kriging and thin plate splines for mapping climate variables.pdf

8/11/2019 Kriging and thin plate splines for mapping climate variables.pdf

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JAG Volume 3 - Issue 2 - 2001

Kriging and thin plate splines for mapping climatevariables

Eric P J Boer , Kirsten M de Beursl and A Dewi Hartkampz

1 Wageningen University and Research Center, Mathematical and Statistical Models Group, Dreyenlaan 4, 6703 HA Wageningen,The Netherlands (e-mail: E.P.J.BoerQato.dlo.nI; K.deBeursQesrinl.com)

2 Natural Resources Group, International Maize and Wheat Improvement Center (CIMMYT), Lisboa 27, Apartado Postal 6-641,06600 Mexico D.F., Mexico (e-mail: [email protected])

KEYWORDS: elevation, interpolation techniques, maxi-mum temperature, Mexico, precipitation

ABSTRACT

Four forms of kriging and three forms of thin plate splines arediscussed in this paper to predict monthly maximum temperatureand monthly mean precipitation in Jalisco State of Mexico.

Results show that techniques using elevation as additional infor-mation improve the prediction results considerably. From these

techniques, trivariate r egression-kriging and trivariate thin platesplines performed best. The results of monthly maximum temper-ature are much clearer than the results of monthly mean precipi-

tation, because the modeling of precipitation is more trouble-some due to higher variability in the data and their non-Gaussian

character.

INTRODUCTION

Statistical interpolation techniques are commonly applied

in Geographical Information Systems (GIS). Data collect-ed on a sparse Tablenetwork of measurement points are

interpolated to a regular grid of points. Burrough &McDonnell [ 1998, p. 1581 show a table with characteris-tics of ten classes of interpolation techniques. In papers

published recently [Goodale et a/, 1998; Dirks et al,1998; Pardo-lguzquiza, 1998; Goovaerts, 20001, a com-parison is made between several of those interpolation

techniques. An important question is often how addi-

tional information can be used to increase the predictionaccuracy. In this paper, several ways of including addi-

tional information classified into two widely used classesof interpolation techniques - thin plate splines and krig-ing - will be discussed.

Climate variables provide an essential input for cropgrowth simulation models. Climate maps (surfaces) canbe generated from a network of measurement stations -with measurements of precipitation, temperature, solarradiation, etc.- through interpolation. Accuracy of pre-

*Corresponding author

dictions of weather conditions at interpolated sites isimportant for crop growth simulation. Rosenthal et a/119981, for example, state that the greater variability intheir crop growth simulation results is most likely due to

the relatively coarse grid for spatial interpolation of pre-

cipitation.

The International Maize and Wheat Improvement Center

(CIMMYT) aims to improve productivity and sustainabili-

ty of smallholder maize and wheat systems in developingcountries. Crop growth simulation models are used toevaluate the opportunities and limitations of these pro-

duction systems [eg, Hartkamp et al, 19981. Climatemaps can provide essential input to crop models. Thelong-term monthly mean precipitation and long-term

monthly maximum temperature, measured over a sparse

network of climate stations in the Jalisco State of

Mexico, will be used as a case study in this paper. ADigital Elevation Model (DEM) can be used as additionalinformation to increase the prediction accuracy of the cli-

mate maps [de Beurs, 19981.

The main purpose of this study is to find an optimal way

of including elevation data of the area into the interpo-lation techniques to increase the prediction accuracy of

the climate maps. In total, four forms of kriging and

three forms of thin plate splines will be presented.

MATERIALS AND METHODSDATA SETS

Two data sets are considered in this paper. The first dataset consists of long-term (2 19 years, from 1940-1990)

monthly mean precipitation values at 193 measurementstations. The second set consists of 136 long-term (2 19years, from 1940-I 990) monthly maximum temperaturedata. These data were extracted from ERIC [IMTA, 19961for a square area (600 km x 600 km, called D), coveringthe state of Jalisco, Mexico. In this study, only themonths April, May, August and September are consid-ered, because for these months the correlation (Pearson

coefficient) between elevation and precipitation is

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Geostatistics and thin plate splines

greater than 0.5. Figure 1 shows the measurement sta-tions and a DEM of the area.

