incorporating elevation in rainfall interpolation in tunisia using geostatistical methods

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Incorporating elevation in rainfall interpolation inTunisia using geostatistical methodsHaifa Feki a , Mohamed Slimani b & Christophe Cudennec c d ea Laboratoire Sciences et Techniques des Eaux, Institut National Agronomique de Tunisie,1082, Tunis-Mahrajne, Tunisieb Dpartement Gnie Rural, Eaux et Forts, Institut National Agronomique de Tunisie, 1082,Tunis-Mahrajne, Tunisiec Agrocampus Ouest, UMR 1069 Sol Agro et Hydrosystme Spatialisation, F-35000, Rennes,Franced INRA, UMR 1069 Sol Agro et Hydrosystme Spatialisation, F-35000, Rennes, Francee Universit Europenne de Bretagne, France

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To cite this article: Haifa Feki, Mohamed Slimani & Christophe Cudennec (2012): Incorporating elevation in rainfallinterpolation in Tunisia using geostatistical methods, Hydrological Sciences Journal, DOI:10.1080/02626667.2012.710334

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1Hydrological Sciences Journal Journal des Sciences Hydrologiques, iFirst, 2012

Incorporating elevation in rainfall interpolation in Tunisia usinggeostatistical methods

Haifa Feki1, Mohamed Slimani2 and Christophe Cudennec3,4,5

1Laboratoire Sciences et Techniques des Eaux, Institut National Agronomique de Tunisie, 1082 Tunis-Mahrajne, [email protected] Gnie Rural, Eaux et Forts, Institut National Agronomique de Tunisie, 1082 Tunis-Mahrajne, Tunisie3Agrocampus Ouest, UMR 1069 Sol Agro et Hydrosystme Spatialisation, F-35000 Rennes, France4INRA, UMR 1069 Sol Agro et Hydrosystme Spatialisation, F-35000 Rennes, France5Universit Europenne de Bretagne, France

Received 26 January 2011; accepted 16 January 2012; open for discussion until 1 April 2013

Editor D. Koutsoyiannis; Associate editor C. Onof

Citation Feki, H., Slimani, M., and Cudennec, C., 2012. Incorporating elevation in rainfall interpolation in Tunisia using geostatisticalmethods. Hydrological Sciences Journal, 57 (7), 121.

Abstract This paper compares the performance of three geostatistical algorithms, which integrate elevation as anauxiliary variable: kriging with external drift (KED); kriging combined with regression, called regression kriging(RK) or kriging after detrending; and co-kriging (CK). These three methods differ by the way by in which thesecondary information is introduced into the prediction procedure. They are applied to improve the prediction ofthe monthly average rainfall observations measured at 106 climatic stations in Tunisia over an area of 164 150 km2using the elevation as the auxiliary variable. The experimental sample semivariograms, residual semivariogramsand cross-variograms are constructed and fitted to estimate the rainfall levels and the estimation variance at thenodes of a square grid of 20 km 20 km resolution and to develop corresponding contour maps. Contour diagramsfor KED and RK were similar and exhibited a pattern corresponding more closely to local topographic featureswhen (a) the network is sparse and (b) the rainfallelevation correlation is poor, while CK showed a smooth zonalpattern. Smaller prediction variances are obtained for the RK algorithm. The cross-validation showed that theRMSE obtained for CK gave better results than for KED or RK.

Key words monthly average rainfall; semivariogram; cross-variogram; kriging with external drift; regression-kriging;co-kriging; Tunisia

Incorporation de laltitude pour linterpolation des pluies en Tunisie en utilisant les mthodesgostatistiquesRsum Nous comparons les performances de trois mthodes gostatistiques qui permettent dintgrer laltitudecomme variable auxiliaire: le krigeage avec drive externe (KED), la rgression-krigeage (RK) et le cokrigeage(CK). Ces trois mthodes se diffrencient par la manire avec laquelle linformation secondaire est introduitedans la procdure destimation. Elles sont appliques pour amliorer lestimation des prcipitations moyennesmensuelles mesures par 106 stations climatiques rparties sur toute la Tunisie, de superficie 164 150 km2, enutilisant laltitude comme variable auxiliaire. On a construit et ajust les semi-variogrammes exprimentaux sim-ples, les semi-variogrammes des rsidus et les variogrammes croiss pour estimer les prcipitations et la variancedestimation au niveau des nuds dune grille carre dune rsolution de 20 km 20 km, et pour dvelopperles cartes des isohytes correspondantes. Les cartes de pluie obtenues par le KED et la RK sont similaires etprsentent un aspect trs li aux variations topographiques lorsque (a) le rseau est peu dense et (b) la corrlationpluiealtitude est mdiocre. Les cartes obtenues par CK prsentent quant elles des isohytes lisses. Les valeursdes variances destimation les plus faibles sont obtenues par la RK. La validation croise montre que la racine delcart quadratique moyen obtenue pour le CK est meilleure que pour le KED ou pour la RK.

Mots clefs pluie moyenne mensuelle; semi-variogramme; variogramme crois; krigeage avec drive externe; rgression-krigeage; cokrigeage; Tunisie

ISSN 0262-6667 print/ISSN 2150-3435 online 2012 IAHS Presshttp://dx.doi.org/10.1080/02626667.2012.710334http://www.tandfonline.com

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1 INTRODUCTION

Rainfall maps have a wide range of applications,such as water resources management and hydrologicalsimulation. Geostatistical multivariate techniques arewidely used for the prediction of precipitationamounts over a given area by accounting for sec-ondary information sampled over the same area toimprove the quality of the maps.

In some studies, radar-rainfall data have beenused in conjunction with measurements at raingaugesto map precipitation (Creutin et al. 1988). However,the bulk of studies have made use of a cheaper,widely-available data source, the digital elevationmodel (DEM), in exploiting the relationship betweenprecipitation amount and elevation.

In fact, topographic relief has marked effects onprecipitation, which, generally, increases with eleva-tion. This increase is due to the fact that hills arebarriers to moist airstreams, forcing the air to rise,and they act as high-level heat sources on sunnydays. The latter causes convective clouds to form overhills preferentially, resulting in showery precipitation.Weather stations tend to be sited at low elevationandmay thus underestimate the regional precipitation.For these reasons, especially, in areas of high topo-graphic relief, it will often be insufficient to use datafrom the nearest weather station to characterize theamount and spatial distribution of precipitation over alarge-scale study area. Goovaerts (2000) reported thatthe techniques which used elevation data generallyout-performed ordinary kriging (OK). That is, whereprecipitation amount and elevation are correlated lin-early, estimates informed by elevation data are oftenmore accurate than those made using the precipitationdata alone. Studies have been conducted in a range ofdifferent locations and environments around the worldand they suggest that incorporating elevation into theinterpolation procedure will often be beneficial.

