokrigig (CoK) The CoK estimate of p B (x o ) is give by: pˆb(x o )= m i p(x i )+ j y(x j ) (8) where {y(x j );,...,m} are experimetal data o altitude. The weights i, j are obtaied as solutio of the CoK system: m m j p (x i, x j )+ k py (x i, x k )+ 1 = p(x i, B(x o )) k=1 m j py (x k, x j )+ t Y (x k, x t )+ 2 = py (x k, B(x o )) t=1 j =1 j =0,..., k=1,...,m (9) with estimatio variace: ˆ 2= m i p(x i, B(x o ))+ k py (x k, B(x o ))+ 1 (10) k=1 where: 1, 2, Lagrage parameters; p (h), the variogram fuctio of the climatological mea; Y (h), variogram of altitude; py (h), cross-variogram of climatological mea ad altitude. The bar over the variogram deotes the mea variogram betwee a poit ad a support B Krigig with a exteral drift (KED) I KED the mathematical expectatio of the climatological raifall mea radom fuctio P(x) is expressed as a liear fuctio of altitude Y(x): E[P(x)]=a 1 +a 2 Y(x) (11) The estimate pˆb(x o ) has the form give i Equatio (4), with the weights i obtaied as solutios of the KED system (Ahmed ad De Marsily, 1987; Hudso ad Wackeragel, 1994):

5 GEOSTATISTICAL METHODS AND CLIMATOLOGICAL RAINFALL 1035 j p (x i, x j ) y(x i )= (x i, B(x o )) j y(x j )=y(x o ) with estimatio variace:,..., (12) ˆ 2= i p(x i, B(x o )) y(x o ) (13) where all the otatio has bee defied previously. It is clear from Equatio (12) how the values of the coefficiets a 1 ad a 2 are ot eeded; it is ot ecessary to kow them i order to apply KED. I order to apply KED it is ecessary to kow the value of the mea altitude of the block cetered at each experimetal locatio, as well the mea altitude of the block to be estimated. While CoK requires the variograms of climatological mea ad raifall as well as the crossvariogram, KED ad OK require oly the variogram of the climatological mea. 3. CASE STUDY The study area is the Guadalhorce river basi, located betwee logitude 4 ad 6 west ad betwee latitude 36 ad 38 orth (Figure 2). The area of the basi is discretized to squares of 4 km side, ad is 2864 Km 2. There are 51 raigauge statios located withi or aroud the river basi, as depicted i Figure 3; i the figure the coordiates are give i km with a arbitrary origi. The climatological mea (mea aual raifall calculated from historical records coverig more tha 20 years i all cases), ad the altitude are kow for every raigauge statio. Figure 4 shows the climatological mea plotted agaist altitude; the liear correlatio coefficiet is O average, the climatological mea icreases with altitude statistically. The physical explaatio of this correlatio coefficiet is complex because there are differet meteorological factors that ifluece the raifall patter i this basi, amog them the wid patters ad the effect of the Mediterraea Sea i the south of the basi (Figure 2). Figure 5 shows a cotour map of the mea altitude over blocks of 4 4 km calculated from a topographical map by averagig the poit altitudes of a regular grid of poits over each block. The highest altitudes are located at the wester ad easter borders of the river basi. The expected aual resources iput i the basi is obtaied by the evaluatio of pˆb(x i ) for N=179 blocks of 4 4 Km 2 i Equatio (2). The cotour maps of the estimates usig Thiesse polygos, OK, CoK ad KED are show i Figure 6(a d), respectively. I the figures, the climatological raifall mea is give i meters ad the isohyet iterval is 50 mm (0.05 m). The average areal climatological mea over the whole river basi is 586.9, 591.1, ad mm for the four methods, respectively. Thus the estimates of the expected aual iput of water resources are 1681, 1693, 1738 ad 1812 hm 3 obtaied for Thiesse, OK, CoK ad KDE, respectively. I decreasig order of the magitude of their estimates the estimators are KED, CoK, OK ad Thiesse; KED gives 4% more aual iput resources tha CoK, 7% more tha OK ad 8% more tha the Thiesse method; CoK gives 3% more iput resources tha OK ad the Thiesse method. The differece betwee OK ad the Thiesse method is less tha 1%. Figure 6 shows a great similarity betwee the Thiesse estimates (Figure 6(a)) ad OK estimates (Figure 6(d)). The isohyet map of KED (Figure