Figure 2 shows scatterplots of long-term monthly maxi-mum temperature (T,,,) and long-term monthly meanprecipitation Pmea, against elevation for August. Thecorrelation between T,,,, and elevation is -0.7 for Apriland May and -0.9 for August and September. For Pmeanthese values are 0.6 (April and May), -0.5 (August) and -0.6 (September). The scatterplot of P,,,, shows statisti-cally less attractive features.

INTERPOLATION TECHNIQUESInterpolation techniques can be divided into determinis-tic and stochastic models. Kriging technique is based on

JAG Volume 3 - I ssue 2 - 2001

stochastic models while the method of thin plate splinesis a deterministic interpolation technique with a local sto-chastic component. It is well known that under certainconditions these two interpolation techniques are equiv-alent to one another [Kent & Mardia, 19941. In thispaper, however, at least the function for modeling thespatial correlation is chosen differently. The modeling of

the trend and the neighborhood used for prediction candiffer too.

Let the actual meteorological measurements be denotedas z(st), Z(Q),..., z(sn), where si = (xi,y$ is a point in 0,Xi and yi are the coordinates of point si and IZ is equal tothe number of measurement points. The elevation at apoint s in the area D will be denoted as q(s). A mea-

A

L

0 200 400 600 Kllometera

FIGURE : The meteorological stations [IMTA, 19961 and a DEM [USGS,19971 of Jalisco State, situated northwest of Mexico City

. .:m.L.

. . . .. .

L

.. . ..

.

0 500 1004 1500 2000 2504

El WH I l

.

. ... .+ . .=-a .

. ..c .

n

.. . :.. . . . .

c = . ..I 9

n m. .I ... 9,

. . ,.+ .A# $. m .=

. .= +., . 1

.._: .=.. $-$. m 8. . # . . ..+. +

0 500 loo0 1500 200f J 2500

El WatiO l

FIGURE 2: SCatterplOtS for Tmax and fmean against elevation for August.

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Geostatistics and thin plate splines JAG Volume 3 - Issue 2 - 2001

surement considered as realization of a stochastic vari-able will be denoted by a lower case. Therefore, q(s) can

be considered as a realization of stochastic variable Q(S)(ie, outcome of a set of stochastic variables that have

some spatial locations and whose dependence on eachother is specified by some probabilistic mechanism).

Bivariate thin plate splineWahba [I9901 described the theory of thin plate splines.

In the case of bivariate thin plate splines, the measure-

ments z(Si) are modeled as:

Z(Si) = f(Si) + E(R), i = 1, . .) 7% (1)

where f is an unknown deterministic smooth function

and e(Si) are random errors. Commonly, it is assumed

that e(Si) are realizations of zero mean and uncorrelatedrandom errors.

The function f can be estimated by minimizing

2 [4Si) - f(Si)12 + XJz(f)

i l

where J~fl is a measure of smoothness of J1 calculated

by means of the following integral:

Jz(f) = 7 7 { (g)2+2(g)2+ ($,)dx& (3)-m--m

and h is the so-called smoothing parameter which regu-

lates the trade-off between the closeness of the function

to the data and the smoothness of the function. Thesmoothing parameter h can be estimated by generalized

cross validation.

The minimization problem (2) is solved with?as the lin-

ear combination

f(s) = kaj4j(s)2 W(h) (4)j=l i=l

where Qj are polynomials, $1 = 1, $2 = x and $3 = y; andY = hz In(h) of the Euclidean distance hi between s and

Si (h, = J(x--x~)~ +(Y -yiJ2). The coefficients Uj and bi inEquation (4) can be calculated by solving a linear system

of order n.

Partial thin plate splineThe bivariate thin plate spline model can be enlarged toa partial thin plate spline model by incorporating addi-tional information (q) into (1). This is done by adding anunknown linear combination of known functions intoJ:The measurements are modeled in the following way, on

the basis of scatterplots in Figure (2):

z(Si) = S(R) + AdSi) + P2Q2(Si) + 4 )~ i = 1, . .) 72 (5)

where the function f(S) = g(s) + 81 q(s) + 82 q*(s) is thefunction to be estimated, g(s) being an unknown smooth

function and PI and 82 are parameters with unknown

value. The function g and the parameters fit and pz can

be estimated by minimizing:

$ [z(Si) - S(Q) - PlJ(Si) - P2q2(%)]2 + M (S). (6)

giving the same solution structure as for bivariate thin

plate splines.

