Comparison of stochastic and deterministic methods for mapping groundwater level spatial variability in sparsely monitored basins

Comparison of stochastic and deterministic methodsfor mapping groundwater level spatial variability in sparselymonitored basins

Ε. A. Varouchakis & D. T. Hristopulos

Received: 20 May 2011 /Accepted: 11 January 2012 /Published online: 8 February 2012# Springer Science+Business Media B.V. 2012

Abstract In sparsely monitored basins, accurate map-ping of the spatial variability of groundwater levelrequires the interpolation of scattered data. This paperpresents a comparison of deterministic interpolationmethods, i.e. inverse distance weight (IDW) and mini-mum curvature (MC), with stochastic methods, i.e. ordi-nary kriging (OK), universal kriging (UK) and krigingwith Delaunay triangulation (DK). The study area is theMires Basin of Mesara Valley in Crete (Greece). Thissparsely sampled basin has limited groundwater resour-ces which are vital for the island’s economy; spatialvariations of the groundwater level are important fordeveloping management and monitoring strategies. Weevaluate the performance of the interpolation methodswith respect to different statistical measures. The Spartanvariogram family is applied for the first time to hydro-logical data and is shown to be optimal with respect tostochastic interpolation of this dataset. The three stochas-tic methods (OK, DK and UK) perform overall betterthan the deterministic counterparts (IDWand MC). DK,which is herein for the first time applied to hydrologicaldata, yields the most accurate cross-validation estimate

for the lowest value in the dataset. OK and UK lead tosmooth isolevel contours, whilst DK and IDW generatemore edges. The stochastic methods deliver estimates ofprediction uncertainty which becomes highest near thesoutheastern border of the basin.

Keywords Environmental monitoring . Geostatistics .

Kriging . Groundwater level . Ungauged basin .

Fractional Brownian motion

Introduction

The accurate mapping of groundwater levels in an aqui-fer is important for effective management andmonitoringdecisions. However, the number and spatial distributionof hydraulic headmeasurements are not always sufficientto accurately represent the groundwater levels in a givenaquifer. Estimates at unsampled locations can be obtainedby applying geostatistical and deterministic interpolationmethods to the available data. This study aims to com-pare the performance of stochastic versus deterministicmethods for mapping groundwater level in areas withsparsely distributed measurements and to specify addi-tional observation locations where denser sampling isneeded.

Interpolation methods routinely used for groundwaterlevel mapping include inverse distance weighting (IDW)(Gambolati and Volpi 1979; Philip and Watson 1986;Rouhani 1986; Buchanan and Triantafilis 2009; Sun etal. 2009) and stochastic methods such as ordinary

Environ Monit Assess (2013) 185:1–19DOI 10.1007/s10661-012-2527-y

Ε. A. Varouchakis :D. T. Hristopulos (*)Department of Mineral Resources Engineering,Technical University of Crete,Chania, Greecee-mail: [email protected]

Ε. A. Varouchakise-mail: [email protected]

kriging (Olea and Davis 1999; Prakash and Singh 2000;Desbarats et al. 2002; Theodossiou and Latinopoulos2006; Ahmadi and Sedghamiz 2007; Abedini et al.2008; Yang et al. 2008; Kholghi and Hosseini 2009;Nikroo et al., 2009; Sun et al. 2009; Dash et al. 2010)and universal kriging (Delhomme 1978; Sophocleous etal. 1982; Aboufirassi and Marino 1983; Sophocleous1983; Pucci and Murashige 1987; Kumar et al. 2005;Ahmadi and Sedghamiz 2007; Brus and Heuvelink2007; Gundogdu and Guney 2007; Kumar 2007; Sunet al. 2009).

Deterministic interpolation methods use closed-form mathematical formulas (IDW) or the solution ofa linear system of equations (minimum curvature) tointerpolate the data. The weights assigned to eachsample value depend only on the distance betweenthe sample point and the location of the interpolatedpoint. Deterministic methods are categorized as globaland local: Global methods use the entire dataset forprediction at each point, whilst local methods use datain a neighbourhood around the interpolation point.Deterministic methods can be either exact or inexactinterpolators (Webster and Oliver 2001). Finally, theydo not generate measures of estimate uncertainty.

Stochastic methods employ the spatial correlationsbetween values at neighbouring points. The mostwidely used stochastic method is kriging (Krige1951; Matheron 1963, 1971). Kriging estimates arelinear combinations of the data with weights that fol-low from the no-bias constraint (i.e. zero mean esti-mation error) and the minimization of the mean squareerror. Due to these properties, kriging is called a bestlinear unbiased estimator. The kriging weights aredetermined from a model semivariogram, which isobtained by fitting the empirical semivariogram totheoretical models or by means of the maximum like-lihood estimation method (Kitanidis 1997; Ahmed2007). The semivariogram measures the spatial corre-lation as a function of the distance between datapoints. Kriging is computationally intensive when ap-plied to large datasets (Webster and Oliver 2001), butthe computational complexity is not a problem forsparsely sampled areas.

Kriging allows the estimation of interpolation uncer-tainties (Deutsch and Journel 1992). In the absence of anugget term (e.g. measurement errors or unresolvedfluctuations), kriging is an exact interpolator at the mea-surement points (Delhomme 1974; Tonkin and Larson2002; Ahmed 2007). Optimal results are obtained if the

probability distribution of the data is jointly normal andstationary in space (spatially homogeneous). Krigingactually refers to a family of interpolators, the mostcommon of which are ordinary kriging (OK) and uni-versal kriging (UK). A newly established version iskriging with Delaunay triangulation (DK) (Hessami etal. 2001)

This article compares the interpolation performanceof OK, UK and DK with the deterministic methodsIDW and minimum curvature (MC). The dataset usedinvolves groundwater levels in a sparsely gauged ba-sin. Measuring the relative performance of differentinterpolators is important for environmental monitor-ing. Two spatial interpolation comparison exerciseshave been organized by the Radioactivity Environ-mental Monitoring Group of the Joint Research Centreof the European Commission (Dubois 1998; Duboisand Galmarini 2005). These exercises focused on ra-dioactivity monitoring in the European continent andin particular on automatic (i.e. without user involve-ment) mapping.