6 1036 E. PARDO-IGÚZQUIZA Figure 2. Locatio of Guadalhorce river basi i the south of Spai 6(d)) has greater similarity with the topography (Figure 5) ad the CoK isohyet map also resembles the topography but to a lesser extet tha KED. For the applicatio of CoK ad KED, the values used have bee 309 of mea altitudes over blocks 4 4 km which correspods to a grid of blocks over the area represeted i Figure 3, after discoutig the blocks over the sea with altitude 0. The experimetal variograms of climatological mea ad altitude ad the experimetal cross-variogram betwee climatological mea ad altitude, as well as the fitted models used i the differet krigig systems, are show i Figure 7(a c). The correspodet models are: p (h)= Sph(h) 40 (14) Y (h)= Sph(h) 40 (15) py (h)= Sph(h) 40 (16) where c Sph(h) a deotes a spherical variogram with sill c ad rage a. These variograms have bee estimated by fittig a model to the oparametric variogram estimates show i Figure 7. The variograms have bee modelled by the same spherical structure i order to apply the liear model of coregioalizatio i CoK (Jourel ad Huijbregts, 1978). The models fitted to the oparametric estimates (Equatios 14 16) are i agreemet well with the parameters estimated by maximum likelihood (Pardo-Igúzquiza, 1997), the results of which are show i Table I. From the results i Table I several cosideratios may be extracted: The Akaike iformatio criterio (Akaike, 1974) suggests that the models without ugget effect for both regioalized variables, climatological mea ad topography are more plausible; ad, eve whe the ugget effect is estimated, its value is very small i relatio to the variace.

7 GEOSTATISTICAL METHODS AND CLIMATOLOGICAL RAINFALL 1037 Figure 3. Guadalhorce river basi ad locatio of the 51 raigauge statios Figure 4. Climatological raifall mea versus altitude

8 1038 E. PARDO-IGÚZQUIZA Figure 5. Cartography of the mea altitude over block (4 4 Km) support The maximum likelihood estimates for the rage are 39.3 km with stadard error of 5.33 km, ad 34.7 km with stadard error of 4.5 km for climatological raifall ad altitude, respectively. The maximum likelihood estimates for the variace are m 2 with stadard error of m 2 ad km 2 with stadard error of km 2. The model Equatio (14) for the climatological raifall variogram is coheret with the model estimated by maximum likelihood. For topography the model suggested by maximum likelihood has a smaller rage but the model give i Equatio (15) is still withi the 95% cofidece iterval for the rage estimated by maximum likelihood. The sample size (51 locatios i Figure 3) is large from a hydrological poit of view; to have 51 raigauges iside a 80 78=6240 km 2 (area of the rectagle of Figure 3) is a high desity etwork. From a statistical poit of view, it is still a adequate umber of data; Pardo-Igúzquiza (1997) shows how with more tha 30 data, the maximum likelihood estimator gives good results for the iferece of spatial covariace parameters. What is more importat, a stadard error is attached to the estimated parameters. The ucertaity of the climatological raifall mea estimates are assessed by the estimatio variace, which is ot available for the Thiesse method. A represetatio of the square root of the estimatio variace (the S.D. of the estimatio error) kow as krigig error, or stadard error, has bee represeted i Figure 8(a c) for OK, CoK ad KED, respectively. The stadard error of the global average (the average for the whole river basi) climatological mea is 79.86mm, mm ad mm for OK, CoK ad KED, respectively (see Appedix 1). The stadard errors for the global average are almost the same, but whe we cosider the local stadard errors, as ca be see from Figure 8(a c), the differeces are oticeable. KED gives a larger krigig error tha CoK ad OK, while OK gives a krigig error larger tha CoK. The latter case is perfectly logical because CoK uses additioal iformatio o altitude as well as the variogram of altitude, ad the cross- variogram betwee climatological mea ad altitude; a decrease of the ucertaity i the estimates is to be expected.

