Modeling Time Series of Ground Water HeadFluctuations Subjected to Multiple Stressesby Jos R. von Asmuth1,2, Kees Maas1,2, Mark Bakker1,2, and Jorg Petersen3

AbstractThe methods behind the predefined impulse response function in continuous time (PIRFICT) time series

model are extended to cover more complex situations where multiple stresses influence ground water head fluctu-ations simultaneously. In comparison to autoregressive moving average (ARMA) time series models, the PIRFICTmodel is optimized for use on hydrologic problems. The objective of the paper is twofold. First, an approach ispresented for handling multiple stresses in the model. Each stress has a specific parametric impulse response func-tion. Appropriate impulse response functions for other stresses than precipitation are derived from analytical solu-tions of elementary hydrogeological problems. Furthermore, different stresses do not need to be connected inparallel in the model, as is the standard procedure in ARMA models. Second, general procedures are presentedfor modeling and interpretation of the results. The multiple-input PIRFICT model is applied to two real cases. Inthe first one, it is shown that this model can effectively decompose series of ground water head fluctuations intopartial series, each representing the influence of an individual stress. The second application handles multipleobservation wells. It is shown that elementary physical knowledge and the spatial coherence in the results of mul-tiple wells in an area may be used to interpret and check the plausibility of the results. The methods presented canbe used regardless of the hydrogeological setting. They are implemented in a computer package namedMenyanthes (www.menyanthes.nl).

IntroductionTransfer function noise (TFN) models are a conve-

nient tool for modeling the evolution of a wide range ofvariables. The general theory of time series analysis (Boxand Jenkins 1970) originally stems from the statisticalsciences. Because of their statistical background, theso-called autoregressive moving average (ARMA) timeseries models can be applied to all sorts of data, as longas the behavior of the system to be modeled is sufficiently

linear or can be linearized by transforming the data. Timeseries models are especially useful for modeling systemswhose behavior cannot, or not easily, be described interms of physical laws and properties (e.g., economicaldata). In addition, TFN models are often used in hydro-logy and other sciences because they are relatively easy toconstruct and at the same time they can yield very accu-rate predictions.

When an ARMA type TFN model is applied toa data set, the so-called model order has to be specified.The model order includes the number of autoregressiveand moving average parameters in both the deterministicand stochastic parts of the model and the delay time ofthe transfer function. Box and Jenkins devised an iterativemodel identification procedure to guide the modeler infinding the optimal model order. First, an initial modelorder is chosen based on statistical criteria like the cross-correlation function between the explained and explana-tory variables. Second, the parameters of the model areestimated by minimizing the variance of the innovationsor one-step-ahead prediction error using an optimization

1Corresponding author: Water Resources Section, Faculty ofCivil Engineering and Geosciences, Delft University of Technology,P.O. Box 5048, 2600 GA, Delft, The Netherlands; 131-30-6069512; fax 131-30-6061165; jos.von.asmuth@kiwa.nl

2Kiwa Water Research, Department of Water Systems Man-agement, P.O. Box 1072, 3430 BB, Nieuwegein, The Netherlands.

3Nature-Consult, Hackelbrink 21, D-31139, Hildesheim,Germany.

Received April 2007, accepted August 2007.Copyright 2007 The Author(s)Journal compilation2007National GroundWaterAssociation.doi: 10.1111/j.1745-6584.2007.00382.x

30 Vol. 46, No. 1GROUND WATERJanuaryFebruary 2008 (pages 3040)

algorithm. Third, the adequacy of the model is checkeddiagnostically using statistical criteria like the auto- andcross-correlation functions of the innovations. If themodel does not yet meet the criteria, the model order isupdated and the procedure is repeated until the modelerfinds the results satisfactory. A disadvantage of thisapproach is that the results of the model identificationprocedure can be ambiguous (Hipel and McLeod 1994)and the process itself is rather heuristic and can be knowl-edge and labor intensive (De Gooijer et al. 1985).

Von Asmuth et al. (2002) presented the principles ofa new type of TFN model that is optimized for use onhydrologic problems and operates in a continuous timedomain. In this approach, the discrete transfer functionused in ARMA models is replaced by a simple analyticalexpression that defines the impulse response function.The resulting class of models is referred to as predefinedimpulse response function in continuous time (PIRFICT).Von Asmuth et al. (2002) showed that PIRFICT modelsovercome a series of limitations of ARMA TFN models,including the use of irregular or high-frequency data andthe modeling of systems with a long memory. In addition,application of the PIRFICT model does not requirea Box-Jenkins style model identification procedure. Sincethe transfer functions are confined a priori to physicallyplausible behavior, there is no need to identify theorder of the transfer functions on statistical grounds.Therefore, application of the model is standardized,which facilitates implementation in a computer packagesuch as Menyanthes (www.menyanthes.nl). When thePIRFICT method was introduced (Von Asmuth et al.2002), we restricted ourselves to the case of a singleinput/output series. Here, we will extend the method tocover more complex, real world situations where multiplestresses influence head fluctuations at one or multipleobservation wells simultaneously.