Cornford (2005) emphasised the problems of inter-preting comparative interpolation studies. First, theirresults do not admit generalization and are often con-tradictory. In addition, for a single dataset, several orall assessed methods may exhibit similar performance.Hence, the choice of the “optimal” interpolation meth-od is dictated by other factors, such as computationalspeed, implementation cost, scaling with data size andthe ability to make probabilistic predictions (estimatesof the prediction error). Van den Boogaart (2005)agrees that comparative studies based on one or twodatasets can be misleading and that a uniformly opti-mal method for all kinds of datasets does not exist. Hepoints out that the performance and utility of themethods should be assessed in terms of decision-making requirements (e.g. concerning outliers, estima-tion variances) and its adaptability to the complexityof the specific dataset (e.g. sparse data, presence oftrends) and not only in terms of mean square errors(Van den Boogaart 2005). Myers (2005) emphasisesthe use of clear software standards, common hardwareconfigurations and an extensive set of performancemeasures to allow the duplication of the reportedresults by others.

In light of the above remarks, we use the sameprogramming environment for all the methods so thatthe results are directly comparable. The methods aredescribed in detail including the values of user-defined

2 Environ Monit Assess (2013) 185:1–19

parameters to allow reproduction of our results byothers. The performance of the interpolation methodsfor the Mires Basin dataset is based on cross-validationmeasures, uncertainty estimation ability, methodo-logical specifications (search neighbourhood, differ-entiability, contour map effects) and adaptability tothe dataset statistics (size of data, outliers). Theresults obtained in this paper are useful for map-ping groundwater level spatial variability in basinswith similar characteristics and more generally inenvironmental monitoring applications that involvespatially distributed data (e.g. air pollutants, groundwaterquality, soil contaminants, etc.).

Materials and methods

Area of study

Mires Basin of the Messara Valley (Fig. 1) is locatedon the island of Crete in Greece. The basin is a down-faulted trough, roughly 14 km long and on average3 km wide. It is filled with alluvial sediments ofQuaternary age, which form an interbedded sequenceof gravels, gravely sands, sands, silts, silty sands andclays (Donta et al. 2006). The basin has marginalgroundwater resources that are extensively used foragricultural activities and human consumption. An

Fig. 1 a Map of Greece showing the Mesara Valley and theMires Basin locations. b Topographic map showing thegroundwater head monitoring locations (dots) in the Mires

Basin along with the corresponding surface elevation andthe temporary river path

Environ Monit Assess (2013) 185:1–19 3

extensive network of pumping stations has been in-stalled since 1984, turning the dry cultivation of olivetrees to drip-irrigated. As a result, productivity hasrisen at the cost of an alarming drop, approximately35 m during the last 30 years, of the groundwaterlevel. Accurate prediction of groundwater level spatialvariations is crucial for the integrated management ofwater resources in the basin and for the prevention ofpossible desertification effects.

The basin has been consistently monitored over thelast 30 years for groundwater level variations, rainfalland surface runoff by the Department of Water Resour-ces Management (DWRM) of the region of Crete. How-ever, the monitoring of spatial variations is poor. Since1984, a number of official boreholes have operated inthe Mires Basin. The data used in this study consist of70 hydraulic head measurements obtained in April 2003(October–April is the wet period of the hydrologicalyear). The data have been provided by the Administra-tion of Land Reclamation of the Prefecture of Crete.This is the only period for which a full set of recordedhead values exists. The sampling locations are unevenlydistributed and mostly concentrated along a temporaryriver that crosses the basin. The range of hydraulic headsvaries between an extreme low of 9.4 metres above sealevel (m a.s.l.) and 62 m a.s.l. The sample statisticalmeasures are presented in Table 1. Since 2003, theregular biannual monitoring of the operating boreholeshas been replaced by continuous monitoring of twotelemetric stations placed in two boreholes selected bythe DWRM.

Interpolation methods

In the following, si (i ¼ 1; . . . ;N ) denotes the sam-pling points, z(si) are the head values (in m a.s.l.) atthese points, and s0 denotes an estimation point, whichis assumed to lie inside the convex hull of the sam-pling network. For mapping purposes, it is assumedthat s0 moves sequentially through all the nodes of the

mapping grid. Below, we briefly review five linearinterpolation methods: two deterministic (IDW, MC)and three stochastic (OK, UK, DK). In linear interpo-lation methods, it holds that

zðs0Þ ¼X

fi:si2S0g lizðsiÞ ð1Þ

where S0 is the set of sampling points in the searchneighbourhood of s0. The neighbourhood is empiricallychosen so as to optimize the cross-validation measures.

Inverse distance weight

The estimation with the IDW method is given bymeans of the equation

zðs0Þ ¼X

fi:si2S0gd�ni ;0P

fi:si2S0g d�ni ;0

!zðsiÞ ð2Þ

where di,0 is the distance between the estimation pointand the sampling points and n>0 is the power exponent;usually, n02 is used. IDWassigns larger weights to datacloser to the estimation point s0 than to more distantpoints. Higher values of n increase the impact of valuesnear the interpolated point, whilst lower values of nimply more uniform weights. As it follows from Eq. 2,the weights add up to 1. IDW is an exact and convexinterpolation method (Hengl 2007). In addition, it isvery fast, straightforward and computationally non-intensive (Webster and Oliver 2001). According toEq. 2, as the distance of si from s0 increases, the respec-tive weight is reduced. IDW’s disadvantages are thearbitrary choice of the weighting function and the lackof an uncertainty measure (Webster and Oliver 2001).

