use of universal function approximation in variance-dependent surface interpolation method: an...

Journal of Hydrology (2007) 332, 16–29

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Use of universal function approximation invariance-dependent surface interpolationmethod: An application in hydrology

Ramesh S.V. Teegavarapu *

Department of Civil Engineering, Kentucky Water Resources Research Institute, University of Kentucky,161 Raymond Building, Lexington, KY 40506-0281, USA

Received 1 August 2005; received in revised form 16 May 2006; accepted 16 June 2006

Summary Variance dependent stochastic interpolation approaches such as kriging are widelyrecognized as standard stochastic methods for interpolation of geophysical and hydrologic vari-ables. Deterministic weighting and stochastic interpolation methods are the most frequentlyused methods for estimating missing rainfall values at a gage based on values recorded at allother available recording gages. Traditional kriging has a major limitation due to the needfor an a priori definition of a mathematical function for a semivariogram that might fit the sur-face to be interpolated. Use of the universal function approximator, artificial neural network(ANN), as a replacement to fitted authorized semivariogram model within ordinary kriging isinvestigated in the current study. The revised ordinary kriging is used for estimation of missingprecipitation data at a rainfall gaging station based on data recorded at all other available gag-ing stations. Historical daily precipitation data obtained from 15 rain gaging stations from atemperate climatic region, Kentucky, USA, is used to test the improvised method and deriveconclusions about the efficacy of this method. Results suggest that use of universal functionapproximator such as ANN within a kriging has several advantages over ordinary kriging.ª 2006 Elsevier B.V. All rights reserved.

KEYWORDSSpatial interpolation;Stochastic interpolation;Universal functionapproximation;Semivariogram;Ordinary kriging;Missing precipitationrecords;Artificial neural networks

0d

Introduction

Spatial interpolation or surface generation is an importantand essential task for estimation of geophysical and hydro-logical variables that vary in space and time. Simulation of

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* Tel.: +1 859 257 1302; fax: +1 859 257 4404.E-mail address: [email protected].

hydrologic processes requires an extension of point mea-surements to a surface representing the spatial distribu-tion of the parameter or input (Vieux, 2001). Spatialinterpolation is often necessary due to limited samplingof these variables in space, and need for understandinga specific process over larger spatial domains where obser-vations are not available at those scales due a variety ofreasons. Two main reasons are: (1) lack of data due to

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Use of universal function approximation in variance-dependent surface interpolation method 17

malfunction of hydrometric, meteorological and geophysi-cal recording instruments, and (2) non-availability of mea-surements at a scale required for modeling hydrologicalprocesses due to prohibitive cost associated with exhaus-tive sampling or monitoring. Traditional deterministic dis-tance-based (e.g., inverse distance weighting method) andstochastic variance dependent (e.g., kriging) interpolationmethods are commonly used in the field of hydrology forspatial interpolation. Teegavarapu and Chandramouli(2005) discussed the limitations and advantages of thesemethods in a study aimed at estimating missing precipita-tion records using spatial interpolation methods. They alsosuggest conceptually acceptable revisions to the inversedistance-based method and provide new, improved andimprovised approaches for estimating missing precipitationrecords. The main objective of this study is to propose andevaluate a variant of ordinary kriging for estimating miss-ing precipitation records.

Missing data relevant to hydrological processes is a com-mon problem encountered by practicing hydrologists andmodelers. Precipitation data are considered vital for hydro-logic analysis and design of water resources systems. Theproblem of missing precipitation data arises due to a varietyof reasons and these include systematic and random errors(ASCE, 1996; Larson and Peck, 1974; Vieux, 2001) intro-duced in the measurement process. Systematic errors inrain gage measurements can be of various types: water lossduring measurement, adhesion loss on the surface of thegage, raindrop splash from the collector (Vieux, 2001).These errors are critical as they affect the continuity ofrainfall data and ultimately influence the results of hydro-logic models that use rainfall as input.

Estimating missing data is one of the most importanttasks required in many hydrological modeling studies. Sev-eral deterministic, stochastic and data-driven methods areavailable for such task. Traditional weighting methods thatbelong to a class of deterministic spatial interpolation tech-niques such as inverse-distance (Simanton and Osborn,1980; Wei and McGuinness, 1973; ASCE, 1996), non-lineardeterministic and stochastic interpolation methods (e.g.,kriging) have been applied in the past for spatial interpola-tion of rainfall data. Singh and Chowdhury (1986) compared13 rainfall estimation methods and found that isohyetalmethod yielded higher estimates of mean daily, monthlyareal rainfall than other methods in the area of their study.Tung (1983) compared 5 methods used for estimating pointrainfall and indicated that arithmetic average and inverse-distance methods did not yield desirable results for moun-tainous regions. Normal ratio approach and inverse distanceweighting methods (ASCE, 1996) have been applied for esti-mating missing precipitation data. In the field of quantita-tive geography IDWM is used for the purpose of spatialinterpolation (Sullivan and Unwin, 2003). Several variantsof IDWM were derived and adopted by researchers with a fo-cus on the weighting schemes. Hodgson (1989) modifiedIDWM to adopt a learned search approach that reduces thenumber of distance calculations. To incorporate topograph-ical aspects, Shepard (1968) proposed a modified IDWM thatis referred to as a barrier method. Alternative methods suchas regression (conventional least-squares) and time seriesanalysis are also used for estimating missing precipitationvalues (Salas, 1993). Daly et al. (1994) developed a regres-

sion model that uses spatial variables (i.e., climate data,elevation, topography and proximity to coastal area) forestimation of precipitation. All the above discussed meth-ods belong to the class of deterministic spatial interpolationtechniques.

Variance dependent surface interpolation methods thatbelong to the general family of kriging have been appliedfor several geophysical interpolation problems in hydrology(Holawe and Dutter, 1999; Kyriakidis et al., 2004; Vieux,2001; Grayson and Gunter, 2001). These stochastic interpo-lation methods are based on the principle of minimizingestimation variances at points where measurements arenot available. Kriging in various forms is applied for estimat-ing missing precipitation data and areal precipitation frompoint measurements. Ashraf et al. (1997) compared interpo-lation methods (kriging, inverse distance and co-kriging) toestimate missing values of precipitation. They indicate thatkriging interpolation method provided the lowest root meansquare error (RMSE).

Co-kriging of radar and rain gage data studies were per-formed by Seo et al. (1990a,b) and Seo (1996) and Krajewski(1987) to estimate mean areal precipitation and interpola-tion of rainfall data. Seo (1998) studied real-time estimationof rainfall fields using radar and rain gage data. These stud-ies took advantage of spatial correlations between rainfalland radar data measurements to improve interpolation viaa variant of kriging referred to as co-kriging.