This paper builds on the work of Slimani et al.(2007), but the analysis is expanded in using mul-tivariate geostatistical methods to assess the impactof incorporating elevation into the monthly aver-age rainfall prediction across the whole of Tunisiaby comparing three interpolation techniques: krigingwith external drift (KED), regression kriging (RK)and co-kriging (CK).

2 STUDY AREA AND DATA

Tunisia is a Mediterranean coastal region (Cudennecet al. 2007) located in the north of Africa with an areaof 164 150 km2. The physical geography of the region

is quite heterogeneous (Fig. 1). Mountains in the northare separated by small plains where some wadis, suchas Oued Mejerda (Jebari et al. 2012), flow. The plainsare progressively replaced by steppes, then by orientalcoasts that stretch from Cap Bon to the Libyan border.The coasts (1300 km) are cut by deep gulfs and thereare many islands. Towards the south, the mountainouschains are lower (Sahara). Climatologically, Tunisia issituated in the geographical transition zone betweenhumid temperate climate and the arid Saharan cli-mate. Although, the general climate is arid, we candistinguish three different climatic regions: (a) thenorth is characterized by a Mediterranean climatewith a humid and sub-humid nuance; (b) the centreand east coast region with a semi-arid climate; and(c) the south, where the climate becomes Saharan.The precipitation is characterized by a marked northsouth spatial distribution, with strong hydrologicalconsequences (Cudennec et al. 2005, 2007). In fact,in the north, the total of the annual rainfall means isbetween 400 and 1000 mm and it may reach 1500 mmper year over the Kroumirie Mountains. The rain-fall season extends from September to April with arainfall maximum during the winter months. In thecentre of Tunisia, the mean rainfall varies from 200 to400 mm per year and it is characterized by a sig-nificant variability from one year to another. Thesouth of Tunisia is characterized by a Saharan climate,where rainfall is erratic. For further details on themeteorological and physical aspects, please refer toSlimani et al. (2007), Feki (2010) and Baccour et al.(2012).

Monthly rainfall data for 106 raingauges inTunisia for the period 19611990 were collectedfrom the global directorate of water resources of theTunisian agricultural ministry (Direction gnrale desresources en eau, DGRE) and the national instituteof meteorology (Institut national de la meteorology,INM). Monthly average rainfall and inter-annual rain-fall were calculated for a common observation periodfor each raingauge station and statistical analysis car-ried out (see Table 1). Almost all variables showeda large distribution of data around the mean valueand quite high relative standard deviation (RSD). Thelocations of observation stations are shown in Fig. 1.Most of the raingauges are located in areas with lowelevation (Fig. 2). Elevations were also available ateach raingauge location and the coefficient of deter-mination, r2, for elevation against precipitation foreach month is given as the last column of Table 1.A digital elevation model (DEM) at 20 km 20 kmresolution (Fig. 3) was used as the secondary data setfor KED and RK.

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Rainfall interpolation in Tunisia using geostatistical methods 3

Fig. 1 (a) Map of Tunisia and (b) location of rainfall measurements.

Table 1 Summary statistics for monthly average rainfall.

Month Minimum (mm) Maximum (mm) Mean (mm) Standard deviation(mm)

r2 (vselevation)

RSD

September 2.46 71.67 32.72 12.12 0.02 0.37October 8.42 149.83 49.16 23.82 0.007 0.48November 4.03 201.49 47.15 32.64 0.0003 0.69December 5.50 262.51 52.60 42.20 0.002 0.80January 6.95 243.59 51.79 39.99 0.003 0.77February 5.82 198.72 43.22 32.04 0.008 0.74March 11.01 159.71 40.56 21.97 0.08 0.54April 5.35 138.90 33.75 19.70 0.09 0.58May 3.06 86.76 22.16 12.89 0.24 0.58June 0.56 31.58 10.73 7.07 0.41 0.65July 0 24.88 3.34 3.20 0.25 0.95August 0 25.42 8.93 5.70 0.29 0.63

3 METHODOLOGY

The geostatistical approach is based on thetheory of regionalized variables (Matheron 1970).It assumes that spatial samples are considered as the

realization of a random spatial process. This allowsthe use of a powerful statistical instrument for spatialestimation: the semivariogram (Feki and Slimani2006).

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10% between 6001020 m

20% between 200600 m

70% between 0200 m

Fig. 2 Distribution of the raingauge numbers with class ofaltitude.

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Fig. 3 Digital elevation model of Tunisia (20 km 20 km).

Let z(si) and z(si) represent realizations of ran-dom variables Z(si) and Z(si) at particular pointssi within a field S. The intrinsic hypothesis (Chauvet1999) assumes that, for a random variable Z(si): (i)the expected value of Z(si) does not depend on theposition si; and (ii) the variance of [Z(si) Z(si + h)]does not depend on the position si in S for any sep-aration vector h. Then the semivariogram functiongives a measure of the spatial correlation of a ran-dom variable or variables as a function of separation

distance. Sample semivariograms and cross-variog-rams were estimated by the function:

(h) = 12n(h)

n(h)i=1

[Z(si + h)

Z(si)][Z(si + h) Z(si)](1)

where is the semivariance of Z (when = ),or the cross-semivariance of Z and Z (when = )at a separation distance h; and n(h) is the number ofpairs of points in a distance interval (h + h).

The cross-variogram must be part of a linearmodel of co-regionalization (LMC). A LMC requiresthat every structure in the cross-variogram or othermodel of spatial continuity be included in the spatialcontinuity model for primary and secondary vari-ables. A mathematical authorized model may then befitted to the experimental variogram and the coeffi-cients of this model can be used for kriging. In vari-ogram models where the semivariance reaches a finitevariance and levels out, the maximum is referred toas the sill. This is termed a bounded model (sphericaland exponential models). Models that do not reach asill are termed unbounded (power model). Unboundedvariograms may indicate a large-scale trend (that is,systemic increase or decrease) in the values of theproperty characterized across the region of study. Allvariogram parameters are introduced into algorithmsusing code written by Matlab computer programs.These algorithms concern the KED, RK and CKmethods and the theory of each method is describedin the Appendix.