Minimum curvature

MC interpolation is based on theminimization of the total

square curvature of the surface z(s), i.e.Rds r2zðsÞ½ �2,

subject to the data constraints. In MC, the interpolatedsurface can be viewed as a thin linear elastic plate pinned

Table 1 Statistical measures of the hydraulic head data

zmin (m a.s.l.) z0.25 (m a.s.l.) z0.50 (m a.s.l.) mz (m a.s.l.) z0.75 (m a.s.l.) zmax (m a.s.l.) bσz (m a.s.l.) bsz bkz9.40 20.50 24.25 32.05 40.20 62.00 12.40 0.81 2.58

zmin minimum value, z0.25 first quartile, z0.50 median, mz mean, z0.75 third quartile, zmax maximum value, bσz standard deviation, bszskewness coefficient, bkz kurtosis coefficient


to the data values at the sampling points. The estimate isobtained by solving the biharmonic partial differentialequation (Briggs 1974; Sandwell 1987), i.e.

@2

@x2þ @2

@y2

� �@2zðsÞ@x2

þ @2zðsÞ@y2

� �¼ 0 ð3Þ

conditioned by the data values z(si). The interpolatingfunction z(s) honours the observed data and tends to aplanar surface as the distance between the interpolationpoint and the observations increases. Typical applicationsof MC include interpolating hydrocarbon (oil) depths(Cooke et al. 1993), interpolating gravitometric and mag-netometric geophysical data for mineral exploration(Mendonca and Silva 1995; Kay and Dimitrakopoulos2000) and mapping the earth’s surface (Yilmaz 2007).

The MC method often suffers from oscillations dueto the presence of outliers in the data or due to verylarge gradients. This problem can become important ifthe dataset is relatively small. The MC interpolation isbased on the Green’s function, gm, of the biharmonicequation, which satisfies r4gmðs� s0Þ ¼ dðs� s0Þ,where dðs� s0Þ is the Dirac delta function. The 2DGreen’s function is given by gmðdÞ ¼ d2 ln d � 1ð Þ(Sandwell 1987; Wessel 2009). The MC estimate isthen expressed as follows:

zðs0Þ ¼XNi¼1

wi gmðdi;0Þ ð4Þ

The weights wi are determined by solving the follow-ing linear system at the N data locations.

zðsiÞ ¼XNj¼1

wj gmðdi;jÞ ð5Þ

where j ¼ 1; . . . ;N and di,j are the distances between

the sample points di;j ¼ si � sj�� .

Ordinary kriging interpolation

The OK method assumes that z(s) is a random functionwith a constant but unknown mean. The OK estimatez(s0) at s0 is calculated based on a weighted sum of thedata

zðs0Þ ¼X

fi:si2S0g liziðsiÞ ð6Þ

The weights li in Eq. 6 are obtained by minimizing themean square estimation error conditionally on the zero-

bias constraint (Cressie 1993), and they depend on thesemivariogram model, gz(r) (Deutsch and Journel1992).

The empirical semivariogram, bgðrÞ, is defined asthe average square difference of the data values be-tween points separated by the lag vector r. If there areno distinct anisotropies, the omnidirectional empiricalsemivariogram is estimated, e.g. using the Matheronmethod-of-moments estimator

bgðrÞ ¼ 1

2NðrÞXNðrÞ

i¼1

zðsiÞ � zðsi þ rÞ½ �2n o

ð7Þ

where N(r) is the number of pairs at lag r (Deutsch andJournel 1992). bgðrÞ is then fitted to a model functiongz(r).

The kriging weights li are given by the followingðN0 þ 1Þ � ðN0 þ 1Þ linear system of equations:

Xfi:si2S0g li gz ðsi; sjÞ þ μ ¼ gz ðsj; s0Þ;

j ¼ 1; . . . ;N0

ð8Þ

Xfi:si2S0g li ¼ 1 ð9Þ

where N0 is the number of points within the searchneighbourhood of s0, gzð si; sjÞ is the semivariogrambetween two sampled points si and sj, gzð sj; s0Þ is thesemivariogram between sj and the estimation point s0,and μ is the Lagrange multiplier enforcing the no-biasconstraint.

The OK estimation variance is given by the follow-ing equation, with the Lagrange coefficient μ compen-sating for the uncertainty of the mean value:

σ2Eðs0Þ ¼

Xfi:si2S0g ligzðsi; s0Þ þ μ ð10Þ

Universal kriging interpolation

In certain cases, the data exhibit a global trend over thestudy area. It is possible to incorporate in kriging atrend (drift function) modelling the global behaviour.The resulting estimation algorithm is known as “uni-versal kriging” (UK) and was proposed by Matheron(1969). UK requires the drift function mz(s) and thesemivariogram of the residuals ez(s) (Goovaerts 1997).The trend is usually approximated by linear or higher


order polynomials of the space coordinates (Ahmed2007). The drift function is given by

mzðsÞ ¼XKk¼1

akfkðsÞ ð11Þ

where fk(s) are the basis functions and ak are the driftcoefficients (Goovaerts 1997). The UK estimator ofthe hydraulic head is expressed as follows:

zðs0Þ ¼ mzðs0Þ þX

fi:si2S0g lieðsiÞ

¼ mzðs0Þ þX

fi:si2S0g li zðsiÞ � mzðsiÞ½ � ð12Þ

where li (i ¼ 1; . . . ;N ) are the UK weights, e(si) is theresidual at si, and mz(s0) is the drift at s0.