9 GEOSTATISTICAL METHODS AND CLIMATOLOGICAL RAINFALL 1039 Figure 6. Cartography of the areal average climatological raifall mea estimated by: (a) Thiesse; (b) OK; (c) CoK; ad (d) KED

10 1040 E. PARDO-IGÚZQUIZA Figure 7. Experimetal (*) ad model (cotiuous lie) for: (a) climatological mea; (b) altitude; ad (c) cross-variogram betwee climatological mea ad altitude

11 GEOSTATISTICAL METHODS AND CLIMATOLOGICAL RAINFALL 1041 Table I. Variogram parameters estimated by maximum likelihood Regioalized variable Rage S.E. Nugget S.E. Variace S.E. NLLF effect AIC Raifall NA NA Raifall Altitude NA NA Altitude S.E.: stadard error; NLLF: egative loglikelihood fuctio; NA: ot applicable. Rage i km; variace ad ugget effect i m 2 ad km 2 for raifall ad ugget effect, respectively. AIC: Akaike iformatio parameter; AIC= log L+2k=NLLF+2k, where L: likelihood fuctio, NLLF: egative loglikelihood fuctio ad k is the umber of parameters fitted to the model (equal to 3 (mea, rage, variace) ad 4 whe the ugget effect is fitted (mea, rage, variace, ugget effect). I the KED system give i Equatio (12), the variogram of the climatological mea p (h) is used, though the variogram that should be used is the variogram of the ukow residuals r(x i )=p(x i ) a 1 a 2 y(x i ). This approximatio does ot modify the estimates of p(x), but overestimates the variace of the estimatio error (Ahmed ad De Marsily, 1987). 4. CROSS-VALIDATION RESULTS Cross-validatio is frequetly used to validate or compare variogram models. The procedure, as is well-kow, cosists of the estimatio of each experimetal value usig the remaider of the experimetal iformatio (i.e. excludig itself). This makes the calculatio of the true error i the estimatio procedure possible: i =pˆ (x i ) p(x i ) (17) where: is the error; pˆ (x) is the estimated value ad p(x) is the true value. Thus, repeatig this estimatio for the umber of experimetal data, the followig cross-validatio statistics may be calculated. Mea error: ME= 1 i mea squared error MSE= 1 i 2 (19) mea stadardized square error MSSE= 1 i 2 ˆ i2 (20) It is assumed that if the variogram model is correct, ME should be almost zero, MSE a small value ad MSSE close to 1. It should be oted that ME is useless, as ME=0 is imposed i the krigig system by the ubiased coditios, o matter which variogram model is used. The value of cross-validatio has bee criticized by some authors (Jourel ad Rossi, 1989). If cross-validatio works efficietly, it ca be used for estimatio purposes. This is the miimum iterpolatio error method of Lebel ad Basti (1985), which etails choosig the values of the variogram parameters that give miimum MSE ad MSSE equal to 1. Pardo-Igúzquiza (1998) shows that whe five variogram estimators are compared, the oe usig cross-validatio (the previous method) gives the worst results. Therefore cross-validatio must be used with cautio, small differeces i the cross-validatio results may be give just by samplig variability. The cross-validatio statistics for OK, CoK ad KED with the 51 experimetal data of climatological (18)

12 1042 E. PARDO-IGÚZQUIZA Figure 8. Cartography of the krigig error for: (a) OK; (b) CoK; ad (c) KED