The interpolation methods differ principally bythe way in which the secondary variable is introducedin the prediction procedure. In fact, for KED (Fig. 4),the following procedure is used with the data foreach month: (i) ordinary least square (OLS) regres-sion (altitude vs precipitation) is conducted aroundeach data location (raingauge) and estimates of thedrift are made at the point locations and on a regu-lar grid; (ii) the residuals are obtained at each datalocation (that is, the differences in observed and esti-mated precipitation amount are obtained); (iii) thevariogram is estimated from the residuals and a modelis fitted; and (iv) estimates of precipitation are madeat the locations of the DEM cells with variogrammatrix extended by the elevation matrix. For RK(Fig. 5), the estimator can be decomposed as a two-stage process: a generalized least-square regression of

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KED

Calculate the weights and the Lagrange parameters:

Calculate estimates on all grid points:

Calculate uncertainty

Plot isohyet rainfall maps and the associated uncertainty maps

Evaluate the sample variance/covariance matrix of data points:

Evaluate the sample variance/covariance matrixbetween data points and grid points:

Estimate the drift by theordinary least squares (OLS) regression

Subtract drift of the main variable for residues evaluation

Model the variogram of residuals:(model type, nugget, range and sill)

Grid definition (20 20 km resolution)

Inputs: x: longitude, y: latitude, zp: elevation at raingauge station, z: rainfall, Y: DEM)

Fig. 4 Explanatory diagram of the KED procedure.

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Estimate the drift by the ordinary least squares (OLS) regression

Subtract drift of the main variable for residues evaluation

RK


Inputs: x: longitude, y: latitude, zp: elevation at raingauge station, z: rainfall, alt: DEM)




Re-estimate generalized least squares(GLS) coefficient and calculate residuals

Calculate the drift by the GLS method:

Add to residuals

Evaluate the sample variance/covariance matrix of data points si:

Evaluate the sample variance/covariance matrix between

data points si and grid points s0:

Model the variogram of residuals (model type, nugget, range and sill)



OK

Fig. 5 Explanatory diagram of the RK procedure.

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the primary variable with the auxiliary variables fol-lowed by a simple kriging of the regression residuals(Castelier 1993, Hengl et al. 2007). This was origi-nally suggested by Odeh et al. (1995), who named itregression kriging, whereas Goovaerts (2000) usesthe term kriging after detrending. The CK (Fig. 6)is another method of multivariable geostatistics need-ing the determination of the cross-variogram besidesthe simple variogram of the principal and the sec-ondary variable, which is often a delicate operation(Pardo-Igzquiza et al. 2007).

The advantage of KED is that the equations aresolved at once, while the advantage of RK is thatthere is no danger of instability as with the KED sys-tem, according to Goovaerts (2000). Some authorsmake different assumptions and skip some compu-tational step(s) so that the products of RK and KEDmight not differ at the end. For example, Bourennaneand King (2003) made an assumption that the var-iogram of residuals is equal to the variogram oftarget variable, which is a simplification. In this case,the KED prediction map will look more similar tothe OK map. Other authors (e.g. Odeh et al. 1995)used only ordinary least-squares estimate of the drift,which is also sub-optimal, but is a shorter solution.Kitanidis (1997) confirms that, in practice, a singleiteration for the estimation of GLS residuals can beused. In this paper, estimates of monthly precipitationin Tunisia were made using the techniques detailedabove, and these were compared using: (a) a visualexamination of the maps derived through estimationvia a regular grid, and (b) a cross-validation proce-dure. The paper has several innovative components;elevation and precipitation relationships have beenrelatively little studied in Tunisia. One specific contri-bution of the paper is in illustrating how rainfall variesregionally with elevation.

4 RESULTS AND DISCUSSIONS

The analysis of the data comprised four steps: (i)a general trend surface analysis; (ii) a general andlocal regression analysis; (iii) structural analysis; and(iv) estimation. In step (i), the desire is to iden-tify any large-scale trends in precipitation valuesacross Tunisia. The estimation of the variogram forKED is based on the assumption that there is sucha large-scale trend in precipitation amounts. So, ifthe trend surface analysis indicates a large-scaletrend is present, then this suggests that the approachemployed using KED is valid. In step (ii), the concernwas to assess relationships between precipitation and

elevation both for the whole country (use all avail-able paired data) and locally (use only subsets of thedata within a moving window). If elevation and pre-cipitation are related locally, the KED is potentiallybeneficial. In step (iii), spatial variation in precipi-tation is characterized by estimating the variogram.Fitting a model to the variogram enables incorpora-tion of information on spatial variation in the prop-erty into the interpolation procedure through krigingalgorithms (step (iv)). The four months of October,January, March and June are used for illustrativepurposes for the four seasons successively (autumn,winter, spring and summer).

4.1 Trend surface analysis

In order to assess evidence for long-range trends inprecipitation amount, a general trend surface anal-ysis was conducted. First-, second- and third-orderdata polynomials were fitted to the precipitation data.Precipitation amount was the dependent variable andthe independent variables for a polynomial of order1 were x and y, for order 2 they were x, y, x2, y2,xy and for order 3 they were x, y, x2, y2, xy, x3, y3,x2y, xy2, with x and y the distances in the westeastand southnorth directions, respectively.

In Table 2, the r2 values are given for first-,second- and third-order polynomials fitted to the pointdata for each month. The figures indicate that a first-order and second-order polynomial accounts for morethan 40% of the variation in precipitation amountfor 10 months. For a third-order polynomial, it ismore than 40% for only 8 months, because Februaryand April are exceptions. For the months of Julyand August, the first-, second- and third-order poly-nomials account for less than 40% of the variationin precipitation and, except for these two months,the r2 values indicate that there is evidence of alarge-scale trend in precipitation amount for mostother months. Given our knowledge of precipitationpatterns in Tunisia, this is not surprising, since precip-itation amount is consistently greater in the north andless in the south. If this trend is due to elevation, thenthe use of KDE, RK and CK informed by elevationdata is likely to be beneficial.

4.2 General and local regression: elevation vsprecipitation

The coefficient of determination, r2, for elevationagainst precipitation for each month is given inTable 1. Scatter plots of elevation vs precipitation for

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CK

Evaluate the general variance/covariance matrix between

data points:

Model the sample covariance matrix of the principal variable C (model type, nugget, range and sill)


Inputs: Principal variables: rainfall (Z), XZ: longitude, Y Z: latitude Secondary variables: altitude (Z), XZ: longitude, YZ: latitude

XZ YZ Z

XZ YZ Z

Model the cross-covariance matrix between the principal and the secondary variable C (model type, nugget, range and sill)

Model the sample covariance matrix of the secondary variable C (model type, nugget, range and sill)

Evaluate the sample and cross-variance/covariance matrix

between data points and grid points:





=

Fig. 6 Explanatory diagram of the CK procedure.