The kriging weights are determined by the solutionof the following (N0 þ KÞ � ðN0 þ K) linear systemof equations, where N0 is the number of points withinthe search neighbourhood of s0,

Xfi:si2S0g li gz ðsi; sjÞ þ

XKk¼1

fkðsjÞμk ¼ gz ðsj ; s0Þ;

j ¼ 1; . . . ;N0

ð13ÞX

fi:si2S0g lifkðsiÞ ¼ fkðs0Þ; k ¼ 1; . . . ;K ð14Þ

where gzð si; sjÞ is the semivariogram of the residualsbetween two sampled points si and sj, gzð sj; s0Þ is thesemivariogram of the residuals between a sampledpoint sj and the estimation point s0, and μk are theLagrange multipliers for each basis function. Thekriging variance is given by the following equation(Goovaerts 1997):

σ2Eðs0Þ ¼

Xfi:si2S0g ligzðsi; s0Þ þ

XKk¼1

fkðs0Þμk ð15Þ

Kriging with Delaunay triangulation

Kriging with DK uses the Delaunay triangles to deter-mine the search neighbourhood S0 around the estima-tion point. The kriging equations in DK are identicalto OK (Hessami et al. 2001). DK reduces the

computational cost of kriging and ensures that theestimate depends only on data in each point’s imme-diate neighbourhood.

The Delaunay triangulation (e.g. Fig. 2) is the dualgraph of the Voronoi diagram for the samplinglocations si, i ¼ 1; . . . ;N . The latter is a set ofpolygons Pi, each of which is centred at si andcontains all the points that are closer to si than toany other data point. The Delaunay triangulation isformed by drawing line segments between Voronoivertices if their respective polygons have a commonedge (Okabe et al. 1992; Mulchrone 2003; Ling etal. 2005). The Delaunay triangle containing theestimation point s0 is located using the «T-search1»function of Matlab® (Matlab v.7.5). The vertices ofthe triangle T0 containing s0 are the first-orderneighbours of s0. Second-order neighbours are de-termined from the vertices of the triangles adjacentto T0 that do not belong to T0 (Hessami et al. 2001;see Fig. 2). The number of second-order neighboursranges between 1 and 3. If the search neighbour-hood only includes the first-order neighbours, theCPU time is reduced, but the precision of theestimates is lower (Hessami et al. 2001).

Semivariogram estimation

The omnidirectional empirical semivariogram of the hy-draulic head fluctuations (after trend removal in UK) isdetermined using the method of moments. Anisotropy isnot modelled since the directional semivariograms (notshown here) do not exhibit significant anisotropic de-pendence. The empirical semivariogram is fitted withisotropic classical models (Table 2) such as the exponen-tial, Gaussian, spherical, power law and linear models(Deutsch and Journel 1992); the Matérn model (Matérn1960; Stein 1999; Pardo-Iguzquiza and Chica-Olmo2008); and the new family of Spartan variograms(Hristopulos 2003; Hristopulos and Elogne 2007).

Spartan (SP) semivariograms have been successfullyapplied to various environmental datasets (Elogne et al.2008; Elogne and Hristopulos 2008; Hristopulos andElogne 2009). Herein, they are for the first time appliedto hydrological data. The Spartan covariance functionsof the fluctuation–gradient–curvature (FGC) SSRF

1 The T-search function will be replaced in future Matlab®releases by DelaunayTri class.


model in d03 dimensions are expressed as follows(Hristopulos and Elogne 2007):

CzðhÞ ¼

η0e�hb2

2pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffijη21 � 4j

p sinðhb1Þhb1

� �; for η1j j < 2

η0e�h

8p; for η1 ¼ 2

η0ðe�hw1 � e�hw2 Þ4pðw2 � w1Þh

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffijη21 � 4j

p ; for η1 > 2

8>>>>>>>>><>>>>>>>>>:

ð16Þ

In the above, η0 is the scale factor; η1 is the

rigidity coefficient, b1;2 ¼ 2� η1j j1=2 2= , w1;2 ¼ η1�jðΔj 2= Þ1=2, Δ ¼ η21 � 4

�� 1=2; ξ is a characteristic length;

h ¼ r x= is the normalised lag vector; and h ¼ jhj is itsEuclidean norm. The exponential covariance is recov-ered for η102, whilst for η1j j < 2, the product of theexponential and hole effect (damped sine) model isobtained. A covariance function that is permissible inthree spatial dimensions is also permissible in twodimensions (Christakos 1991). Hence, Eq. 16 can beused in two dimensions, albeit it does not correspond tothe FGC-SSRF 2D covariance (which does not have aknown analytical expression).

For each of the above theoretical models, wedetermine the optimal semivariogram parametersusing the least squares method (Tables 3 and 4).We implemented this method by means of the«fminsearch» Matlab® function which is based onthe Nelder–Mead minimization algorithm. The se-lection of the “optimal semivariogram model” is

Fig. 2 Delaunay triangulation of monitoring sites in MiresBasin. The vertices of the enclosing triangle (dark colour) thatcontains the estimation point s0 are the first-order neighbours of

s0; the vertices of the three adjacent triangles (grey colour) thatdo not belong to the enclosing triangle provide the second-orderneighbours of s0

Table 2 List of theoretical semivariogram functions

Semivariogram models

Exponential gzðrÞ ¼ σ2z 1� exp � rj j

x

� h iGaussian gzðrÞ ¼ σ2

z 1� exp � r2

x2

� h iSpherical gzðrÞ ¼ σ2

z 1:5 rj j=x� 0:5 rj j=xð Þ3h i

θ x� rj jð Þ; if x� rj j < 0; θ ¼ 0; else if x� rj j > 0; θ ¼ 1

Power law gzðrÞ ¼ c rj j2H ; 0 < H < 1 (c is the coefficient and H the Hurst exponent)

Linear gzðrÞ ¼ c rj jMatérn gzðrÞ ¼ σ2

z 1� 21�v

ΓðvÞrj jx

� vKv

rj jx

� n o(v>0 is the smoothness parameter, Γ ð�Þ is the gamma function, Knð�Þ

is the modified Bessel function of the second kind of order ν)

σ2 is the variance, rj j is the Euclidean norm of the lag vector r, and ξ is the characteristic length


based on the results of the leave-one-out cross-validation (see below).