13 GEOSTATISTICAL METHODS AND CLIMATOLOGICAL RAINFALL 1043 Table II. Cross-validatio statistics ME MSE MSSE OK CoK KED mea are show i Table II. The KED approach gives the most cogruet results, with the smallest MSE ad the MSSE closest to 1. The values of MSSE for OK ad CoK suggest that the sill of the variograms was uderestimated, but that the estimatio variace obtaied by KED was ot. If the distributio of the errors is Gaussia (which is geerally the case, at least approximately by the cetral limit theorem), 95, 68, 38 ad 19% cofidece itervals for the climatological raifall mea estimates may be costructed as [pˆ 2ˆ ], [pˆ ˆ ], [pˆ 0.5ˆ ] ad [pˆ 0.25ˆ ], where pˆ is the estimate ad ˆ is its krigig error. It ca the be calculated i which percetages from amog the previous radom itervals the true values are cotaied. Because there are 51 experimetal data, the expected umber of times that the radom itervals will cotai the true values are 48, 35, 19 ad 10 for the previous itervals, respectively. The results for OK, CoK ad KED are show i Table III. The best results are obtaied with KED. Other represetatios of the cross-validatio results are give i Figures 9 ad 10. I Figure 9(a d), the estimated values versus the true values of climatological mea are represeted for Thiesse, OK, CoK ad KED, respectively. The poits should be aroud the 45 lie. Agai the best results are obtaied for KED with a liear correlatio coefficiet of The liear correlatio coefficiet for Thiesse, OK ad CoK are 0.75, 0.74 ad 0.77, respectively. Figure 10(a d) show the errors give by equatio (17) versus the estimated values. The magitude of the errors should be idepedet of the magitude of the estimated values. All the methods yield the larger errors whe the magitude of the estimate is very high, but KED gives roughly the best results. 5. CONCLUSIONS Amog the geostatistical methods used i the estimatio of the areal average climatological raifall mea o a river basi i the south of Spai, ordiary krigig uses oly values o the raifall while cokrigig ad krigig with exteral drift use both iformatio o raifall ad o topography. Cokrigig has more requiremets tha krigig with a exteral drift, as two direct variogram models ad a crossvariogram model eed to be fitted istead of oly oe direct variogram model i krigig with a exteral drift ad ordiary krigig. Cosiderig the cross-validatio results, krigig with a exteral drift seems to give the most coheret results i the case study preseted; therefore, the estimates of the expected total resources iput over the Guadalhorce river basi give by krigig with a exteral drift ca be cosidered the most appropriate. Table III. Results from radom cofidece itervals [ẑ2ˆ ] (%) [ẑˆ ] (%) [ẑ0.56ˆ ] (%) [ẑ0.25ˆ ] (%) Theoretical 45 (95) 35 (68) 19 (38) 10 (19) OK 42 (82) 29 (56) 18 (35) 8 (15) CoK 41 (80) 24 (47) 15 (29) 11 (21) KED 47 (92) 32 (62) 20 (39) 11 (21)

14 1044 E. PARDO-IGÚZQUIZA Figure 9. Estimated value versus true value for: (a) Thiesse method; (b) OK; (c) CoK; ad (d) KED

15 GEOSTATISTICAL METHODS AND CLIMATOLOGICAL RAINFALL 1045 Figure 10. Error vs. estimated value for: (a) Thiesse method; (b) OK; (c) CoK; ad (d) KED

16 1046 E. PARDO-IGÚZQUIZA ACKNOWLEDGEMENTS The work of the author has bee supported by a Europea Commuity Traiig Fellowship i the Departmet of Meteorology (Readig Uiversity, UK) as part of the Traiig ad Mobility of Researches (TMR) programme. The author would like to thak the reviewers for providig very costructive commets. APPENDIX A. CALCULATION OF THE ESTIMATION VARIANCE OF THE GLOBAL MEAN FROM COMBINATION OF LOCAL ESTIMATION VARIANCES Figure A1 shows a area V composed of N elemetary uits of equal area u i : N V= u i (A1) Over the whole area V, there is defied a regioalized variable Z that is oly kow i experimetal locatios. There are observatios that are assumed as realizatios of radom variables {Z i, i= 1,...,}. Let Z V be the global mea that is give by Z V = 1 N Z V= 1 N N u Z i N where Z V = Z u i ad Z iu is the mea value of Z i the i-thm cell of size (or support) u ad estimated as a weighted sum of the experimetal data o poit support: Z iu = i k Z k (A3) k=1 i where k is the weight give to the k-thm experimetal data i the estimatio the mea value of Z over the support u i ad calculated as the solutio of the krigig system. Krigig also provides the variace of the estimatio error: s i2 =var(r iu )=var(z iu Z iu )=E{[Z iu Z iu ] 2 } Although a ubiased estimate of global mea i Equatio (A2) ca be easily foud from the local estimates i Equatio (A3) as: Z V = 1 N Z V= 1 N N u Z i (A2) (A4) (A5) Figure A1. Global domai V as sum of N=6 equal elemetary domais of support u ad =5 experimetal values () of puctual support

