real-time forecasting of water table depth and soil moisture profiles

Advances in Water Resources 29 (2006) 692–706

www.elsevier.com/locate/advwatres

Real-time forecasting of water table depth and soil moisture profiles

Ate Visser a, Roelof Stuurman a, Marc F.P. Bierkens a,b,*

a TNO-Netherlands Institute of Applied Geoscience, P.O. Box 80015, NL-3508 TA Utrecht, The Netherlandsb Department of Physical Geography, Utrecht University, P.O. Box 80115, NL-3508 TC Utrecht, The Netherlands

Received 9 September 2004; received in revised form 11 July 2005; accepted 12 July 2005Available online 15 November 2005

Abstract

We present a method for real-time forecasting of water table depth and soil moisture profiles. The method combines a simpleform of data-assimilation with a moving window calibration of a deterministic model describing flow in the unsaturated zoneand local as well as regional drainage. The local drainage level is calibrated on-line using a moving window calibration. Assigningmore weight to the last available measurements then yields a form of model adaptation that is in between on-line calibration anddata-assimilation (i.e. a simplified form of Newtonian nudging). Five-day hydrological forecasts are performed based on 5-dayweather forecasts, while on-line observations of phreatic level and soil moisture content are assimilated on a daily basis. Advantagesof the proposed method are that it improves the real-time forecasts compared to off-line calibration and ordinary moving windowcalibration and that it yields physically consistent soil moisture profiles.� 2005 Elsevier Ltd. All rights reserved.

Keywords: Water table depth; Soil moisture; On-line forecast; Calibration; Data-assimilation

1. Introduction

In areas where the phreatic surface is close to the landsurface, its fluctuation has a large impact on agriculturalyield as well as on the functioning of ecosystems [2].Consequently, in groundwater dependent agro-ecosys-tems, reliable models for forecasting phreatic surfacedepth can be invaluable tools for successful managementof agricultural and ecological resources. Models used forforecasting can be fully physically mechanistic, com-pletely empirical (black box), or a combination of both.Examples of empirical models are autoregressive exoge-nous (ARX) models (which can partly be physicalbased, [9]), threshold autoregressive self-exciting open-

0309-1708/$ - see front matter � 2005 Elsevier Ltd. All rights reserved.doi:10.1016/j.advwatres.2005.07.011

* Corresponding author. Address: TNO-Netherlands Institute ofApplied Geoscience, P.O. Box 80015, NL-3508 TA Utrecht, TheNetherlands.

E-mail addresses: [email protected] (A. Visser), [email protected] (R. Stuurman), [email protected] (M.F.P. Bierkens).

loop (TARSO) models [10] or transfer function noise(TFN) models [3,11,12]. In this paper, next to watertable depth, we also aim to predict the soil moisture pro-file. Even though multivariate time series models havebeen used to model soil moisture content at a limitednumber of depths [1], for a complete description of thesoil moisture profile a physically mechanistic model ismore convenient. Therefore, a physically mechanisticmodel is used in this paper.

Physically based models have been used extensivelyto predict both water table and soil moisture states.For instance, Knotters and Van Walsum [11] and Knot-ters and De Gooijer [10] have calibrated the unsaturatedzone model SWATRE to generate water table depthregimes. Wendroth et al. [12] used a Kalman Filter toimprove the prediction of soil moisture profiles fromsurface measurements, whereas Hoeben and Troch [7]showed the application of the Kalman Filter to soilmoisture profile prediction from satellite data in a theo-retical case. These examples show that the gap between

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A. Visser et al. / Advances in Water Resources 29 (2006) 692–706 693

deterministic modelling and time series analysis is nar-rowing, as more complex models are used and the tech-niques used for modelling groundwater fluctuations shifttowards data-assimilation.

Most of the work cited above focuses on off-linecalibration and validation. However, availability ofon-line measurements, e.g. using TDR, pressure trans-ducers, GSM and internet technology, makes on-lineforecasting possible. In this paper, we introduce amethod for on-line forecasting of water table depthand soil moisture profiles using a deterministic modelfor saturated–unsaturated flow, a simple form of data-assimilation, on-line calibration, and weather forecasts.On-line forecasts can be used for both drought and floodwarnings, enabling more timely and accurate opera-tional water management. On-line calibration anddata-assimilation provides substantially better forecaststhan off-line calibration, as we will show later.

The article is composed as follows. Section 2describes the data set, the unsaturated zone model used,the on-line calibration and data-assimilation methodand the numerical experiments performed. The resultsare presented and discussed in Section 3, after whichthe main conclusions are given in Section 4.

2. Materials and methods

2.1. Data and test site

The available data consisted of 20 years (1980–2000)of daily measurements of the phreatic surface depthand 20 years of daily averaged meteorological datafrom the same period (precipitation and Makkink refer-ence crop evapotranspiration [4]). From this data set,

Table 1Observed soil horizons, with Staring Centre code and corresponding Mualemcontent hr, saturated hydraulic conductivity ksat, empirical parameters a (air eparameter soil moisture retention curve)

Top [cm] Bottom [cm] Staring Center code hs [cm3/cm3] h

0 40 B2 0.43 040 90 O2 0.38 090 200 O1 0.36 0

Table 2Data sets from De Bilt (Netherlands) used in this study

Data set Groundwater Soil moisture M

Set 1 Daily groundwaterlevels 1980–2000;observed manually

Not available Da1

Set 2 Daily averaged groundwaterlevels observed witha pressure transducer from13-11-2003 to 16-2-2004

Daily averaged valuesof soil moisture contentobserved with TDR at 25,50, 75, 100 cm below surfacefrom 13-11-2003 to 16-2-2004

Fpb(f1

three sets of 5-year periods were selected: 1982–1986,1987–1991 and 1992–1996. These three periods can beconsidered as representing the meteorological range atthe site, i.e. relatively dry, average and relatively wet.The three data sets were selected for reasons of testingmodel performance in varying circumstances (robust-ness).

