e c o l o g i c a l m o d e l l i n g 2 0 7 ( 2 0 0 7 ) 22–33

avai lab le at www.sc iencedi rec t .com

journa l homepage: www.e lsev ier .com/ locate /eco lmodel

Impact of time series data on calibration and predictionuncertainty for a deterministic hydrogeochemical model

Thorjørn Larssena,∗, Tore Høgasena, B. Jack Cosbyb

a Norwegian Institute for Water Research, Gaustadalleen 21, 0349 Oslo, Norwayb University of Virginia, Charlottesville, VA 22904-4123, USA

a r t i c l e i n f o

Article history:

Published on line 20 April 2007

Keywords:

MAGIC

Acidification modelling

Recovery

Markov Chain Monte Carlo

Model calibration

Data availability

Time series data

a b s t r a c t

Model calibration is fundamental in applications of deterministic process-based models.

Uncertainty in model predictions depends much on the input data and observations avail-

able for model calibration. Here we explored how model predictions (forecasts) and their

uncertainties vary with the length of time series data used in calibration. As an example

we used the hydrogeochemical model MAGIC and data from Birkenes, a small catchment

in southern Norway, to simulate future water chemistry under a scenario of reduced acid

deposition. A Bayesian approach with a Markov Chain Monte Carlo (MCMC) technique was

used to calibrate the model to different lengths of observed data (4–29 years) and to estimate

the prediction uncertainty each calibration. The results show that the difference between

modelled and observed water chemistry (calibration goodness of fit) in general decreases

with increasing length of the time series used in calibration. However, there are consider-

able differences for different time series of the same length. The results also show that the

uncertainties in predicted future acid neutralizing capacity were lowest (i.e. the distribu-

tion peak narrowest) when using the longest time series for calibration. As for calibration

success, there were considerable differences between the future distributions (prediction

uncertainty) for the different calibrations.

data and model structure are closely related in model applica-

1. Introduction

Deterministic simulation models are often used to mimichydrological, hydrogeochemical or other complex environ-mental systems and to make predictions under differentscenarios. Such models produce certain output parameters asa function of a group of input parameters.

There are several sources of uncertainty in predictions fromsuch models (Funtowicz and Ravetz, 1990; Saloranta et al.,2003). One important source of uncertainty is related to the

quality, quantity and representativity of input data and obser-vations used in calibration. Uncertainty of this origin is oftenconsidered (by the modellers, at least) to be the largest con-

∗ Corresponding author. Tel.: +47 22185100; fax: +47 22185200.E-mail address: thorjorn.larssen@niva.no (T. Larssen).

0304-3800/$ – see front matter © 2007 Elsevier B.V. All rights reserved.doi:10.1016/j.ecolmodel.2007.03.016

© 2007 Elsevier B.V. All rights reserved.

tributor to the overall uncertainty (Bloschl and Grayson, 2000;Uhlenbrook and Sieber, 2005). Another type of uncertainty isrelated to the model structure and the extent to which themodel properly describes the system to be modelled. Modelsare by necessity simplified descriptions of the natural system,and the challenge when selecting or building a model is toinclude the important drivers for the problem to be inves-tigated, but yet not over parameterize (Perrin et al., 2001;Snowling and Kramer, 2001). The uncertainties related to input

tions. Additional uncertainty, usually impossible to quantify,is of an epistemological nature, such as sudden dramaticchanges in the system, or system processes not included in

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he model (for example wildfires, landslides, volcanic erup-ions) (Funtowicz and Ravetz, 1990). The focus in this papers the prediction uncertainty related to the length of the timeeries data available for model calibration.

Calibration of input parameters through fitting the cor-esponding output parameters to observational data isundamental in dynamic, process-oriented models used forrediction. Model predictions depend strongly on model cal-

bration, and calibration depends on available observationalata. Some recent studies discuss the relationship betweenncertainty in predictions and length of calibration data timeeries in hydrological modelling (Anctil et al., 2004; Brath etl., 2004; Xia et al., 2004). Much less work has been done onncertainty assessment and time series data with hydrogeo-hemical models, although it has been demonstrated thatncreasing numbers of observational data at different pointsn time will significantly reduce uncertainties in predictionsLarssen et al., 2004).

A principal approach to quantifying uncertainty in modelredictions is to give some sort of probability distribution tohe input parameters to the deterministic model and therebybtain probability distributions rather than single values asodel outputs. Several approaches have been developed for

ncertainty assessment in which Bayesian techniques areombined with conceptual hydrological catchment modelsBates et al., 2003; Kuczera and Parent, 1998; Makowski et al.,002; Thiemann et al., 2001). For biogeochemical or hydrogeo-hemical models as well as other types of integrated wateresources modelling, such approaches are less developed, butre now becoming increasingly available (Larssen et al., 2006).