Application methodology

The groundwater level in Mires Basin is mapped usingthe stochastic and deterministic interpolation methodsdescribed above. All the methods are implemented byan original code developed and run in the Matlab®programming environment (Matlab v.7.5 on MicrosoftWindows XP). This approach allows control of themodel parameters and a straightforward comparisonof the results. To avoid numerical instabilities, wenormalise the coordinates of the study area in theinterval [0, 1].

A leave-one-out cross-validation is performed to as-sess the accuracy of the spatial predictions. This ap-proach consists of removing one datum at a time fromthe dataset and estimating its value based on the remain-ing data. The interpolated values are compared to theirmeasured counterparts using the global performancemeasures listed in Table 5 (Isaaks and Srivastava1989; Goovaerts 1997; Leuangthong et al. 2004;Ahmadi and Sedghamiz 2008).

Results and discussion

Global cross-validation measures

Table 6 presents the results for the cross-validationmeasures defined above for each of the interpolationmethods studied. IDW is applied using inverse squaredistance weights (n02). This exponent value is widelyused in geohydrology and also provides more accurateresults for the Mires Basin than the other values. Theoptimum search neighbourhood consists of the fourclosest observation points to the estimation location.The MC method, implemented based on Eqs. 4 and 5,uses the entire dataset for prediction.

Figure 3 shows the empirical semivariogram and itsfit with the optimal Spartan and power law models thatprovide similar cross-validation results for the OK andDK methods. The empirical semivariogram does notapproach a sill, which is interpreted as a lack ofstationarity within the study area. The power lawmodel is non-stationary, whilst the Spartan model isstationary but approaches the sill outside the studyarea. The cross-validation measures obtained withthe above semivariogram models and with the best-fit Matérn model, which gives slightly inferior results,

Table 3 Optimal estimates ofsemivariogram model parametersobtained by a least squares fit tothe experimental semivariogramof the data (columns 2–4)

The search radius defines theneighbourhood used in the OKpredictor (column 5)aNumber of first- and second-order neighbours used by DK ateach estimation point

Model Sill ξ Otherparameters

OK search radius(normalised units)

DK no. ofneighboursa

Exponential 133 0.30 NA 0.38 4–6

Gaussian 160 0.28 NA 0.38 4–6

Spherical 150 0.63 NA 0.59 4–6

Power law 538 NA 2H01.31 0.59 4–6

Linear 331 NA NA 0.38 4–6

Matérn 440 0.94 v00.92 0.59 4–6

Spartan 184 0.46 η101.12 0.46 4–6

Table 4 Optimal estimates ofsemivariogram model parametersobtained by a least squares fit tothe experimental semivariogramof the residuals (columns 2–4)

The search radius defines theneighbourhood used in the UKpredictor (column 5)

Model Sill ξ Other parameters UK search radius(normalised units)

Exponential 142 0.34 NA 0.38

Gaussian 211 0.35 NA 0.38

Spherical 137 0.69 NA 0.59

Power law 500 NA 2H01.39 0.59

Linear 300 NA NA 0.38

Matérn 236 0.66 v00.87 0.59

Spartan 169 0.75 η101.07 0.59


are shown in Table 6. The results obtained with UKusing the same variogram models are also shown inTable 6.

In OK, the Spartan semivariogram model (Fig. 3)gives the most accurate estimates in terms of meanabsolute error (MAE), i.e. 3.37 ma.s.l., compared withthe power law model, which is a close second with3.58 ma.s.l. The Spartan model is also superior withrespect to the other estimation measures (Table 6). DKhas a MAE of 3.48 ma.s.l. respectively for both theSpartan and power law semivariograms. However, asshown in Table 6, the validation measures obtained

with the Spartan model are overall slightly better. TheSpartan semivariogram is thus used in OK and DKinterpolations. For OK, a search radius equal to thecharacteristic length (0.46 in normalised units) yieldsthe best cross-validation results (Table 3). DK is ap-plied using the first- and second-order neighbours ofthe estimation point (Table 3), resulting in higheraccuracy.

For the application of UK, the drift is approximatedbymzðsÞ ¼ k1xþ k2yþ k, where k1029.83, k20−11.14and k023.13 are the drift coefficients (constants) and s0(x,y) are the space coordinates of the data. This is fol-lowed by a calculation of the semivariogram of theresiduals. Leave-one-out cross-validation (Table 6)shows that the Spartan model (Fig. 4) delivers the mostaccurate results with respect to MAE, i.e. 3.40 ma.s.l.,and performs overall better than the other “near-opti-mal” semivariogram models; the power law modelcomes second with 3.50 ma.s.l. Therefore, the Spartansemivariogram is applied in the UK interpolation. Theoptimum search radius used with the Spartan model isequal to 0.59 (normalised units), which is somewhatshorter than the estimated characteristic length (Table 4).The power law model in both OK and UK is alsoapplied using an optimum search radius equal to 0.59(Tables 3 and 4).