The groundwater data were obtained from an obser-vation well located on the main meteorological field ofthe Royal Netherlands Meteorological Office (KNMI)at De Bilt (Central Netherlands near the city of Utr-echt), which is also the source of the meteorologicaldata. The field lies at the edge of an ice-pushed ridge thatis a remnant form the glaciers that covered the north ofthe Netherlands during the Saalien ice age. Because ofits proximity, the ridge is expected to be a source ofregional groundwater fluxes, influencing the phreaticsurface [3]. The soil at the test site was divided into threehorizons. For each horizon Mualem–Van Genuchtenparameters were obtained from the Dutch pedotransferfunction system developed at the Staring Centre [13], asshown in Table 1.

On-line measurements of water table depth and soilmoisture were available for a period of 3 months(November 13, 2003 to February 16, 2004). Soil mois-ture was measured at 25 cm below surface (as an averageof two measurements 1 m apart), and 50 cm, 75 cm and100 cm below surface. To remove any instrumentalnoise and within-day variation, 1-h interval measure-ments were averaged to obtain daily values. For thisperiod, measurements of precipitation and necessaryvariables to calculate Makkink reference evapotranspi-ration [4], as well as 5-day meteorological forecasts ofthese variables were available. Table 2 gives an overviewof the available data.

–Van Genuchten parameters: saturated water content hs, residual waterntry value), I (shape parameter unsaturated conductivity) and n (shape

r [cm3/cm3] ksat [cm/d] a [1/cm] I [–] n [–]

.02 9.65 0.0227 �0.983 1.548

.02 15.56 0.0214 0.039 2.075

.01 13.21 0.0224 0 2.167

eteo Remark

aily averaged precipitationnd evapotranspiration980–2000

Three data subsets of 5 yearseach were selected for reasonsof comparison

or each day, daily averagedrecipitation and evapotranspiration,oth observed and 5-day forecastsrom weather forecasts) from3-11-2003 to 16-2-2004

Groundwater levels and soilmoisture data from on-linemeasurements; On-line meteoobservations and 5-dayforecasts from KNMI website

694 A. Visser et al. / Advances in Water Resources 29 (2006) 692–706

2.2. Modelling water table depth and soil moisture content

For modelling water table depth and soil moistureprofiles the Soil Water Atmosphere Plant model(SWAP) [6] was used, which applies Richards� equation,in combination with the Mualem–Van Genuchten soilhydraulic parameters, to describe the flow of water inthe unsaturated zone. A linear relationship betweenlocal drainage flux qd and phreatic surface height h

was used, parameterised by the surface water level ofthe nearest drainage ditch or water course, hs, and thedrainage and infiltration resistance, c:qd = (h � hs)/c.The regional flux from/to the deeper groundwater, i.e.the upward or downward seepage flux qs, was assumedto be constant in time here, although it might have aseasonal fluctuating component. Due to preferentialflow, Richards� equation is not able to adequately modelthe dynamics of groundwater level fluctuations in allsoils. Therefore, a mobile fraction, fm, was used to com-pensate for preferential flow of percolating water. Themobile fraction concept is similar to a double porosityconcept. It reduces the domain where actual flow occursto a fraction fm. The immobile fraction of the domain,1 � fm, exchanges water with the mobile domainthrough diffusion only. During initial testing of themodel, it turned out that the unsaturated conductivitiesobtained from the pedotransfer function system devel-oped at the Staring Centre [13] were somewhat toosmall. Therefore, the saturated conductivity ksat, of allhorizons was multiplied by a single factor kf, whichwas calibrated together with the other parameters. Theshape parameters of the unsaturated conductivity func-tions were not changed. This left five parameters (hs, c,qs, fm and kf) to be estimated by calibration.

2.3. Calibration, forecasting and data-assimilation

2.3.1. Off-line calibrationInitially, the five parameters hs, c, qs, fm and kf were

calibrated off-line using a Gauss–Marquardt–Levenbergalgorithm and the groundwater data from 1982 to1996 only (Data set 1 of Table 2). The additional useof measurements on soil water content (Data set 2 ofTable 2) also allowed for calibration of the Van Genuch-ten parameters a and n, which may strongly improve soilwater profile predictions. As a and n influence satura-tion, and indirectly conductivity, it was necessary to alsore-calibrate fm and kf in order to capture the dynamicsof the groundwater fluctuations. To reduce the numberof free parameters in this case, the second and third soillayer were taken together.

2.3.2. Forecasting

Before explaining the various ways by which on-linemeasurements are used while forecasting, it is useful tobriefly describe what we mean by using the model in fore-

casting mode. In forecasting mode, observations of mete-orological input (precipitation and evapotranspiration)are used to advance the model to the current day, i.e.to reconstruct the state (groundwater level and soil mois-ture profile) at the current day as accurately as possible.Next, weather forecasts are used as input to the model,such that with the current predicted state as initial condi-tion, the state is forecasted for the coming days.