The hydrogeochemical case study presented in this papers recovery from acidification in response to reduced acideposition. The case is particularly suitable for illustratinghe variation in uncertainty with the observation data timeeries because long-term (30+ year) monitoring data exist, andidely tested and well-documented models are available. Data

rom the Birkenes catchment in southern Norway are used.he deterministic model is the widely used hydrogeochemi-al model MAGIC (Model for Acidification of Groundwater inatchments) (Cosby et al., 1985, 2001). MAGIC is mostly used

or predicting future development in water chemistry as aesult of changing inputs of sulphur and nitrogen from thetmosphere (acid rain). Here we present predictions of futureater chemistry based on model calibrations made with dif-

erent time series of model calibration data.The objectives are to illustrate how (1) the calibration suc-

ess and (2) the uncertainties in model predictions may varyith different lengths of time series data used for model cal-

bration. In order to utilize the information in the time seriesbservations, we combine the deterministic model (MAGIC)ith a stochastic model for the observed output data and esti-ate all unknown parameters by Bayesian computations. We

sed a Markov Chain Monte Carlo (MCMC) technique for cali-rating the MAGIC model to different sub-sets of the Birkenestream chemistry data. We then compare (1) the agreementetween simulated and observed data for each calibration

nd (2) the probability distributions of predicted future waterhemistry under a future deposition scenario for each calibra-ion. Different sub-sets of 4–5 years length and 11 years lengths well as the full 29-year series were used for calibration to

2 0 7 ( 2 0 0 7 ) 22–33 23

examine the effects of different time series of observations oncalibration success and uncertainty in predictions.

2. Material and methods

2.1. The MAGIC model and the calibration routine

The MAGIC model is used to predict long-term effects of aciddeposition on soils and surface water chemistry (Cosby et al.,1985, 2001). The model has been extensively used at a range ofdifferent sites and applications (see appendix in Cosby et al.,2001).

The predictions are largely driven by atmospheric depo-sition of the major ions sulphate (SO4

2−), nitrate (NO3−),

chloride (Cl−), ammonium (NH4+), calcium (Ca2+), magne-

sium (Mg2+), sodium (Na+), potassium (K+) and hydrogen (H+).MAGIC calculates for each year the concentrations of ionsin soil solution and surface water under the assumption ofsimultaneous reactions involving sulphate adsorption, cationexchange, dissolution–precipitation–speciation of aluminium,and dissolution–speciation of inorganic and organic carbon.MAGIC accounts for the mass balance of major ions in the soilby book-keeping the fluxes from atmospheric inputs, chemi-cal weathering, net uptake in biomass, and loss to runoff. Amodel run is initialized in pre-industrial times (1850 is usedin the current application) and driven forward in time bychanging input of ions from deposition. During calibration,the model is run from the initial year to the period in time forwhich observations are available for surface water chemistryand soil chemistry. In traditional calibrations of MAGIC, wherethe input parameter uncertainty is not estimated, two sets ofinput parameters are adjusted to produce a match betweensimulated and observed soil and water variables:

• The weathering rate of the four base cations Ca2+, Mg2+,Na+, and K+, the sum of which is the total soil base cationweathering (BCw).

• The relative amounts of the individual exchangeable basecations Ca2+, Mg2+, Na+, and K+ in the soil, the sum of whichis the total initial soil base cation saturation (BS0). The sub-script refers to the year the model is initiated, i.e. 1850.

Calibration is then done by adjusting BCw and BS0 untiloutput parameters match observed data for water chemistryand soil base saturation in the calibration year or years.

In the traditional calibration of the MAGIC model, the val-ues for all input parameters, except BCw and BS0, are treatedas known. In the Bayesian approach, however, all input param-eters and output parameters are technically interpreted asunknown, ideal or effective values which are representativeof the whole catchment and a whole year. They are to somedegree measurable, but cannot be specified exactly in practice,since the corresponding measurements (input data and out-put data) typically will be point measurements in space andoften also in time.

comprise 28 parameters assumed independent of time,including lake and catchment characteristics and soil chem-ical and physical characteristics (see Table 1 for a detailed