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Table 2 Precipitation amount against x and y: values of r2

for polynomials of order 1, 2 and 3.

Month Order 1 Order 2 Order 3

September 0.464 0.575 0.524October 0.526 0.560 0.404November 0.439 0.640 0.528December 0.436 0.638 0.731January 0.470 0.678 0.759February 0.469 0.658 0.005March 0.429 0.596 0.404April 0.537 0.643 0.225May 0.573 0.706 0.485June 0.515 0.595 0.677July 0.247 0.200 0.275August 0.312 0.379 0.348

the four illustrative months are given in Fig. 7. Thesmallest r2 for all months is for December (0.002) andthe largest is for June (0.41). Therefore, the assump-tion of a (general) linear relationship between eleva-tion and precipitation for the spring and the summermonths is not unreasonable. However, KED and RKare informed by this relationship locally, and gen-eral regression analyses provide only an overview.Therefore, the relationship between elevation andprecipitation was examined locally. Maps of the coef-ficient of determination in five data neighbourhoods(that is the nearest observations to a given station areincluded in the regression and the coefficients of thefitted model are attached to the station location) forthe four illustrative months are given in Fig. 8. Thecoefficient of determination varies locally with clus-ters of large values visible in the north for October,

January and March. Clearly, the evident patternswill vary as the data neighbourhood changes. Theseresults suggest that using elevation as a secondaryvariable in estimation will increase the accuracy ofestimates in some locations (that is, those locationswhere r2 is large). In contrast, in places where r2 issmall, OK is likely to provide estimates as accurateas those provided by RK and KED. For the month ofJune, the large values of the coefficient of determina-tion are visible at high altitude (the Dorsal in the northand the Dhahar toward the south).

4.3 Structural analysis

Omni-directional variograms were estimated fromthe precipitation data for each month and those ofOctober, January, March and June are shown in Fig. 9.The parameters of the models fitted to the variogramsof all months are detailed in Table 3. Almost allthe general variations of the monthly average rain-fall (11 months out of 12) are modelled by the powertype of variogram, which highlights the presenceof trend or drift caused by the different directionaldrifts. In fact, in the eastwest, northwestsoutheastand northeastsouthwest directions, variogram mod-els are mostly of power type. Therefore, the generaldrift seems to be the result of the combination ofthe three secondary trends caused by the eastwest,northwestsoutheast and northeastsouthwest direc-tions. Finally, in the northsouth direction, variogrammodels are bounded except for the transition months(from May to August), which indicates a seasonal

(a) (b)

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2000R2 = 0.0074 R

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Fig. 7 Scatter plots of monthly rainfall against elevation for: (a) October, (b) January, (c) March and (d) June.

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Fig. 8 Coefficient of determination: elevation vs precipitation in five data neighborhoods (see text for explanation) for: (a)October, (b) January, (c) March and (d) June.

trend. The most important nugget effect values ofthe average monthly rainfall concern the northwestsoutheast direction, then the eastwest one. Theyare the most weak in the northsouth direction. Thegeneral variograms present nugget effect values that

vary between the directional nugget effect values forJanuaryAugust. The nugget effect values are themost marked in the northwestsoutheast direction inthe winter months, which corresponds to the princi-pal trend direction generating the greatest variations

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Fig. 9 Simple semivariograms: (a) October, (b) January, (c) March and (d) June. The experimental variogram is representedby dots () and the model by a solid line ().

compared to the other directions. There is a significantspatial variation, which matches the winter seasonof frontal disturbances coming from the northwestencountering the mountainous chains of the Dorsal ina perpendicular direction.

Variograms were also estimated for the resid-uals of the regression (between elevation and pre-cipitation amount). It is clear that these variograms(Fig. 10) have very similar aspects to the generalvariograms (Fig. 9). Thus, the variogram of residu-als from the regression represents variation remainingeven after the trend has been removed. The relation-ship between rainfall and elevation is also highlightedusing the cross-variograms (Table 4). Except for thewinter months, represented by January, the influenceof elevation on the spatial distribution of rainfall isobserved in all the other seasons: autumn, spring andsummer represented respectively by October, Marchand June (Fig. 11). In fact, the corresponding cross-variograms are well structured. For October, the rain-fall gradient and elevation are inversely proportional,in contrast to March and June, where they evolve inthe same direction. During the rainy season, repre-sented by January, the cross-variogram shows rather

a cloud of points. These latter confirm the absenceof the influence of elevation on rainfall during thisseason, as the perturbations are wide-ranging and canreach thousands of kilometres. Therefore, rainfall dis-tribution is indifferent to the orographic obstacles. Forother seasons, rainfall is of convective type (espe-cially in summer), thus in direct correlation withthe relief. The variogram features of elevation forthe 12 monthly average rainfalls are summarized inTable 5.

4.4 Data mapping

All the variogram parameters are introduced into thedifferent kriging algorithms to obtain monthly rainfallmaps and the corresponding error estimation maps.The precipitation maps (Fig. 12(a)) obtained by KEDshow that during the rainy months, the pattern of pre-cipitation contours take the aspect of the bends of theDEM in the regions with very weak densities of sta-tions: the chain of the Dhahar in the south and in theregions where rainfall and elevation are little corre-lated: the high altitudes of the Dorsal in the north.In fact, in these regions we notice that the KEDmodel

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Table3Pa

rametersof

thesimplevariog

rams.

Mon

thDirection

Mod

eltype

Parameters

Mon

thDirection

Mod

eltype

Parameters

C0

C0

Sep.

GPo

wer

210

00.09

0.95

Mar.

GPo

wer

6240

0.5

0.94

EW

Sph

erical

195

064

4915

2867

EW

Power

1440

0.5

0.98

NES

WPo

wer

165

00.06

1NES

WPo

wer

1920

0.5

0.91

NS

Exp

onential

091

5015

1693

NS

Exp

onential

035

520

2120

51NWS

EExp

onential

750

8400

1619

93NWS

EPo

wer

7200

0.38

0.99

Oct.

GPo

wer

968

80.21

51

Apr.

GPo

wer

2730

0.4

0.94

EW

Sph

erical

114

048

779

1659

93EW

Power

1950

0.35

0.98

NES

WSph

erical

398

834

768

1473

93NES

WPo

wer

4680

0.1

1NS

Sph

erical

171

033

060

7600

0NS

Exp

onential

026

749

2200

00NWS

EExp

onential

228

047

507

1593

39NWS

EPo

wer

2730

0.35

0.98

Nov.

