Embed Size (px)
Citation preview
00do
*
1
Journal of Hydrology 376 (2009) 428–442
Contents lists available at ScienceDirect
Journal of Hydrology
journal homepage: www.elsevier .com/locate / jhydrol
Assessing parameter, precipitation, and predictive uncertainty in a distributedhydrological model using sequential data assimilation with the particle filter
Peter Salamon *, Luc Feyen 1
Land Management and Natural Hazards Unit, Institute for Environment and Sustainability, DG Joint Research Centre, European Commission, Via Enrico Fermi 2749,TP 261, 21027 Ispra (VA), Italy
a r t i c l e i n f o
Article history:Received 18 December 2008Received in revised form 9 April 2009Accepted 18 July 2009
This manuscript was handled by K.Georgakakos, Editor-in-chief, with theassistance of Ashish Sharma, AssociateEditor
Keywords:Sequential data assimilationParticle filterDistributed hydrological modelUncertainty
22-1694/$ - see front matter � 2009 Elsevier B.V. Alli:10.1016/j.jhydrol.2009.07.051
Corresponding author. Tel.: +39 332 786013; fax: +3E-mail addresses: [email protected] (P. Salamon), lTel.: +39 332 789258.
s u m m a r y
Sequential data assimilation techniques offer the possibility to handle different sources of uncertaintyexplicitly in hydrological models and hence improve their predictive capabilities. Amongst the differenttechniques, sequential Monte Carlo or particle filter methods offer the capability to handle non-linear/non-Gaussian state-space models while preserving the spatial variability of updated state variables, bothdesirable features when assimilating data in distributed hydrological models. In this work we apply theresidual resampling particle filter to assess parameter, precipitation, and predictive uncertainty in thedistributed rainfall–runoff model LISFLOOD. First, we compare estimated posterior parameter distribu-tions with results of the Shuffled Complex Evolution Metropolis global optimization algorithm obtainedusing identical input data for the Meuse catchment and considering parameter uncertainty only. Bothapproaches result in well identifiable posterior parameter distributions and provide a reasonable fit tothe observed hydrograph. The resulting posterior distributions, however, vary considerably in shape,location, and scale, most likely caused by the different assumptions made in the output error model.An evaluation of the predictive distributions illustrates that predictive uncertainty is significantly under-estimated for both approaches when accounting for parameter uncertainty only. A second case studyshows that considering additionally precipitation uncertainty not only increases the spread of the poster-ior parameter distributions but may also result in a completely different location and/or shape of the pos-terior distributions. Evaluation of the posterior precipitation multiplier distribution reveals that nooverall systematic bias exists in the precipitation grids and that particle filtering is a suitable tool toquantify and reduce precipitation uncertainty. Furthermore, considering precipitation and parameteruncertainty leads to an improvement in model predictive capabilities, especially for the high flow peri-ods. However, the remaining underestimation of predictive uncertainty also indicates that model struc-tural uncertainty is equally important, in spite of using a physically-based distributed hydrological modelthat should theoretically provide an improved description of the hydrological system dynamics.
� 2009 Elsevier B.V. All rights reserved.
Introduction and scope
The increase in quantity and quality of meteorological andhydrological data has led to the development of more complex,spatially distributed hydrological models in order to improve theability to predict hydrological events. Whereas the first lumpedconceptual rainfall–runoff models were applied mainly to forecastdischarges in small and midsize catchments, their application tolarge river basins depended on many assumptions, primarily theuniformity of meteorological forcing data and model parameters.It is not surprising that such assumptions often decrease theaccuracy in runoff prediction (e.g., Smith et al., 2004).
rights reserved.
9 332 [email protected] (L. Feyen).
Physically-based spatially distributed models have the capacityto make use of high-resolution input data and thus provide a spa-tial response to the distributed inputs. However, despite theirincreasing complexity and an improved usage of the diversity ofavailable data, a variety of obstacles (e.g., the difference in spatio-temporal scale between the model and measurements; measure-ment errors; or the simplification of physical processes withinthe model) still introduce a significant amount of uncertainty intothe model predictions. To quantify and reduce this uncertainty,tools are required that are capable not only of including meteoro-logical forcing and static data (e.g., topography, land use, and soiltype), but also other observations (e.g., soil moisture and dis-charges). Under these premises data assimilation techniques areincreasingly being used to merge in an optimal manner the varioussources of data and their corresponding uncertainties into rainfall–runoff models (e.g., Weerts and El Serafy, 2006; Clark et al., 2008).
P. Salamon, L. Feyen / Journal of Hydrology 376 (2009) 428–442 429
Due to the ability to explicitly treat the various sources ofuncertainty, sequential data assimilation algorithms can also beused to estimate model states and parameters simultaneously.Watershed model calibration mostly accounts only for uncertain-ties in streamflow measurement data (e.g., Sorooshian et al.,1993) and parameter estimates (e.g., Vrugt et al., 2003; Feyenet al., 2007, 2008). Uncertainties arising from errors associatedwith forcing data (e.g., rainfall, temperature), initial conditions ofstate variables, as well as model structural errors are typicallynot handled in a clear manner by these calibration routines. Recentpublications (e.g., Moradkhani et al., 2005a,b; Vrugt et al., 2005,2006) have shown the positive impact on parameter estimationand model predictions when employing sequential data assimila-tion techniques for parameter calibration.
The most frequently used methods for hydrologic data assimila-tion are Kalman filtering, variational data assimilation, and particlefiltering. These methods update/improve river basin states such asgroundwater storage or soil moisture by extracting informationfrom observations up to the current time and propagating it to statevariables. One of the earliest state-space filtering methods appliedto hydrological models is the extended Kalman filter (e.g., Kitanidisand Bras, 1980a,b). In this approach local approximations of thenon-linear system dynamics are employed to recursively infer thestate variables. Unfortunately, instabilities or even divergencemay be produced as second- and higher-order derivatives are ne-glected by the local approximations (Evensen, 1994). Furthermore,the computational costs are significant, especially for high-dimen-sional state-vector problems, such as spatially distributed models.Another variation of this technique is the ensemble Kalman filter(e.g., Evensen, 2006; Reichle et al., 2002a,b; Moradkhani et al.,2005a; Vrugt et al., 2005). Here, an ensemble of model states, gen-erated from random input perturbations, is propagated and thenupdated based on the Kalman gain. The ensemble Kalman filter iscomputationally efficient, but it is based on the assumption thatall probability distributions involved are Gaussian. Another disad-vantage of the ensemble Kalman filter with respect to spatially dis-tributed models is that when assimilating point data such asdischarge observations, the updated states are average values ofthe upstream area. As such, they do not preserve the spatial distri-bution pattern obtained by the model simulation (Kim et al., 2007).Theoretically this problem can be circumvented by updating allindividual cell values using state augmentation (Hendricks Frans-sen and Kinzelbach, 2008). Unfortunately, this is likely to decreasecomputational efficiency of the EnKF significantly.