Deraisme et al. (2001) used kriging with external driftand collocated cokriging for interpolation of rainfall inmountainous areas. Goovaerts (2000) used information fromdigital elevation model within kriging for spatial interpola-tion of rainfall. A recent study of precipitation data in Aus-tria by Holawe and Dutter (1999) illustrates the utility ofkriging and variogram parameters in understanding of spa-tial and temporal structure of the precipitation. Krigingwithin in a Bayesian framework was used by Handcock andWallis (1994) to model and spatial and temporal variationsof mean temperature in the northern United States. Kyriaki-dis et al. (2004) used kriging for simulation of daily precip-itation time series at a regional scale. These studies suggestthat kriging with variations is a viable stochastic techniquefor interpolation of precipitation data at different spatialand temporal scales.

Kriging remains one of the most preferred stochastic sur-face interpolation techniques (Webster and Oliver, 2001;Sullivan and Unwin, 2003). However, it is plagued by severallimitations. Selection of a semivariogram model, assign-ment of arbitrary values to sill and nugget parameters,and distance intervals, and the computational burden in-volved in interpolation of surfaces, are a few difficultiesassociated with this method.

Artificial neural networks (ANNs) are data-driven ap-proaches that rely on learning relationships between depen-dent and independent variables to predict variables ofinterest. An ANN with one hidden layer and appropriatenumber of neurons has the ability to approximate any con-tinuous function. This ability of function approximation ofANN is confirmed via mathematical proofs provided inde-pendently by Cybenko (1989) and Hornik et al. (1989).Therefore ANNs are generally referred to as universal func-tion approximators. Application of artificial neural networks(ANNs) as universal function approximators has gained

Distance, d

Range, a co

c1Y(d)

Sill

Nugget

Figure 1 Typical parameters (nugget and sill) of asemivariogram.

18 R.S.V. Teegavarapu

enormous interest in the hydrology and water resources re-search community for application to a number of hydrolog-ical prediction problems (ASCE, 2001a,b). A wealth ofliterature is available on the application of neural networksin the general area of water resources and hydrology. A fewapplications of artificial neural networks in hydrology aremainly related to streamflow (Govindaraju and Rao, 2000;Panu et al., 2000) and rainfall prediction (e.g., Frenchet al., 1992; Teegavarapu and Mujumdar, 1996). Frenchet al. (1992) used a feed forward neural network with backpropagation training algorithm for forecasting rainfall inten-sity fields at a lead-time of 1 h with the current rainfall fieldas input. Teegavarapu and Mujumdar (1996) used ANN for 10day ahead rainfall prediction. Panu et al. (2000) have devel-oped ANN-based methods for infilling of missing values instreamflow time series. All these studies have proved thatANNs can be used for spatial and temporal extension andprediction of rainfall and streamflow time series. Anexhaustive review of literature of all works relevant to theapplication of neural networks in hydrology would be quitean undertaking. However, the reader is directed to anexhaustive state-of-the-art review conducted by the ASCETask committee (ASCE, 2001a,b).

The current study is aimed to assess and evaluate the useof universal function approximator such as ANN to replacethe semivariogram model in an ordinary kriging to estimatemissing precipitation data. Ordinary kriging and its concep-tually accurate improvised forms (i.e., UOK and UOKC) aredeveloped and used in the current study to estimate missingprecipitation records at a gaging station. The paper is orga-nized as follows. A brief introduction to traditional ordinarykriging is provided first followed by discussion about its lim-itations. Use of ANN to develop a data-driven model for thesemivariogram is described and explained next. Finally, theapplication of universal function approximation-based or-dinary kriging for estimating missing rainfall records is dis-cussed and the results, analysis and conclusions arepresented.

Ordinary kriging

Kriging (Journel and Huijbregts, 1978; Isaaks and Srivastava,1989; Vieux, 2001; Webster and Oliver, 2001) is an optimalsurface interpolation method based on spatially dependentvariance. The degree of spatial dependence is generally ex-pressed as a semivariogram in kriging. The general expres-sion that is used to estimate the semivariogram is given by

cðdÞ ¼ 1

2nðdÞXdij¼dðhi � hjÞ2 ð1Þ

where c(d) is the semivariance which is defined over obser-vations, hi and hj lagged successively by distance d. Surfaceinterpolation using kriging depends on the selected semi-variogram model, and the experimental or hypothesizedsemivariogram that must be fitted with a mathematicalfunction or model. Depending on the shape of semivario-gram several mathematical models are possible that includelinear, spherical, circular, exponential and Gaussian. A typ-ical spherical semivariogram is shown in Fig. 1. Three com-monly used semivariogram models namely, spherical,exponential and Gaussian are defined by Eqs. (2)–(4).

cðdÞ1 ¼ C0 þ C11:5d

a� 0:5

d

a

� �3" #

ð2Þ

cðdÞ2 ¼ C0 þ C1 1� exp � 3d

a

� �� ð3Þ

cðdÞ3 ¼ C0 þ C1 1� exp �ð3dÞ2

a2

!" #ð4Þ

The parameters, C0, d and a are referred to as the nugget ornugget effect, distance and range, respectively as shown inFig. 1. The nugget is the semivariance at zero distance andsill is the constant semivariance beyond the range value, a.The summation of C0 and C1 is referred to as the sill and thesemivariance at range, a, is equal to the sill value. The val-ues of C0 and C1 are obtained by trial and error procedure. Inkriging the weights are based not only on the distance be-tween the measured points and the prediction location,but also on the overall spatial arrangement among the mea-sured points and their values. Ordinary kriging is referred toas a linear unbiased exact estimator. The weights mainly de-pend on the fitted model (i.e., semivariogram) to the mea-sured points. The general equation for estimating missingvalue, hm, at a point in space is given by

hm ¼XNi¼1

kihi ð5Þ

where, ki is the weight obtained from the fitted semivario-gram, and hi is the value of observation at location i. Theobserved data is used twice, once to estimate the semivari-ogram and then to interpolate the values. To avoid any sys-tematic bias in the kriging estimates, a constraint onweights is enforced that is given by Eq. (6).

XNi¼1

ki ¼ 1 ð6Þ

The estimation variance at a specific point in space is givenby

r2m ¼

XNi¼1

kicm;i þ w ð7Þ

where r2m is the variance, cm,i is the variance between the

points m and i, and w is the Lagrangian multiplier used insolving kriging equations to minimize estimation variance.


The success and applicability of kriging for spatial inter-polation depend on several factors. These factors are: (1)the number of spatial data; (2) location of sampling pointsin space; (3) selection of fitted or experimental semivario-gram model and (4) success in cross-validation process.Voltz and Webster (1990) suggest that to compute a vario-gram with reasonable confidence 100 data points for isotro-pic case and 200–400 data points for anisotropic case areneeded. Sullivan and Unwin (2003) indicate that preferredfunctional forms for the semivariogram are non-linear andcannot be estimated easily using standard regression soft-ware. The robustness of kriging heavily relies on the properselection of a semivariogram model (Grayson and Gunter,2001).