Two important aspects of dealing with complex datasets are addressed in this paper. The first one is the treat-ment of different types of stresses within the model.Different stresses require different parametric impulseresponse functions. Von Asmuth et al. (2002) used thePearson type III function for modeling the effect of pre-cipitation surplus. Here, we will introduce analytical sol-utions of elementary hydrogeological schematizations asguides to develop appropriate impulse response functionsfor other stresses. We will also show that from a physicalpoint of view, different stresses do not always have to beconnected in parallel and get a separate transfer function,as is the standard procedure in ARMA TFN modeling.The second aspect deals with the interpretation andchecking of the plausibility of the results. While the timeseries literature commonly involves the analysis of indi-vidual time series, using the PIRFICT approach, one cananalyze and process all available series of heads in anarea in batch. Given that they are part of the same hydro-logical system, the results of neighboring observationwells may show a spatial coherence, which yields valu-able extra information as regards the properties of the sys-tem and the plausibility of the results. In this paper,however, all series are still modeled separately and the re-sulting spatial patterns are analyzed a posteriori. Future

research will include methods to impose spatial coher-ences a priori in the model.

This paper is organized as follows. First, we discusshow different types of stresses are dealt within the model.We illustrate the approach by analyzing a single seriesbeing influenced by precipitation, evaporation, groundwater withdrawal, and river-level fluctuations. Second,results are presented for a case where data are availablefrom multiple observation wells. A discussion and con-clusions are given at the end of the paper.

Methodology

From a Single to a Multiple Input ModelThe basic equation of an ARMA model, which is

discrete in time, is equivalent to the following convolu-tion integral in continuous time (Von Asmuth et al.2002):

hit Z t2N

Rishit 2 sds 1

where hi [L] is the predicted head at time t attributable tostress i. Ri is the value of stress i at time t, and hi is thetransfer or impulse response function of stress i. InARMA time series models, multiple stresses are dealtwith by connecting them in parallel and assigning a sepa-rate transfer function to each. The output series is thenobtained by summing the separate effects of all stresses.For the case where a number of N stresses influence thehead, the equations of a continuous time TFN model maybe written as follows:

ht XNi1

hit 1 d 1 nt 2

where h is the observed head at time t, d is the localdrainage level relative to some reference level [L], and nis the residual series [L].

Several main types of stresses can be distinguished.These types include precipitation (p), evaporation (e),ground water withdrawal (or injection) (w), surface waterlevel (s), barometric pressure (b), and (hydrological) inter-ventions (m). Please note that tidal fluctuations, on whoseeffect a large volume of paper is devoted, are included inthe s type. From a physical point of view, a ground watersystem is likely to respond to different types of stresses dif-ferently, but there are also certain stresses that will causea very similar response. In the latter case, separate stressesdo not necessarily need separate response functions. Forinstance, the effect of evaporation e on the head h is essen-tially the same as that of precipitation p, but it is negative,and may be modeled as follows:

het Z t2N

2 esfhpt 2 sds 3

where hp is the response of the system to precipitationand f is a reduction of e as compared to the reference

J.R. von Asmuth et al. GROUND WATER 46, no. 1: 3040 31

evaporation series. The evaporation factor f is a parameterthat depends on soil and land cover and should not beconfused with the crop factor (e.g., Penman 1948), as thelatter is used for a crop that is optimally supplied withwater, while f also incorporates the average reduction ofthe evaporation due to actual soil water shortages. Forsake of simplicity, we consider f to be constant. Althoughit may actually be a function of time, the use of an aver-age reduction factor is in fact already an improvementupon traditional ARMA modeling practice, where f isoften simply ignored. In the case of the barometric pres-sure b, which codetermines the loading on a (semi)-confined aquifer, we propose to use the time derivative inthe convolution:

hbt 2Z t2N

dbsds

hbt 2 sds 4

in order to be able to use an impulse response functionhb that behaves similar to the other stresses, as a stepchange of b will result in a quick increase of h, followedby a slow decay whenever the aquifer is not completelyconfined.