Variational data assimilation is commonly employed in numer-ical weather prediction but has not been used frequently in hydro-logical models up to now (e.g., McLaughlin, 2002; Seo et al., 2003).It involves the minimization of a cost function, which representsthe aggregated error over the entire assimilation window. Theminimization problem is normally solved by using a linearized ver-sion of the hydrological model. Unfortunately, the development ofsuch an adjoint model is a complicated task, especially for distrib-uted rainfall–runoff models. The variational methods are computa-tionally less expensive than the extended Kalman filter (Liu andGupta, 2007). However, as variational methods work over a giventime window containing a sequence of observations, they are moreapt for smoothing problems than for real-time data assimilationwhere observations arrive continuously in time.
Particle filtering is a recursive Bayesian filter based on MonteCarlo simulation. The key idea is to represent posterior probabilitydistribution functions by a set of randomly drawn samples, calledparticles, with associated weights (Arulampalam et al., 2002). Par-ticle filtering has the advantage of being applicable to non-linear,non-Gaussian state-space models. Furthermore, particle filteringperforms updating on the particle weights rather than on statevariables directly, which renders it ideal for assimilating data in
spatially distributed models. As a particle represents a set of spa-tially distributed state variables, the spatial distribution is main-tained after updating. A common problem in particle filtering isweight degeneracy, where after some iterations most of the parti-cles have a zero weight. This problem can be alleviated by increas-ing the number of particles at the cost of an increase incomputational demand and/or by resampling of the particles(e.g., Doucet et al., 2001). However, as pointed out by Gordonet al. (1993), a proper treatment of the process noise is crucial inorder to avoid that the resampling procedure leads to sampleimpoverishment, i.e., many particles having high weights becausethey are selected many times.
Despite the variety of data assimilation approaches and its pro-ven benefits for uncertainty analysis and model prediction, only inrecent years more research concerning the application of thesetechniques to rainfall–runoff models has been published especiallyconcerning the use of particle filters (e.g., Moradkhani et al., 2005b;Weerts and El Serafy, 2006; Smith et al., 2008). Most of the work hasfocused on lumped hydrological models, whereas very few studieshave applied data assimilation techniques in combination with spa-tially distributed rainfall–runoff models (e.g., Kim et al., 2007;Blöschl et al., 2008; Clark et al., 2008). Moreover, all of the studiesthat have used distributed models have applied the ensemble Kal-man filter to update model states. No study has yet appeared thatuses the particle filter for data assimilation in a spatially distributedmodel, despite its suitability for non-linear, non-Gaussian state-space models, and its characteristic to retain spatial relations byupdating particle weights rather than state variables.
In this paper we present an application of the particle filtermethod to the spatially distributed rainfall–runoff model LIS-FLOOD (Van der Knijff et al., 2008). We illustrate its performancefor inferring parameter estimates and compare the results withthe parameter distributions obtained using global optimization.Furthermore, we analyse the impact of precipitation uncertaintyon the inference of parameter distributions and on streamflowforecasting. The paper is organized as follows. ‘‘Methodology”briefly presents the theory of particle filtering and of the residualresampling technique aimed at reducing the degeneracy problemin sequential Monte Carlo methods. In ‘‘Hydrological model, data,and study area description” and ‘‘Implementation of the particlefilter with LISFLOOD” LISFLOOD and the study area are describedas well as a brief technical implementation of the particle filterwith the model. ‘‘Quantifying input and observation errors” out-lines how the streamflow measurement error and the uncertaintyin the precipitation fields are characterised. ‘‘Results” illustrate theresults of the different simulations and the summary with conclu-sions is presented in ‘‘Summary and conclusions”.
Methodology
In this section we briefly describe the theory of Bayesian filter-ing and the approximation of the residual resampling particle filterto the optimal Bayesian solution. For a more detailed descriptionand discussion of Bayesian filtering in general and its specificapplication to hydrological models the reader is referred to Doucetet al. (2001) and Moradkhani et al. (2005a,b).
Bayesian filtering
To define the problem of Bayesian filtering consider the evolu-tion of a discrete-time dynamic state-space model which can beformulated as follows:
xkþ1 ¼ f ðxk; h; ukÞ þxkþ1 xkþ1 � Nð0;Qkþ1Þ ð1Þykþ1 ¼ hðxkþ1; hÞ þ mkþ1 mkþ1 � Nð0;Rkþ1Þ ð2Þ
430 P. Salamon, L. Feyen / Journal of Hydrology 376 (2009) 428–442
here xk 2 Rnx denotes the nx dimensional state vector of the systemat time k with the initial Probability Density Function (PDF) pðx0Þ,which evolves over time as a first order Markov process accordingto the conditional PDF pðxkþ1jxkÞ. The ny dimensional observationvector ykþ1 2 Rny is conditionally independent given xk+1 and theobservation Eq. (2) is represented by the PDF pðykþ1jxkþ1Þ. Thenon-linear operators f : Rnx ! Rnx and h : Rnx ! Rny express thesystem transition in response to the forcing data uk, time invariantmodel parameters h, as well as the forecasted state variables. Theindependent random vectors xk+1 and mk+1 represent the modeland the measurement error and consist of a white noise sequencewith mean zero and variance Qk+1 and Rk+1, respectively.
The principal goal of the state estimation problem in Bayesianfiltering is to construct the posterior PDF pðxkþ1jykþ1Þ based on allthe available information. The posterior filtering density consti-tutes the complete solution to the sequential data assimilationproblem and allows calculating any ‘‘optimal” estimate of the state,e.g., the conditional mean. The optimal state at time k + 1 given theobservations up to time k can hence be obtained with the posteriorPDF via Bayes rule:
pðxkþ1jykþ1Þ ¼pðykþ1jxkþ1Þpðxkþ1jykÞ
pðykþ1jykÞð3Þ
Since the system state at each time step is just dependent on theprevious state, the forecast density pðxkþ1jykÞ obtained with theChapman–Kolmogorov equation can be simplified to the followingequation:
pðxkþ1jykÞ ¼Z
xk
pðxkþ1jxkÞpðxkjykÞdxk ð4Þ
The normalizing factor in Eq. (3), also denominated as predic-tive distribution, is written as follows:
pðykþ1jykÞ ¼Z
xkþ1
pðykþ1jxkþ1Þpðxkþ1jykÞdxkþ1 ð5Þ
The Kalman filter provides an analytical solution to the Eqs. (3)–(5), provided that the state-space model of Eqs. (1) and (2) repre-sents a Gaussian linear system. Unfortunately, most hydrologicapplications are non-linear and non-Gaussian and the multidimen-sional integration typically makes a closed-form solution intracta-ble. A different approach to solve Eqs. (3)–(5) for these cases is theapplication of the sequential Monte Carlo or particle filter method(Doucet et al., 2001) as outlined in the following section.