2.3.3. On-line calibration and forecasting

When applying the model in forecasting mode using asingle parameter set, it becomes clear that seasonal var-iation is not captured by the model. For instance,parameters such as surface water level hs and seepageflux qs will vary with season. To account for this varia-tion, a moving window on-line calibration of a singleparameter was used (either hs, or qs; see hereafter) toimprove forecasts by accounting for seasonal fluctua-tions. The moving window calibration amounts to recal-ibrating the single parameter based on observations ofthe last n-days (n is the window size) by minimizingthe criterion:

JðkÞ ¼Xn

i¼0

R2ðk � iÞ ð1Þ

with J(k) the value of the criterion at day k, R(k � i)being the residuals to the measurements taken i daysago. On each day k, the parameter (hs or qs) obtainedfrom the last calibration is used for forecasting ahead,i.e. for days k + 1, k + 2, etc. (with weather forecastsas input).

2.3.4. Adding data-assimilation

For short lead-time forecasting with a dynamicmodel, two criteria are important: (1) adequate parame-ters and (2) a close approximation of the initial condi-tions. The moving window calibration takes care ofthe first criterion (adequate parameters), but does notguarantee that the model state predicted at time k isclose to the real state, i.e. the initial conditions neededfor forecasting times k + 1, k + 2, etc. In fact, we foundthat the residual errors of the calibrated series to the lastavailable measurements were often considerable, intro-ducing an error in the initial condition for the forecast.To reduce this error, the objective function for themoving window calibration was changed by giving extraweight to the last measurements of the n-day calibrationperiod. This will force the calibration method to �worktowards� the last measurements. We define k as the mea-sure for the extra weight (i.e. k = 0 assigns equal weightto all measurements and k = 1 assigns only weight to themeasurements on the last day). The new objective func-tion J then becomes:

JðkÞ ¼ ð1� kÞ �Xn

i¼0

R2ðk � iÞ þ k � R2ðkÞ ð2Þ

Tab

le3

Pro

per

ties

and

dat

ase

tsu

sed

inth

em

od

elex

per

imen

tsin

this

pap

er

Exp

erim

ent

no

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mil

atio

nF

ive-

day

fore

cast

Rem

ark

1h

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fm,

kf

fro

mgr

ou

nd

wat

erd

ata

1982

–199

6N

oN

oU

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go

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met

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ata

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m19

82to

1996

Bre

aku

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mp

aris

on

2c,

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fm,

kf

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nd

wat

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ata

1982

–199

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om

a31

-day

mo

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ind

ow

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bra

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nu

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nd

wat

erd

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1982

–199

6

Yes

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sin

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nd

wat

erd

ata

1982

–199

6U

sin

go

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met

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ata

fro

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1996

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on

3c,

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fm,

kf

fro

mgr

ou

nd

wat

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1982

–199

6h

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om

a31

-day

mo

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ind

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bra

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sin

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nd

wat

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1982

–199

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syn

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icm

easu

rem

ent

no

ise

add

ed

Yes

,u

sin

ggr

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nd

wat

erd

ata

1982

–199

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ith

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icm

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rem

ent

no

ise

add

ed

Usi

ng

ob

serv

edm

eteo

dat

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om

1982

to19

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syn

thet

icer

rors

add

ed

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aku

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ree

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iod

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4c,

qs,

fro

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ou

nd

wat

erd

ata

1980

–200

0;fm

,K

san

dV

anG

enu

chte

np

aram

eter

sa

and

nu

sin

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nd

wat

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om

13-1

1-20

03to

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-200

4.

hs

fro

ma

31-d

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1-20

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4

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go

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ine

gro

un

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ater

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/or

soil

mo

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ata

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-11-

2003

to16

-2-2

004

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actu

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ther

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cast

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om

13-1

1-20

03to

16-2

-200

4


Again, on each day k the parameter (hs or qs) obtainedfrom the last calibration is used for forecasting ahead,i.e. for days k + 1, k + 2, etc., except that the initial con-ditions are better approximated if k > 0. In the limit, fork = 1, the on-line calibration procedure reduces to aform of data-assimilation called ‘‘Newtonian nudging’’[8], where the primary goal is state reconstruction atevery time step k. In this case, the parameter (hs or qs)is tuned in such a manner that the predicted water tablelevel and soil moisture profile at time k will resemble theobservations as closely as possible.

It can thus be seen that the on-line minimisation ofEq. (2) constitutes a form of model adaptation that isin between calibration and data-assimilation: assigningmore weight to the last measurement (large k) willreduce the initial error (the model will work towardsthe observations at time k), but introduces a largerparameter error, because parameters are only based ona single observation.

2.3.5. On-line calibration, data-assimilation and

forecasting (k = 0 and k = 1 combined)Using Eq. (2) forces one to weigh between accurate

parameter estimates and accurate prediction of initialconditions. However, if the results of a k = 0 calibrationand a k = 1 calibration are combined, we can have thebest of both worlds: on each day k, two calibration runsare performed, one with k = 0 yielding accurate parame-ter estimates and one with k = 1 yielding an accurate pre-diction of the (initial) state at time k. The forecasts fordays k + 1, k + 2, etc. are then based on a model run thatuses the parameters from the k = 0 calibration and theinitial conditions (water table level and soil moisture con-tents at day k) produced by the k = 1 calibration.

2.4. Model experiments and runs

To test the combined on-line calibration and data-assimilation method for on-line forecasting a number ofmodel experiments were performed. Table 3 gives anoverview of the different experiments performed and thedata sets used for calibration and data-assimilation. Fourdifferent types of experiments were performed:

1. Off-line calibration using groundwater level data;2. On-line (moving window) calibration, data-assimila-

tion and forecasting using groundwater level data;3. Sensitivity analysis for on on-line calibration, data-

assimilation and forecasting using groundwater leveldata;

4. On-line calibration, data-assimilation and forecast-ing using on-line observations of groundwaterlevel and soil moisture content and on-line weatherforecasts.