24 e c o l o g i c a l m o d e l l i n g 2 0 7 ( 2 0 0 7 ) 22–33

Table 1 – Input data with prior distributions used in the calibration

Input parameters Unit Prior distributiona

Constant parametersFixed soil parameters

Depth M Uniform (0.3, 0.5)Porosity % Uniform (40, 60)Bulk density kg m−3 Uniform (696, 850)Cation exchange capacity mequiv. kg−1 Uniform (95, 117)Sulphate adsorption half saturation �equiv. L−1 Uniform (1, 500)Sulphate adsorption maximum mequiv. kg−1 Uniform (0, 50)Aluminium dissolution constant Log 10 Uniform (6, 11)Temperature ◦C Fixed (5)CO2 partial pressure % Uniform (0.50, 2)Dissolved organic C concentration �mol L−1 Uniform (0, 250)Nitrification (% of incoming flux) % Fixed (100)Ammonium uptake (% of incoming flux) % Uniform (0, 100)Nitrate uptake (% of incoming flux) % Uniform (0, 100)

Soil base cation weathering (BCw)Ca2+ weathering mequiv. m−2 year−1 Uniform (0, 100)Mg2+ weathering mequiv. m−2 year−1 Uniform (0, 100)Na+ weathering mequiv. m−2 year−1 Uniform (0, 100)K+ weathering mequiv. m−2 year−1 Uniform (0, 100)

Initial soil base cation saturation (BS0)b

Exchangeable Ca2+ % Uniform (0.1, 50)Exchangeable Mg2+ % Uniform (0.1, 50)Exchangeable Na+ % Uniform (0.1, 50)Exchangeable K+ % Uniform (0.1, 50)

Surface water characteristicsRelative area water: catchment % Fixed (0.01)Water retention time Year Fixed (0)Aluminium dissolution constant (KGibb) Log10 Uniform (6, 11)Temperature ◦C Uniform (3, 10)CO2 partial pressure % Uniform (0.05, 0.2)Dissolved organic C concentration (SDOC) �mol L−1 Uniform (0, 250)Nitrification (% of incoming flux) % Fixed (100)

Time dependent parameters xtc

Runoff amount m year−1 Normal (z1t, �1 z1t)

Deposition concentrationsCa2+ �equiv. L−1 Normal (z2t, �2 z2t)Mg2+ �equiv. L−1 Normal (z3t, �2 z3t)Na+ �equiv. L−1 Normal (z4t, �2 z4t)K+ �equiv. L−1 Normal (z5t, �2 z5t)NH4

+ �equiv. L−1 Normal (z6t, �2 z6t)SO4

2− �equiv. L−1 Normal (z7t, �2 z7t)Cl− �equiv. L−1 Normal (z8t, �2 z8t)NO3

− �equiv. L−1 Normal (z9t, �2 z9t)

a Prior distributions are selected from the available knowledge on each parameter. In cases where an expert judgment would suggest upperand lower limits, we used uniform distributions.

b The sum of four exchangeable base cations (Ca2+, Mg2+, Na+, K+) are constrained to be ≤100.c The observed runoff amount in year t is z1t. (The MAGIC model uses observed runoff and contain no hydrological model.) The observed

deposition concentration in year t for component i (i = 2, . . ., 9) is zit. The distributions for all components two to nine are exactly correlated atthe same time with correlation 1. All other prior distributions are independent. The prior distribution for xt has mean equal to zt and standard

stimaestim

deviation proportional to zt, where the proportionality factors are eas hyper-parameters (i.e. parameters of the prior distribution) to beshape and rate equal to 0.01). See Larssen et al. (2006) for details.

list). In this application weathering of base cations and netuptake of ions by vegetation were assumed constant over time.

In addition, there are nine time dependent input parametersconsisting of annual values of runoff amount and depositionconcentrations (Table 1). The output parameters from MAGICconsist of five parameters related to soil (pH and exchangeable

ted from data. The two proportionality factors �1 and �2 are treatedated with vague priors (1/�2

1 and 1/�22 being gamma distributed with

base cations) and nine parameters describing surface waterchemistry, all of which are variable over time.

In order to utilize the information in the time series ofobservations, we combined the deterministic model (MAGIC)with a stochastic model for the observed output data, andestimated all unknown parameters by Bayesian computa-

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ions using an MCMC scheme. This involved specifying priorrobability distributions for the input parameters (based onhe observed distributions if available) and likelihood for theutput data. The prior distributions for the input parame-ers implicitly define corresponding prior distributions forhe output parameters which combined with the data like-ihood give posterior distributions for all parameters. Theosterior distributions were calculated by running the deter-inistic model repeatedly with parameter sets generated by

he MCMC scheme which adjusts the distributions based onhe differences between simulated and observed values. Theesults were analyzed after the algorithm had converged. Thepproach allows a formal management of uncertainties inoth parameters and data, and their impact on model resultsan be explicitly shown. The MCMC algorithm and its linko the MAGIC model are described in detail in Larssen et al.2006).