The residual resampling particle filter
The key idea of the particle filter is to represent the posteriorPDF by a set of random drawn samples, called particles, with asso-ciated weights. With a large number of particles this Monte Carlocharacterization approximates the posterior PDF as (Arulampalamet al., 2002).
pðxkþ1jykþ1Þ �XNp
i¼1
wikþ1dðxkþ1 � xi
kþ1Þ ð6Þ
where xikþ1 and wi
kþ1 denote the ith particle and its weight, respec-tively, Np is the number of particles, and d denotes the Dirac deltafunction. The weights are normalized so that
PNp
i¼1wikþ1 ¼ 1 and
are obtained through importance sampling. Since it is usuallyimpossible to sample from the true posterior PDF, an alternativeis to sample from a proposal distribution, also called importancedensity, denoted by qðxkþ1jykþ1Þ. The weights can then be assignedaccording to (e.g., Moradkhani et al., 2005a,b)
wikþ1 /
pðxikþ1jykþ1Þ
qðxikþ1jykþ1Þ
ð7Þ
The recursive weight updating in a sequential case is then de-rived as (e.g., Moradkhani et al., 2005a,b)
wikþ1 ¼ wi
k
pðykþ1jxikþ1Þpðxi
kþ1jxikÞ
qðxikþ1jxi
k; ykþ1Þð8Þ
The selection of the importance density is an important issue(e.g., Doucet et al., 2001; Arulampalam et al., 2002) as it signifi-cantly influences the filter performance. Generally, the transitionprior is used as the proposal distribution, where
qðxikþ1jxi
k; ykþ1Þ ¼ pðxikþ1jxi
kÞ ð9Þ
Substituting (9) into (8), the weight updating simplifies to
wikþ1 ¼ wi
kpðykþ1jxikþ1Þ ð10Þ
A common problem with the sequential importance samplingfilter is the sample impoverishment, where after several iterationsonly very few particles have non-zero importance weights. Thisalso means that large computational effort is dedicated to updatingparticles whose contributions to the approximation of the poster-ior PDF is basically negligible. The simplest approach to limit theimpact of degeneracy is to increase the number of particles. Unfor-tunately, this is often impractical because of the increase in com-putational demand. Instead, in practice a resampling procedure isapplied to alleviate the degeneracy problem. Resampling basicallyduplicates particles having high normalized weights and discardsparticles with low normalized weights, while keeping the totalnumber of particles unchanged. A variety of resampling algorithmshas been described in the literature, such as sampling importanceresampling (Rubin, 1988), residual resampling (Liu and Chen,1998), stratified resampling (Kitagawa, 1996), deterministicresampling (Kitagawa, 1996), and systematic resampling (Carpen-ter et al., 1999). In this work we use the residual resampling algo-rithm as it is computationally cheaper than the other algorithmsand it provides a smaller variance in comparison to the samplingimportance resampling scheme (Liu and Chen, 1998).
Particle filtering is used here for joint state-parameter estima-tion; hence in addition to the state variables, parameter variablesare also resampled. This is done using the following proposal den-sity (Moradkhani et al., 2005a,b)
qðhkþ1jhk; ykþ1Þ ¼ pðhkþ1jhkÞ ð11Þ
One side effect of resampling is that by selecting particles with ahigh weight many times, the diversity amongst the particles maybe lost when the process noise in Eq. (1) is small (Gordon et al.,1993). This effect is known as sample impoverishment and canbe avoided by perturbing forcing data, model parameters, and/orstate variables. In this work we perturbed resampled parametersat each successive assimilation step following Moradkhani et al.(2005a,b)
hikþ1 ¼ hi
kðresampledÞ þ eik ei
k � Nð0;VarhkÞ ð12Þ
here eik is a random noise, normally distributed with zero mean and
variance Varhk. Varh
k is the variance of parameter particles at time kbefore resampling. In order to avoid model instabilities by havinglarge changes in the perturbed parameter samples and to also as-sure a minimum process noise by preventing the variance of param-eters to collapse to very small values, Varh
k is confined by upper andlower limits. Various test runs have shown that using a minimumvariance corresponding to the 95% confidence interval of 2% of theinitial parameter uncertainty range and a maximum variance corre-sponding to the 95% confidence interval of 16% of the initial param-eter uncertainty range provide reasonable boundaries to fulfil theabove mentioned conditions. When accounting for uncertainty inthe forcing data we also perturbed precipitation data. This is de-scribed in more detail in ‘‘Precipitation uncertainty”.
P. Salamon, L. Feyen / Journal of Hydrology 376 (2009) 428–442 431
A further problem in joint state-parameter estimation is thatthe model response to a change in parameters, especially with re-gard to parameters controlling the subsurface flow components ina spatially distributed model, will not have immediate effect on thedischarge simulated at the downstream outlet. This problem canbe avoided giving the model a sufficiently large response time be-fore updating states and parameters. Naturally, this response timedepends strongly on the catchment characteristics. As dischargetime series are usually measured at a high-frequency, this will inmost cases, however, lead to omitting large quantities of observeddata. We therefore calculated particle weights at each observationtime step within the response time interval. Each particle was as-signed the median of the weights over all observation time-stepswithin the response time interval. Particle weights were then nor-malized to one and used for the resampling procedure. This ap-proach has the advantage of making full use of the available dataand at the same time giving more stable parameter estimates.
A detailed description of the residual resampling particle filteras used in this study is given below. Note that here we updatedmodel states and parameters after seven daily model time-steps,which can be considered as a sufficient model response time tochanges in model parameters for our particular study area. In thefollowing, the current and subsequent assimilation steps are de-noted by k and k + 7, respectively.
A. Prediction step
(1) Initialize the system by sampling the parameters hik
from a uniform distribution for Np particles, wherei ¼ 1; . . . ;Np, and perform a warming up of the hydro-logical model to obtain the associated initial states xi
k.(2) Assign the particle weights uniformly wi
k ¼ 1=Np.(3) Propagate the Np model states xi
k and parameters hik
forward in time to obtain the model predicted statesxi
kþl using the hydrologic model, where l ¼ 1; . . . ;7.
B. Update step
(1) Obtain updated particle weights for each time stepwithin the assimilation step assuming a Gaussianobservation noise with variance Rkþl according to (e.g.,Moradkhani et al., 2005a,b; Weerts and El Serafy, 2006)
wikþl ¼
exp � 12Rkþl
ykþl � hikþl
h i2� �
PNp
i¼1 exp � 12Rkþl½ykþl � hi
kþl�2
� � l ¼ 1; . . . ;7 ð13Þ
where hikþl is the simulated discharge of the ith particle at
time k + l, and yikþl is the observed discharge at the same time
step.
(2) Calculate particle median weights wikþ7;med over all the
available discharge measurements during the assimi-lation step.
(3) Normalize particle weights so thatPNp
i¼1wikþ7;med ¼ 1.
C. Residual resampling step
Fig. 1. Schematic overview of the LISFLOOD model. P = precipitation; Int = inter-ception; EWint = evaporation of intercepted water; Dint = leaf drainage; ESa = evap-oration from soil surface; Ta = transpiration (water uptake by plant roots);INFact = infiltration; Rs = surface runoff; D1,2 = drainage from top- to subsoil;D2,gw = drainage from subsoil to upper groundwater zone; Dpref,gw = preferential
(1) Generate Ni ¼ bNpwikþ7;medc copies of each particle,
where the operator bc takes the integer part of itsargument.