These four steps are described in more detail hereafter.


2.4.1. Off-line calibration using groundwater level data

As a first exercise the parameters hs, c, qs, fm and kf

were calibrated using a historical set of daily groundwa-ter level data collected between 1982 and 1996. Soilphysical parameters were assigned to the three horizonsusing pedotranfer functions ([13] see Table 1). To checkthe robustness of the model the 15-year period wasdivided into three 5-year periods (i.e. 1982–1986,1987–1991 and 1992–1996). The parameters hs, c, qs,fm and kf were calibrated separately on the three (5year) sets of groundwater data. A 1-year warm-upperiod was used before every 5-year calibration run toeliminate the adverse effects of the unknown initial con-ditions of the soil moisture profile. Goodness-of-fit errorstatistics were calculated in terms of mean error ME(bias) and root means squared error RMSE (accuracy)for each of the 5 periods. These goodness-of-fit statisticsare indicative of the forecasting error when the model isused in an open-loop mode (without using measure-ments for on-line calibration and data-assimilation).Note that the errors so calculated do not include errorsin weather forecasts, as observed meteorological dataare used instead of forecasted ones. The contributionof weather forecast error, which may be considerable,is treated separately in a sensitivity analysis hereafter.

2.4.2. On-line calibration, data-assimilation and

forecasting using groundwater level data

The groundwater levels in the study area show a dis-tinctive seasonal behaviour. This is also the case for thesurface water levels and patterns of regional groundwa-ter flow. Consequently, it can be expected that a consid-erable gain in forecasting performance is achieved whenre-calibrating surface water height hs or bottom flux qs

on-line. To test this, a moving window calibration ofhs (keeping the other parameters fixed at their bestestimate for the particular 5-year period, i.e. from theoff-line calibration) was carried out, while performing5-day forecasts (i.e. on-line calibration of hs was pre-ferred over qs; see Section 3). The observed groundwaterlevel data from 1982 to 1996 (divided into three periods)were used for on-line calibration, with observed meteo-rological data from this period as input. By trying differ-ent window lengths a period of 31 days was chosen asthe one giving the best results. As explained in Section2.3, when criterion (Eq. (2)) is used with increasing val-ues of k, one moves from on-line calibration to data-assimilation. In case of small k emphasis is put onobtaining the correct parameters, while increasing k putsmore emphasis on obtaining the correct initial state val-ues. Here, different values of k were tried, as well as acombination method of both k = 0 and k = 1 (see alsoSection 2.3). The proposed methods were applied tothree 5-year periods making up the 15-year data seriesand their performance was compared with the resultsfrom off-line calibration.

2.4.3. Sensitivity analysis for on-line calibration,

data-assimilation and forecasting using groundwater

level data

As explained before, when testing the forecasting per-formance of the model using observed meteorologicaldata, the effect of errors in the weather forecasts is nottaken into account. Comparing the output of suchmodel runs with observed data thus provides a measureof the combined effect of model error and measurementerror. It is however difficult to separate model error andmeasurement error using real data. To test the effect ofmeasurement error alone as well as the effect of errorsin weather forecasts (i.e. input errors) a sensitivity studywas performed using stochastic simulation. Separateanalysis of the effect of weather forecast errors providesa measure of the maximum accuracy that can beachieved (assuming that errors in weather forecasts areinevitable). Separating the effect of measurement errorfrom model error will subsequently indicate how possi-ble improvements can best be achieved, i.e. by improv-ing the model itself or by more precise observationmethods.

2.4.3.1. Measurement error. To simulate measurementerror, white noise (with a standard deviation of 1 cmas derived from experience) was added to the modelledtime series of phreatic head. Five realisations of mea-surement noise were added to the modelled series tomake the �measurement noise series�. The modelled ser-ies was treated as �reality�, whereas the five realisationswere treated as observations with error. Five-day fore-casts were made using the noisy observations for cali-bration and data-assimilation and compared with the�real� series. The same methods of calibration anddata-assimilation as described above (off-line calibrationand on-line calibration/data-assimilation with varying k)were compared.

2.4.3.2. Errors in weather forecasts. Comparing 5-dayweather forecasts of precipitation with actual weatherdata gave insight into the error margins around theweather predictions. Time dependency in the errors ofthe weather forecast was neglected. It turned out thatthe magnitude of the error in the precipitation forecastwas proportional to the value of the actual precipitation.The residual error had a mean of 1.62 (mm) and a stan-dard deviation of 2.45 (mm). The resulting error process(e) was thus defined as a function of the precipitation (P)as

eðP Þ ¼ �0:75 � P þ Nð1:62; 2:45Þ ð3Þ

This noise model generally underestimates large stormsand tends to predict continuous drizzle, as is observedin real weather forecasts. For each 5-day forecast period,five different realisations for the precipitation predictionwere generated. Negative precipitation predictions were

Table 4Calibration of parameters from three 5-year periods of groundwaterlevel measurements and goodness-of-fit statistics

Parameters 1982–1986 1987–1991 1992–1996

fm [–] 0.5070 0.4876 0.5690hs [cm] �144.7 �144.7 �142.4c [d] 124.3 112.5 111.4qs [cm/d] 0.0718 0.0971 0.0879kf [–] 2.07 1.79 2.40