Choosing semivariogram models and fitting them to dataremain among the most controversial topics in geostatistics(Webster and Oliver, 2001). One reason for the difficulty isdue to the number of authorized models that are availableto choose from. Sullivan and Unwin (2003) indicate thatthe selection of the experimental semivariogram and amathematical model is to some extent a ‘‘black-art’’ thatneeds careful analysis informed by good knowledge of thevariables being analyzed. Cross-validation is often used asa procedure for selecting semivariogram models. The mostwidely used and authorized semivariogram models in ordin-ary kriging provide unique solutions for weights that are re-quired for the interpolation. However, selection ofparameters for these models is a tedious trial and errortask. Recent work by Jarvis et al. (2003) points to the useof an artificial intelligence system for the selection of anappropriate interpolator-based on task-related knowledgeand data characteristics. In the current study, an artificialneural network model is used to replace the semivariogram.The motivation to use an artificial intelligence techniquesuch as neural network in the current study is partly basedon their work.

Authorized semivariogram models

Experimental semivariogram models are generally used tosummarize the variance associated with spatial data. Thesevariogram models are referred to as authorized models(Webster and Oliver, 2001) which are bounded models inwhich the variance has a maximum value known as the sill.One condition should be fulfilled before any functional formcan be used for fitting the semivariogram for the data in thekriging. The condition is given by

WTCW P 0 ð8Þ

where W is the weight matrix, C is covariance matrix thatincludes covariances between the observation points andthose between the observation points and the location atwhich an estimate is required. The condition in Eq. (8) whenfulfilled confirms the existence of one and only one uniqueand stable solution and suggests that the matrix ‘‘C’’ satis-fies a mathematical condition known as positive definite-ness. Only authorized variogram models (e.g., spherical,exponential or Gaussian) satisfy the condition of positivedefiniteness. Selection of a semivariogram model other thanan authorized one may result in non-unique and unstablesolution (i.e., weights). However, Isaaks and Srivastava

(1989) indicate that it is possible to concoct a new mathe-matical model or functional form for semivariogram andto verify its positive definiteness, and finally use it withinkriging for interpolation.

Universal function approximation-basedordinary kriging

The use of an artificial neural network model to approxi-mate the semivariogram in kriging is referred to as universalapproximation-based ordinary kriging (UOK) in the currentstudy. The conceptual difference between traditional ordin-ary kriging (OK) and UOK is shown in Fig. 2. Measures of spa-tial variability of observed data is generally expressed bysemivariogram in the field of geostatistics. However, manyother measures are also possible as suggested by Deutschand Journel (1998). They discuss 10 different measures ofspatial variability and indicate that practitioners shouldspend more time on data analysis and variogram modelingthan on all kriging and simulation runs combined. In the cur-rent study spatial variability is also expressed as a functionof correlation between point observations in space. Ding-man (2002) suggests the use of spatial correlation coeffi-cients to characterize relationships among observed datain space. The plot of correlation coefficients and the dis-tance is referred to as a correlogram in the current study.The use of a correlogram in ordinary kriging and artificialneural networks for modeling the semivariogram is referredto as a universal function approximation-based ordinary kri-ging using correlogram, UOKC, in the current study.

Universal function approximator: Artificialneural network

The neural network architecture used in the current studyas universal function approximator is shown in Fig. 3. Thearchitecture has three layers (input, hidden and output).The configuration of a neural network includes determiningthe number of nodes in the hidden layer, and the connectionweights. The interconnection weights are shown in Fig. 3 asWj1 and W0

j1. The subscript, j, is the number of hidden layerneurons with the maximum value equal to n0. The networkis made up of interconnected set of simple information pro-cessing elements (nodes). The nodes are arranged in a mul-ti-layer system without any connections between the nodesof the same layer. The number of nodes in the input layer isbased on the number of input arrays, while the number ofnodes in the output layer is equal to the number of modeloutputs. The optimal number of nodes in the hidden layercan be determined by trial and error procedures or by usingbest network search methods provided by some software(e.g., Braincel, 1998). The significance of the hidden layernodes is that they add a degree of flexibility to the perfor-mance of the network and enhance its capability to deal ro-bustly and efficiently with inherently complex non-linearrelations (Shamseldin, 1997). Details of ANN architecturesand the back propagation training algorithm can be foundin Freeman and Skapura (1991).

In the current study, within the neural networks model avariant of backpropagation algorithm, referred to asbackpercolation is used for training. Backpercolation is a

a

b Universal function approximation based Kriging

Developmentof semivariogram

Selection of mathematicalfunction

Interpolation

Traditional ordinary Kriging

Developmentof Variogram cloud or semivariogram or correlogram cloud

UniversalApproximationofmathematicalfunction

Interpolation

Input layer

Hidden layer

Output layer

Distance VarianceOr semivariance

Figure 2 Comparison of traditional ordinary kriging and universal function approximation-based kriging.

VarianceDistance

W11

J = no

wo11

wo21

J = 1

Hidden layer

Output layerInput layer wo31

W12

W13

Figure 3 Artificial neural network architecture with input, hidden and an output layer.


learning algorithm which works in conjunction with the tra-ditional backpropagation used for training of feedforwardnetworks. In backpercolation algorithm, the weights arenot changed according to the error of the output layer asin backpropagation, but according to a unit error that iscomputed separately for each unit. This procedure effec-

tively reduces the amount of training cycles needed. Inthe present study, Braincel (1998) which uses backpercola-tion algorithm in its training is used. The software also pro-vides a facility to select the best network configuration orarchitecture of the neural network once an upper limit onthe number of neurons in each hidden layer is provided.


Application of UOK and UOKC

The ordinary kriging and its variants, UOK and UOKC areused to estimate missing rainfall data at a base station(i.e., Lexington, Kentucky). Data at the base station are as-sumed to be missing for the purpose of testing these estima-tion methods. Historical daily rainfall data from year 1971–2002 available at 15 rainfall gaging stations in the state ofKentucky, USA, are used for analysis. These gaging stationsare shown in Fig. 4 and are numbered for convenience. Thedata used in the current study are compiled and provided byKentucky Agricultural Weather Center, University of Ken-tucky. When using UOK and UOKC, approximately 67% ofthe historical data (7800 days) are used for training the arti-ficial neural networks and 33% of data (3900 days) are usedfor testing the methods. GSLIB (Deutsch and Journel, 1998),a geo-statistical software library, is used to apply ordinarykriging with three experimental semivariogram models gi-ven by Eqs. (2)–(4).

The ordinary kriging is applied to the test data using atrial and error method of selecting the parameters for thesemivariogram. The performance of OK, UOK and UOKCmethods are compared using widely recognized and com-monly used error measures, root mean squared error(RMSE), mean absolute error (MAE) and a goodness-of-fitmeasure criterion, correlation coefficient (q), based on ac-tual and estimated rainfall values at the base station.