(2) Calculate the normalized residual weights wikþ7;res
according to
wikþ7;res ¼
ðNpwikþ7;med � NiÞ
Np �PNp
i¼1Ni
ð14Þ
flow to upper groundwater zone; Duz,lz = drainage from upper- to lower ground-
water zone; Quz = outflow from upper groundwater zone; Qlz = outflow from lowergroundwater zone; Dloss = loss from lower groundwater zone. Note that snowmelt isnot included in the figure (even though it is simulated by the model) (after Van derKnijff et al., 2008).(3) Construct an empirical cumulative distribution func-tion using wi
kþ7;res and sample it Np �PNp
i¼1Ni times toobtain the remaining particles.
D. Perturbation
(1) Stop criterion: if k is equal to the desired number oftime-steps, stop; otherwise:(2) Perturb parameter sets for each particle according to
Eq. (12) and return to step A (2).
Hydrological model, data, and study area description
In this work results are presented for the Meuse catchment up-stream of Borgharen (see Fig. 2), covering an area of approximately21,000 km2. Due to the relatively flat topography with elevationsranging from 50 m to 700 m and its geographical location theMeuse is mainly fed by rain all year round. Hence, the flows gener-ally show a strong seasonal cycle with high flows during the winterperiod and low flows during the summer period. The predominantland use types are forest, agriculture, moor and heath and the soillayers are mostly impervious, resulting in a relatively fast responsetime of discharges to precipitation. The model setup employeduses a 5-km grid resolution and a daily time step. Daily observeddischarges for the Borgharen gauging station were available forthe calibration (from 1/10/1992 to 30/09/1995) and validation(from 10/1/1989 to 30/09/1992) periods. The first year of bothperiods was used as a model spin up.
We use the rainfall–runoff model LISFLOOD (Van der Knijffet al., 2008) which has been developed to predict floods in largescale river basins. The model is raster based and is implementedusing a combination of the PCRaster dynamic modelling language(e.g., Karssenberg, 2002) and the Python scripting language, facili-tating the handling of large data sets. A schematic overview of themodel with its different components is illustrated in Fig. 1. Since it
Fig. 2. Overview of the Meuse catchment with gauging station Borgharen (afterFeyen et al., 2007).
Table 1Prior uncertainty ranges associated with calibration parameters in LISFLOOD.
Parameter Lower bound Upper bound
Upper Zone Time Constant (UZTC) 1 10Lower Zone Time Constant (LZTC) 50 5000Groundwater Percolation Value (GWPV) 0.01 0.5Xinanjiang parameter b (Xb) 0.05 0.5Power Preferential Bypass Flow (PPF) 5 15
432 P. Salamon, L. Feyen / Journal of Hydrology 376 (2009) 428–442
is spatially distributed, the model is capable to account for the spa-tial variability in forcing data, e.g., precipitation, and input param-eters, e.g., soil and land use properties. A more detailed descriptionof the LISFLOOD model can be found in Van der Knijff et al. (2008).
Most input parameters and variables of LISFLOOD are estimateda priori if possible. Here, we employ the Soil Geographical Databaseof Europe (King et al., 1994), the HYPRES database on hydraulic soilproperties (Wösten et al., 1999), and the CORINE Land Cover data-base (European Environment Agency, 2000) to obtain all necessarysoil and land use related parameters. The meteorological inputtime series used to drive the model were obtained from the Mete-orological Archiving and Retrieving System (MARS) (Rijks et al.,1998). Ordinary kriging with daily observations of on average 23stations was employed to generate the interpolated precipitationgrids for the study area.
Although LISFLOOD is based on physics to a certain extent, someprocesses are only represented in a conceptual way and thus cannotbe directly measured. Currently, LISFLOOD requires five lumpedparameters to be estimated by calibration against measured dis-charges (see Table 1). The Upper Zone Time Constant (UZTC) andLower Zone Time Constant (LZTC) determine the residence timeof water in the respective groundwater zones, and thus controlthe amount and timing of the outflow from these storages. TheGroundwater Percolation Value (GWPV) controls flow from theupper to the lower groundwater zone. The Xinanjiang parameterb (Xb) is an empirical shape parameter in the Xinanjiang model(Zhao and Liu, 1995), which influences the fraction of saturated areawithin a grid cell. The Power Preferential Bypass Flow parameter(PPF) relates preferential flow with the relative saturation of thesoil. For the purpose of comparing parameter distributions obtainedin the study of Feyen et al. (2007), similar upper and lower boundsof the prior parameter distributions were used here (Table 1).
Fig. 3. Scatterplot of the estimated error deviation of streamflow measure-ments according to Eq. (11) using approximately 9 years of data from the Borgharengauging station. The solid line shows the fit of the spline function through thedata which was used to compute the likelihood function for the updatingprocedure.
Implementation of the particle filter with LISFLOOD
Particle filtering updates the weight of particles rather thanstate variables directly. In the case of LISFLOOD, each particle con-tains a set of 12 spatially distributed state variables maps, whichhave to be propagated and resampled as outlined in ‘‘The residualresampling particle filter”. In order to limit the effects of degener-acy and sample impoverishment and to obtain well defined poster-ior parameter distributions an ensemble of 3200 particles wasemployed for all simulations. Due to the computational demandsthe residual resampling particle filter was implemented using par-allel computing in a similar manner as presented in the work byVrugt et al. (2006). We employed a Local Area Multicomputer–Message Passing Interface (LAM/MPI) distributed computing inter-face for the Python programming language (Nielsen, 2007).Whereas weight updating, resampling, and parameter perturba-tion was performed on the master node, the propagation of theparticles was distributed onto the separate slave nodes. The simu-lations in this work were performed using 17 processors with afrequency of 3.2 GHz. The CPU time required for the joint state-parameter estimation with the residual resampling particle filterfor a 3-year simulation period with a daily time step and a param-eter and state updating every 7th day took approximately 20 h.
Fig. 4. Development of parameter uncertainty bounds and hydrograph prediction uncertainty for the calibration period (10/01/1993–09/30/1995). Observed discharges arerepresented by dark crosses. The grey shaded area in the hydrograph denotes the 95 percentile confidence interval for the prediction uncertainty resulting from parameteruncertainty. Shaded areas in the parameter uncertainty bounds correspond to 95, 90, and 10 percentile confidence interval.
P. Salamon, L. Feyen / Journal of Hydrology 376 (2009) 428–442 433
Fig. 5. Hydrograph prediction uncertainty for the validation period (10/01/1990–09/30/1992). Observed discharges are represented by dark crosses. The grey shaded areadenotes the 95 percentile confidence interval for the prediction uncertainty resulting from parameter uncertainty.
Fig. 6. Predictive QQ plots comparing the results of the residual resampling particlefilter (crosses) with the result of the global optimization algorithm SCEM-UA(diamonds) obtained by Feyen et al. (2007) for the calibration period (a) and thevalidation period (b).