Error measures

Constant drainage level

ME [cm] 0.065 0.038 0.117RMSE [cm] 7.36 6.97 11.10u [–] 0.929 0.933 0.969

Fluctuating drainage level

ME [cm] 0.476 0.821 �1.480RMSE [cm] 6.36 6.28 11.00u [–] 0.893 0.901 0.963

Mobile fraction fm, drainage level hs, drainage resistance c, bottomflux qs, lag-one temporal autocorrelation u.


set to zero; the resulting overestimation of total/averageprecipitation was neglected here. The predicted ground-water level was compared to a forecast based on the trueprecipitation data using the exact initial conditions andmodel parameters (i.e. the model runs with observedmeteorological data were treated as reality). The differ-ence therefore is the forecast error due to errors inweather predictions only. The effect of weather forecasterror on forecast performance was tested on the samemethods of calibration and data-assimilation as de-scribed above (off-line calibration and on-line calibra-tion/data-assimilation with varying k).

2.4.4. On-line calibration, data-assimilation andforecasting using on-line observations of groundwater

level and soil moisture content and on-line weather

forecasts

Finally the ‘‘best’’ method was tested on 3 months ofon-line water table depth and soil moisture measure-ments combined with actual weather forecasts from thisperiod. During this period the forecasting method wasused both on water table depth alone and on water tabledepth and soil moisture measurements, this to investi-gate the added value of the soil moisture data on predic-tions of water table depth and soil moisture content.

3. Results

3.1. Off-line calibration and forecasting usinggroundwater level data

Table 4 shows parameter sets as obtained by calibrat-ing SWAP to groundwater level observations of threeperiods (1982–1986, 1987–1991 and 1992–1996). Ascan be seen there are considerable differences betweenthe three periods, both in terms of fitted parameters aswell as the error measures. All three series show astrongly autocorrelated residual error. This suggests thata considerable accuracy improvement can be expectedwhen using a moving window calibration method, asan important part of this correlation is likely to accruefrom ignoring seasonal behaviour. The differences indrainage resistance and bottom flux are possibly causedby changes in the local and regional drainage systems.

3.2. Choice of parameter for on-line (moving window)

calibration and nudging

The model has five �free� parameters. Both hs and qs

are directly related to the absolute value of the watertable depth. The parameters fm, c, and kf mostly relateto the temporal variation of water table depth and soilmoisture. The latter three parameters were thereforeexcluded for use in moving window calibration, asproper calibration would require a certain level of

dynamic (i.e. considerable precipitation) within the cali-bration period, which might not appear during long dryperiods. Choosing between hs and qs for moving windowcalibration is difficult. They cannot be calibrated both,because they are strongly negatively correlated, so onehas to choose between them. Although the regionalgroundwater flux qs may have a seasonal component,sudden changes of its value are not expected (suddenchanges in drainage level are more reasonable). More-over, calibrating the drainage level gave slightly betterresults. Therefore we decided to use the drainage levelas a parameter to calibrate on-line.

3.3. On-line calibration, data-assimilation and

forecasting using groundwater level data

Fig. 1 shows the average of 15 years of calibrateddrainage levels using a moving window calibration of31 days. It reflects actual changes in drainage level, butalso compensates for model errors (model insufficiency)and possible seasonal changes in regional groundwaterflux. Using even a rough approximation of the averageof the fluctuating drainage levels, also shown in Fig. 1,as input for a modelled time series already improvesthe fit to the measurements when compared to a mod-elled series using fixed drainage level (see also Table4). Again, this suggests that a forecasting model basedon a moving window calibration will give better resultsthan a forecasting model using a fixed drainage level.

The results of using a moving window calibration areshown in Table 5 and Fig. 2. The error statistics in thetable and figure show average forecasting errors forthe 5-day forecasting period. So, for k = 0, for everytime step the parameters hs of the model was re-cali-brated using the last 31 observations. Next the model

Fig. 1. Fifteen-year average of calibrated drainage level hs (black dots) obtained by a moving window calibration, together with the sinusoidapproximation (grey line).

Table 5Average ME (cm) and RMSE (bold) (cm) of 5-day forecasts fordifferent methods of calibration on groundwater level data

Method Real data

1982–1986 1987–1991 1992–1996

Offline (constant hs) 0.14 0.07 �0.057.29 6.99 11.10

Moving window (k = 0) 0.03 0.14 0.465.81 5.71 6.16

k = 0.1 0.10 0.14 0.265.16 4.79 5.54

k = 0.25 0.03 0.05 0.144.81 4.56 5.46

k = 0.5 �0.03 �0.02 0.094.99 4.71 5.64

k = 1 �0.03 �0.05 0.085.02 4.75 5.67

k = 0 and k = 1 combined 0.00 0.34 0.144.74 4.60 5.27


was used to forecast 5 time steps ahead. Comparing theforecasts with the observations yielded 5 forecastingerrors. Repeating this for all time steps N thus yielded5N forecasting errors from which the error statistics inTable 5 and Fig. 2 were calculated. This procedurewas repeated for increasing values of k, giving moreweight to the last observation. Also, the combinedmethod was tried.

Fig. 2 and Table 5 clearly show the improvements ofthe forecasts by moving window calibration, indicatedby a reduction in root mean square error of about20%. As can be seen, the RMSE for the forecast periodis reduced further by introducing data-assimilation,i.e. by giving more weight to the last availablemeasurement.