Case study area

The case study area comprises of the eastern part of thestate of Kentucky. The state wide average annual precipita-tion-based on data from 1971 to 2003 varied between76.2 cm (30 in.) and 193 cm (76 in.) with values higher than127 cm (50 in.) in the southeastern region and lower than107 cm (42 in.) in the northeastern part. The statistics ofrainfall data at different stations with the same historicalrecord length are given in Table 1. The Cumberland (or

Figure 4 Location of precipitation measurin

Appalachian) Plateau dominates the eastern third of Ken-tucky and contains the highest point, Black mountain, at1263 m above mean sea level. The Bluegrass Region(north-central) is a series of hills fronting the Ohio River.The far western corner includes the Mississippi River floodplain with the lowest elevation (78 m) in the state. Thestate with mean elevation of 229 m is dominated by theOhio River forming its northern borders, and the Cumber-land and Tennessee River systems, and their many spin-offlakes. Other major rivers include the Kentucky, Licking,and Mississippi, along its western border with the state ofMissouri.

Results and analysis

Universal function approximation-based ordinary kriging(UOK) is applied for estimating missing precipitation data(33% of total data) and its performance is compared withthat of ordinary kriging with three separate semivariogrammodels used in this study. The variant of OK, UOKC, is alsoused for estimating missing data. A variogram cloud is firstcreated to understand the spatial variability of the data. Avariogram cloud (Chauvet, 1982) is a plot of variance com-puted based on observations at any two points and distancebetween them. The variogram cloud and correlogram areshown in Figs. 5 and 6, respectively. It can be observed fromFig. 6 that the spatial correlations decrease as the distancebetween the observation points decreases. In the currentcontext these results also agree with the general assump-tion that correlation of rainfall amounts between stationsdecreases rapidly with increasing distance between stations(Osborn et al., 1979).

Webster and Oliver (2001) indicate that in principle avariogram model can be fitted to a variogram cloud. A totalof 15C2 (105) data points are possible based on 15 stationsand average variance values are used in the developmentof the variogram cloud in the current study. The neural net-work model is trained based on 85 observations and tested

g stations in the state of Kentucky, USA.

Table 1 Statistics of observed daily and annual rainfall values at different stations

Station �x1 S1 �x2 S2

Bardstown 0.986 1.290 125.456 23.602Berea 0.904 1.158 119.019 21.460Bowling Green 1.052 1.430 129.931 22.489Buckhorn 0.861 1.062 118.562 19.403Campbellsville 1.064 1.410 132.268 24.305Covington 0.810 1.087 108.575 16.068Cumberland 0.892 1.120 125.280 22.278Grayson 0.864 1.074 107.513 14.709Hardinsburg 1.024 1.283 122.428 19.093Jackson 0.889 1.133 125.042 20.554Lexington 0.881 1.209 116.271 20.947London 0.785 1.097 118.087 20.340Louisville 0.912 1.245 114.691 18.880Somerset 1.016 1.290 129.080 19.667Williamstown 0.765 1.074 113.373 17.493

�x1 – Mean rainfall (cm), daily.S1 – Standard deviation (cm), daily.�x2 – Mean rainfall (cm), annual.S2 – Standard deviation (cm), annual.

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0 50 100 150 200 250 300

Distance (mile)

Var

ian

ce

Figure 5 Variogram cloud based on historical precipitationdata.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 50 100 150 200 250 300

Distance (mile)

Co

rrel

atio

n C

oef

fici

ent

Figure 6 Correlogram cloud based on historical precipitationdata.

Table 2a Regression-based functional forms of fittedvariogram models

Regression model Functional form

Linear c(d) = 0.0031 (d) + 0.0448Logarithmic c(d) = 0.0192 Ln(d) + 0.0315Power c(d) = 0.0376 (d)0.2965

Exponential c(d) = 0.0466 e0.0463(d)

d: scaled distance.


on 20 observations. As the number of data points is limitedto 105, binning process (Chang, 2004) generally used in kri-ging is not used in the development of the semivariogramthat defines average values of variance in several distancelags. The spatial data available for the development of a

semivariogram or variogram is inspected for systematic vari-ations in the mean values of the observed historical data atall the stations (Olea, 1999). This inspection proceduretranslates to checking the normality of average values ofobserved precipitation data. It is evident from mean dailyprecipitation values provided in Table 1 that the intrinsichypothesis of a non-existence of spatial trend is valid. Pre-liminary data analysis was carried out for checking the nor-mality conditions in an earlier study (Teegavarapu andChandramouli, 2005). The main assumption made in the cur-rent study is that the covariance between any two data val-ues at any two locations will depend only on the distancebetween them and not on the direction. Therefore the ef-fect of anisotropy on variogram parameters or propertiesis neglected.

Artificial neural networks and four different traditionalregression models (linear and non-linear) are used to modelthe variogram cloud. Functional forms (linear, logarithmic,power and exponential) of these models are given in Table2a. The architecture of ANN used in the current study is gi-ven in Fig. 3. The input neuron is the distance and outputneuron of the ANN provides the variance. The hidden layerneurons are obtained by the best net search approach(Braincel, 1998). The neural network training is done usinga back-percolation training algorithm with learning rate of


0.1 (Rumelhart and Mclelland, 1986; Haykin, 1994). Thelearning rate is a factor that determines the amount bywhich the connection weight is changed according to errorgradient information. The learning rate can significantly im-pact the training and final results (Haykin, 1994; Maier andDandy, 1998). In hidden and output layers, a sigmoidal acti-vation function is used for modeling the transformation ofvalues across the layers.

The ANN architecture used in the current study is a feed-forward network and consists of one hidden layer with 3 hid-den layer neurons in case of UOK and 1 neuron in UOKC. Thearchitecture for ANN is represented as 1-3-1 for UOK and1-1-1 for UOKC. The ANN architecture 1-3-1 suggests one in-put neuron (distance), three hidden layer neurons and oneoutput neuron (variance). Results related to performanceof these models and ANN using RMSE and MAE are shownin Table 2b. It is evident from the performance measuresthat ANN performs better than the four traditional regres-sion models when two error measures are considered. It isimportant to note that the error measures calculated areaverage values. A 1% difference in RMSE value suggests onan average one model is either over predicting or under pre-dicting variance values by 1%. Any improvement, howeverminute it is, in the variance estimation can be consideredsignificant as interpolation by kriging relies on the estima-tion of variance. The nugget value provided by ANN is0.043 which is approximately equal to the value that canbe read from the variogram cloud in Fig. 5. The value ofnugget is obtained by providing a distance value of zero asan input to ANN model. The nugget is one of the crucialparameters of the semivariogram that needs to be esti-mated accurately to ensure proper reliable estimates ob-tained via interpolation. The condition relevant to positivedefiniteness discussed in Section ‘‘Universal functionapproximation-based ordinary kriging’’ is checked prior tothe application of the ANN model as a semivariogram inthe kriging.