434 P. Salamon, L. Feyen / Journal of Hydrology 376 (2009) 428–442
Quantifying input and observation errors
Streamflow measurement error
A proper specification of the likelihood function used to extractinformation from the measured streamflow data is crucial for theperformance of particle filtering. Unfortunately, informationdescribing the statistical properties of the measurement error isusually not available, as is also the case here, and estimating theseproperties has not proven to be simple. Due to the non-linear nat-ure of the rating curves used to transform water levels into dis-charges, large flows tend to have larger error variances comparedto smaller flows (heteroscedasticity) (Sorooshian and Dracup,1980). Therefore, the most common approach found in the litera-ture to describe the statistical properties of the measurement erroris to assume a Gaussian distribution having zero mean and a para-metric function for the error variance which relates the magnitudeof the flow to the variance (e.g., Thiemann et al., 2001; Clark et al.,2008). The parameters of the variance function are then usuallyestimated by the modeller or are approximated during calibration.
Here, we use an alternative nonparametric approach as sug-gested by Vrugt et al. (2005, 2006). The error deviation
ffiffiffiffiffiffiffiffiRkþl
pis
estimated as
ffiffiffiffiffiffiffiffiRkþl
p¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2u
u
� ��1
ðDuykþlÞ2
sð15Þ
where Du denotes the difference operator applied u times. This non-parametric error deviation estimation, which is applied locally tothe time series, assumes that the time sequence of discharge valuesis sufficiently smooth, the sampling interval is high compared to thetemporal scale of streamflow dynamics, and that streamflow erroris heteroscedastic. Fig. 3 illustrates the scatterplot of the estimatederror deviation (u ¼ 4) versus the observed discharges usingapproximately 9 years of data from the Borgharen gauging station.A spline function was fit through the scattered data, which was thenemployed to derive the variance of the streamflow measurementerror at each time step as a function of observed streamflow.
Precipitation uncertainty
Bias in the representation of spatially distributed precipitationoriginates from various sources: errors occurring during the mea-surement at a station, e.g., wind, wetting, drifting, evaporation,instrument and/or human error; errors due to the difference inmeasurement and model grid scale; and errors due to the interpo-lation technique selected. As such, statistical properties of the errorassociated with interpolated precipitation fields based on sparseprecipitation measurements are not trivial to derive. A variety of
P. Salamon, L. Feyen / Journal of Hydrology 376 (2009) 428–442 435
approaches have been developed that consider many of the errorsources (e.g., Lanza, 2000; Clark and Slater, 2006). Unfortunately,these methods require either the availability of a considerableamount of additional data, e.g., long historic time series of mea-surements, and/or are very site specific.
More recently, various authors have analysed precipitationuncertainty using a precipitation multiplier model (e.g., Kavetskiet al., 2002; Kavetski et al., 2006a,b; Vrugt et al., 2008). For example,by calibrating not only model parameters but also precipitationmultipliers, which represent the systematic error in rainfall forcingdata, Vrugt et al. (2008) demonstrated that for the two watershedsanalysed the predictive capabilities of the hydrological model weresignificantly improved when accounting additionally for precipita-tion uncertainty. In this work we adopted a similar approach toquantify precipitation uncertainty. Precipitation ensembles aregenerated employing a precipitation multiplier according to
p0kþl ¼ pkþl þ pkþlupk l ¼ 1; . . . ;7 ð16Þ
where upk 2 ½�1;1� is the precipitation multiplier and pk+l are the
interpolated precipitation grids derived from the rain gauge obser-
Fig. 7. Comparison of posterior parameter distributions of the five LISFLOOD parametersstep (black outlined bins) and the global optimization algorithm SCEM-UA (grey shaded
vations. Note that upk is constant for the time-steps within each
assimilation step, but time-variant throughout the entire assimila-tion period. The precipitation multipliers are then updated in a sim-ilar manner as described above for the state and parametervariables. To obtain the initial multipliers we sampled a Gaussiandistribution with mean zero and a standard deviation of 0.15. Thestandard deviation is slightly smaller than the average value foundby Vrugt et al. (2008) in order to account for the spatial variabilityof the precipitation grids. After the resampling, precipitation multi-pliers were perturbed as is shown in Eq. (12) for the parameters,using a normally distributed random noise with mean zero and astandard deviation of 0.03.
Results
Case study 1: estimating posterior parameter distributions using theresidual resampling particle filter
The first case study focuses on the estimation of LISFLOODparameters considering only parameter uncertainty. Initial param-
obtained using the residual resampling particle filter after the last assimilation timebins) in the study of Feyen et al. (2007).
436 P. Salamon, L. Feyen / Journal of Hydrology 376 (2009) 428–442
eter sets are obtained by drawing 3200 samples from a uniformdistribution with prior parameter ranges as illustrated in Table 1.Fig. 4 presents the development of the parameter uncertaintybounds and the hydrograph for the calibration period of 2 years.In agreement with other studies (e.g., Moradkhani et al., 2005b) asignificant reduction of parameter uncertainty is observed for allparameters after the high flow period (between 12/01/1993 and01/01/1994), reflecting the high information content of the majorflow event at that time. However, the three parameters related tothe quick flow components of the model (Xb, UZTC, PPF) show areduction of uncertainty already previously, indicating that smallevents and even low flows contain sufficient information in thecase of only considering parameter uncertainty to constrainparameter distributions in a distributed model. Close inspectionof the hydrograph illustrates a reasonable conformity between ob-served and simulated time series. Yet, not all measurements liewithin the prediction uncertainty bounds suggesting that the attri-bution of uncertainty to only the parameter estimates is insuffi-cient. This becomes especially visible for the validation periodwhich uses the posterior parameter distributions obtained fromthe last sequential data assimilation step in the calibration period(Fig. 5). Here, the model is overpredicting the low flow periodswhile providing good fits to most of the high flow events.
To analyse in more detail whether the predictive uncertainty asshown in the calibration and validation hydrographs (Figs. 4 and 5)is consistent with the observed data we use predictive QQ plots.Predictive QQ plots provide a simple and informative summaryof the performance of the probabilistic predictions (e.g., Thyeret al., 2009; Laio and Tamea, 2007). Fig. 6 compares the predictiveQQ plots for the calibration and validation period of the residualresampling particle filter and the global optimization algorithmSCEM-UA (Vrugt et al., 2003). The latter parameter distributionshave been derived using an identical model setup and input forcing(Feyen et al., 2007). Furthermore, both calibration methods con-sider only parameter uncertainty. Both approaches show an under-estimation of the predictive uncertainty especially during thevalidation period. The particle filter performs slightly better thanthe SCEM-UA approach, which is probably due to the use of a lessrestrictive output error model. However, the plots indicate thatoverall the output error models of both approaches assume a
Fig. 8. Partial autocorrelation of residuals during calibration considering parameteruncertainty only (triangles) and considering parameter and precipitation uncer-tainty (crosses). Dashed line shows the 99% probability limits provided by the Rstatistical package (R Development Core Team, 2008).
streamflow measurement error range which is too small and thatconsidering parameter uncertainty only is clearly not sufficient toquantify predictive uncertainty. Furthermore, residual errors dur-ing calibration of the Meuse catchment using SCEM-UA illustrateda strong autocorrelation (Feyen et al., 2007), suggesting that theassumption of statistically independent errors has to be rejected.Fig. 8 shows that the independence assumption of the output errormodel used for the particle filter is clearly violated for lag 1. Theviolation of the statistical independence assumption for both out-put error models results most likely in a further underestimationof the prediction limits.