Trivariate thin plate splineHutchinson [I9981 shows that there is another way of

incorporating the covariable elevation into bivariate thin

plate splines. Namely, replacing the bivariate function

j(si) in (1) by t nvariate function &si,qj):

z(Si7 qi) = f(si, qi) + E(Si7 qi), i=l , . .n (7)

where qi is the elevation on the location Si. The functionJ~(fl is enlarged by several terms [see Wahba, 19901 and

the minimization problem is solved in the same way asfor bivariate thin plate splines. The solution can be writ-

ten as:

P(s) aj4j(S) + cbiQ(hi)j=l i=l

(8)

where @j are polynomials, $1 = 1, $2 = x, $3 = y and $4= q; and Y = h. The Euclidean distance hi between (s,q)and (si,qJ is calculated by

hi = J(Z - Xi)2 + (y - yi)2 + (q - q$ .

Scaling becomes important [Hutchinson, 19981, because

X, y and q can be expressed in different units and the

scale of variation can vary in different directions We fol-

lowed the suggestion of Hutchinson [I9981 to calculatethe generalized cross validation (GCV) on different scal-ings of elevation and select the scaling with the lowest

value of the GCV.

Ordinary kriging

The principles of ordinary kriging are well explained else-

where [Isaaks & Srivastava, 1989; Cressie, 1993;Wackernagel, 19951. The measurements are modeled in

the following way:

z(Si) = f(Q) + E(Si)r i = 1,2,...,n (9)

where, in this case,flsJ are considered as realizations ofrandom function F in point Si, which may contain adeterministic function p(s) = E{F s)} to model possibletrends; E(Si) are realizations of zero mean and uncorre-lated random errors. The trend p(s) is assumed to beequal to an unknown constant p.

The spatial correlation between the measurement pointscan be quantified by means of the semivariance function:

y(s, h) = +r[F(s) - F(s + h)] (IO)

where we assume that h is the Euclidean distance

between two points. Assume that the trend is constant

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and y(s,h) is independent of S. A parametric function isused to model the semivariance for different values of h.In this research, the spherical model - c Sph(a) - is used.

y(h) =

(

c{;(:)-;(k)}, Olhla(11)

c, h>a

where c is the scale parameter of the semivariance func-tion and a is a parameter which determines the so-calledrange of spatial dependence. The random errors (and/orthe spatial nugget random function) have a variance CO.For the stochastic variable 2 the following semivariancefunction is used: CO Nug(0) + c Sph(a).

The interpolated value at an arbitrary point SO n D is therealization of the (locally) best linear unbiased predictorof F(se) and can be written as weighted sum of the mea-surements.

is1

where the weights wi are derived from the kriging equa-tions by means of the semivariance function; n is thenumber of measurement points within a radius frompoint SO in this study we have taken a radius of 240 km).The parameters of the semivariance function and thenugget effect can be estimated by the empirical semi-variance function. An unbiased estimator for the semi-variance function is half the average squared differencebetween paired data values.

I(h) = & $y[%(Si,%(Si q2 (13)a=1where n(h) is equal to the number of data pairs of mea-surement points separated by the Euclidean distance h.

Ordinary cokriging

Cokriging makes use of different variables, modeled asrealizations of stochastic variables. In this study, eleva-tion - f&) - of the area D is used as covariable to pre-dict values of T,, and P,,,,. The spatial dependence ischaracterized by two semivariance functions yzz(s,h),,-&,h) and the cross-semivariance function:

Y&, h) = ;E {[Z(S) - Z(S + h)][Q(s) - Q(s + h)] (14)

To ensure that the variance of any possible linear combi-nation of the two stochastic variables is positive, a so-called linear model of coregionalization is applied. Thismodel implies that each semivariance and cross-semivari-ante function must be modeled by the same linear com-bination of semivariance functions [Isaaks & Srivastava,19891. Furthermore, the matrix of coregionalizationshould be positive semi-definite. A nested semivariancefunction is used with a nugget and two spherical semi-variance functions with different ranges. The cross-semi-

variance function can be estimated by the empirical

cross-semivariance function

%,@) = 2n(h) i=l~~~~(Si)-~(.i+h)][q(sl)-q(si+h)] (15)

where n(h) is the number of data pairs where both vari-ables are measured at an Euclidean distance h.