The functional form ultimately derived from ANN archi-tecture may not be as simple in structure (mathematicalform) as any authorized functional form (i.e., spherical orexponential). The neural network architecture (hidden lay-ers and neurons in each layer) will define and affect thefunctional form that will ultimately represent the experi-mental variogram. The functional form in case of ANN ingeneral is an additive form of non-linear and linear combi-nation of weights and sigmoidal activation functions. Byusing different architectures, several covariance matrices

Table 2b Performance measures of several fitted modelsto variogram using test data

Model RMSE MAE

ANN 0.003 0.005

RegressionLinear 0.006 0.007Logarithmic 0.007 0.008Power 0.005 0.007Exponential 0.005 0.006

RMSE: root mean squared error, MAE: mean absolute error.

can be derived which can be checked for conditions thatcan guarantee the existence of unique and stable solutions.Webster and Oliver (2001) suggest and illustrate the use ofAkaike’s information criterion (AIC) to select a parsimonioussemivariogram model. A parsimonious model in case of ANNcan be obtained by limiting the number of neurons (i.e., de-grees of freedom) in the hidden layers to reduce the over-fitor over-training (Maier and Dandy, 1998) of the ANN. Theover-training process will reduce the generalization capabil-ities of ANN. However, it should be noted that a minimumnumber of hidden layer neurons are required for ANN toapproximate the function.

In case of ordinary kriging three different variogrammodels are tested in the current study. These are spherical,exponential and Guassian models as expressed in Eqs. (2)–(4). The parameters (nugget and sill) are estimated by trialand error and after several trials those that resulted in thebest performance from several such trials, using error mea-sures, RMSE, MAE and correlation coefficient (q) are pre-sented in Table 3. Also the performance of OK modelsimproved with the increase in the ratio of nugget to sill(C0/(C1 + C0)). Variograms with different proportions of nug-get and sill give rise to different kriging weights and vari-ances, and hence different estimates (Armstrong, 1998).The best results from ordinary kriging using three differentvariogram models and UOK and UOKC are presented in Table4. It is evident from the results in Table 4 that the ordinarykriging with spherical semivariogram model resulted in thebest performance compared to other two authorized semi-variogram models (exponential and Gaussian) when all threeerror measures are considered. The performance of UOK ismarginally better than OK when q performance measure isused.

A comparison can be made between the weights assignedby UOK to stations and the correlation coefficients derivedbased on observations among stations. The comparison isshown in Fig. 7a. A plot of weights and correlation coeffi-cients as shown in Fig. 7b suggests that there is no strongrelationship between correlation coefficients and weights,contrary to what one might expect. It is important to notethat in case of UOK, the sum of weights should be equalto 1, a condition specified to avoid any systematic bias inthe estimation of weights in the kriging. It is interesting tocompare weights assigned by UOK to the stations and thedistance between any station and the base station. A com-parison of weights and normalized distances are providedin Fig. 8. It is evident from Fig. 8 that relatively higherweights are assigned to few stations that are nearest tothe base station (i.e., Lexington) than those that are far-thest from the base station. The magnitudes of weights ob-tained using kriging are influenced by distances amongstations and point of interest. It is also evident from Fig. 8approximately 60% of the total weight is assigned to thenearest 6 rainfall gaging stations to the base station. Similarconclusions were reported by other researchers (Sullivanand Unwin, 2003; Armstrong, 1998) who adopted ordinarykriging for surface interpolation.

Inspection of the layout of stations (Fig. 3) and weightsprovide interesting insights into estimation process usingkriging. Higher values of weights are associated with sta-tions 1 and 8 compared to station 5 which is located south-west of station 8. The lower magnitude of weight may

Table 3 Semivariogram parameters (nugget and sill) and performance measures for three different fitted semivariogram modelsused in ordinary kriging

Semivariogram model Nugget (C0) Sill (C0 + C1) q RMSE MAE

Spherical model 0.000 0.100 0.704 0.251 0.0920.200 1.200 0.725 0.240 0.0880.200 1.100 0.726 0.239 0.0890.100 0.500 0.727 0.239 0.087

Exponential model 0.000 0.100 0.725 0.239 0.0880.100 0.500 0.730 0.238 0.0880.200 0.800 0.730 0.238 0.0880.400 1.500 0.730 0.238 0.088

Guassian Model 0.000 0.100 0.628 0.303 0.4340.200 1.100 0.684 0.262 0.0970.400 1.800 0.688 0.256 0.0960.300 1.300 0.690 0.259 0.096

RMSE: root mean squared error, MAE: Mean absolute error.

Table 4 Performance of UOK, UOKC and ordinary kriging(OK) approach with three different semivariogram models

Method q RMSE MAE

OK1 0.730 0.233 0.088OK2 0.731 0.238 0.088OK3 0.695 0.256 0.094UOK 0.732 0.238 0.092UOKC 0.725 0.237 0.092

OK1: Ordinary kriging (Spherical model).OK2: Ordinary kriging (Exponential model).OK3: Ordinary kriging (Guassian model).

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bFigure 7 (a) Comparison of weights and correlation coeffi-cients associated between each station and the base stationand (b) plot of correlation coefficients and weights.


indicate that the station 5 is not providing more informationthan station 8 in the interpolation. It is also interesting tonote that even though station 10 is farther away from basestation in comparison to station 8 the weight assigned byUOK is higher to station 10 than that is assigned to station8. However, a review of correlation coefficients suggestthat the correlation between observations at station 10and base station is highest in comparison to correlation be-tween the base station and stations 5 or 8. The intersampledata relationships are taken into account by the kriging pro-cess in the estimation of optimal weights. Also the stationsnearest to the base station screen the effect of observationsat locations farther away from the base station.

A comparison of weights obtained from UOK and UOKC isshown in Fig. 9. The weights derived for UOK and UOKC arealmost equal in magnitudes for all except for few stations asshown in Fig. 9. The estimates of precipitation depths byUOK and UOKC are almost equal in value in most of the timeintervals considered in the study. This is evident from thescatter plot shown in Fig. 10 and also from mean absoluteerror values shown in Table 4. It is interesting to note thatboth the models one using the variance and the other usingcorrelation to explain spatial dependence of variability inpoint observations produce similar results. The maximumprecipitation depth estimated by UOK and UOKC at the basestation is less than 8.89 cm while the maximum depth actu-

ally observed in the testing period was 11.63 cm. The esti-mation variance at the location of the base stationcalculated based on Eq. (7) is equal to 0.354 (cm2) forUOK. The variance is positive and it satisfies the mainrequirement of ordinary kriging.

The estimated and observed precipitation depths at thebase station by OK, UOK and UOKC are shown in Fig.10a–c. It is evident from these figures that all the threemethods underestimate precipitation depths when the

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Figure 9 Comparison of weights derived based on variogramand correlogram.


observed values of precipitation are higher than 7.62 cmand therefore produce higher errors. The residual errorplots for estimation by OK and UOK are shown in Fig. 11aand b. Both the plots indicate that there is no bias or sys-tematic trend in the estimation by OK and UOK. The averagevalues of errors are close to zero and there is no timedependency of residuals.