After calibration, predictions were made of future develop-ent of water chemistry under different scenarios for changes

n atmospheric deposition. The results can then be used, fornstance, in a policy context to support selection of measuresor emission reduction.

In this particular application, the calibration routine wasun 20,000 times for each set of observations, generating the0,000 parameter files for the MAGIC model. The first 10,000uns included a burn-in phase of the algorithm and wereiscarded. The last 10,000 runs were used to generate therobability distributions and confidence bands (or credibleand, in Bayesian terminology) presented here. Due to theon-Gaussian nature of the distributions produced by theserocedures, our confidence bands refer to ±percentile distri-utions around the median result.

.2. Data and prior distributions

he Birkenes catchment is located in southern Norway, about0 km inland of Kristiansand. The catchment area is 0.41 km2

nd the elevation 200–300 m. The vegetation is mainly 100-ear old Norway spruce with some pine and birch and anndergrowth of mosses, blueberry and fern. Podzolic andrown-earth soils have developed from a shallow layer oflacial till on granitic bedrock. Peaty deposits have developedn poorly drained sites in the catchment. On the slopes, well-rained thin organic layers on gravel or bedrock are common.he catchment is drained by three small second-order streamshich converge about 150 m above the lower catchment bor-er. A station for sampling precipitation and air is locatedbout 500 m north of the catchment and operated by Theorwegian Institute for Air Research (Aas et al., 2003). Theatchment is substantially influenced by sea salt depositionhich has considerable influence on the stream water chem-

stry.Data from the Birkenes catchment represent an unusually

ong and complete time series of precipitation and surfaceater chemistry. Measurements of major anions and cationsave been recorded since 1974 for both bulk deposition (daily)

nd runoff chemistry (weekly) (Aas et al., 2003; SFT, 2004).n the current application, the chemical composition of thennual deposition and the runoff amount were calculatedrom daily or weekly observations.

2 0 7 ( 2 0 0 7 ) 22–33 25

In addition to these time series, data describing physi-cal and chemical properties of the catchment are available(Table 1). These input data include soil depth, soil porosity, soildensity, and cation exchange capacity (CEC). Representativevalues for the catchment are obtained by averaging over sev-eral measured soil profiles. At the Birkenes catchment overallestimates of the soil properties are based on samples from foursoil profiles, each sampled in five depths, analyzed in 1992 and2000 (SFT, 1993, 2001; Stuanes et al., 1995).

Measurements used for model calibration are the rela-tive amount of exchangeable base cations in the soil (basesaturation) and the annual average stream water concentra-tions of main anions and cations, available from 1974 to 2002(except for 1979 and 1984) (SFT, 2004). These annual valuesare obtained as discharge-weighted averages of weekly runoffchemistry. Selected data are shown in Figs. 1–4 together withmodel results.

The long-term historic deposition of pollutants wastaken from Schopp et al. (2003), who used a European-scale atmospheric dispersion and transportation model (theEMEP model) to calculate average values for sulphur andnitrogen deposition for 50 km × 50 km grid cells. Since thegrid cell average was not directly applicable for the smallcatchment, the values were scaled to match the mea-sured annual deposition at Birkenes over the period 1974–2002.

The forecast deposition scenario applied is based onthe current legislation in Europe related to S and N emis-sions. Deposition calculations from emissions are also forthe forecasts calculated with the EMEP model (Schopp et al.,2003).

Prior distributions for the input parameters were con-structed from available information for each parameter(Table 1). The prior distributions for parameters such as soildepth and porosity were based on measurements and thuswere rather narrow. On the other hand, weathering rates (BCw)and initial base saturation (BS0) which are difficult to mea-sure, were given rather wide prior distributions. Weatheringrates and initial base saturation for each of the four basecations (Ca2+, Mg2+, Na+ and K+) were calibrated separately,but are presented here as summed values. BCw and BS0 arethe most important calibrated parameters in applications ofthe MAGIC model. The prior distributions for the time depen-dent parameters (deposition amount and composition) wereestimated from the observations as a part of the calibrationalgorithm.

Results from model calibrations to six different data setsare presented: three short (4–5 years) periods (1980–1983,1988–1992, 1998–2002), two medium (11-year) periods(1980–1990 and 1990–2000) and one long (29-year) period(1974–2002). All calibrations were run independently, andwith all the same input data. Goodness of fit between sim-ulated and observed data for each calibration is expressedas the Root Mean Squared deviations over all years in thecalibration period divided by the average of the observedvalues over the calibration period.