The cross-validation measures (Table 6) show thatno method performs significantly better than the

Table 5 List of cross-validation measures used to compare thetrue and estimated values of the hydraulic head

Mean absolute error "MA ¼ 1N

PNi¼1 z�ðsiÞ � zðsiÞj j

Bias "BIAS ¼ 1N

PNi¼1 z

�ðsiÞ � zðsiÞMean absolute relativeerror

"MAR ¼ 1N

PNi¼1

z�ðsiÞ�zðsiÞzðsiÞ

�� Root mean squareerror

"RMS ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1N

PNi¼1 z�ðsiÞ � zðsiÞ½ �2

q

Linear correlationcoefficient R ¼

PN

i¼1zðsiÞ�zðsiÞ½ � z�ðsiÞ�z�ðsiÞ½ �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPN

i¼1

zðsiÞ�zðsiÞ½ �2r ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPN

i¼1

z�ðsiÞ�z�ðsiÞ½ �2r

In the following, z*(si) is the estimated head at point si, obtainedby removing z(si) from the dataset and interpolating the remain-

ing data, whilst zðsiÞ denotes the spatial average of the data andz�ðsiÞ the spatial average value of the estimates

Table 6 Cross-validation measures for the stochastic and deterministic interpolation methods investigated

Method MAE (m a.s.l.) BIAS (m a.s.l.) MARE RMSE (m a.s.l.) R

IDW 3.45 −0.17 0.15 5.58 0.89

MC 4.01 0.10 0.17 6.18 0.87

DK-SP 3.48 0.10 0.15 5.47 0.89

DK-P 3.48 0.14 0.15 5.52 0.87

DK-M 3.63 −0.08 0.15 5.74 0.89

OK-SP 3.37 0.02 0.14 5.15 0.91

OK-P 3.58 0.07 0.15 5.46 0.90

OK-M 3.80 0.02 0.16 5.84 0.89

UK-SP 3.40 0.13 0.14 5.23 0.91

UK-P 3.50 0.09 0.15 5.54 0.89

UK-M 3.80 0.09 0.15 5.78 0.89

Results obtained with the three “optimal” (in terms of cross-validation measures) semivariogram models are presented. Optimal valuesare shown in italics

IDW inverse distance weighted, MC minimum curvature, DK kriging with Delaunay triangulation, OK ordinary kriging, UK universalkriging, SP Spartan semivariogram, P power law semivariogram, M Matérn semivariogram, MAE mean absolute error, MARE meanabsolute relative error, RMSE root mean square error, R linear correlation coefficient


others. OK gives uniformly the best results for themean errors and the correlation coefficient, followed,for most measures, by UK, DK and IDW in the ordermentioned here. OK has clearly the lowest bias, veryclose to zero. MC generates a bias similar to DK andUK, but lower than IDW; however, its performance isinferior with respect to the other validation measures.

The Spartan semivariogram model provides themost accurate cross-validation estimates for the threestochastic methods investigated. In terms of MAE,OK-SP gives the most accurate estimate, followed byUK-SP and DK-SP. The bias of OK-SP is very close tozero, whilst it is worse for DK-SP and UK-SP, in theorder stated. MARE and R are similar for OK-SP andUK-SP, but slightly lower for DK-SP. Finally, OK-SPyields the lowest RMSE, followed by UK-SP and DK-SP. Overall, OK-SP provides the most accurate esti-mation measures compared with the other stochasticand deterministic methods.

Lowest-value estimation

In addition to the global cross-validation results inves-tigated above, a statistic of interest is the estimationaccuracy of the lowest groundwater level, i.e. 9.4 ma.s.l. DK-SP gives the most accurate estimate, i.e. 29 ma.s.l. DK estimates the lowest level 17% more accu-rately than OK-SP and UK-SP, which yield 33 ma.s.l.,and 17.6% more accurately than IDW, which yields33.15 ma.s.l. In contrast, the highest level of 62 ma.s.l. is accurately estimated by both the stochastic anddeterministic interpolation methods. The superior per-formance of DK with respect to the lowest-value esti-mation is due to its local averaging property.

DK is herein applied using both the first-order andsecond-order neighbours of the estimation point. Atthe location of the minimum, up to six neighbours areused in DK. The maximum distance from the neigh-bours (0.11 normalised units) is shorter than the esti-mated optimal radius for OK-SP and UK-SPinterpolations (0.46 and 0.59 normalised units, respec-tively). In order for OK-SP and UK-SP to approachthe DK-SP optimal estimate (29 ma.s.l.), they shouldbe applied with a smaller estimation neighbourhood.Using circular neighbourhoods, the OK-SP and UK-SP neighbours of the minimum value location do notcoincide with the DK-SP neighbours (see Fig. 5).Therefore, the optimal local neighbourhood used byDK-SP is not feasible for OK-SP and UK-SP. IDWyields optimal global cross-validation results if appliedwith a circular neighbourhood that encloses the fournearest neighbours but delivers an estimate of 33.15 ma.s.l. for the lowest value.

In light of the above, we compare the minimumvalue estimation by means of OK-SP, UK-SP andIDW with the same radii (0.05–0.13 normalised units;

Fig. 3 Plot of omnidirectional experimental semivariogram ofgroundwater level data (stars), the optimal Spartan model (param-

eter estimates: variance, bσ2 ¼ 184; characteristic length, bx ¼ 0:46(normalised units); stiffness coefficient,bη1 ¼ 1:12) and the optimal

power lawmodel (bc ¼ 538, 2bH ¼ 1:31). Numbers of pairs used ateach lag distance are also shown on the plot

Fig. 4 Plot of omnidirectional semivariogram of residuals (stars)and optimal Spartan model (parameter estimates: variance,

bσ2 ¼ 169; characteristic length,bx ¼ 0:75 (normalised units); stiff-ness coefficient, bη1 ¼ 1:07). Residuals are derived by removing alinear drift. Numbers of pairs used at each lag distance are alsoshown on the plot


see Table 7). The leave-one-out cross-validationresults are shown in Table 7 using the Spartansemivariogram, which provides the most accurateestimates of the minimum compared with the othermodels tested for all the methods (DK, OK andUK) and all search radii used (OK and UK). Mostof the estimates in Table 7 are better than thoseobtained using “globally optimal” interpolationradii, which are derived by minimizing the meanabsoluter error over all the points.