In general many spatial interpolation methods fail in esti-mation of missing rainfall data at a station in two situations:(1) when rainfall is measured at all the other stations and no

Figure 10 Observed and estimated precipitation depths: (a) ordordinary kriging using correlogram (UOKC) and (c) universal functio

rainfall occurred in reality at the base station and (2) whenrainfall is measured at the base station and no rainfall ismeasured or occurred at all the other stations. In the formercase all the three methods, OK, UOK and UOKC provide neg-ative or non-zero estimates depending on the sign of de-rived weights, while in the latter case all the threemethods generate zero values for the estimates. A remedialstrategy to address the first situation is proposed here. This

inary kriging (OK), (b) universal function approximation basedn approximation based ordinary kriging.

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bFigure 11 Estimation error (estimate � observed) based on precipitation estimates by (a) ordinary kriging (OK) and (b) universalfunction approximation based kriging (UOKC).


strategy is based on the assumption that the station that isclosest to the base station and the base station will experi-ence similar precipitation magnitudes and patterns. Theobservations at a station that is closest to the base stationby Euclidean distance or a station selected based on stron-gest correlation are used to modify the estimates providedby UOK interpolation. Two stations, one based on Euclideandistance (i.e., Berea) and another based on spatial correla-tion (i.e., Louisville) are used to replace the non-zero valuesestimated by UOK at base station (i.e., Lexington) by zerovalues on those days when no precipitation is recorded atthose stations.

An experiment is conducted by using this strategy afterthe initial precipitation estimates are obtained from UOK.Performance measures, q (correlation coefficient), RMSEand MAE using data at Berea and Louisville after revisingthe estimates at base station (i.e., Lexington) are 0.711,0.243, 0.089 and 0.741, 0.235 and 0.079, respectively. Useof data from rainfall gaging station at Louisville results inbetter performance compared to that when using data fromstation at Berea. This experiment suggests that distance isnot always a measure of spatial correlation between obser-vations recorded at any two stations. In general, distance-based weighting methods suffer from one major conceptuallimitation based on the fact that Eucledian distance is notalways a definitive measure of the correlation among spatial

point measurements. Also, the interpolation methods fail toestimate missing values correctly, if errors are introducedinto the measurement process of rainfall at one or morerainfall stations. These are artefacts of interpolation tech-niques that cannot be avoided or eliminated all togetherin many situations.

Considering the limitations of spatial interpolation anddistance-based weighting techniques used for estimatingmissing precipitation data a simple approach of replacingmissing values at the base station using the recorded valuesfrom a nearest station (based on distance and correlation) isviable. To assess the utility of this simple approach, all themissing precipitation records at the base station are re-placed by recorded values from stations Berea and also fromLouisville via two experiments. The performance measures,q, RMSE and MAE, using values from Berea and Louisville,are 0.566, 0.301 and 0.103, and 0.667, 0.286 and 0.099,respectively. These experiments indicate that in spite oftheir limitations, stochastic spatial interpolation and deter-ministic distance-based weighting methods provide betterestimates of missing data than simpler methods. The mainreason for their success is the ability to utilize informationfrom several points in space and embed the spatial variabil-ity of the underlying physical or hydrological process viatrue (e.g., correlation; variance) or surrogate measures(e.g., distance) to provide desired estimates. Yang and


Holder (2000) support similar arguments in relation to theimprovement of performance of estimation methods withan increase in the number of points in space used forinterpolation.

Positive kriging

In case of UOK or UOKC estimation, negative weights maycause negative precipitation values at the base stationwhere estimates are needed when these weights are at-tached to high precipitation values. Chiles and Delfiner(1999) indicate that even a fairly crudely determined setof weights can give excellent results when applied to data.However, caution should be exercised if negative weightsexist as a part of solution. It is evident from Fig. 9 that thereare two negative weights (for stations 13 and 14) for UOKCand one negative weight for UOK (for station 13), respec-tively. In general positive weights are not a requirementfor sufficient and necessary condition for positive estimatesin spatial interpolation using ordinary kriging. Negativeweights can be eliminated by using a procedure referredto as ‘‘positive kriging’’ (Barnes and Johnson, 1984). A var-iant of positive kriging is used in this study to restrict thekriging weights to positive values. The objective functionused by Barnes and Johnson (1984) is the estimated varianceand the one used in the current study is the difference be-tween the observed and estimated precipitation depths(interpolated values using weights) over a specific time per-iod. The approach described by a mathematical program-ming (optimization) formulation is given by

Minimize

Xnj¼1

XNi¼1ðkih

jiÞ � hj

m

!2

ð9Þ

Subject to :

XNi¼1

ki ¼ 1 ð10Þ

ki P 0 8i ð11Þ

where hjm is the observed value of the precipitation at the

base station, hji is the observed precipitation value at a spe-

cific station, j, N is the number of stations excluding thebase station and n is the number of days. The objectivefunction (Eq. (9)) is used to minimize the difference be-tween kriging estimate and observed value of precipitationvalue over a period of n days. Constraint defined by inequal-ity 11 will ensure that all the weights are positive.

The formulation is solved using Microsoft Excel solverwith initial weights obtained from the UOK. The solver usesa generalized reduced gradient (GRG) non-linear optimiza-tion algorithm for solution. The optimal positive weightsobtained based on the formulation are 0.090, 0.127,0.000, 0.002, 0.000, 0.030, 0.062, 0.000, 0.000, 0.159,0.096, 0.289, 0.134 and 0.012 for stations 1, 2, 3,. . .,14,respectively. Historical data (i.e., 7800 days) is used forestimation of weights using the optimization formulation(Eqs. (9)–(11)) and estimation is carried out for the testingdata. The RMSE, MAE and q values for testing period are0.222, 0.080 and 0.763, respectively. It is interesting tonote that four of the weights are zero indicating no effect

of observations recorded at these four stations on the esti-mates. The final weights obtained based on the solution ofthe formulation (Eqs. (9)–(11)) can be reproduced if onlyand if the weights obtained UOK are used as starting valuesas a part of initial solution using Excel solver. If a conditionis enforced on the weights which are zero with a lowerbound of 0.01, then the optimized weights obtained are0.077, 0.122, 0.010, 0.010, 0.010, 0.015, 0.051, 0.010,0.010, 0.157, 0.092, 0.291, 0.134 and 0.010. The RMSE,MAE and q values for testing period using these weightsare 0.223, 0.081 and 0.763, respectively. The non-negativeweights obtained through the formulation described abovewill guarantee positive variance at the point of interestbased on the Eq. (7). However, the estimation may bebiased and not as same as the one originally provided byordinary kriging.