Fig. 7 presents a comparison of the posterior parameter distri-butions obtained with the residual resampling particle filter andSCEM-UA. For all parameters except the UTZC parameter the pos-terior distributions vary considerably in shape, location andspread. For example, SCEM-UA assigns the GWPV parameter a dis-tribution which is close to zero. As this parameter controls the flowfrom the upper zone groundwater storage into the lower zone, it isclear that as a consequence the LZTC parameter does not influencethe simulations at all as the lower zone storage is practically al-ways empty. The smaller Xb values obtained by the particle filtertechnique lead to a higher flux into the upper groundwater zoneand less surface runoff which is then probably compensated bythe increased PPF values, resulting in less preferential flow to thegroundwater zone for the same moisture content. Additionally,the higher GWPV values compensate the increased flux into theupper groundwater zone by a larger flow into the lower zone stor-age. One of the principal reasons for the observed divergence inestimated parameter distributions are probably the differentassumptions made in the error model for the likelihood function.Whereas SCEM-UA assumes a normally distributed error modelwith a constant variance, we assume here a heteroscedastic errormodel as outlined in ‘‘Streamflow measurement error”. AlthoughFeyen et al. (2007) used a Box–Cox transformation to account fornon-normality and heteroscedastic errors, the differences inassumptions in the formulation of the likelihood function can havea significant effect on posterior parameter distributions (Bates andCampbell, 2001).
Yet, comparison of the simulated hydrographs shows that bothapproaches provide similar fits for the calibration and validationhydrographs (Figs. 4 and 5 here and Figs. 11 and 12 in Feyenet al. (2007)), thus supporting the equifinality thesis of Beven(2006). This is somewhat surprising as one could assume thatequifinality of models should disappear as improved hydrologicalmodels and an increased quantity and quality of observations the-oretically provide sufficient restrictions on the calibration problem.Results here, however, indicate that even when using a relativelycomplex, partly-physically based rainfall–runoff model and a con-siderable amount of spatially distributed input data, the calibrationproblem is still not adequately constrained to provide a unique setof parameter distributions.
Case study 2: an analysis of precipitation uncertainty and its impacton parameter estimations
The second case study focused on the quantification of precipi-tation uncertainty and its impact on the estimated parameter dis-tributions. Precipitation ensembles were generated as outlined in‘‘Precipitation uncertainty” for the 3200 particles employed duringthe assimilation process. Fig. 9 presents the development of theparameter uncertainty bounds and the hydrograph for the calibra-tion period of 2 years. In contrast to the previous case study a sig-nificant reduction of parameter uncertainty for most parametersoccurs only after the major event between 12/01/1993 and 01/01/1994, whereas the uncertainty for the Xianjiang parameter isonly reduced after the second high flow event. Smaller events
Fig. 9. Development of parameter uncertainty bounds and hydrograph prediction uncertainty for the calibration period (10/01/1993–09/30/1995) accounting for errors inthe precipitation input data. Observed discharges are represented by dark crosses. The grey shaded area in the hydrograph denotes the 95 percentile confidence interval forthe prediction uncertainty resulting from parameter uncertainty. Shaded areas in the parameter uncertainty bounds correspond to 95, 90, and 10 percentile confidenceinterval.
P. Salamon, L. Feyen / Journal of Hydrology 376 (2009) 428–442 437
438 P. Salamon, L. Feyen / Journal of Hydrology 376 (2009) 428–442
and low flow periods preceding the high flow phase apparently donot contain sufficient information to constrain parameter uncer-tainties when precipitation uncertainty is added. Furthermore,
Fig. 10. Comparison of posterior parameter distributions of the five LISFLOOD parameterprecipitation uncertainty.
Fig. 11. Development of the uncertainty bounds of the precipitation multiplier up forbounds correspond to 95, 90, and 10 percentile confidence interval.
precipitation uncertainty leads not only to wider parameteruncertainty bounds (Fig. 10). Except for the LZTC, which has a moreample distribution with a small shift in the mean, in comparison to
s obtained with (black outlined bins) and without (grey shaded bins) accounting for
the calibration period (10/01/1993–09/30/1995). Shaded areas in the uncertainty
Fig. 12. Histogram of the average posterior precipitation multiplier distribution forthe calibration period (10/01/1993–09/30/1995).
P. Salamon, L. Feyen / Journal of Hydrology 376 (2009) 428–442 439
the distribution obtained considering only parameter uncertainty,all other PDFs show a significant change in the location and/orshape. This indicates that care has to be taken when relating cali-brated model parameters to invariant properties of the underlyingcatchment and transferring them to ungauged basins as is done inregionalization studies. Although a recent study (Thyer et al., 2009)has shown that posterior parameter distributions considering notonly parameter but also other sources of uncertainty (e.g., precip-itation) are more robust, i.e., they provide a more suitable param-eter set for regionalization, it is likely that in this case hereposterior parameter distributions change when accounting addi-tionally for model structural errors. Furthermore, if posteriorparameter distributions are transferred to an ungauged catchment,the input error model needs to be adapted to the specific catch-ment, in order to produce reliable estimates.
Considering the hydrographs for the calibration and validationperiods shown in Figs. 9 and 13, an improved predictive capabilityby a better embracement of the measured discharges is observable.Note that the hydrograph for the validation period was generatedusing the posterior parameter distributions obtained after the lastassimilation step and employing a precipitation uncertainty char-acterised by the average posterior distribution of the precipitationmultiplier shown in Fig. 12. However, particularly the hydrographfor the validation period still shows some overprediction duringlow flow periods. This is likely caused by the extraction of waterupstream of Borgharen for the summers of 1991 and 1992 (Feyenet al., 2007) and/or other structural errors in the model which havenot been accounted for here. A more detailed analysis of the pre-
Fig. 13. Hydrograph prediction uncertainty for the validation period (10/01/1990–09/30/by dark crosses. The grey shaded area denotes the 95 percentile confidence interval for
dictive uncertainty reveals that accounting for precipitation uncer-tainty improves the predictive distribution as was expected, butthat the overall enhancement in model predictive capability duringvalidation period is relatively small (see Fig. 14 a and b). However,when analysing only flows greater than 200 m3/s a significantimprovement in predictive capability when considering parameterand precipitation uncertainty becomes visible (see Fig. 14c and d).Clearly, accounting for an error in precipitation enhances mostlyhigh flows, whereas the low flow periods are principally unaf-fected. Furthermore, analysing the partial autocorrelation of theparticle filter allowing for parameter and precipitation uncertaintyshows that the correlation for lag 1, though statistically significant,is lower than in the previous case study (see Fig. 8). This suggeststhat the independence assumption of the output error model is notstrongly violated and hence no negative effect on the predictiveuncertainty is expected.