Results are shown for the stream water concentrations ofSO4

2− and acid neutralizing capacity (ANC). SO42− is the most

important driver for acidification at Birkenes, while ANC is thevariable most commonly used to link water acidification with

26 e c o l o g i c a l m o d e l l i n g 2 0 7 ( 2 0 0 7 ) 22–33

Fig. 1 – SO42− concentrations in the stream from 1970 to 2005. Volume-weighted annual average observations are shown as

dots; filled dots indicate years used in the calibration and open dots years not used. The solid line shows the median of themodelled values; dotted lines indicate the 90% confidence band (or credible band, in Bayesian terminology) for the modelled

data.

to biological response (Lien et al., 1992). ANC is defined as:

2+ 2+ +

ANC = [Ca ] + [Mg ] + [Na ]

+ [K+] − [SO42−] − [NO3

−] − [Cl−]

where units are �equiv. L−1.

3. Results and discussion

Fig. 1 shows the observed and modelled SO42− stream water

concentrations when different time periods of stream waterchemistry data are used to calibrate the model. The 90%confidence bands were much narrower for the time period

e c o l o g i c a l m o d e l l i n g 2 0 7 ( 2 0 0 7 ) 22–33 27

Fig. 2 – ANC in the stream from 1970 to 2005. Volume-weighted annual average observations are shown as dots; filled dotsi ot usv dible

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ndicate years used in the calibration and open dots years nalues; dotted lines indicate the 90% confidence band (or cre

o which the model was calibrated than for the period forhich data were not used. The confidence band was consid-

rably narrower when the full series of observations was usedFig. 1).

An importance difference between the different calibra-

ions is the simulation of the general trend in the observations.he observed SO4

2− concentrations clearly show a decreasingrend and the different calibrations matched this trend witharying success. Among the three calibrations using the short

ed. The solid line shows the median of the modelledband, in Bayesian terminology) for the modelled data.

series, the best agreement with the observed trend was for thecalibration period 1988–1992 (Fig. 1). For the calibration peri-ods 1980–1983 and 1998–2002, the model reconstructed a lowerslope than observed.

Both of the two calibrations using the 11-year series repro-

duced the trends in observations without clear bias (Fig. 1).However, the calibration based on 1990–2000 gave much higherSO4

2− concentrations in the beginning of the 1970s and alsogave larger fluctuations between years.

28 e c o l o g i c a l m o d e l l i n g 2 0 7 ( 2 0 0 7 ) 22–33

Fig. 3 – SO42− concentrations in the stream from 1850 to 2100. Volume-weighted annual average observations are shown as

dots; filled dots indicate years used in the calibration and open dots years not used. The solid line shows the median of themodelled values; dotted lines indicate the 90% confidence band (or credible band, in Bayesian terminology) for the modelled

data.

Fig. 2 shows the annual average observed and modelledANC in stream water for the different model calibra-

tions. As was the case for SO4

2− concentrations, theANC confidence band was narrower for the time periodfor which data were used to calibrate. However, sinceANC was calculated from seven separate ion concentra-

tions, the confidence band was relatively wider than forSO 2−.

4

There was a significantly increasing trend in the observedANC values, although not as strong as for SO4

2−. The varia-tion in ANC from year-to-year may be considerable and largerthan the long-term trend driven by the decreasing acid deposi-

e c o l o g i c a l m o d e l l i n g 2 0 7 ( 2 0 0 7 ) 22–33 29

Fig. 4 – ANC in the stream from 1850 to 2100. Volume-weighted annual average observations are shown as dots; filled dotsi ot usv dible

tc

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ndicate years used in the calibration and open dots years nalues; dotted lines indicate the 90% confidence band (or cre

ion. ANC in the 1990s was somewhat over-predicted for manyalibrations (Fig. 2).

The goodness of fit for the model for each calibrationeriod is shown in Table 2. The best fit for both SO4

2−

nd ANC was for the calibration where all the data aresed. The calibrations with the 11-year time series in gen-ral produced better fits than with the 4–5-year series.owever, for ANC one of the short series (1998–2002) gave

ed. The solid line shows the median of the modelledband, in Bayesian terminology) for the modelled data.

better fit than one of the medium long series (1980–1990).

The different calibrations gave different predictions forfuture as well as past development in ion concentrations.