DK-SP estimation based on the first-order neigh-bours (i.e. the vertices of the triangle enclosing thelowest-value location) provides the same accuracy(31.93 ma.s.l.) as OK-SP, but inferior than UK-SP(see Table 7). By increasing the search radius of OK-SP, UK-SP and IDW, the second-order neighbours areprogressively included (see Fig. 5). The cross-validation performance of OK-SP, UK-SP and IDWimproves (see Table 7) approaching the optimal of

DK-SP (29 ma.s.l.) as the neighbourhood of the latteris closely matched (i.e. for a search radius of 0.11normalised units). The best estimate is obtained withUK-SP (31.34 ma.s.l.).

The optimal radius for the lowest value estimation isnot generally suitable for OK, UK and IDW interpola-tion because it generates search neighbourhoods that donot include any neighbours around some of the datapoints. The smallest search radius that leads to at leastone neighbour for each data point is equal to 0.13(normalised units). However, this value delivers inferiorcross-validation measures for OK-SP, UK-SP and IDWcompared with the respective optimal radii. A radius of0.13 (normalised units) provides a better IDW estimate(32.27 ma.s.l.) of the minimum than the optimal neigh-bourhood (33.15 ma.s.l.). In contrast, the OK-SP andUK-SP estimates (33.23 and 33.15 ma.s.l., respectively)are inferior to those obtained with the “globally optimal”interpolation radii (33 ma.s.l).

Fig. 5 First- and second-order neighbours (full black circles) ofextreme low value (at location marked by a star) of the datasetlocated using Delaunay triangulation. The circles centred on the

estimation point enclose the neighbouring points for specificsearch radius (0.05–0.11 corresponding to normalised units)used for OK, DK and IDW for the extreme low value calculation

Table 7 OK-SP, UK-SP and IDW estimates (in metres above sea level) of the extreme low value in the dataset using search radii(normalised units)

Search radius 0.05 (2) neighbours 0.06 (3) neighbours 0.09 (4+) neighbours 0.11 (5,6+) neighbours 0.13 (5,6+) neighbours

OK-SP 32.61 31.93 32.36 31.50 33.23

UK-SP 32.25 31.71 32.31 31.34 33.15

IDW 33.80 33.23 32.87 31.98 32.27

The radius 0.11 leads to a neighbourhood similar to DK which generates the most accurate estimate, 29 ma.s.l., of the extreme lowvalue. The numbers in parentheses denote the number of Delaunay neighbours present inside the corresponding search radius, whilst (+)denotes the presence of other neighbours as well (see Fig. 5). Delaunay neighbours (symbolised with full black circles in Fig. 5) are thevertices of the enclosing triangle (dark colour) and of the three adjacent triangles (grey colour)

SP Spartan semivariogram


Isolevel contour maps of hydraulic head

Next, we generate isolevel contour maps of thegroundwater surface in the basin. We use interpolatedvalues of the hydraulic head on a 100×100 grid (ac-tual cell size, 114×47 m). Only grid points inside theconvex hull (7,317 grid points) of the sampling net-work are given numerical values to ensure that theinterpolated field is based on sufficient information.The contour maps generated are shown in Figs. 6, 7, 8,9, 10, 11, 12 and 13.

IDW and DK contours are rougher than those esti-mated by means of the other methods. This is due tothe fact that both methods use a small number ofneighbours, leading to considerable variation of the

estimates. MC, OK and UK lead to smoother contours.The smoothness of MC contours is due to the assump-tion of an underlying differentiable function. OK andUK yield very smooth contours because their esti-mates are based on observations within a neighbour-hood defined by the large characteristic length (50–75% of the area’s extent in normalised units).

OK, DK and UK interpolation maps are derivedusing the non-differentiable Spartan model. The powerlaw semivariogram, which is also non-differentiable,gives results similar to the Spartan model. The third bestis a non-differentiable Matérn model with smoothnesscoefficient v<1 (0.92 and 0.87 for original data andresiduals, respectively). Similarly, a non-differentiablesemivariogram (spherical model) was used for the

Fig. 6 Isolevel contour mapof estimated groundwaterlevel in Mires Basin usingIDW

Fig. 7 Isolevel contour mapof estimated groundwaterlevel in Mires Basin usingMC


hydraulic head in a different study (Fasbender et al.2008). We propose an explanation for the non-differentiability of the groundwater level surface. Thedata reflect the surface formed by the upper boundary ofthe saturated zone. We suggest that the height of thiszone is determined by a deposition–removal process:locally varying increments of water are added (e.g. dueto precipitation) and removed (e.g. due to pumping andevapotranspiration) from the aquifer. Hence, the heightat any given time results from the superposition of (bothpositive and negative) random increments. If the incre-ments are approximately Gaussian, this process isexpected to generate a non-stationary fractional Brow-nian motion (fBm) pattern (Mandelbrot and Van Ness1968). In surface hydrology, fBm processes have beenused to model the level of water reservoirs (Feder 1988).Hence, it is not coincidental that the fBm power law

semivariograms are very close to the best-performingmodel in the OK, DK and UK cross-validation proce-dures. Of course, for a finite-sized basin, the purelypower law fBm dependence should be truncated by thedomain size. The non-differentiability of the groundwa-ter level explains the poor performance of MC, whichassumes a differentiable hydraulic head function, incomparison to the other methods.

Estimation variance

Stochastic interpolation methods quantify the kriging(error) variance which determines the precision of theestimates. The map of kriging standard deviation (krig-ing error) can be used to identify locations where theestimates have high uncertainty and further sampling isneeded (Prakash and Singh 2000; Fatima 2006;

Fig. 8 Isolevel contour mapof estimated groundwaterlevel in Mires Basin usingDK with the Spartansemivariogram model

Fig. 9 Isolevel contour mapof kriging standarddeviation for groundwaterlevel in Mires Basin usingDK with the Spartansemivariogram model


Theodossiou and Latinopoulos 2006; Yang et al.2008).