GPo

wer

2420

00.43

41

May

GPo

wer

850

0.14

0.96

EW

Sph

erical

330

011

0000

2000

00EW

Power

00.13

61

NES

WSph

erical

067

393

1320

00NES

WPo

wer

1190

0.05

61.01

NS

Exp

onential

092

040

1562

00NS

Power

1190

0.05

61

NWS

ESph

erical

1210

011

0000

1636

51NWS

EPo

wer

00.12

11

Dec.

GPo

wer

3060

00.83

1Jun.

GPo

wer

800

0.02

31

EW

Power

1439

41.8

0.96

EW

Power

850

0.00

351.2

NES

WPo

wer

900

01.62

0.92

NES

WPo

wer

550

0.02

71

NS

Exp

onential

014

6592

1616

93NS

Power

300

0.02

51

NWS

EPo

wer

2158

81.54

0.99

NWS

EPo

wer

500.03

31

Jan.

GPo

wer

1280

01.58

0.95

Jul.

GPo

wer

825

0.00

20.97

EW

Power

1280

01.52

0.96

EW

Purenu

gget

1100

--

NES

WPo

wer

320

01.6

0.92

NES

WPurenu

gget

1012

--

NS

Exp

onential

015

3600

2200

00NS

Power

308

0.02

0.88

NWS

EPo

wer

1760

01.50

40.98

NWS

EPo

wer

132

0.02

0.9

Feb.

GPo

wer

990

01.1

0.94

Aug

.G

Sph

erical

264

3300

2000

00EW

Power

659

60.65

1EW

Power

561

0.01

61

NES

WPo

wer

439

61.1

0.96

NES

WPo

wer

561

0.01

31.02

NS

Exp

onential

089

100

2200

00NS

Power

396

0.02

1NWS

EPo

wer

1319

31.04

0.97

NWS

EPo

wer

462

0.01

71.01

Note:

G:g

eneral;E

W:e

astwest;NWS

E:n

orthwestsoutheast;NES

W:n

ortheastsou

thwest.

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(a) (b)

(c) (d)

120 000

Omnidirectional

Omnidirectional Omnidirectional

Omnidirectional

100 000

80 000

60 000

40 000

20 000

00 40 000 80 000 120 000 160 000 200 000

|h|240 000 280 000 320 000 360 000

(|h|)

320 000

280 000

240 000

200 000

160 000

120 000

80 000

40 000

00 40 000 80 000 120 000 160 000 200 000 240 000 280 000 320 000 360 000

90 000

81 000

72 000

63 000

54 000

45 000

36 000

27 000

18 000

9 000

0

(|h|)

0 40 000 80 000 120 000 160 000 200 000

|h|

240 000 280 000 320 000 360 000

6 300

5 600

4 900

4 200

3 500

2 800

2 100

1 400

700

00 40 000 80 000 120 000 160 000 200 000

|h|240 000 280 000 320 000

Fig. 10 Residual semivariograms: (a) October, (b) January, (c) March and (d) June. The experimental variogram isrepresented by dots () and the model by a solid line ().

Table 4 Parameters of the cross-variogram.

Month Model type Parameters

C0

September Power 1760 0.03 0.95October Power 0 0.03 1November Power 0 0.0064 1December Power 5100 0.0015 1January Power 4320 0.023 0.95February Power 3894 0.038 0.94March Power 6580 0.08 0.94April Power 4620 0.08 0.94May Power 3220 0.08 0.96June Power 700 0.06 1July Power 252 0.03 0.97August Spherical 0 6 800 200 000

allows the dominance of the external drift that is thealtitude. For the dry months, we rediscover the samepattern of precipitation contours as the maps obtainedby OK (Slimani et al. 2007). The same result wasobtained by Tapsoba et al. (2005), who indicate that,in the poorly sampled regions, the spatial organizationof rain values reflects, to various degrees, that of thetopography. Rossiter (2005, 2007) confirms this resultwhile mentioning that, if the principal and the auxil-iary variables are little correlated, the estimation byKED resembles the drift.

From a visual check, the plotted rainfall mapsobtained by RK (Fig. 12(b)) are very similar to

those obtained by KED. This result is confirmed bythe studies conducted by Hengl et al. (2007). Themodel of RK always shows, for the rainy months(OctoberApril), the dominance of the elevation val-ues over precipitation in the poorly-sampled zonesand the little correlated zones. In contrast to thepreceding maps, those obtained by CK (Fig. 12(c))show a smooth zonal pattern during the humid sea-son. Goovaerts (2000) found the same result whileestimating annual averaged rainfall: the CK does notimprove, nor damages, the results obtained by the OK(Slimani et al. 2007); this is due to the weak corre-lation between the principal and secondary variablesduring this season. The relief introduction does notchange the general aspect of the maps significantlyin the well-sampled regions. The maps obtained byCK, during the spring and summer, present few dif-ferences from that obtained by OK (Slimani et al.2007). This is the visible effect of elevation dur-ing the low-rain season found in the structuralanalysis.

Independently of the season and of the predic-tion method, uncertainty associated with the rainfallestimation (Fig. 13) tends to be highest in placeswhere there are few or no sample data (southwest andextreme south). The most notable uncertainty valuesconcern the winter, which is illustrated by January(Fig. 13 (a-2), (b-2), (c-2)), since, during this season,

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(a) (b)

(c) (d)

Omnidirectional

4 000

2 000

0

2 000

4 000

6 000

8 000

10 000

12 000

14 0000 40 000 80 000 120 000 160 000 200 000

|h|

240 000 280 000 320 000 360 000

(|h|) Omnidirectional

18 000

15 000

12 000

9 000

6 000

3 000

0

3 000

6 0000 40 000 80 000 120 000 160 000 200 000

|h|

240 000 280 000 320 000 360 000

(|h|)

Omnidirectional

24 000

21 000

18 000

15 000

12 000

9 000

6 000

3 000

00 40 000 80 000 120 000 160 000 200 000

|h|

240 000 280 000 320 000 360 000

(|h|)Omnidirectional

20 000

18 000

16 000

14 000

12 000

10 000

8 000

6 000

4 000

2 000

00 40 000 80 000 120 000 160 000 200 000

|h|

240 000 280 000 320 000 360 000

(|h|)

Fig. 11 Cross-semivariograms: (a) October, (b) January, (c) March and (d) June. The experimental variogram is representedby dots () and the model by a solid line ().