For SO4

2−, the modelled initial concentrations (in 1850) var-ied only slightly between the different calibrations (Fig. 3).This is because the calibrated parameters affecting the ini-tial SO4

2− concentrations were given a rather narrow initial

30 e c o l o g i c a l m o d e l l i n g

Table 2 – The difference between the median modelledvalues and observations, calculated as the square root ofthe sum of squares of the observed minus modelledvalues divided by the average value for the observations

Calibration period [SO42−] ANC

4–5 years1980–1983 0.56 −2.101988–1992 0.34 −2.161998–2002 0.56 −1.57

11 years1980–1990 0.29 −1.861990–2000 0.35 −1.36

29 years1974–2002 0.21 −1.21

distribution (Table 1). Whereas the atmospheric deposition

of anthropogenic origin is the main driver for the long-termchanges in SO4

2−, the initial concentration in the model arisesfrom SO4

2− in deposition originating from sea salts and fromweathering of S-bearing minerals in the soil.

Fig. 5 – Posterior distributions of modelled SO42− and ANC in thr

different sets of calibration data.

2 0 7 ( 2 0 0 7 ) 22–33

For ANC there were considerable differences between thecalibrations with respect to initial conditions (Fig. 4). Thehighest modelled median ANC in 1850 was 61 �equiv. L−1

(with confidence band 50–73 �equiv. L−1), from the calibra-tion using 1980–1983 observations, while the lowest value was34 �equiv. L−1 (with confidence band 28–43 �equiv. L−1), fromthe calibration using 1990–2000 observations. The medianvalue 1850 ANC for the calibration using the full time serieswas 42 �equiv. L−1. The difference between the calibrations ismainly driven by the calibrated values for base cation weath-ering and the initial base saturation.

The distributions of SO42− concentrations and ANC for the

various calibrations are compared in Fig. 5 for three differentforecast years. The calibration using the full time series hadthe narrowest distribution for both SO4

2− and ANC, showingthat the longer time series of calibration data limits the widthof the distribution of the forecasts (i.e. reduces the uncertainty

in forecasts).

For SO42−, there was much overlap between the distribu-

tions, although the distributions using short and medium longdata series were wider (Fig. 5). The distribution for the calibra-

ee different forecast years (2020, 2050, 2100) for the

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ion using 1990–2000 data was considerably different from thethers, as it has a particularly wide peak range (but similar0% confidence range). Also, it peaked at higher SO4

2− con-entrations. The reason for this behaviour can be seen fromhe observations used in this particular calibration; there arelearly two slopes in the declining trends (before and after990–1991), with a very steep decline in the second periodFig. 1). This change in slope will give the calibration routine a

ider range of acceptable SO42− concentrations.

As the SO42− deposition was kept constant in the model

orecast from 2020 and the SO42− adsorption is low, there was

ery little change in the modelled stream water SO42− con-

entrations from 2020 onwards. Although, the SO42− inputs

re kept constant from 2020, the ANC still increases due touild-up of the depleted exchangeable base cation pool fromeathering and thereby slowly increasing capacity to neutral-

ze incoming acid.The distributions of the forecasted ANC were clearly dif-

erent for several of the calibrations (Fig. 5). The 1990–2000alibration clearly gave the lowest ANC values, while all otheralibrations gave higher values than the one using all data.he particularly low distribution of the 1990–2000 calibrationeflects the high SO4

2− concentrations forecasted for this cal-bration.

The most important calibrated parameters for the mod-lled ANC are base cation weathering (BCw), initial baseaturation (BS0), aluminium dissolution constant (KGibb) and

oil solution DOC concentration (SDOC). As realistic mea-urements corresponding to these parameters are eithermpossible or difficult to measure, they are given wide priors.he calibration procedure then narrows down the distribu-

ig. 6 – Posterior distributions of selected calibrated input paramase saturation), BCw (base cation weathering), SDOC (DOC conceissolution constant). Prior distributions for each ion in BS0 and

2 0 7 ( 2 0 0 7 ) 22–33 31

tions. High weathering rates correspond with initial low basesaturation and vice versa (Fig. 6). The combination of thesetwo parameters determines the level and the slope of thebase cation concentrations and hence the ANC. The calibra-tion using all observations had a peak density weatheringrate of 51 mequiv. m−2 year−1 and an initial base saturation of41%. The calibration using 1990–2000 observations had lowerBCw and higher BS0 than the others. This means that cationexchange will occur faster than for the others which waspromoted in this particular calibration due to the very steepdecline in SO4

2−, but not in ANC seen in the observations forthe last half of the 1990s. This means that the main responseto the fast SO4

2− decline was a decline also in base cationconcentration, driven by changing ion exchange. The calibra-tion using the full time series also matches this time periodreasonably well (although not as well as the 1990–2000 calibra-tion), while the other calibrations tend overestimate the ANCin this period. With the exception of the 1990–2000 calibration,all other calibrations had higher BCw and lower BS0 than thecalibration using the full time series.