The interpolated value at an arbitrary point SO n D is therealization of the (locally) best linear unbiased predictorof F(so) and can be written as weighted sum of the mea-surements:

f(scl)= 2 WliZ(Si) + 2 ZjP(Sj)i-1 j=l

(16)

where ml is the number of measurements of Z(S) takenwithin a radius (of 240 km) from SO out of the modelingdata set), m2 are the number of meteorological stationswithin a radius of 240 km from SO (out of the modelingand validation set). The weights wli and ~2~ can be

determined using the two semivariance functions andthe cross-semivariance function.

Regression-kriging

Odeh et al [I9951 compared, among other techniques,three forms of regression-kriging (comparable with krig-ing with external drift). The idea of regression-kriging, inthis paper, is that we characterize the trend componentp(s) of the model for the random function F(S) as anunknown linear combination of known functions (regres-sion model). In ordinary kriging the trend component ismodeled as constant; in the usual form of universal krig-

ing the trend component is modeled as a polynomial ofa certain degree. In our application the trend is modeledas:

0 +&q(S) + h?(S) (17)

The interpolated value at location se can be calculated bya linear combination of the regression model and aweighted sum (ordinary kriging) of regression residuals

Z*(Si) = (Si) - 8 - bljlq(Si) - bq2(Si).

This results in:

P(%) = s + Wq(So) + &12(SO) + e Wz*(%)i=l

(18)

The difficulty of this form of regression-kriging, and ofuniversal kriging in general, is that the parameters of theregression model and the parameters of the semivariancefunction of the spatial correlated regression residualsshould be estimated simultaneously [Laslett &McBratney, 19901. Under the assumption of normality,the parameters can be estimated by restricted maximumlikelihood (REML), which is one of the techniques to esti-mate the parameters of the regression model and theparameters of the semivariance function simultaneously

[Gotway & Hartford, 19961.

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Trivariate regression-kriging

Finally, trivariate regression-kriging will be introduced.Trivariate regression-kriging is a form of regression-krig-ing, where trivariate ordinary kriging is applied on theregression residuals. The trend is chosen equally as intrivariate thin plate splines. The interpolated value at alocation so can be calculated by:

(19)

The weights wi are determined by the semivariance func-tion, which is a function of the Euclidean distancebetween two points (si, qi) and (s, q). The units are ofdifferent order and scaling becomes important, the samescaling is used as for trivariate thin plate splines. In thiscase, REML is not applied because of limitations of thesoftware used. The residual semivariance function is nowestimated from the OLS regression residuals.

Comparison of interpolation techniquesTo compare the interpolation techniques, the originaldata set is divided into a modeling data set and a valida-tion set of 25 measurement points. The 25 points are notchosen randomly, but are selected by the authors, sothat the area is still reasonably covered by measurementpoints. Five validation sets are chosen from each dataset. Each validation set contains different measurementpoints from the original data sets. Predictions on thelocations of the validation points - f(si) - and the mea-sured values at these locations - Z(Si) - are compared bythe two criteria: the Mean Square Error (MSE) and the

Maximal Prediction Error (MPE).

(20)

MSE = @(si) - z(si)]a=1

(21)

where n, (= 25) is the number of validation points.

RESULTSThe automatic calculation procedure of thin plate splinesallows a straightforward analysis of these techniques.There is no need for any prior estimation of the spatial

dependence of measurement points. ANUSPLIN[Hutchinson, 19971 is used to perform the analyses. Fortrivariate thin plate splines it is useful to optimize the ele-vation scale [Hutchinson, 19981. Therefore the squareroot generalized cross validation for trivariate thin platesplines is determined at different scales of elevation(meter, decameter, hectometer and kilometer).Decameter is the optimal scaling for Tmax and kilometer

for Pm,,,. This way of scaling is found to be sufficient,because no major differences between the GCV of twosuccessive scales are found. Trivariate regression-krigingis applied with the same scaling as trivariate thin plate

splines.