Interventions are defined as structural changes toa hydrologic system, like forest clearing, construction ofditches or drainage systems, and so on. In general, thenature of an intervention determines the manner in whichit should be modeled. For example, if the interventioncauses a sudden change in actual evaporation on time tm,such as a forest clearing, it may be modeled as follows:

hmt Z t2N

mskhpt 2 sds 5

where m(t) 0 for t < tm and m(t) 1 for t > tm, and k isa parameter representing the change caused by the inter-vention. Another example is a change in the level ofa floodgate. This intervention itself acts as an indepen-dent stress on the system, and, in such cases, a newresponse function hp should be estimated. It is noted thatit is possible that the hydrogeological properties of thesystem are changed significantly by an intervention. Inthat case, the response of the system to all stresses ischanged and Equation 1 becomes:

hit Ztm2N

Rishi1t 2 sds 1Z t

tm 1 t

Rishi2t 2 sds

6where hi1 and hi2 are the response functions before andafter the intervention, respectively. A transition period oft, during which the system shifts from one state toanother, should be omitted from the data. The length ofthis period depends on the response time of the system.

The Impulse Response Functions for DifferentTypes of Stresses

Under a wide variety of hydrogeological settings, theresponse of an impulse of precipitation surplus may be

simulated accurately with a Pearson type III distributionfunction (PIII) (Von Asmuth et al. 2002):

hpt Aantn21exp2 at

n 7

where A, a, and n are parameters that define the shape ofhp. The choice of this impulse response function wasbased on physical arguments, and it was shown that thePIII function was (at least) as effective as an ARMA typetransfer function of optimal order. Although the PIII func-tion is very flexible, it proves to be less effective for non-distributed types of stress. For ground water withdrawals(stress type w) and surface water level fluctuations (type s),we propose to use response functions inspired on analyti-cal solutions of simple hydrogeological schematizations.Notice that the crucial issue in selecting a parametricimpulse response function is whether the range of shapesit can take is sufficient to approximate the true responseof the system accurately. In this sense, our viewpoint isthat of time series analysis, where the only assumptionsregarding the transfer model are that the system is linearand that the model order is adequate. Here, we assumethat the functions chosen can capture the essential behaviorof the stress type regardless of the exact geohydrologicalsetting. A systematic comparison of the performance ofdifferent impulse response functions for different types ofstresses, however, falls beyond the scope of this paper andwill be dealt with in an upcoming paper.

For withdrawals (stress type w), we choose the wellformula of Hantush (1956) as a blueprint. The Hantushformula assumes a fully penetrating well in an aquifer ofinfinite extent, with transmissivity T [L] and storage coef-ficient S [2], covered by a storage-free aquitard withresistance c [T]. While a standard pumping test yieldsa step response function, here we are looking for animpulse response function, which is the derivative of thestep response function with respect to time:

hwt 2 14pTt

exp 2r2S

4Tt2

t

cS

8

where r denotes the distance between the observation andthe pumping well. Since we intend to use this formula inother hydrogeological settings as well, we convert thisequation to the following parametric impulse responsefunction:

hwt 2 ctexp 2

a2

b2t2 b2t

9

where a, b, and c are just parameters that no longer havea transparent physical meaning (except for cases whereHantush assumptions happen to be satisfied). For surfacewater fluctuations (stress type s), we choose the so-calledpolder function of Bruggeman as a blueprint (Bruggeman1999). This function represents a sudden unit increase ofthe water level at the boundary of a one-dimensional semi-confined aquifer of semi-infinite extent. The derivativegives the impulse response function hs, the response toa very short rise and fall of the surface water level:

32 J.R. von Asmuth et al. GROUND WATER 46, no. 1: 3040

hst 2 14pkDt3

x2S

r exp 2 x2S4kDt

2t

cS

10

where x denotes the distance between the surface waterfeature and the observation well. We convert the physicalparameters to the abstract parameters a9, b9, and a scalingfactor c9, so that hs becomes:

hst 2 c9pb92

a92t3

s exp 2 a92b92t

2 b92t

11

For stress type b (the barometric pressure), we chose thePIII function, that is, the same impulse response functionas used for precipitation and evaporation, because pressureis a distributed type of stress, too. The adequacy of a timeseries model based on these impulse response functionsmay be checked with the aid of the methods described andillustrated in the coming paragraphs. The practical use-fulness of these impulse response functions has been de-monstrated through application to thousands of wells inEurope, Australia, and North and South America.