The distributions for calibrations using 4–5 and 11 years ofdata had considerably wider peaks than the calibration usingthe full data series. For BCw all distributions overlapped toat least a small extent. For BS0 the calibration to 1980–1984had particularly low values and for 1980–1990 particularlyhigh values. The distribution for the calibration using thefull data series are towards the lower range for BCw and the

higher range for BS0. This is because only the long time seriesincludes both slopes in the observed SO4

2− concentrations.For the soil solution concentration of organic anions

(expressed as charge; denoted SDOC) and the dissolution con-

eters for the different sets of calibration data: BS0 (initialntration in soil solution) and KGibb (the aluminiumBCw, as well as SDOC and KGibb are given in Table 1.

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32 e c o l o g i c a l m o d e l

stant for aluminium in the soil (KGibb), there are considerabledifferences between the different calibrations. The posteriordistributions for both parameters are rather large for most cal-ibrations, and clearly narrower for the calibration using thefull time series (Fig. 6). This shows that when the calibrationis limited by longer time series data, a more limited range ofthese calibrated parameters are possible.

4. Concluding remarks

For most model applications only a few years, or even only1 year of data are available for calibration. Natural variationbetween years will in such cases mask parts of long-termtrends that the model is built to predict. Uncertainty in modelpredictions will be considerably higher than if a long periodof observations is available for calibration. In the examplesshown here, even an 11-year data record gave considerableuncertainty in predictions. The results were also substantiallydifferent depending on which time period was used.

The case used as example in this analysis, SO42− and ANC

at the Birkenes catchment, the observations have a rathernoisy time trend. This increases the uncertainty when usingshorter time series.

For many applications of hydrogeochemical models only afew years or even only 1 year of data are available for calibra-tion. Natural variation between years will in such cases maskparts of long-term trends that the model is built to predict.Uncertainty in model predictions from a calibration based ona short period of observed data will be considerably higherthan if a long period of observations is available for calibra-tion. In the examples shown here, even an 11-year data recordgave considerable uncertainty in predictions, and the uncer-tainty estimates varied substantially depending on the lengthof the time period used for calibration.

Model predictions will always have uncertainties, althoughthey are commonly ignored in many model applications. Wehave presented a statistical framework for automatic cali-bration and uncertainty estimation in deterministic modelsusing a Bayesian approach based on an MCMC algorithm forcombing information from observations with the determinis-tic model. The approach makes it possible to present modelprediction results as probability distributions rather than sin-gle values.

In most model applications, in particular those linked todecision support or policy making, uncertainty analyses arecommonly ignored. Uncertainty analysis is often consideredcomplicated and maybe even be seen as undermining theconclusions from the model simulations because one actu-ally reveals that the results are uncertain. In the developmentof our approach we have been mindful that, although themethod itself is technically complex, the final outputs fromthe analysis should be easy to communicate in a straightfor-ward manner.

Many policy makers and other stake holders have experi-ence in dealing with risks. If results from model applications

are presented as probability distributions rather than singlenumbers, those using the results may be able to combinethe probabilities with the consequences and act accordingto the resultant perceived risk. In this way models can be

2 0 7 ( 2 0 0 7 ) 22–33

even more powerful tools in support of decision making andassessments.

Acknowledgements

This work was financed in part by the Research Council ofNorway and the Norwegian Institute for Water Research. Wethank Magne Aldrin, Ragnar Huseby, and Richard Wright forhelpful comments.

e f e r e n c e s

Aas, W., Tørseth, K., Solberg, S., Berg, T., Manø, S., Yttri, K.E., 2003.Monitoring of Long Range Transported Air Pollution andPrecipitation-Atmospheric Deposition 2002. The NorwegianPollution Control Authority (SFT).

Anctil, F., Perrin, C., Andreassian, V., 2004. Impact of the length ofobserved records on the performance of ANN and ofconceptual parsimonious rainfall-runoff forecasting models.Environ. Model. Softw. 19, 357–368.

Bates, S., Cullen, A., Raftery, A., 2003. Bayesian uncertaintyassessment in multicompartment deterministic simulationmodels for environmental risk assessment. Environmetrics14, 355–371.