17 GEOSTATISTICAL METHODS AND CLIMATOLOGICAL RAINFALL 1047 the calculatio of the variace of the estimatio error of the global mea is ot so straightforward, i.e. because the estimatio variaces of the differet u uits are correlated the: var(r V )=var(z V Z V )=E{[Z V Z V ] 2 } 1 N N 2 s i The calculatio of var(r V ) is a little more cumbersome that just the arithmetic mea of estimatio variaces. It is clear that (A6) var(r V )= 1 N 2 var(r V* )= 1 N 2 var(z V Z V )= 1 N 2 E{[Z V Z V ] 2 } (A7) where var(x) is variace of the variable X. It the requires oly some algebra to show that: N var(r* V )=C VV + N k=1 l=1 N ki lj C kl 2 k=1 ki C kv where C VV is the mea covariace iside the support V; C kl is the covariace betwee Z k ad Z l ; C kv is the mea covariace betwee Z k ad the support V. If the covariace model is pure ugget effect, i.e. there is o correlatio betwee the experimetal samples, for the covariace fuctios: C VV =0 C kv =0 (A9) C kl =0ifkl C kl =var(z) ifkl ad for the weights: (A8) ki = 1 i, k (A10) which gives ad the var(r V * )=N 2 var(z) 2 (A11) var(r V )= var(z) (A12) which is the classical statistical result that gives the variace of the mea of a radom variable. The fact that the observatios are correlated icreases the stadard error (squared root of the estimatio variace) of the mea. I the example preseted i the text, the stadard error for the global mea, evaluated by ay of the krigig procedures preseted, is aroud 80 mm while usig stadard theory cosiderig that they are idepedet observatios we have, with =51, ad var(z) =0.0313, that the stadard error is / m=25 mm. REFERENCES Ahmed, S. ad De Marsily, G Compariso of geostatistical methods for estimatig trasmissivity usig data o trasmissivity ad specific capacity, Water Resourc. Res., 23(9), Akaike, H A ew look at the statistical model idetificatio, IEEE Tras. Auto. Cotrol, 79, Delhomme, J.P. ad Delfier, P Applicatio du krigeage a l optimisatio d ue campage pluviometrique e zoe aride, i Proceedigs, Symposium o Desig of Water Resources Projects with Iadequate Data. Vol. 2, UNESCO, Madrid, pp Hudso, G. ad Wackeragel, H Mappig temperature usig krigig with exteral drift: theory ad a example from Scotlad, It. J. Climatol., 14, Jourel, A.G. ad Huijbregts, Ch Miig Geostatistics. Academic Press, New York, 600 p. Jourel, A.G. ad Rossi, M.E Whe do we eed a tred model i krigig?, Math. Geol., 21(7), Lebel, T. ad Basti, G Variogram idetificatio by the mea-square iterpolatio error method, J. Hydrol., 77, Mathero, G Priciples of geostatistics, Eco. Geol., 58, Pardo-Igúzquiza, E., MLREML: a computer program for the iferece of spatial covariace parameters by maximum likelihood ad restricted maximum likelihood, Comput. Geosci., 23(2), Pardo-Igúzquiza, E Maximum likelihood estimatio of spatial covariace parameters, Math. Geol., 30(1), Thiesse, A.H Precipitatio for large areas, Mo. Wea. Re., 39,