The semivariance functions for ordinary kriging are esti-mated by weighted least squares with GSTAT [Pebesma &Wesseling, 19981. For cokriging the semivariance func-tions, by means of the linear model of coregionalization,are estimated by COREG [Bogaert et al, 19951. The resid-ual semivariance function for trivariate regression-krigingis estimated in a relatively simple way. First the trend is

estimated by ordinary least squares (OLS), followed bythe estimation of the spatial variability of the regressionresiduals. For regression-kriging, where the semivariancefunction depends only on x and y, the parameters of theregression model and the parameters of the semivariancefunction are estimated simultaneously by the REMLoption of PROC MIXED in SAS [Littell et al 19961. Figures3 and 4 show some examples of fitted semivariance func-tions (with models and parameter values) for T,,,,, andP mean for cokriging and trivariate regression-kriging.

Only the recorded elevations of the meteorological sta-

tions are used for interpolation with ordinary cokriging.We used a DEM of the area but prediction accuracyincreased substantially as just the recorded elevation atthe point to be predicted (validation point) was available,as for all other interpolation techniques. Tables 1 and 2show the results of all 7 interpolation techniques for 5validation sets for T,,, and Pm,,, respectively.

The results of Tables and 2 demonstrate the benefit ofusing the covariable elevation. Especially for Tmax hedifferences of interpolation with elevation and withoutelevation are convincing. This is due to the high correla-

tion between elevation and Tma,. Comparing regression-kriging, cokriging, trivariate regression-kriging, trivariatethin plate splines and partial thin plate spline for Tmaxshows an advantage for the two interpolation techniqueswhich made use of three-dimensional coordinates(trivariate). The differences between the results of theinterpolation techniques are less clear for Pm,,,. Onlyfor validation sets 1 and 2 did the trivariate interpolationtechniques perform more accurately concerning the MSE.

The prediction results of Pm,,, for validation set 1 forAugust and September are very poor for bivariate thin

plate splines and partial thin plate splines. This is mainlycaused by two validation points in the South-East of thearea, which have a large prediction error. Probably, thisis caused by a local trend at adjacent measurement sta-tions.

DISCUSSIONIn this paper 7 interpolation techniques are discussed, 5including and 2 excluding elevation as additional infor-mation. The two techniques excluding elevation perform,especially for Tmax onsiderably less accurately than the

techniques including elevation. The MSE and MPE values

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Geostatistics and thin plate splines JAG Volume 3 - issue 2 - 2001

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PURE 3: Estimated semivariance functions for cokriging and trivariate regression-kriging for Tmm,upper right: elevation; Lower left: cross semivariance function between T

April, validation set I. Upper left:

fut??ion TmM for trivariate regression-kriging.and elevation; Lower right: residual semivariance

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italics

Interuolation techniaue

ordinsry kriging vlreunion-~ig~g

cokrigingtrivariate regression-krigingbivsriate thin plate splinestfivariate thii plate splinespartial thin plate spliieaordinary kriging v2regression-krigingcokrigingtrivariate regression-krigingbivariate thin plate splinestrivariate thin plate splinespartial thii plate splinesordinary kriging v3regression-krigingcokriging

trivariate regression-krigingbivariate thii plate splinestrivariate thin plate splinespartial thin plate splinesordinary kriging v4reunion-~i~ngcokrigingtrivariate regression-krigingbivariate thii plate splinestrivariate thin plate splinespartial thin plate splineaordiiarykriging v5regression-krigingcokrigingtrivariate region-~i~ng

bivariate thin plate spliiestrivariate thin plate splinespartial thin plate splines

MSE MPE


TABLE 1: Results of the Mean Square Error (MSE) and the Maximum Prediction Error(MPE) for 5 validation sets (VI-v5) of a long-term monthly maximum temperature (I,,) .The values with the lowest MSE and MPE of the 7 interpolation techniques are written in

wr may aw sw10.3 9.8 8.2 8.4

4.1 3.8 2.2 2.1

5.0 4.9 4.0 3.4

5.5 3.9 4.1;:I 4.0 2.9 2.9

5.9 5.6 5.0 5.2

10.5 9.9 8.8 8.34.8 4.5 3.1 3.04.8 5.2 3.4 3.77.2 5.9 3.9 3.94.5 4.2 2.0 2.04.3 3.7 1.9 1.83.5 3.5 2.1 2.08.4 7.3 4.9 5.03.8 4.0 1.8 1.8