Parameter EstimationThe first step in the parameter estimation process is

the evaluation of the model equations using an initial esti-mate of the parameters, which will result in a time seriesof model errors or residual series. When modeling hydro-logical data, the value of a model residual at a certaintime is often correlated with its value at earlier times, soresiduals cannot simply be modeled as a set of indepen-dent Gaussian deviates. Here, the model residuals aremodeled with a separate noise model, which is given by:

nt Z t2N

/t 2 sdWs 12

where / is the noise impulse response function and W isa continuous white noise (Wiener) process [L]. This isequivalent to an AR(1) model embedded in a Kalmanfilter under the pure prediction scenario in discrete time,when the response function / is exponential (Von Asmuthand Bierkens 2005). In that case, the one-step-ahead pre-diction error or innovation series m of the noise modelmay be obtained as follows:

mt nt 2 nt 2 texp2at 13The noise model is important for the parameter estima-tion process and for dealing with irregularly spaced data,but also for prediction, forecasting, and stochastic simula-tion purposes. For further details, we refer to Von Asmuthand Bierkens (2005), as the parameter estimation processitself is not influenced by the fact that the transfer modelcontains multiple stresses.

General Modeling ProcedureThe model identification procedure devised by Box

and Jenkins (1970) is that, first, the model order is

specified, second, the parameters are estimated, and third,the model results are checked. The adequacy of the modelresults may be checked with statistical criteria like theautocorrelation and cross-correlation functions of theinnovation series m. The autocorrelation function indicateswhether the white noise assumption holds, which is a pre-requisite of the algorithms used for the estimation of themodel parameters and their covariance; it also indicateswhether the order of the noise model is adequate. Thecross-correlation functions between the innovation seriesm and the different input series indicate whether there arepatterns left in the innovation series that could be ex-plained by the input series. This gives an indication of theadequacy of the order of the transfer functions but also ofpossible nonlinearity in the relationships. If the modeldoes not meet these so-called diagnostic checks, themodel order is updated. This procedure is repeated untilthe optimal model order is identified (Box and Jenkins1970).

As stated earlier, there is no need for identifying theorder of the transfer model on statistical grounds in thePIRFICT approach, as the impulse response functions arechosen on physical grounds and span a whole range ofARMA model orders (Von Asmuth et al. 2002). In bothcases, however, the modeler also has to identify thestresses that influence the ground water dynamics, decidewhich stresses to use, and check the results. Stresses thatare not incorporated in the model can lead to erroneousresults when their dynamical behavior is correlated withone or more of the other (already considered) stresses. Onthe other hand, the model may have difficulty in uniquelyidentifying all influences when there is a high number ofstresses, a lack of pronounced dynamics of the stresses, ora low quality or scarcity of the data. The diagnosticchecks introduced by Box and Jenkins are devised toassess whether the TFN model is adequate and optimal ina statistical sense, which remains important especially forthe noise model, but does not exclude the possibility thatthe model results are influenced by noncausal correla-tions. Here, we propose the use of plausibility checks toguide the modeler in assessing whether the results of thetransfer model are physically realistic. The resulting mod-eling procedure is illustrated in Figure 1.

The plausibility checks include the model residuals,the evaporation factor f, the local drainage base d, and themoments of the impulse response functions and theirstandard deviations. The model residuals, uniting all fac-tors that are not accounted for by the model, are animportant aid in identifying possible unknown stressesthat may be a source of model distortions. Nonrandompatterns of the residuals in space or time reveal the factthat there are still stresses missing in the model. The pat-terns themselves often give enough information to pin-point the nature and location of the missing stresses. Theevaporation factor f is important, as the seasonal cycle inthe evaporation is often present in other natural or anthro-pogenic stresses such as ground water withdrawals foragricultural or drinking water purposes as well. Regard-ing the drainage base, an estimate that is too low or toohigh may be caused by the influence of stresses that arenot incorporated in the model or that are not well

J.R. von Asmuth et al. GROUND WATER 46, no. 1: 3040 33

quantified. This can easily happen with stresses that donot show pronounced dynamics, such as more or less con-stant seepage or withdrawal rates. In addition, the mo-ments of the impulse response functions of the differentstresses provide relevant information. Moments can beused to characterize the functioning of the ground watersystem and can be related to its geohydrologic properties(Von Asmuth and Maas 2001; Von Asmuth and Knotters2004). In contrast to physical parameters that are definedonly in the context of a certain schematization, momentsare related to common statistical terms and are more gen-erally applicable. The jth moment of an impulse responsefunction is defined as follows:

Mj ZN2N

t jhtdt 14

where M0 represents the area under the impulse responsefunction, l M1/M0 is the mean of the impulse responsefunction, and r2 M2/M0 2 l2 is the variance. Matchingof moments is a common technique for solving differen-tial equations, amongst others in transport modeling (e.g.,Yu et al. 1999; Luo et al. 2006).