Bloschl, G., Grayson, R., 2000. Spatial observations andinterpolation. In: Grayson, R., Bloschl, G. (Eds.), SpatialPatterns in Catchment Hydrology. Cambridge UniversityPress, Cambridge, UK, pp. 51–81.

Brath, A., Montanari, A., Toth, E., 2004. Analysis of the effects ofdifferent scenarios of historical data availability on thecalibration of a spatially-distributed hydrological model. J.Hydrol. 291, 232–253.

Cosby, B.J., Ferrier, R.C., Jenkins, A., Wright, R.F., 2001. Modellingthe effects of acid deposition: refinements, adjustments andinclusion of nitrogen dynamics in the MAGIC model. Hydrol.Earth Syst. Sci. 5, 499–518.

Cosby, B.J., Hornberger, G.M., Galloway, J.N., Wright, R.F., 1985.Modelling the effects of acid deposition: assessment of alumped parameter model of soil water and streamwaterchemistry. Water Resour. Res. 21, 51–63.

Funtowicz, S.O., Ravetz, J.R., 1990. Uncertainty and quality inscience for policy. In: Theory and Decision Library A:Philosophy and Methodology of the Social Sciences. Kluwer,Dordrecht, p. 15.

Kuczera, G., Parent, E., 1998. Monte Carlo assessment ofparameter uncertainty in conceptual catchment models: themetropolis algorithm. J. Hydrol. 211, 69–85.

Larssen, T., Cosby, B.J., Høgasen, T., 2004. Uncertainties inpredictions of surface water acidity using the MAGIC model.Water Air Soil Pollut.: Focus 4, 125–137.

Larssen, T., Huseby, R.B., Cosby, B.J., Høst, G., Høgasen, T., Aldrin,M., 2006. Forecasting acidification effects using a Bayesiancalibration and uncertainty propagation approach. Environ.Sci. Technol. 40, 7841–7847.

Lien, L., Raddum, G.G., Fjellheim, A., 1992. Critical loads of acidityto freshwater-fish and invertebrates. NIVA-report 2732/1992.Norwegian Institute for Water Research, Oslo.

Makowski, D., Wallach, D., Tremblay, M., 2002. Using a Bayesianapproach to parameter estimation; comparison of the GLUE

and MCMC methods. Agronomie 22, 191–203.

Perrin, C., Michel, C., Andreassian, V., 2001. Does a large numberof parameters enhance model performance? Comparativeassessment of common catchment model structures on 429catchments. J. Hydrol. 242, 275–301.

n g

S

S

S

S

S

e c o l o g i c a l m o d e l l i

aloranta, T.M., Kamari, J., Rekolainen, S., Malve, O., 2003.Benchmark criteria: a tool for selecting appropriate models inthe field of water management. Environ. Manage. 32, 322–333.

chopp, W., Posch, M., Mylona, S., Johansson, M., 2003. Long-termdevelopment of acid deposition (1880–2030) in sensitivefreshwater regions in Europe. Hydrol. Earth Syst. Sci. 7,436–446.

FT, 1993. The Norwegian monitoring programme for long-rangetransported air pollutants. Annual Report 1992, No. 533/93.The Norwegian Pollution Control Authority (SFT), Oslo,Norway.

FT, 2001. The Norwegian monitoring programme for long-range

transported air pollutants. Annual Report-Effects 2000, No.834/01. The Norwegian Pollution Control Authority (SFT), Oslo,Norway.

FT, 2004. The Norwegian monitoring programme for long-rangetransported air pollutants. Annual Report-Effects 2003, No.

2 0 7 ( 2 0 0 7 ) 22–33 33

913/2004. The Norwegian Pollution Control Authority (SFT),Oslo, Norway.

Snowling, S., Kramer, J., 2001. Evaluating modelling uncertaintyfor model selection. Ecol. Model. 138, 17–30.

Stuanes, A.O., Abrahamsen, G., Røsberg, I., 1995. Acidification ofsoils in five catchments in Norway. Water Air Soil Pollut. 85,635–640.

Thiemann, M., Trosset, M., Gupta, H., Sorooshian, S., 2001.Bayesian recursive parameter estimation for hydrologicmodels. Water Resour. Res. 37, 2521–2535.

Uhlenbrook, S., Sieber, A., 2005. On the value of experimental datato reduce the prediction uncertainty of a process-oriented

catchment model. Environ. Model. Softw. 20, 19–32.

Xia, Y., Yang, Z., Jackson, C., Stoffa, P., Sen, M., 2004. Impacts ofdata length on optimal parameter and uncertainty estimationof a land surface model. J. Geophys. Res. Atmos. 109, D07101,doi:10.1029/2003JD004419.