5.0 4.3 1 7 1.8

;.5 ;.o ::s

2.1

1.97.5 7.3 5.0 5.25.8 5.3 2.1 2.37.1 6.2 1 7 2.07.4 7.2 5.0 5.43.9 4.2 2.1 1.95.1 5.0 2.3 2.69.4 3.7 1.5 1.87.5 7.6 5.8 6.13.8 3.7 1.6 1.83.9 4.1 2.0 2.26.8 6.4 5.7 5.43.8 2.4 1.2 1.14.6 4.3 2.0 1.72.0 2.0 1.3 1.2

7.4 7.1 8.3 7.52.6 3.6 1.8 1.72.6 2.3 1 1 1 0

are lower when elevation is used as additional informa-tion for prediction, especially when the correlationbetween the two variables is high.

From the techniques which include elevation, trivariatethin plate splines and trivariate regression-kriging seemto perform best. Cokriging, is the most time-consuminginterpolation technique to implement in this study.Therefore, in this case study, cokriging is not preferable.The main reason for cokriging having relatively poor pre-diction results is the fact that a linear relation is assumedbetween climate variable and elevation. The other tech-niques (including elevation) used in this paper, do notassume such relation because regression models withmore regressors and trivariate techniques are used.

The results of Tmax are much clearer to interpret than theresults of P . There is much more variability in theP Mean predictions resulting from a higher variability in thedata. The correlation between the climate variable and

w

5.3 5.0 3.7

may aug

3.7

sep

5.7 5 1

6.1 5.8 5.2 5.45.0 4.9 4.2 4.2

5.4 5.3

5.5 5.6 4.3 4.54.6 4.9 3.8 8.76.1 5.7 5.8 5.24.5 4.4 4.3 4.15.1 5.2 3.9 4.46.9 6.7 5.3 5.65.9 5.7 3.3 3.75.5 5.4 3.0 2.94.9 4.7 3.3 3.26.6 6.4 5.6 6.05.4 5.5 3.5 3.4

5.3 5.2 3.7 3.95.9 6.0 4.4 4.5

5.2 5.3 3.8 4.06.6 5.9 5.2 5.15.3 6.1 4.3 4.46.4 6.2 9.5 4.56.4 6.6 6.4 6.83.7 4.0 4.1 4.25.6 5.9 5.1 5.63.9 4.0 2.9 3.56.5 6.6 6.4 6.84.3 4.2 2.7 .I3.7 4.1 3.9 4.46.8 7.0 5.7 5.54.4 3.8 2.8 2.45.5 5.0 3.3 2.83.5 3.6 2.2 2.1

6.7 7.0 6.8 6.25.3 5.6 2.8 2.63.5 3.4 2.8 2.7

elevation is lower, which causes smaller differencesbetween including and excluding elevation. In general,precipitation data are clearly non-Gaussian. Although atransformation can be considered, this has been reportedto have disadvantages for local estimation [Roth, 19981.

Beek et al [I9921 stress the importance of interpolationtechniques for crop growth simulation. In this paper moreadvanced forms of kriging and thin plate splines areapplied. Especially, the trivariate forms of kriging and thinplate splines performed well. The main advantage of thinplate splines over kriging is the operational simplicity ofthis technique, which can be very important from a prac-tical point of view. The kriging procedure requires moreeffort and experience. For Tmax the predictions results oftrivariate regression-kriging are slightly more accuratecompared to the results of trivariate thin plate splines, butthe differences are small. Within the geostatistical frame-work trivariate regression-kriging, as described in thispaper, seems to be an attractive option.

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TABLE 2: Results of the Mean Square Error (MSE) and the Maximum Prediction Error (MPE)for 5 validation sets (~14) of a long-term monthly mean precipitation (P,,,). The valueswith the lowest MSE and MPE of the 7 interpolation techniques are written in italics.