Fig. 11 presents the development of the uncertainty bounds ofthe precipitation multiplier. No consistent bias or seasonalitypattern is noticeable for the precipitation multipliers during thecalibration period and the variance of the posterior distributionsis reduced in comparison to the chosen prior distribution afteronly a few assimilation steps. This is also visible in the histogramof the average posterior distribution illustrated in Fig. 12. Themean of �0.012 of the average posterior distribution indicatesthat the precipitation grids typically correspond to the actualprecipitation. Furthermore, the posterior standard deviation of0.101 of the precipitation multiplier illustrates a significantreduction in precipitation uncertainty. However, as precipitationmultipliers are applied to the entire precipitation catchmentand are conditioned only to the discharge measurements at themost downstream point of the catchment, this does not implythat bias at a smaller scale does not occur as over/underestima-tion of precipitation in smaller areas could be averaged out dueto the catchment size. The results of this case study are in linewith other studies that have analysed precipitation uncertaintyusing rainfall–runoff models (e.g., Kuczera et al., 2006; Vrugtet al., 2008). Nevertheless, a quantitative comparison betweenthese studies in order to attribute the uncertainty reduction toa specific reason is difficult due to the hydrological and meteoro-logical differences of the catchments, the different modelapproaches, as well as if and how other sources of uncertaintyhave been incorporated. In this specific case posterior precipita-tion uncertainty is likely to compensate for the non-considerationof other types of forcing error (e.g., error in temperature) andmodel structural error. In addition, the treatment of precipitationas a spatially distributed variable most likely also leads to asmaller uncertainty then when treating precipitation as an aver-aged value in a lumped model.
1992) accounting for precipitation uncertainty. Observed discharges are representedthe prediction uncertainty resulting from parameter and precipitation uncertainty.
Fig. 14. Predictive QQ plots comparing the results of the residual resampling particle filter accounting for parameter uncertainty only (diamonds) and accounting forparameter and precipitation uncertainty (crosses) for the calibration period (a and c) and the validation period (b and d).
440 P. Salamon, L. Feyen / Journal of Hydrology 376 (2009) 428–442
Summary and conclusions
The application of sequential data assimilation techniques inhydrological models has recently received an increased attentiondue to their ability of explicitly treating various sources of uncer-tainty and thus improving model predictive capabilities. Amongstthe different techniques, the sequential Monte Carlo method, alsoknown as the particle filter, appears to be the most suitable forsequential data assimilation in spatially distributed models. Thisis principally due to two features: (1) its applicability to non-linear,non-Gaussian models which is frequently the case in hydrologicalapplications; (2) as the filter updates particle weights instead ofstate variables directly, it implicitly preserves the spatial variabil-ity of all state variables in contrast to other assimilation tech-
niques. In this work we have investigated the use of a residualresampling particle filter to assess parameter, precipitation, andpredictive uncertainty in the distributed hydrological model LIS-FLOOD for the Meuse catchment upstream of the Borgharen gaug-ing station.
Simulations considering only parameter uncertainty illustratedthat prior parameter distributions converged to well identifiableposterior parameter PDFs providing a reasonable agreementbetween the observed and simulated discharges. However,comparing the results with parameter distributions obtained fromthe global optimization algorithm SCEM-UA for the same catch-ment and identical input data illustrated that posterior parameterPDFs are significantly different. This is most likely due to thedifferent assumptions made in the error model for the likelihood
P. Salamon, L. Feyen / Journal of Hydrology 376 (2009) 428–442 441
function of the two approaches. Yet, as both approaches providesimilar fits to the observed hydrographs, these findings suggestthat the hypothesis of equifinality is not only valid for simple,lumped conceptual models but also for relatively complex,partly-physically based, spatially distributed models. Further-more, a detailed analysis of the predictive distributions showedthat both approaches underestimate the predictive uncertaintysignificantly, illustrating that accounting for parameter uncer-tainty only during calibration is insufficient to properly quantifypredictive uncertainty.
In a second case study the impact of precipitation uncertaintyon parameter estimation was investigated. Comparison of the pos-terior parameter PDFs with the case where no forcing data errorwas considered showed a clear difference in the characteristics ofthe posterior distributions. The parameter distributions not onlyillustrate an increased variance but also a change in location andshape, indicating the strong effect of precipitation uncertainty onparameter estimates. Quantification of the predictive uncertainty,especially concerning high flow periods, was enhanced whenaccounting additionally for an error in precipitation. However, re-sults also illustrated that predictive uncertainty was still not fullyquantified indicating the importance of model structural error,even when employing a distributed, partly-physically based hydro-logical model. An evaluation of the precipitation multiplier showedthat in general precipitation grids correspond well with the actualprecipitation and that precipitation uncertainty can be quantifiedand reduced using the herein presented particle filtering approach.
In this work we illustrated that the particle filter is easily appli-cable to a distributed model and can explicitly account for differentsources of uncertainty. However, there are a few important topicsthat require further research. Firstly, the specification of the threemain sources of uncertainty in hydrologic models, i.e., forcing data,parameter, and model structural uncertainty, is generally difficultto address as the magnitude and statistical properties can vary sig-nificantly based on, for example, specific catchment characteristics,the type of measurement used to obtain data, or the spatiotempo-ral frequency at which forcing data is available. Hence, further re-search is needed to evaluate these effects on the different sourcesof uncertainty. Furthermore, we have assumed that streamflowmeasurement error is well specified using a Gaussian distribution.This assumption might not hold in practice and more detailed re-search on the properties of streamflow measurement error andsuitable likelihood functions to describe this error is desirable.
Acknowledgments
The hydrological and meteorological data for this study havebeen provided by the RIZA (Arnhem), Rijkswaterstaat DirectieLimburg (Maastricht), Dienst Hydrologisch Onderzoek (Brussels),DIREN (Nancy), KMI (Brussels) and KNMI (De Bilt). We also wishto acknowledge Stefano Venturini, Mauro del Medico, Rutger Dan-kers, and Konrad Bogner of the Land Management and Natural Haz-ards Unit at the Joint Research Centre for their input to this paper.
References
Arulampalam, M.S., Maskell, S., Gordon, N., Clapp, T., 2002. A tutorial on particlefilters for online nonlinear/non-Gaussian Bayesian tracking. IEEE Trans. SignalProces. 50 (2), 1740188.
Bates, B.C., Campbell, E.P., 2001. A Markov chain Monte Carlo scheme for parameterestimation and inference in conceptual rainfall–runoff modeling. Water Resour.Res. 37 (4), 937–947.
Beven, K., 2006. A manifesto for the equifinality thesis. J. Hydrol. 320, 18–36.Blöschl, G., Reszler, C., Komma, J., 2008. A spatially distributed flash flood
forecasting model. Environ. Modell. Soft. 23, 464–478.Carpenter, J., Clifford, P., Fearnhead, P., 1999. An improved particle filter for non-
linear problems. IEEE Proc. Radar Son. Nav. 146 (1), 2–7.Clark, M.P., Slater, A.G., 2006. Probabilistic quantitative precipitation estimation in
complex terrain. J. Hydrometeorol. 7 (1), 3–22.