The error maps (Figs. 9, 11 and 13) identify thelocations of the Mires Basin with the largest krigingstandard deviation (SD). The south and east borders ofthe basin can benefit from further sampling according toOK (SD06–7.5 ma.s.l.) and UK (SD05–6.5 ma.s.l.).DK shows a standard deviation range approximatelybetween 6 and 8 ma.s.l. at the same locations, but alsosimilar values along the west border. The fact that DK isbased only on three to six neighbouring points oftenresults in higher kriging variances than OK or UK. InMires Basin, most estimation points have more than sixneighbours in their UK andOK search neighbourhoods,thus reducing the OK and UK variances with respect toDK. UK delivers the lowest standard deviation as itincludes a linear trend function that reduces variability

compared with OK. Interpolation with the Spartansemivariogram model delivers the lowest standard de-viation for all three (OK, UK, DK) interpolation meth-ods tested.

Semivariogram validation

Cross-validation studies mostly focus on univariatemeasures of performance, such as the ones describedabove. Stochastic interpolation methods also allow acomparison of the empirical semivariogram with thatobtained from the interpolated values, thus testing theaccurate reproduction of spatial continuity by interpo-lation. In Figs. 14 and 15, we compare (1) the exper-imental semivariogram of the observations, (2) theoptimal theoretical model, (3) the experimental semi-variograms obtained from OK (Fig. 14) and DK

Fig. 10 Isolevel contourmap of estimatedgroundwater level in MiresBasin using OK with theSpartan semivariogrammodel

Fig. 11 Isolevel contourmap of kriging standarddeviation for groundwaterlevel in Mires Basin usingOK with the Spartansemivariogram model


(Fig. 15) interpolation estimates, and (4) the respectiveoptimal models. Figure 16 presents the experimentalsemivariograms of the observations and the UK inter-polation estimates. Optimal theoretical models are notpresented because the semivariogram fit is performedfor the residuals. In all cases, the semivariogram of theestimates shows a very similar structure to that of thedata. However, the former exhibit an overall lowervariability, tending to have lower values than the em-pirical semivariogram of the data. This behaviourreflects the smoothing effect of interpolation.

General remarks

Stochastic and deterministic methods for the interpo-lation of groundwater levels have been used in other

studies before. Below, we briefly describe how thisstudy differs from previous ones.

To our knowledge, this is the first application ofDK to groundwater level interpolation. First- andsecond-order neighbours are used in DK to improveestimation accuracy. In contrast with studies that em-ploy only first-order neighbours, we clearly presentand apply the methodology for locating second-orderneighbours. This paper also presents the recently de-veloped Spartan semivariogram model for environ-mental applications. In the present study, this modelis shown to be optimal for interpolation. The OK-SPand UK-SP methods employed in this manuscriptapply the Spartan semivariogram model for the firsttime to hydrological data.

We compare three stochastic versus two deterministicmethods for mapping groundwater level that have not

Fig. 12 Isolevel contourmap of estimatedgroundwater level in MiresBasin using UK with theSpartan semivariogrammodel

Fig. 13 Isolevel contourmap of kriging standarddeviation for groundwaterlevel in Mires Basin usingUK with the Spartansemivariogram model


heretofore been compared on the same dataset. The casestudy in our manuscript investigates the performance ofwell-known methods with respect to interpolation in asparsely gauged basin (Mires Basin). To our knowledge,the groundwater level in Mires Basin has not beenmodelled with geostatistical methods.

The method comparison is based on cross-validationmeasures which include global statistical quantities, theaccuracy of the minimum value estimate, estimationvariance and semivariogram reproduction. The impactof search neighbourhood effects on the validation results

is analysed in detail. The comparison is conducted in theMatlab® programming environment using a code devel-oped by one of the authors, as opposed to commercialsoftware packages. This approach provides increasedflexibility and common ground for comparison.

Conclusions

This paper presents a comparison of stochastic (OK,UK, DK) and deterministic (IDW, MC) interpolationmethods for groundwater level monitoring in sparselygauged areas. For the hydraulic head data from MiresBasin (Crete, Greece), we established that the OK andUK interpolation methods overall perform best withrespect to various cross-validation measures, whilstDK and IDW show similar performance. However,no method is significantly superior to the others. Theisolevel contours generated by DK and especially byIDW are rough. The stochastic methods provide guid-ance for the location of additional monitoring sitesbased on the values of the kriging variance. Sincethe size of the Mires Basin dataset is relatively small,computational limitations are irrelevant. For largedatasets, computational time and memory usage foreach method should also be investigated.

The three-parameter Spartan semivariogram modelis herein applied for the first time to hydrological dataand yields the optimal cross-validation performanceamong the investigated models. In addition, it deliversthe estimates with the lowest standard deviation. TheSpartan model is non-differentiable. We interpret this

Fig. 14 Comparison of groundwater level semivariograms: data(stars), OK estimates using Spartan (SP) semivariogram(circles), along with optimal SP model fits to data (dashed line)and to OK estimates (continuous line)

Fig. 15 Comparison of groundwater level semivariograms: data(stars), DK estimates using Spartan (SP) semivariogram(circles), along with optimal SP model fits to data (dashed line)and to DK estimates (continuous line)

Fig. 16 Comparison of omnidirectional groundwater levelsemivariograms of data (stars) and UK estimates (circles)


property as the result of a deposition–removal processthat leads to an fBm-like behaviour of the groundwaterlevel surface. We also show that DK provides the bestcross-validation estimate for the extreme low valuedue to its localised nature of interpolation.

Acknowledgements We would like to thank Professor GeorgeKaratzas, Department of Environmental Engineering, TechnicalUniversity of Crete, Greece, for many helpful discussions.

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Comparison of stochastic and deterministic methods for mapping groundwater level spatial variability in sparsely monitored basins