Some researchers adopt a strategy in which they elimi-nate the observation point or points that generate negativeweights and repeat the entire ordinary kriging process to de-rive new set of weights. However, this procedure will not al-ways guarantee positive weights after the revised set ofspatial observations are used for kriging. The mathematicalprogramming formulation described in Eqs. (9)–(11) mayguarantee feasible (i.e., local) or global optimum solution.The solution is sensitive to initial values assigned to theweights and the historical data length (i.e., number ofyears) used for minimizing the objective function value.The former aspect is due to limitations of GRG algorithmused for optimization and the latter relates to effect of datalength on the relative distribution of weights among the sta-tions and on the value of objective function value. Eventhough the variant of positive kriging used in the currentstudy guarantees positive weights, it should be noted thatthe main limitation of mathematical programming formula-tions that often violate physical reality with mere imposi-tion of bounds on variables cannot be avoided.

General remarks

Ordinary kriging is computationally intensive when a surfaceinterpolation problem with several spatial data points isconsidered. The time required to develop a kriging interpo-lation model is generally invested in the development andassessment of semivariogram model parameters and cross-validation to accept a specific model for an interpolationproblem. In case of UOK and UOKC, time is invested in thetraining and testing of ANN model and finally checking theconditions that need to be fulfilled to obtain unique and sta-ble weights for calculations of kriging estimates. It can beargued that the selection of the number of hidden layersand the number of neurons is a trial and error process thathas similar disadvantages of ordinary kriging. This is truewhen the optimum number of neurons and the hidden layersare obtained by trial and error approach and not by auto-matic best neural network search method (Braincel, 1998)adopted in the current study. The computational time incase of UOK depends on the architecture of the networkand the training parameters.

In case of ordinary kriging, several semivariogramsshould be experimented and checked using cross-validationprocess before the best semivariogram can be selected for


spatial interpolation process. The evaluation of semivario-gram model also involves selection of appropriate parame-ters that ultimately define the nature of the function. Incase of UOK or UOKC, a split sample method that dividesthe variogram cloud data into two sets (training and testingsets) is used. Therefore in UOK or UOKC, the selection of thesemivariogram model is not based on quantification of per-formance measures that indicate how well the interpolationperformed but how well the variogram model can explainthe spatial variability of the data. Use of variogram cloudfor function approximation using ANN is justified as thenumber of data points is limited to 105 in the current study.The number of training data used from variogram cloud forthe development of variogram model using ANN affects theperformance of kriging. Also, the training data selectedshould be representative of variances at several separationdistances and not confined to a specific distance range.

Weights in UOK and UOKC are derived based on a func-tional form obtained through ANN and fitted for the vario-gram. Appropriate selection of semivariogram is importantas it indirectly affects the performance of these methodsin estimation. Also, the finite number of trials conductedin the current study may not conclusively indicate that theperformance of UOK is inferior to OK or other interpolationmethods. The current study is an initial attempt to capturethe spatial structure of a measurable variable using a uni-versal function approximator as opposed to the use of oneof the authorized traditional functional forms generallyadopted in kriging. Exhaustive studies need to be conductedto test the efficacy of replacing the semivariogram modelwith ANN model before any conclusive recommendationscan be made about the approach attempted in this study.Estimation of missing precipitation data is considered forthe evaluation of universal function approximation-basedkriging in the current study. The approach presented in thisstudy can also be extended to spatial interpolation of otherhydrological and geophysical variables. Hydrologists andmodelers should not loose focus of the underlying assump-tions of kriging and conditions relevant to minimum positivevariance guaranteed by it while assuming universal functionapproximator such as ANN within kriging as a quick panacea.

Conclusions

Universal functional approximator such as artificial neuralnetworks (ANN) is used for fitting a semivariogram modelusing the raw data in ordinary kriging to estimate missingprecipitation data. The use of ANN eliminates the need forthe pre-defined authorized semivariogram models to cap-ture the spatial variation of data, and the trial and errorprocess involved in estimation of semivariogram parame-ters. Utility of universal function approximation-based kri-ging is assessed and demonstrated through an applicationto a case study in which missing daily precipitation data ata rain gage station are estimated. Results suggest that per-formance of universal approximation function based krigingin estimating missing precipitation data are comparable tothose achieved by traditional ordinary kriging with pre-se-lect semivariogram models. However, caution should beexercised in using the universal function approximation ap-proach as a replacement for a semivariogram model. The

functional form created by ANN may not always result inan authorized model that provides unique and stable solu-tion, and satisfies the conditions of ordinary kriging.

Acknowledgements

The author thanks the Kentucky Agricultural Weather Cen-ter, University of Kentucky, for providing the data requiredfor the research study reported in this paper. The authorsincerely thanks the two anonymous reviewers for theirobjective and constructive criticism that led to theimprovement of the manuscript.

References

ASCE, 1996. Hydrology Handbook, second ed. American Society ofCivil Engineers ASCE, New York.

ASCE, 2001a. Task committee on artificial neural networks inhydrology, Artificial Neural Networks in Hydrology. I. Preliminaryconcepts. Journal of Hydrologic Engineering, ASCE 52, 115–123.

ASCE, 2001b. Task committee on artificial neural networks inhydrology, Artificial Neural Networks in Hydrology. II HydrologicApplications. Journal of Hydrologic Engineering, ASCE 52, 124–137.

Ashraf, M., Loftis, J.C., Hubbard, K.G., 1997. Application ofgeostatisticals to evaluate partial weather station network.Agricultural Forest Meteorology 84, 255–271.

Armstrong, M., 1998. Basic Linear Geostatistics. Springer Verlag,Germany.

Barnes, R.J., Johnson, T.B., 1984. Positive kriging. In: Verly, G.,David, M., Journel, A.G., Marechal, A. (Eds.), Geostatistics fornatural resources conservation, 1. Reidel, Dordrecht, Holland,pp. 231–244.

Braincel, 1998. Software Users Manual. Promised Land Technolo-gies, USA.

Chang, K.T., 2004. Introduction to Geographic Information Systems,third ed. McGraw Hill, 432 pp.

Chauvet, P., 1982. The variogram cloud. In: Johnson, T.B., Barnes,R.J. (Eds.), The Proceedings of the 17th APCOM Symposium.Society of Mining Engineers, New York, pp. 757–764.

Chiles, J.-P., Delfiner, P., 1999. Geostatistics: Modeling SpatialUncertainty. Wiley, New York.

Cybenko, G., 1989. Approximation by superpositions of a sigmoidalfunction. Mathematical Control Signals Systems 2, 303–314.

Daly, C., Neilson, R.P., Phillips, D.L., 1994. A statistical topographicmodel for mapping climatological precipitation over mountain-ous terrain. Journal of Applied Meteorology 33, 140–158.

Deutsch, C.V., Journel, A.G., 1998. Geostatistical Library Softwareand User’s Guide. Oxford University Press, New York.

Deraisme, J., Humbert, J., Drogue, G., Frelson, N., 2001. Geosta-tistical interpolation of rainfall in mountainous areas. In:Monestiez, P., Allard, D., Froidevaux, R., (Eds.), geoENV III –Geostatistics for Environmental Applications.