There is an optimum k of 0.25 when considering theoverall RMSE of the forecast period. Larger values ofk result in large errors in the model parameter, whereassmaller values of k result in large errors in initial values.This effect can be seen in Fig. 3. The effect of a small ini-tial error (k = 1) is mitigated by the growing RMSE as aresult of the larger parameter error such that smaller k�sperform better for larger lead times and vice versa. Thecombination benefits from both a very small initial errorand a better parameter value, and gives the best overallforecast in terms of RMSE.

This is further illustrated in Fig. 4. This figure depictsa situation where the 31-day on-line calibration periodcontains the days from July 5 until August 4. The fore-casting period contains the days of August 5–9. Themodel based on the on-line calibration alone (k = 0,black line) follows the observed values in the past wellbut produces a poor forecast due to poor initial condi-tions on August 4. The model based on the last measure-ment (k = 1, dotted line) performs well initially butbreaks down for larger lead times because of a poorparameter estimate. Using the parameter from the 31-day moving window (k = 0) calibration and the initialconditions from the k = 1 calibration (pure data-assim-ilation) gives the best results (the grey line).

3.4. Sensitivity analysis for on on-line calibration,

data-assimilation and forecasting using groundwaterlevel data

The difference between observations and forecasts forthe on-line calibration and data-assimilation methodspresented in the previous section are due to both modelerrors and measurement errors, while errors in weather

Fig. 2. Total ME and RMSE of forecast period for different methods of calibration (1982–1986 period).

Fig. 3. Evolution of prediction error over forecast period with several values for weighing k and combined method; k determines the weight to thelast observation as in Eq. (2).


forecasts are ignored (by using observed meteorologicaldata as input). To investigate the relative importance ofmodel error, measurement error and errors in weatherforecasts a sensitivity study was performed using simu-lated data.

3.4.1. Measurement error

Table 6 shows the forecast errors that occur due tomeasurement errors only. As explained before, theresults are obtained by perturbing a synthetic (modelled)time series with measurement noise and forecasting thetime series using the various calibration and data-assim-

ilation methods. Clearly the errors due to measurementerrors are much smaller than the total error based oncomparison with the measured data (Section 3.3, Table5). The effect of model error (model insufficiency) onthe error in forecasts is thus much larger than the effectof measurement errors. This difference is not so large forthe case of off-line calibration. This can be explained bythe fact that off-line calibration cannot compensate forthe varying drainage levels that were used in the simu-lated series.

Compared to calibration to observed data (withoutnoise), the optimum value of k has shifted towards zero

Table 7Daily and average ME (cm) and RMSE (bold) (cm) of forecasts ofdifferent lead time based on weather forecasts with simulated errors

Lead time Input noise series

1982–1986 1987–1991 1992–1996

Day 1 �0.04 �0.07 �0.170.49 0.81 2.14

Day 2 �0.30 �0.37 �0.572.20 2.32 3.49

Day 3 �0.59 �0.72 �1.023.64 3.60 4.83

Day 4 �0.87 �1.12 �1.424.90 4.87 6.04

Day 5 �1.12 �1.52 �1.765.99 6.00 7.08

Average �0.59 �0.76 �0.993.95 3.97 5.03

Fig. 4. Illustration of the advantage of the combined method. Benefiting from the initial state from the k = 1 calibration (pure assimilation) and theparameter value of the k = 0 calibration (pure moving window calibration), the combined method starts off well and stays in line with the observedvalues.

Table 6Average ME (cm) and RMSE (bold) (cm) of 5-day forecasts fordifferent methods of calibration on groundwater level series withmeasurement noise

Method Measurement noise series

1982–1986 1987–1991 1992–1996

Offline (constant hs) 0.01 0.05 0.053.65 3.70 3.47

Moving window (k = 0) �0.01 0.00 0.010.89 0.97 0.90

k = 0.1 0.00 �0.01 0.000.79 0.85 0.63

k = 0.25 0.00 �0.01 �0.010.93 0.94 0.95

k = 0.5 0.00 �0.01 0.001.04 1.08 1.06

k = 1 0.00 �0.01 0.001.05 1.06 1.09

k = 0 and k = 1 combined 0.01 0.00 0.010.87 0.87 0.87


(k = 0.1) as the measurement error obscures the true ini-tial conditions of the forecast. However, due to fluctua-

tions of the drainage level within the 31-day calibrationperiod, there is something to be gained from assigningmore weight to the last measurement, as it will focus

Fig. 5. Evolution of distribution of water table elevation forecast error (cm) resulting from input noise (errors in weather forecast) over forecastperiod. Characteristics of input noise result in skewed distribution with long left tail.

Fig. 6. Evolution of the prediction error of water table depth over the forecast period for three sources of error, model error + measurement error,measurement error and input error (error in weather forecast).


Table 8Estimated parameters (mobile fraction fm, drainage level hs, drainageresistance c, seepage flux qs and, ksat multiplication factor kf) using allgroundwater data (1980–2000) and the soil moisture content data from13 November 2003 to 16 February 2004

Parameters

fm [–] 0.4398hs [cm] �139.2c [d] 109.8qs [cm/d] 0.0919kf [–] 9.82


calibration on the last measurement and therefore pro-vides a better estimate of the drainage level for the lastpart of the calibration period. This explains that theoptimum k is not exactly zero.

3.4.2. Errors in weather forecasts

Table 7 shows that the forecast deteriorates quicklywith forecast lead time due to errors in the weather fore-cast. The forecast for the first day is on average verygood, in terms of RMSE, except for the 1992–1996 per-iod. In this period one single storm on a wet soil profileled to overland flow in the control run. This storm wasnot present in the weather forecast resulting in a severeunderestimation of the groundwater table (>20 cm) forthat day.