Interpolation technique

ordii kriig vlregression-krigingcokriig

trivariate regression-krigingbivariate thiu plate splineatrivariate thin plate spliiespartial thii plate splinesordii kriig v2regression-krigingcokrigingtrivariate regression-kriingbivariate thin plate spliiestrivariate thin plate splinespartial thii plate spliiesordinary kriging V

regreasion-krigingcokriigtrivariate regression-krigingbivariate thin plate spliieatrivariate thin plate spliieapartial thii plate spliiesordinary kriging v4regression-krigiugcokrigingtrivariate regession-krigingbivariate thii plate spliiestrivariate thii plate splinespartial thin plate spliiesordiiary kriging v5regression-kriging 6.8 68.7 1891.4 2291.8 8.3 24.4 135.2 135.1cokriging 12.2 91.1 2276.2 2538.7 10.0 27.7 153.8 148.0trivariate regression-kriging 8.4 58.5 2170.4 2723.6 8.3 23.0 149.0 132.5bivariate thii plate splines 7.6 84.3 1607.1 1984. 7.0 21.3 83.3 132.1trivariate thin plate spliies 6.8 64.2 2419.2 2923.4 8.7 25.5 136.6 131.8partial thii Dlate sdines 6.8 67.7 1724.2 2050.3 7.9 24.5 88.2 129.1

MSE

apr may aug36.7 62.4 1251.9 ?23.636.1 64.4 1182.1 1111.235.3 58.1 1132.8 820.0

83.Q 50.6 728.4 520.144.1 78.5 3562.6 3418.233.7 50.8 753.6 525.936.9 67.4 3456.6 3101.4

17.5 158.5 4715.1 4261.513.7 168.0 4712.9 4175.319.3 172.3 4687.3 4062.417.6 170.7 4479.5 9554.d13.8 170.2 4802.2 3834.3la 0 168.7 4492.2 3623.413.3 166.8 5319.8 4037.67.8 55.4 1804.2 1206.73.5 51.9 1683.5 971.05.8 48.3 1652.7 1012.46.7 49.3 1787.4 1252.66.4 55.2 1762.8 1264.23.0 48.5 1665.3 1230.03.4 52.5 1720.1 1397.217.4 60.3 2477.1 3150.9.5 44.9 2467.8 3321.013.6 59.6 2537.4 3517.813.7 47.0 3252.8 4038.015.9 71.6 3069.9 4029.39.5 45.1 3696.9 4137.19.1 42.4 2858.4 3767.88.6 61.5 2193.9 2873.6

ACKNOWLEDGEMENTS

The authors thank A.C. van Eijnsbergen and A. Stein forfruitful discussions at various stages of this work.

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MPE

apr may aug =p26.2 22.4 84.8 80.626.7 22.7. 89.2 78.726.4 23.4 77.3 59.5

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R SUM

Dans cet article on decrit quatre formes de kriging et troisformes de splines bicubiq ues sous forme de plaques minces pourpredire la temperature mensuelle maximale et la precip itati onmensuelle moyenne dans Ietat Jalisc o de Mexico. Les resultatsmontrent que les techniques utilisant Ielevation comme infor-mation additionnelle ameliorent considerablement la prediction.A partir de ces techniques, il savere que la regression-kriging a

trois variables et les splines bicubiques donnaient les meilleursresultats. Les resultats de temperature mensuelle maximale sontplus nets que les resultats de precipitation mensuelle moyenne,parce que la modelisation de precipitation est plus problema-tique a cause de la plus forte variabilite dans les donnees et leurcaractere non-gaussien.

R SUM N

En este articulo se discuten cuatro formas de interpolation porkriging y tres formas de filtro (thin plate splines) para predecir latemperatura maxima mensual y la precip itati on media mensualen el Estado de Jalisc o en Mexico. Se muestra que las tecnicasque utilizan datos altimetricos coma information adicional mejo-ran consi derablemente la predic cidn de resultado s. Entre estas

tecnicas, el kriging de regresion trivariada y 10s filtros (thin platesplines) trivariados dieron 10s mejores resultados. Los resultadosde temperatura maxima mensual son m6s niti dos que 10s resul-tados de precipitacidn media mensual, porque la modelizacionde la precipitation es m s dificultosa debido a mayor variabili-dad de 10s datos y debido a su caracter no-Gaussi ano.

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