Table 5 Experimental semivariogram of elevation.

Elevation

Model C0

September Power 18 720 0.3 0.95October Power 4 800 0.3 1November Power 4 800 0.3 1December Power 4 800 0.3 1January Power 11 040 0.3 0.95February Power 11 040 0.3 0.94March Power 11 040 0.3 0.94April Power 11 040 0.3 0.94May Power 9 600 0.3 0.96June Power 4 800 0.3 1July Power 10 080 0.3 0.97August Spherical 6 240 48 000 2 105

Notes: C0 (110 mm2) nugget effect. (110 mm2) sill (for the spherical model) or slope (for thepower model). (m) range (for the spherical model) or power coefficient (forthe power model).

the spatial variability is the most marked. In the northand centre of Tunisia, the uncertainty means reach amaximum of 22 mm, and become increasingly greatertoward the south, with a maximum of 48 mm inthe extreme south. To the south, we note that theRK method begins to distinguish itself from othertype of kriging, while giving the weakest estimationuncertainty throughout the year.

5 VALIDATION PROCEDURE

To assess the performance of each prediction methodof the monthly average rainfall, cross-validation wasused. This technique allows the comparison of theestimated and the true or actual values using the infor-mation available in our sample data set. The samplevalues are temporarily discarded from the sample dataset; the value is then estimated using the remainingsamples. The estimates are finally compared to thetrue values (Fig. 14). The KED and RK gave the samevalues of coefficient of determination between theactual and the estimated rainfall amounts (Fig. 15).For humid seasons (SeptemberFebruary), the largestcoefficient of determination is given by KED and RKwhereas for the less humid season (April and May),the CK method gave the largest coefficient of deter-mination values. In terms of the RMSE (Fig. 16),CK provides the most accurate estimates for almostall months. The largest RMSE (and thus the mostbiased estimates) are for the winter (December andJanuary).

6 CONCLUSION

The methodology consists of a sequence of stepswhich provide different information about the distri-bution of the considered variable. The major steps are

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100

200

300

400

500

600

700

800

900

1000

1100

4 4.5 5 5.5 6 6.5 7

3.4

3.5

3.6

3.7

3.8

3.9

4

4.1

0

200

400

600

800

1000

1200

1400

1600

1800

longitude (m)

latit

ude

(m)

Isohytes des pluies - Janvier (1/10 mm)

4 5 6 7

3.4

3.5

3.6

3.7

3.8

3.9

4

4.1

100

200

300

400

500

600

700

800

900

longitude (m)

Isohytes des pluies - Mars (1/10 mm)

latit

ude

(m)

4 5 6 7

3.4

3.5

3.6

3.7

3.8

3.9

4

4.1

0

50

100

150

200

250

longitude (m)

latit

ude

(m)

Isohytes des pluies - Juin (1/10 mm)

4 5 6 7

3.4

3.5

3.6

3.7

3.8

3.9

4

4.1

100

200

300

400

500

600

700

800

900

1000

1100

longitude (m)

latit

ude

(m)

Isohytes des pluies - Octobre (1/10 mm)

4 5 6 7

3.4

3.5

3.6

3.7

3.8

3.9

4

4.1

0

200

400

600

800

1000

1200

1400

1600

1800

longitude (m)

latit

ude

(m)


4 5 6 7

3.4

3.5

3.6

3.7

3.8

3.9

4

4.1

100

200

300

400

500

600

700

800

900

1000

1100

1200

longitude (m)

latit

ude

(m)


4 4.5 5 5.5 6 6.5 7

3.4

3.5

3.6

3.7

3.8

3.9

4

4.1

0

50

100

150

200

250

longitude (m)

latit

ude

(m)

Isohytes des pluies - Juin(1/10 mm)

4 4.5 5 5.5

(1)

6 6.5 7

3.4

3.5

3.6

3.7

3.8

3.9

4

4.1

100

200

300

400

500

600

700

800

900

1000

1100

1200

4 5 6 7

3.4

3.5

3.6

3.7

3.8

3.9

4

4.1

10020030040050060070080090010001100120013001400150016001700180019002000

4 4.5 5 5.5 6 6.5 7

3.4

3.5

3.6

3.7

3.8

3.9

4

4.1

200

300

400

500

600

700

800

900

1000

1100

1200

1300

4 4.5 5 5.5 6 6.5 7

3.4

3.5

3.6

3.7

3.8

3.9

4

4.1

50

100

150

200

250

longitude (m)

latit

ude

(m)


4 5 6 7

105

106

106 106 106 106

106 106 106 106

106 106

1061200

1100

1000

3.4

3.5

3.6

3.7

3.8

3.9

4

4.1

longitude (m)

latit

ude

(m)


longitude (m)

latit

ude

(m)


longitude (m)

latit

ude

(m)


longitude (m)

latit

ude

(m)

Isohytes des pluies - Juin (1/10 mm)

(a)

(c)

(b)

105 105 105

105

105 105 105 105

105 105 105

(2) (3) (4)

Fig. 12 Rainfall maps obtained by Uncertainty maps obtained by (a) KED, (b) RK and (c) CK for: (1) October, (2) January,(3) March and (4) June.

the following: compiling a valid data base, monitoringnetwork analysis, structural analysis including variog-raphy, interpolation using geostatistical predictors andcross-validation. The completion of these steps leadsto decision-oriented maps as the final result and torelevant descriptive information provided at each step.

The coefficient of determination, r2, for ele-vation against precipitation for each month is low,with the largest values for the spring and the sum-mer seasons. Therefore, the assumption of a generallinear relationship between elevation and precipita-tion for these seasons is reasonable. However, KED

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longitude (m)

latit

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(m)

4 5 6 7

105

3.4

3.5

3.6

3.7

3.8

3.9

4

4.1

106

110

120

130

140

150

160

170

180

190

200

longitude (m)

latit

ude

(m)

Ecart type - Janvier (1/10 mm)

4 5 6 7

105

3.4

3.5

3.6

3.7

3.8

3.9

4

4.1

106

170

180

190200

210

220

230

240250

260

270

280290

300

310

320

330

longitude (m)

latit

ude

(m)

Ecart type - Mars (1/10 mm)

4 5 6 7

105

3.4

3.5

3.6

3.7

3.8

3.9

4

4.1

106

100

110

120

130

140

150

160

170

180

longitude (m)

latit

ude

(m)

Ecart type - Juin (1/10 mm)