Table 10Average ME and RMSE (bold) of 5-day forecasts when including soil moisdata-assimilation; for groundwater (ME and RMSE in cm), and soil moistu

Use of soil moisture observations No soil moisture ink = 0 and k = 1 calibration

Only soilk = 0 calib

Groundwater [cm] 0.61 0.795.49 5.46

Soil moisture content [%]25 cm �1.01% �0.98%

2.09% 2.05%50 cm �1.91% �1.57%

3.94% 3.37%75 cm �0.04% 0.00%

3.26% 3.33%100 cm �0.15% �0.24%

0.67% 0.48%

Table 9Van Genuchten parameters estimated using soil moisture content data from

Soil parameters

Top [cm] Bottom [cm] hs [cm3/cm3] hr [cm3/cm3]

0 40 0.40a 0.02(0.43)

40 200 0.40a 0.02(0.38)

Saturated water content hs, residual water content hr, saturated hydraulic coa Adjusted manually to measurements; original values in brackets.

Fig. 5 shows histograms of the forecast error due toerrors in the weather forecast. The characteristics ofthe input noise result in a negatively skewed error. Thereason is that the underestimation of large storms resultsin underestimation of groundwater table peaks, whichresults in a few but large negative errors.

3.4.3. Relative importance of error sources

Based on the analysis of the impact of measurementerror and weather forecast error the relative magnitudeof the three error sources (model, measurement andinput error) can be determined for different lead times.Fig. 6 shows the development of the prediction errorresulting from measurement + model error, measure-ment error alone and weather forecast error in simulatedtime series. From Fig. 6 we can conclude that the modelerror is the largest error source, except for lead times lar-ger than 3 days, when the error due to the accumulatingerror in the weather forecast becomes as large as themodel error.

3.5. On-line calibration, data-assimilation andforecasting using on-line observations of groundwater

level and soil moisture content and on-line weather

forecasts

Finally, the hydrological parameters hs, c, qs were cal-ibrated on the complete data set of measured groundwa-

ture measurements in the combined method of on-line calibration andre content at 4 depths (ME and RMSE in %)

moisture inration

Only soil moisturein k = 1 calibration

Soil moisture in boththe k = 0 and k = 1 calibration

1.24 1.435.15 5.02

�0.86% �0.85%1.85% 1.86%�1.84% �1.54%

3.86% 3.41%�0.01% 0.10%

2.98% 3.13%�0.16% �0.19%

0.67% 0.56%

13 November 2003 to 16 February 2004

ksat [cm/d] a [1/cm] I n

94.77 0.0299 �0.983 1.694(9.65) (0.0227) (1.548)

150.36 0.0374 0.039 1.965(15.56) (0.0214) (2.075)

nductivity Ks, empirical parameters a, I and n.

Table 11ME and RMSE (bold) of on-line forecast of groundwater level and soilmoisture profile for different lead times using weather forecasts asinput

Lead time Groundwaterlevel [cm]

Soil moisture content [%]

25 cm 50 cm 75 cm 100 cm

Day 1 1.71 �1.45% �2.77% �0.12% �0.32%3.83 1.84% 3.45% 2.94% 0.41%

Day 2 2.15 �1.43% �2.61% 0.05% �0.33%6.28 1.88% 3.29% 3.29% 0.39%

Day 3 2.38 �1.38% �2.50% 0.33% �0.35%7.84 1.87% 3.47% 3.58% 0.41%

Day 4 3.06 �1.11% �2.17% 0.77% �0.32%8.46 1.72% 3.33% 3.85% 0.40%

Day 5 4.21 �0.83% �1.68% 1.45% �0.29%9.08 1.61% 3.03% 4.27% 0.42%

Average 2.70 �1.24% �2.34% 0.49% �0.32%7.34 1.78% 3.32% 3.62% 0.41%


ter levels available at the site, i.e. from 1980 till 2000.Using the hydrological parameters obtained from thiscalibration, the soil parameters a, n, kf and fm were sub-sequently calibrated using the available soil moisturemeasurements (a period of 96 days—from 13 November2003 to 16 February 2004). This resulted in the param-eter set presented in Tables 8 and 9. Using the calibratedsoil physical parameters ksat, a and n instead of the VanGenuchten parameters taken from the Staring Series [13]dramatically improved the forecast of the soil watercontents.

Fig. 7. Time series of observed and predicted water table depth an

To test the added value of using soil moisture obser-vations in the forecasting scheme, the combined method,i.e. using both the k = 0 calibration and the k = 1 cali-bration (pure data-assimilation), was tested withobserved weather data (excluding input error) for theperiod 13 November 2003 to 16 February 2004. Fourweighing combinations for groundwater and soil watercontent were tested: (1) excluding soil water contentfrom both k = 0 and k = 1 calibration; (2) including soilwater content in the k = 0 calibration, but not in thek = 1 calibration; (3) including soil water content inthe k = 1 calibration, but not in the k = 0 calibration;(4) including soil water content in both the k = 0 andk = 1 calibration. The weight for each soil water contentmeasurement was 100 (1 for each groundwater measure-ment), to compensate for difference in scale and numberof soil water content and groundwater measurements.