Example Applications

Single Head SeriesThe functioning and results of a multiple input PIR-

FICT model on a single well located in the northern partof the province of Limburg (the Netherlands) are pre-sented in this subsection. The well, with the nationalcode 46DP0032, has two screens. We consider only thetop one, which is located 13 m below the surface. Thewell is located on the edge of the floodplain of the riverMeuse in an aquifer that consists of coarse gravelly sandsoverlain by finer sands which originate from river depos-its. In addition to the river-level fluctuations, the head is

influenced by precipitation and evaporation and bya pumping station near the town of Bergen where groundwater is withdrawn for drinking water production.

Time series data of all stresses are available. Theprecipitation and potential evaporation series originatefrom stations of the Royal Dutch Meteorological Insti-tute in Venray and Eindhoven, respectively. The riverlevels were monitored at a dam in Sambeek, downstreamof well 46DP0032; the pumping rates were obtainedfrom the drinking water company of Limburg. Theparameters of all three impulse response functions areoptimized using the methods described in Von Asmuth andBierkens (2005). Here, we will consider the results of thetransfer part of the model and of this individual seriesonly, after the first run of the model. The model resultsand parameter estimates are summarized in Table 1. Inthe table, two parameters are given that define the good-ness of fit: the percentage of variance accounted forR2adj and the root mean squared error. In the definition ofR2adj, the residual variance r

2nt is weighed according to

the variance r2htof the original signal in the followingmanner:

R2adj r2ht 2 r

2nt

r2ht3 100% 15

The results of the time series model are shown inFigure 2, where the measured heads h (dots) and pre-dicted heads

PNi1 hi 1 d(solid) are plotted together in

the upper graph, while hi is plotted for every stress in thegraphs beneath. Thus, the ground water level series is de-composed into four partial series, which show the effectsof the individual stresses. For example, it may be observedthat the recent series of wet years in the Netherlands (1999to 2002) has led to an overall increase in the ground waterlevels. Furthermore, the effect of evaporation does notvary much from year to year but shows a distinct seasonalpattern as it is mainly influenced by temperature and solarradiation. The river level has little effect as it is maintainedby dams, except for high-water events when the riverleaves its channel and enters the floodplain. In those cases,the head responds very quickly and shows distinct peaks.Finally, heads show a gradual decline since 1994 due to theground water withdrawal that started around that time.Note that these distinct differences in dynamic behaviorallow the time series model to distinguish the effects of theindividual stresses. Also note that the frequency of theground water level observations changes around 1992 fromfour times a year to once a week. This does not posea problem for the model, as the impulse response equationsare continuous in time and predictions are not fixed toa certain time discretization.

For each of the different stresses, an impulse responsefunction is estimated (although not always independentfrom the other stresses, see Equations 3 through 5). Theimpulse response function forms the heart of the timeseries model and represents the response of heads toa unit impulse of the stress. The shape of the impulseresponse function depends on the position of the observa-tion well and the properties of the system. As an example,the impulse response function of the level of the river

Identificationof Stresses

ParameterEstimation

PlausibilityChecking

DiagnosticChecking

ParameterEstimation

ModelIdentification

Figure 1. Proposed procedure for modeling time series ofground water heads in complex situations. The procedurefor identifying the noise model is that of Box and Jenkins(1970). When both the plausibility and diagnostic checks aremet, the results can be accepted for further use.

34 J.R. von Asmuth et al. GROUND WATER 46, no. 1: 3040

Meuse is plotted in Figure 3. From the graph, one caninfer that the heads indeed respond quickly to a rise andfall in river level, as the impulse response function (solidline) peaks in less than half a day. Apart from its dynamicresponse, one is often also interested in the stationaryinfluence of a certain stress. This is represented by thestep response function (dashed line), which is the integralof the impulse response function with respect to time; thestep response represents the response of the system toa sudden and then ongoing rise of the river level. Thelevel that the step response function approaches towardinfinity is the zeroth moment (M0), also known as the

gain in the time series analysis literature. Using Equation11, we can obtain M0 for stresses of type s as follows:

M0;s ZN0

hstdt c9exp22a9 16

In the case presented, M0,s for the river Meuse is0.73 m, which means that a river level rise of 1 m willeventually lead to a ground water level rise of 73 cm onthis location. For as long as the assumption of linearityholds, the zeroth moment can be used for quick scenariocalculations, as the multiplication of M0,s and a planned