Dingman, S.L., 2002. Physical Hydrology. Prentice Hall, New Jersey.Freeman, J.A., Skapura, D.M., 1991. Neural Networks: Algorithms,

Applications and Programming Techniques. Addison-Wesley Inc.,USA.

French, M.N., Krajewski, W.F., Cuykendal, R.R., 1992. Rainfallforecasting in space and time using a neural network. Journal ofHydrology 137, 1–37.

Govindaraju, R.S., Rao, A.R., 2000. Neural Networks in Hydrology.Kluwer Academic Publishers., Netherlands.

Goovaerts, P., 2000. Geostatistical approaches for incorporatingelevation into the spatial interpolation of rainfall. Journal ofHydrology 228 (1), 113–129.


Grayson, R., Gunter, B., 2001. Spatial Patterns in CatchmentHydrology: Observations and Modeling. Cambridge UniversityPress, Cambridge.

Handcock, M.S., Wallis, J.R., 1994. An approach to statisticalspatial-temporal modeling of meteorological fields. Journal ofthe American Statistical Association 89 (426), 368–378.

Haykin, S., 1994. Neural Networks: A Comprehensive Foundation.Macmillan Publishing, New York.

Hornik, K., Stinchcombe, M., White, H., 1989. Multilayer feedfor-ward networks are universal approximators. Neural Networks 2(5), 359–366.

Hodgson, M.E., 1989. Searching methods for rapid grid interpola-tion. Professional Geographer 411, 51–61.

Holawe, F., Dutter, R., 1999. Geostatistical study of precipitationseries in Austria: time and space. Journal of Hydrology 219, 70–82.

Isaaks, H.E., Srivastava, R.M., 1989. An Introduction to AppliedGeostatistics. Oxford University Press, New York.

Jarvis, C.H., Stuart, N., Cooper, W., 2003. Infometric and statisticaldiagnostics to provide artificially-intelligent support for spatialanalysis: the example of interpolation. International Journal ofGeographical Information Science 17, 495–516.

Journel, A.G., Huijbregts, C.J., 1978. Mining Geostatistics. Aca-demic Press, New York.

Krajewski, W.F., 1987. Co-kriging of radar and rain gage data.Journal of Geophysics Research 92D8, 9571–9580.

Kyriakidis, P.C., Miller, N.L., Kim, J., 2004. A spatial time seriesframework for simulating daily precipitation at regional scales.Journal of Hydrology 297, 236–255.

Larson, L.W., Peck, E.L., 1974. Accuracy of precipitation measure-ments for hydrologic forecasting. Water Resources Research156, 1687–1696.

Maier, H.R., Dandy, G.C., 1998. The effect of internal parametersand geometry on the performance of back-propagation neuralnetworks: an empirical study. Environmental Modeling andSoftware 13 (2), 193–209.

Olea, R.A., 1999. Geostatistics for Engineers and Earth Scientists.Kluwer Academic Publishers, MA, USA.

Osborn, H.B., Renard, K.G., Simanton, J.R., 1979. Dense networksto measure convective rainfall in the southwestern UnitedStates. Water Resources Research 156, 1701–1709.

Panu, U.S., Khalil, M., Elshorbagy, A., 2000. Streamflow datainfilling techniques based on concepts of groups and neuralnetworks. In: Govindaraju, R.S., Ramchandra Rao, A. (Eds.),Artificial Neural Networks in Hydrology. Kluwer AcademicPublishers, The Netherlands.

Rumelhart, D.E., Mclelland, J.L., 1986. Parallel Distributed Pro-cessing. MIT Press, Cambridge, MA.

Salas, J.D.-J., 1993. Analysis and modeling of hydrological timeseries. In: Maidment, D.R. (Ed.), Handbook of Hydrology. Mc-Graw-Hill, New York.

Seo, D.J., 1996. Nonlinear estimation of spatial distribution ofrainfall – an indicator cokriging. Stochastic Hydrology andHydraulics 10, 127–150.

Seo, D.J., 1998. Real-time estimation of rainfall fields using radarrainfall and rain gage data. Journal of Hydrology 208, 37–52.

Seo, D.J., Krajewski, W.F., Bowles, D.S., 1990a. Stochastic inter-polation of rainfall data from rain gages and radar usingcokriging – 1. Design of experiments. Water Resources Research26 (3), 469–477.

Seo, D.J., Krajewski, W.F., Bowles, D.S., 1990b. Stochastic inter-polation of rainfall data from rain gages and radar using cokriging– 2. results. Water Resources Research 26 (5), 915–924.

Simanton, J.R., Osborn, H.B., 1980. Reciprocal-distance estimate ofpoint rainfall. Journal of Hydraulic, ASCE 106 (HY7), 1242–1246.

Singh, V.P., Chowdhury, K., 1986. Comparing some methods ofestimating mean areal rainfall. Water Resources Bulletin 222,275–282.

Shamseldin, A.Y., 1997. Application of a neural network techniqueto rainfall-runoff modeling. Journal of Hydrology 199, 272–294.

Shepard, D., 1968. A two-dimensional interpolation function forirregularly spaced data. In: Proceedings of the 23rd NationalConference of the Association for Computing Machinery, pp.517–524.

Sullivan, D.O., Unwin, D.J., 2003. Geographical Information Anal-ysis. Wiley, New Jersey.

Teegavarapu, R.S.V., Chandramouli, V., 2005. Improved weightingmethods, deterministic and stochastic data-driven models forestimation of missing precipitation records. Journal of Hydrol-ogy 312, 191–206.

Teegavarapu, R.S.V., Mujumdar, P.P., 1996. Rainfall forecastingusing neural networks. In: Proceedings of IAHR InternationalSymposium on Stochastic Hydraulics I, pp. 325–332.

Tung, Y.K., 1983. Point rainfall estimation for a mountainousregion. Journal of Hydraulic Engineering, American Society ofCivil Engineering ASCE 109 (10), 1386–1393.

Vieux, B.E., 2001. Distributed Hydrologic Modeling using GISWaterScience and Technology Library. Kluwer Academic Publishers,Dordrecht, The Netherlands.

Voltz, M., Webster, R., 1990. A comparison of kriging, cubic splinesand classification for predicting soil properties from sampleinformation. Journal of Soil Science 41, 473–490.

Webster, R.A., Oliver, M.A., 2001. Geostatistics for EnvironmentalScientists. Wiley, Chichester, UK.

Wei, T.C., McGuinness, J.L., 1973. Reciprocal distance squaredmethod, a computer technique for estimating area precipita-tion, Technical Report ARS-Nc-8., US Agricultural ResearchService, North Central Region, Ohio.

Yang, X., Holder, T., 2000. Visual and statistical comparisons ofsurface modeling techniques for point-based environmentaldata. Cartography and Geographic Information Science 27,165–175.

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