The results are shown in Table 10. By including soilwater content in the k = 0 calibration only, the forecastof groundwater and soil water content improved onlyslightly in terms of RMSE, but at the expense of anincreasing ME (bias) of groundwater forecasts. Includ-ing soil water content only in the k = 1 calibration, alsolead to reduction of the RMSE, however yielding aneven larger ME of groundwater forecasts. Including soilwater content in both the k = 0 and the k = 1 calibra-tion, the forecast of groundwater and soil water contentimproved further in terms of RMSE, but again at theexpense of an increasing ME. The increased precision(smaller random error) can be explained by the fact that

d soil moisture content of the 1st, 3rd and 5th forecast day.

Fig. 8. Evolution of prediction error over forecast period of ground-water level and soil moisture profile.


adding information will constrain the groundwater levelforecasts. Apparently, this occurs at the cost of morebias, possibly because the estimated soil physicalparameters are not entirely consistent with the observedwater table level. However, as the increase of precisionwas larger than the increase in bias, using soil mois-ture data in both the k = 0 and k = 1 calibration ispreferred.

Finally we applied the whole scheme for on-line fore-casts of groundwater level and soil moisture profiles—using weather forecasts and the combined on-linecalibration and data-assimilation method (i.e. combin-ing k = 0 and k = 1 calibration) using soil moistureobservations in both the k = 0 and k = 1 calibration

runs. Results are presented in Table 11. Fig. 7 showstime series of observed and predicted water table depthand soil moisture content for different lead times. Pre-dictions of water table depth and shallow soil moisturecontent are close to the observed values. However, theforecasted soil moisture content at 75 cm depth showsvariation whereas the observed soil moisture is constantand close to saturation. Apparently, the soil water reten-tion relationship used in at this depth is not correct. Itseems that the value of the soil physical parameter a(see Table 9) is too large.

The error statistics of soil water content at differentdepths reflect differences in dynamics (see Fig. 8). Theaccuracy of the forecast of 25 cm soil water contentremains similar over the forecast period, reflecting diffu-sion of soil moisture mitigating the error from theweather forecast. The 50 cm soil water content showssimilar behaviour, whereas the 75 cm soil water contentis more influenced by groundwater level fluctuationsthat react quickly to precipitation due to preferentialflow and therefore follows the pattern of groundwatererror statistics, deteriorating over the forecast period.The errors of 100 cm soil water content are small andconstant in time, reflecting the fact that soil moistureis close to saturation for most of the time.

3.6. On-line implementation and dissemination

The application is currently running at the websiteof the Netherlands Institute for Applied Geoscience(http://hydroline.nitg.tno.nl). Every day the systemautomatically performs all the necessary tasks for mak-ing and presenting the groundwater table forecasts: (1)Up-to-date weather data and forecasts are downloadedfrom the website of the Royal Netherlands Meteorolog-ical Institute. (2) The most recent measurements ofgroundwater level and soil moisture condition are down-loaded from the database on the website of the providerof the measuring equipment. These measurements havebeen sent to this database by the measuring equipmentusing an SMS-message (short message service) over theconsumer GSM network. (3) Using this data, bothk = 0 and k = 1 calibration are carried out to producea calibrated drainage level and the initial state for theforecast period. (4) The 5-day forecast is made and pre-sented on the web page in a table and the graph shownin Fig. 9. Also, the calculated soil moisture profiles arepresented in separate graphs on a second page, togetherwith a table showing remaining storage capacity andplant-available soil moisture. (5) Each day, a seasonalensemble forecast is made, i.e. by running the modelfor the next half year 40 times using the current watertable level and soil moisture profile as initial conditionsand 40 individual years of historical meteorological dataas input. The ensemble forecast, which is presented on athird page, thus provides the range of possible hydrolog-

http://hydroline.nitg.tno.nl

Fig. 9. On-line presentation of groundwater level forecast and recent measurements. Also shown is the ‘‘regime curve’’ as background, i.e. a 31-daymoving window average of the past 8 years, the 25–75 percentile (normal), the 10–25 and 75–90 percentile (high and low) and the extreme values.


ical states for the coming months, given the uncertaintyin future weather conditions.

4. Conclusions

We introduced a method for on-line forecasting ofwater table depth and soil moisture profiles. The methodis based on a combination of a moving window calibra-tion of the deterministic model of the unsaturatedzone SWAP and a simple form of data-assimilationusing on-line soil moisture and water table depth mea-surements. On-line meteorological data and weatherforecasts are used as input. The system is currently run-ning on-line.

Moving window calibration improves the forecastcompared to an off-line model, indicated by a decreaseof the root mean square error of approximately 20%.Adding a simple form of data-assimilation improvesthe forecast with another 20% in RMSE. Using soilmoisture measurements in both off-line calibration ofsoil physical parameters and on-line calibration anddata-assimilation also improves the forecast.

From a sensitivity analysis it followed that the modelerror is the largest source of error for lead times up toand including the third forecast day, after which theerror as a result of errors in the weather forecastsbecomes as large.

Further improvements of the method are anticipatedby providing an ensemble forecast based on ensembleweather forecasts. Also, a more advanced data-assimila-tion method, e.g. the ensemble Kalman Filter [5], islikely to improve the method presented here.

Acknowledgements

We would like to acknowledge the Royal Nether-lands Meteorological Institute (KNMI) for providingfor both the historical meteorological data and the dailysupply of 5-day weather forecasts. Also we would like tomention the provider of the on-line measuring equip-ment (Eijkelkamp BV, The Netherlands) for theirassistance and co-operation. The comments of Fransvan Geer (TNO Institute of Applied Geoscience) consid-erably improved an earlier version of this paper. The


paper greatly benefited from the comments of reviewersMartin Knotters and Jetse Kalma.

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real-time forecasting of water table depth and soil moisture profiles

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