4 5 6 7

105

3.4

3.5

3.6

3.7

3.8

3.9

4

4.1

106

20

25

30

35

40

45

longitude (m)

latit

ude

(m)

Ecart type - Octobre (1/10 mm)

4 5 6 7

105

3.4

3.5

3.6

3.7

3.8

3.9

4

4.1

106

120

140

160

180

200

220

240

260

280

300

longitude (m)

latit

ude

(m)


4 5 6 7

105

3.4

3.5

3.6

3.7

3.8

3.9

4

4.1

106

140160180200220240260280300320340360380400420440460480500520540

longitude (m)

latit

ude

(m)


4 5 6 7

105

3.4

3.5

3.6

3.7

3.8

3.9

4

4.1

106

100110120130140150160170180190200210220230240250260270280290

longitude (m)

latit

ude

(m)


4 5 6 7

105

3.4

3.5

3.6

3.7

3.8

3.9

4

4.1

106

35

40

45

50

55

60

65

70

75

80

85

90

95

longitude (m)

latit

ude

(m)


4 5 6 7

105

3.4

3.5

3.6

3.7

3.8

3.9

4

4.1

106

120

140

160

180

200

220

240

260

280

300

longitude (m)la

titud

e (m

)


4 5 6 7

105

3.4

3.5

3.6

3.7

3.8

3.9

4

4.1

106

180200220240260280300320340360380400420440460480500520

longitude (m)

latit

ude

(m)


4 5 6 7

105

3.4

3.5

3.6

3.7

3.8

3.9

4

4.1

106

100110120130140150160170180190200210220230240250260270

longitude (m)

latit

ude

(m)


4 5 6 7

105

3.4

3.5

3.6

3.7

3.8

3.9

4

4.1

106

20

25

30

35

40

45

50

55

60

65

70

(a)

(c)

(b)


(1) (2) (3) (4)

Fig. 13 Uncertainty maps obtained by (a) KED, (b) RK and (c) CK for (1) October, (2) January, (3) March and (4) June.

and RK are informed by this relationship locally, so,the relationship between elevation and precipitationwas examined locally. The coefficient of determina-tion varies locally with seasons and regions. Theseresults suggest that using elevation as a secondaryvariable in estimation will increase the accuracy ofestimates in locations where r2 is large. In addi-tion, cross-variograms show the control of elevationon the spatial distribution of rainfall for all seasonsexcept the winter. The use of elevation as secondaryinformation is therefore justified.

This paper indicates that, for most months, theuse of elevation data to inform estimation of monthly

precipitation in Tunisia is beneficial. Interpolationsof an under-sampled target variable are improved byusing an auxiliary variable in KED, RK and CK.Contour diagrams for KED and RK were similarand exhibited a pattern corresponding more closelyto local topographic features, while CK showed asmooth zonal pattern. Smaller prediction variancesare obtained for the RK algorithm.

Co-kriging provides more accurate estimates,judging by the cross-validation RMSE, than any othertechnique for almost all the months. We demonstratedthat, using relatively simple procedures and widelyavailable secondary data (elevation), it is possible

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(1) (2) (3)

Fig. 14 Correlation between actual values and estimated values obtained by cross-validation: (1) KED, (2) RK and (3) CK.

to provide more accurate estimates of precipitationthan is achievable using standard popular interpola-tion methods. The methods applied herein have thepotential to be applied in a wide range of environ-ments. We showed that, when elevation was used as asecondary variable, the accuracy of estimating precip-itation was increased, depending on the interpolationtechnique.

Finally, the variograms estimated for residualsfrom the regressions have highly similar aspects tothe general variograms. So, the variogram of residuals

from the regression represents variation remainingeven after the trend has been removed. This indicatesthat elevation is not the only factor influencing therainfall distribution, and future work may researchinto the secondary information most indicative for themonthly average rainfall prediction.

Acknowledgements We gratefully acknowledge theINM and the DGRE for providing rainfall data, andan anonymous reviewer for critical reviews of themanuscript.

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1

KED

RK

COK

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

Septe

mber

Octob

er

Nove

mber

Dece

mber

Janu

ary

Febr

uary

Marc

hAp

rilMa

yJu

ne July

Augu

st

Fig. 15 Fluctuation of the coefficients of determination (actual values vs estimated values) by month and by predictionmethod.

Fig. 16 Cross-validation: root mean square errors varying with prediction method and with month.

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Feki, H. and Slimani, M., 2006. Analyse structurale de la pluviomtrieen Tunisie. In: WATMED 3, Troisime confrence interna-tionale sur les ressources en eau dans le bassin mditerranen,Liban.

Goovaerts, P., 2000. geostatistical approaches for incorporating ele-vation into the spatial interpolation of rainfall. Journal ofHydrology, 228, 113129.

Hengl, T., Heuvelink, G.B.M., and Rossiter, D.G., 2007. Aboutregression-kriging: from equations to case studies. Computersand Geosciences, 33, 13011315.

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Jebari, S., et al., 2012. Historical aspects of soil erosion in theMejerda catchment, Tunisia. Hydrological Sciences Journal, 57(5), 901912.

Kitanidis, P.K., 1997. Introduction to geostatisticsapplications inhydrogeology. Cambridge: Cambridge University Press.

Matheron, G., 1970. La thorie des variables rgionalises et sesapplications. Fontainbleau: Ecole Nationale Suprieure desMines de Paris, Cahiers du centre de morphologie mathma-tique, fascicule 5.

Odeh, I.O.A., McBratney, A.B., and Chittleborough, D.J., 1995.Further results on prediction of soil properties from ter-rain attributes: heterotopic cokriging and regression-kriging.Geoderma, 67 (34), 215226.