Table 1Model Results and Parameter Estimates for the Ground Water Head Series Observed in Well 46DP0032

Input series Stress Type Parameter Value r

Venray Precipitation A(M0,p) 299.2 23b 0.003117 0.00033n 0.898 0.025

Eindhovenentry Evaporation f 1.16 0.076P.S. Bergenentry Pumping well a 2.58 2.4

b 0.0522 0.027c 0.00585 0.03M0,w 23.633 1025 5.683 1026

SambeekBoven River a9 0.02527 0.094b9 0.0368 0.13c9 0.7656 0.057M0,s 0.7279 0.0939

Note: R2adj 92.0%. Root mean squared error 0.068 m. Drainage base 11.53 m.

11.5

12

12.5

13

Hea

d

refl

(m)

Results of series 46DP0032PredictionObservations

0.40.60.8

Ris

e (m

)

VENRAY

0.80.60.4

Ris

e (m

)

EINDHOVEN

00.5

1

Ris

e (m

)

SambeekBoven

1980 1985 1990 1995 2000

0.10

0.1

Ris

e (m

)

Date

P.S. BERGEN

Figure 2. The ground water head fluctuations in well 46DP0032 (top) decomposed in four partial series due to (from top tobottom) rainfall, evaporation, river-level fluctuations, and a pumping station. Adding the partial series and the estimateddrainage base d (Table 1) results in the predicted series.

J.R. von Asmuth et al. GROUND WATER 46, no. 1: 3040 35

rise in the river level yields a prediction of the groundwater level rise. Note that the standard deviation of M0,s isreasonable, whereas the standard deviation of parametera9 is large (a similar case is true for M0,w of the groundwater withdrawal). This is caused by the covariancebetween the parameters.

Multiple Head SeriesWe will illustrate the procedure in which the results

from multiple observation wells may be interpreted andchecked in a second application. This application con-cerns a drinking water production area with multipleobservation wells located in the northwestern part ofGermany, in an area called Harlingerland near theTown of Esens. Ground water is withdrawn by the drink-ing water company Oldenburgisch-Ostfriesischer Wasser-verband (OOWV) at a current rate of 9 million m3/year.The 15 pumping wells are clustered in two rows, whichare more or less placed in series (Figure 4). The wellscreens are located at a depth of 25 to 40 m from the soilsurface. The withdrawal rate history (Figure 5) showsa marked increase in the period from 1972 until 1976.

Such a rate change is very important for a reliable esti-mate of the influence of a pumping well, as it will havecaused a marked drawdown. Furthermore, the withdrawalrates show a distinct seasonal cycle caused by factors likethe increased watering of yards in dry periods. The with-drawal rate is obtained by summing the pumping rates ofthe individual wells. Data on the individual wells, whichare controlled separately, are available only since 1992.Anyway, use of the individual pumping rates is notstraightforward, as the number of 15 wells is too high toestimate their influence independently. Ground waterdynamics are recorded at 116 observation wells witha total of 139 screens in a circle with a radius of 5 kmaround the pumping wells. The screens depths range fromless than 1 to more than 90 m from the surface. Part of thewells are monitored every 2 weeks and the other partmonthly. The period in which the wells were monitoreddiffers from well to well, with the earliest measurementsdating back to 1964, while other wells were installed asrecently as 1999. In some of the wells, monitoring stop-ped in 1975, while in some others, the period between1975 and 1998 is missing. The aquifer in the area consistsmainly of sands; in some parts, a clay layer separates thephreatic from the deeper aquifer.

The head fluctuations are modeled with precipitation,reference evapotranspiration, and ground water with-drawals; there are no rivers or other important fluctuatingsurface water in the area. Results of the 139 time seriesmodels are summarized in Table 2, where the minimum,median, and maximum value of the parameters in all 139models are given, along with their 95% confidence inter-val. The median value of R2adj points out that the fit ofmost models is good, although there are clearly outliers inthe results, as the extremes of most parameter estimatesare not within a range that is physically plausible. Thisdoes not mean that the estimates are necessarily biased, asmost extremes are accompanied by large standard devia-tions. For these cases, a confidence interval of 62r in-cludes the value zero, but also a realistic value indicatingthat the quality or quantity of the data are not sufficient todetermine the value of that parameter.

The only estimate in Table 2 that does seem biased isthe minimum value for the evaporation factor f, which is

0 2 4 6 8 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Time (days)

Res

pons

e Fa

ctor

()

Step ResponseImpulse Response95% Conf. Int.