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APPENDIX AKriging with external drift (KED)

Kriging with external drift allows the prediction of avariable Z, known only at a small set of points of thestudy area, through another variable Y , exhaustivelyknown in the same area. We chose to model Z witha random function Z(s) and Y as a deterministic vari-able Y (s) with s a particular point of a field S. Thetwo quantities are assumed to be linearly related, i.e.it is assumed that the expected value of Z(s) is equalto Y (s) up to a constant a0 and a coefficient b1:

E [Z(s)] = a0 + b1Y (s) (A1)

We examine the case of a random function Z(s),whose prediction is to be improved by introducing theshape function Y (s) that provides detail at a smallerscale than the average sample spacing for Z(s). Theestimator Z(s0) at location s0 is a linear combinationof the sample values Z(si) at location si (i= 1, . . . , n):

ZKED(s0) =n

i=1KEDi Z(si) with

ni=1

i = 1 (A2)

We look for an unbiased predictor, that is one with aprediction error that is expected to be zero, E[Z(s0) Z(s0)] = 0, so that:

E[Z(s0)] = E[Z(s0)] (A3)

This equality can be developed into:

E[Z(s0)] =n

i=1iE[Z(si)]

= a0 + b1n

i=1i Y (si)

= a0 + b1 Y (s0)

(A4)

This equation implies that the weights should be onaverage consistent with an exact interpolation of Y (s):

Y (s0) =n

i=1i Y (si) (A5)

The objective function (O) to minimize in this caseconsists of the prediction variance, E2(s0), and twoconstraints:

o = 2E(s0) 1(

ni=1

i 1)

2(

ni=1

i Y (si) Y (s0)) (A6)

where 1 and 2 are Lagrange parameters, and theprediction variance:

2E(s0) = var[Z(s0) Z(s0)]

=n

i=1

nj=1

ijC(si sj)

2n

i=1iC(si s0) + C(0)

(A7)

with C the covariance function of Z.The minimum of this quadratic function is found

by setting the partial derivatives of the objective func-tion O (i,1,2) to zero, leading to the system ofequations:

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nj=1

jC(si sj) 1 2Y (si)

= C(si s0) for i = 1, . . . , nn

j=1j = 1

nj=1

j Y (sj) = Y (s0)

(A8)

with the minimal prediction variance:

2K = C(0) n

i=1iC(si s0)+1 + 2Y (s0) (A9)

The mixing of a second-order stationary randomfunction with a nonstationary mean function mayseem surprising. However, stationarity is a conceptthat depends on scale (Wackernagel 1994). Data cansuggest stationarity for widely-spaced data on Z(s),while they look nonstationary when inspecting thefine detail provided by a function Y (s). Thus KEDconsists of incorporating into the kriging system sup-plementary universality conditions about one or sev-eral external drift variables Yi(s) i = 1, . . . , Mmeasured exhaustively in the spatial domain. Thefunctions Yi(s) need to be known at all locations siof the samples Z(si), as well as at the nodes of theestimation grid. In this method, we assume linearrelationships between the variable of interest and theauxiliary variables at the observation points of thevariable of interest. This assumption is very impor-tant in the prediction using KED method. Thus, if anonlinear function better describes the relationshipsbetween the two variables, this function should firstbe used to transform the data of the auxiliary variable.The transformed data could then be used as externaldrift.

APPENDIX BRegression kriging (RK)

In the case of RK, rainfall at a new, unvisited location(s0) is predicted by summing the predicted drift andresiduals:

Z(s0) = m(s0) + e(s0) (B1)

where the drift m is commonly fitted using linearregression analysis, and the residuals e are interpo-lated using OK:

Z(s0) =p

i=0i qi(s0) +

nj=1

j(s0) e(sj) (B2)

with q0(s0) = 1; where i are the estimateddrift model coefficients, qi(s0) is the ith externalexplanatory variable or predictor at location s0, p isthe number of predictors, j(s0) are weights deter-mined by the covariance function, and e(sj) arethe regression residuals. In matrix notation, the RKmodel is:

Z = qt + (B3)

where is the zero-mean regression residual. Thepredictions are made by:

Z(s0) = qt0 + t0e (B4)

where q0 is vector of p + 1 predictors at s0, isvector of p + 1 estimated drift model coefficients,0 is vector of n kriging weights, and e is vector ofn residuals. The drift model coefficients are prefer-ably solved using the generalized least squares (GLS)(Hengl et al. 2007) estimation to account for spatialcorrelation of residuals.

GLS = (qtC1q)1qtC1z (B5)where q is the matrix of predictors at all observedlocations (n p + 1), Z is the vector of sampledobservations and C is the n n covariance matrixof residuals. The covariances between point pairsC(si,sj), under stationarity assumptions also writ-ten as C(h), are typically estimated by modelling avariogram.

The estimation variance of RK is given by:

2E = (C0 + C1) ct0C1c0 + (q0 qtC1c0)t

(qtC1q)1 (q0 qtC1c0)(B6)

where c0 is the vector of covariances between residu-als at the unvisited and observation locations; and C0and C1 are, respectively, the nugget effect and the sillof the variogram model.

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APPENDIX CKriging and co-kriging

Let z(si) and z(si) represent realizations of ran-dom variables Z(si) and Z(si) at particular pointssi within a field S. It is obvious that the (co)krigingestimator is the best linear unbiased estimator, so:

The (co)kriging estimator Z(s0) is a linear com-bination of the sample values Z(si) and Z(si):

Z(s0) =ni=1

iZ(si) +nj=1

jZ(sj) (C1)

where Z(s0) is the estimate of Z at point s0, nand n are the number of data points of Z and Zused in estimation, and i and j are the associ-ated weights. For ordinary kriging, Z(si) representsthe values of the monthly average rainfall at samplepoint si and the weights j are 0, since only monthlyaverage rainfall contributes to the estimation process.For co-kriging, Z(si) and Z(sj) represent the valuesof monthly average rainfall and elevation at samplepoints si and sj, respectively.

the weights in equation (C1) are determined by mini-mizing the estimation variance:

var[Z(s0) Z(s0)] (C2)subject to the constraint that the estimate be unbiased:

E[Z(s0) Z(s0)] = 0 (C3)This yields the kriging system of equations:

ni=1

i(sm, si) + = (sm, s0) m = 1, . . . , n

ni=1

i = 1(C4)

and the co-kriging system of equations:

ni=1

i(sm, si) +nj=1

j(sm, sj) +

= (sm, s0) m = 1, . . . , nni=1

i(sm, si) +nj=1

j(sm, sj) +

= (sm, s0) m = 1, . . . , nni=1

i = 1

ni=1

j = 1

(C5)

where is the semivariance of Z (when = )or cross-semi variance of Z and Z (when = )at a separation distance h. The , , values areLagrange multipliers.

When deriving the co-kriging system, there is anextra constraint of unbiasedness; therefore, the lastequation in the system is required. Solving this sys-tem of equations for the weights and the Lagrangemultipliers allows the calculation of the point valueestimates Z(s0) by equation (C1), and the estima-tion variance:

2(s0) = +ni=1

i(si, s0)

+nj=1

j(sj, s0)

(C6)

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incorporating elevation in rainfall interpolation in tunisia using geostatistical methods

Documents

eaux et forts

institut national agronomique

sciences hydrologiques

christophe cudennec3

mohamed slimani2

onofcitation feki

registered office

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