Figure 3. Estimated impulse and step response functions ofobservation well 46DP0032 for the level of the river Meuse(measured at dam Sambeek-Boven).

Figure 4. Spatial distribution of the percentage of variance accounted for R2adj, in plan view (left figure). Low values for R2adjare found near the 15 pumping well screens that are clustered in two rows (red dots in right figure).

36 J.R. von Asmuth et al. GROUND WATER 46, no. 1: 3040

very low (upper 95% limit is 0.4). In this case, however,the zeroth moment of the evaporation M0,e which isdefined as follows:

M0;e ZN0

fhptdt fA 17

does have a large confidence interval (6941), as theinfluence of precipitation itself (A) cannot be determinedfrom the data accurately (the probable cause is that theperiod in which data are available for this particular seriesspans only 16 months, with one observation per month).

To determine the reason for the lower R2adj of somemodels, its spatial distribution is plotted (Figure 4). Fromthe figure, it is clear that the model fits are lowest in theimmediate vicinity of the pumping wells. This suggeststhat the influence of the individual wells is not modeledcorrectly, which is indeed the case, as the withdrawal rateof the total wellfield is incorporated in the model, not therates of the individual wells. The individual model fits(Figure 6) give the same indication. Near the wellfield,the head fluctuates wildly when the individual pumpingwells are shut on and off. The predictions, however, fol-low only the general drawdown pattern. At some distance,the predictions fit the head fluctuations much better.

The model fit is also lower in some of the shallowerwells. The head series in those cases show distinct signsof (threshold) nonlinearity (e.g., Knotters and De Gooijer1999). The PIRFICT method can currently handle thresh-old nonlinearity for precipitation and evaporation, but notyet for other stresses, so this was not considered further atthis time.

Next, we focus on the estimated influence of thewellfield, which was an important objective for doingthe time series analysis of this site. Using Equation 9,we find that the zeroth moment of a well is given by(Hantush 1956):

M0;w ZN0

hwtdt 2 2cK02a 18

in which K0 is the modified Bessel function of the secondkind and zeroth order (Abramowitz and Stegun 1964).First, we remove all wells with estimates of M0,w withlarge confidence intervals, as they are of little value andonly blur our view on the other results; we choose a cut-off value of 8 3 105. The 34 series that were disregardedall lack the period from 1972 to 1976 in which the with-drawal rate was increased significantly. In Figure 7, thespatial distribution of the 105 remaining M0,w estimatesand their confidence interval is plotted in two cross sec-tions, one parallel and one perpendicular to the series ofpumping wells. As expected, the M0,w estimates are high-est near the pumping wells; the left row of wells, whichhave deeper screens, causes stronger drawdowns than theright row. In the perpendicular cross section, the patternof M0,w approximates the shape of a drawdown cone. TheM0,w estimates at shallow well screens are clearly lowerthan at deeper well screens; this may be caused by a resis-tance layer or by recharge from ditches or creeks. This isin line with the head series of these observation wells (notshown here), where head differences between higher andlower screens range up to 4 m. Although the model fit islow in the vicinity of the pumping well screens (withina radius of about 200 to 300 m), all in all, the M0,w esti-mates seem nevertheless reasonable. The estimates ofother parameters near the wells, however, are clearlybiased. In Figure 8, cross sections are shown of R2adj andM0,e for the observation wells in the vicinity of the pump-ing wells. The low values of R2adj near the pumping wellscreens correlate with the low values of M0,e. As it is notlogical for the influence of evapotranspiration to be

1965 19791972 1986 1993 2000

1

1.5

2

2.5

3

3.5

4

4.5x104

Flow

rate

(m3 /d

ay)

Date

Withdrawal

Figure 5. Ground water withdrawal history at location Har-lingerland. The flow rate is obtained by summing the ratesof the individual pumping wells.

Table 2Range of Model Results and Parameter Estimates for All 139 Ground Water Head Series

Parameter Minimum (62r) Median Maximum (62r)

R2adj 38.1 85.9 97.5M0,p 59 (61.33 103) 864 6580 (63.53 105)M0,e 172 (6122) 790 7976 (65913)M0,w 1.933 1028 (64.13 1025) 4.503 1025 1.253 1022 (60.20)f 0.09 (60.31) 1.20 4.29 (687)d 23.21 1.26 7.01

J.R. von Asmuth et al. GROUND WATER 46, no. 1: 3040 37

heterogeneous in deeper soil layers, this indicates that theinfluence of evapotranspiration is overestimated andpartly correlates with the effect of the individual pumpingwells.

Discussion and ConclusionsIn this paper, the PIRFICT model for time series

analysis was extended to handle multiple inputs. Forstresses other than areal recharge, analytical solutions ofsimple hydrogeological schematizations were used asa guide to develop appropriate impulse response func-tions. Different stresses are not necessarily connected inparallel in the model, as is the standard procedure inARMA models. A case with a single observation wellwas used to illustrate how the model can effectivelydecompose head series into partial series that each showthe effect of an individual stress. In the example withmultiple observation wells, it was shown that, next to its

a priori use in defining the impulse response functions,physical knowledge is also valuable in checking the con-sistency and plausibility of the model results a posteriori.The parameter values should fall within a range that isphysically plausible. The spatiotemporal patterns ob-served in the variables supply important and independentfeedback on the results, as there is no spatial dependencyimposed on the models. By focusing on the model resid-uals, missing stresses, processes, or other sources of errormay be readily identified. High error levels are a possiblesource of bias in the estimates, as other stresses maypartly compensate missing stresses when their influenceis correlated. In this case, the main source of error wasthe fact that the behavior of the individual pumping wellswas not accounted for. In spite of this, the overall draw-down pattern of the wellfield in total, which is oftenthe factor of interest, was represented well. The spatialdistribution showed that the estimates for the evapora-tion series were clearly biased. Such a problem may be

3

2

1

0

1

2

3

Hea

d

refl

(m)

Results of series F2320151_1

1969 1975 1981 1987 1993 1999

00.5

11.5

22.5

Hea

d

refl

(m)

Results of series B2311630_1

Figure 6. Head observations and predictions of wells F2320151 (near the pumping station) and B2311630 (at some distance),giving an indication of good and bad model fits.

Figure 7. Estimated gains (dots) for the wellfield in the different observation wells presented in west-east (left) and north-south (right) cross sections. The error bars indicate the 95% confidence interval of the estimates.

38 J.R. von Asmuth et al. GROUND WATER 46, no. 1: 3040

corrected by incorporating data on the missing factors inthe model (in this case, the pumping at individual wells).If such data are not available (as in this case), the biascan be reduced by constraining the optimization problemto a realistic range based on the values and patterns in thesurrounding observation wells. Future research willinclude methods to impose a spatial coherence a priori inthe model, so that the individual dynamics of a high num-ber of pumping wells can be incorporated in the modeland the results of neighboring observation wells arelinked and remain plausible.

The presented time series model may be applied todecompose series of head fluctuations into partial series,representing the influence of individual stresses. This en-ables the evaluation of individual effects of a stress, suchas hydrological interventions, pumping wells, climaticchanges, and surface water levels. Furthermore, the pre-sented model may be used for forecasting, gap filling,scenario studies, trend analyses, and optimization andcontrol of hydrogeological systems (using more or lessstandard time series analysis methods; e.g., Hipel andMcLeod [1994]). The PIRFICT approach is particularlysuited for the batch processing of many series, it usesa small number of parameters, and it is not limited byirregular or high-frequency data. Although here we spe-cifically focus on ground water heads, the approach pre-sented can also be applied to (contaminant) transportproblems or for that matter to hydrological problems ingeneral. In the field of transport, the Pearson III functionhas proven to be well usable also (it matches the convec-tion-dispersion equation [e.g., Jury and Roth 1990; Maas1994]), whereas in other fields, care has to be taken toselect appropriate impulse response functions and meth-ods for the processes and stresses occurring there.

An attractive feature of time series models is thatthey are based on relatively few assumptions and the fitsare generally high; the model lets the data more or lessspeak for itself. As such, time series models are a valuabletool for preprocessing ground water level series beforecalibrating a ground water model. Using time series mod-els, missing stresses or series that are influenced by

hydrological interventions may be readily identified.Also, series may be identified that are not suitable formodel calibration, for example, because they representa hydrological feature that is not incorporated in themodel, such as perched water tables. One step further,transient ground water models may be calibrated directlyon the moments of the impulse response functions esti-mated with time series analysis (Van de Vliet andBoekelman 1998; Von Asmuth and Maas 2001), whichrequires the calibration of steady models only. Momentsof the impulse response functions can also be modeledspatially with the analytic element method, creating a pos-sibility for transient modeling with analytic elements(Bakker et al. 2007).

AcknowledgmentsThe authors wish to thank the Provincial Council of

Limburg, the drinking water company OOWV and KiwaWater Research for funding the research. We also thankAndres Alcolea and two anonymous reviewers for theirdetailed and constructive comments, which helped toimprove the paper.

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