Clark, M.P., Ruppa, D.E., Woods, R.A., Zheng, X., Ibbitt, R.P., Slater, A.G., Schmidt, J.,Uddstrom, M.J., 2008. Hydrological data assimilation with the ensemble Kalmanfilter: use of streamflow observations to update states in a distributedhydrological model. Adv. Water Resour. 31 (10). doi:10.1016/j.advwatres.2008.06.005.
Doucet, A., Freita, N.D., Gordon, N., 2001. Sequential Monte Carlo Methods inPractice. Springer, New York. 581 pp.
European Environment Agency (2000). The European Topic Centre on TerrestrialEnvironment: Corine land cover raster database 2000 – 100 m.
Evensen, G., 1994. Sequential data assimilation with a non-linear quasi-geostrophicmodel using Monte-Carlo methods to forecast error statistics. J. Geophys. Res.99 (C5), 10143–10162.
Evensen, G., 2006. Data assimilation: The Ensemble Kalman Filter. Springer. 280 pp.Feyen, L., Kalas, M., Vrugt, J.A., 2008. Semi-distributed parameter optimization and
uncertainty assessment for large-scale streamflow simulation using globaloptimization. Hydrol. Sci. 53 (2), 293–308.
Feyen, L., Vrugt, J.A., Nuallain, B.O., van der Knijf, J., de Roo, A., 2007. Parameteroptimisation and uncertainty assessment for large-scale streamflow simulationwith the LISFLOOD model. J. Hydrol. 332 (3–4), 276–289.
Gordon, N., Salmond, D., Smith, A.F.M., 1993. Novel approach to nonlinear and non-Gaussian Bayesian state estimation. Proc. Inst. Electr. Eng., Part F 140, 107–113.
Hendricks Franssen, H.J., Kinzelbach, W., 2008. Real-time groundwater flowmodeling with the Ensemble Kalman Filter: joint estimation of states andparameters and the filter inbreeding problem. Water Resour. Res. 44 (9),W01419. doi:10.1029/2007WR006097.
Karssenberg, D., 2002. The value of environmental modelling languages for buildingdistributed hydrological models. Hydrol. Processes 16, 2751–2766.
Kavetski, D., Kuczera, G., Franks, S.W., 2002. Confronting input uncertainty inenvironmental modelling. In: Duan, Q. et al. (Eds.), Calibration of WatershedModels, Water Science and Application, vol. 6. AGU, Washington DC, pp. 49–68.
Kavetski, D., Kuczera, G., Franks, S.W., 2006a. Bayesian analysis of input uncertaintyin hydrological modelling: 1. Theory. Water Resour. Res. 42, W03407.doi:10.1029/2005WR004368.
Kavetski, D., Kuczera, G., Franks, S.W., 2006b. Bayesian analysis of input uncertaintyin hydrological modelling: 1. Application. Water Resour. Res. 42, W03408.doi:10.1029/2005WR004376.
Kim, S., Tachikawa, Y., Takara, K., 2007. Recursive updating of distributed statevariables using observed discharges through Kalman filter. Annu. J. Hydraul.Eng. JSCE 51, 67–72.
King, D., Daroussin, J., Tavernier, R., 1994. Development of a soil geographicaldatabase from the soil map of the European communities. Catena 21, 37–56.
Kitagawa, G., 1996. Monte Carlo filter and smoother for non-Gaussian nonlinearstate space models. J Computat. Graph. Stat. 5 (1), 1–25.
Kitanidis, P.K., Bras, R.L., 1980a. Real-time forecasting with a conceptual hydrologicmodel: 1. Analysis of uncertainty. Water Resour. Res. 16 (6), 1025–1033.
Kitanidis, P.K., Bras, R.L., 1980b. Real-time forecasting with a conceptual hydrologicmodel: 2. Application and results. Water Resour. Res. 16 (6), 1034–1044.
Kuczera, G., Kavetski, D., Franks, S., Thyer, M., 2006. Towards a Bayesian total erroranalysis of conceptual rainfall–runoff models: characterising model error usingstorm-dependent parameters. J. Hydrol. 331, 161–177.
Laio, F., Tamea, S., 2007. Verification tools for probabilistic forecasts of continuoushydrological variables. Hydrol. Earth Syst. Sci. 11 (4), 1267–1277.
Lanza, L.G., 2000. A conditional simulation model of intermittent rain fields. Hydrol.Earth Syst. Sci. 4, 173–183.
Liu, J.S., Chen, R., 1998. Sequential Monte Carlo methods for dynamic systems. J. Am.Stat. Assoc. 93 (443), 1032–1044.
Liu, Y., Gupta, H.V., 2007. Uncertainty in hydrologic modelling: toward anintegrated data assimilation framework. Water Resour. Res. 43, W07401.doi:10.1029/2006WR005756.
McLaughlin, D., 2002. An integrated approach to hydrologic data assimilation:interpolation, smoothing, and filtering. Adv. Water Resour. 25, 1275–1286.
Moradkhani, H., Sorooshian, S., Gupta, H.V., Houser, P.R., 2005a. Dual state-parameter estimation of hydrological models using ensemble Kalman filter.Adv. Water Resour. 28, 135–147.
Moradkhani, H., Hsu, K.-L., Gupta, H., Sorooshian, S., 2005b. Uncertainty assessmentof hydrologic model states and parameters: sequential data assimilation usingthe particle filter. Water Resour. Res. 41, W05012. doi:10.1029/2004WR003604.
Nielsen, O.M., 2007. Pypar – Parallel Programming with Python (Version 1.9.3).<http://datamining.anu.edu.au/~ole/pypar/>.
R Development Core Team, 2008. R: A Language and Environment for StatisticalComputing. R Found For Stat. Comput, Vienna, Austria.
Reichle, R., McLaughlin, D., Entekhabi, D., 2002a. Hydrologic data assimilation withthe ensemble Kalman filter. Mon. Weather Rev. 130 (1), 103–114.
Reichle, R., McLaughlin, D., Entekhabi, D., 2002b. Extended versus ensemble Kalmanfiltering for land data assimilation. J. Hydrometeorol. 3 (6), 728–740.
Rijks, D., Terres, J.M., Vossen, P. (Eds.), 1998. Agrometeorological Applications forRegional Crop Monitoring and Production Assessment, Tech. Rep. EUR 17735EN, Eur. Comm. Joint Res. Cent., Ispra, Italy.
Rubin, D.B., 1988. Using the SIR algorithm to simulate posterior distributions. In:Bernardo, J.M. et al. (Eds.), Bayesian Statistics. Oxford Univ. Press, New York, pp.395–402.
Seo, D.J., Koren, V., Calina, N., 2003. Real-time variational assimilation of hydrologicand hydrometeorological data into operational hydrologic forecasting. J.Hydrometeorol. 4 (3), 627–641.
442 P. Salamon, L. Feyen / Journal of Hydrology 376 (2009) 428–442
Smith, M.B., Seo, D.-J., Koren, V.I., Reed, S.M., Zhang, Z., Duan, Q., Moreda, F., Cong, S.,2004. The distributed model intercomparison project (DMIP): motivation andexperiment design. J. Hydrol. 298 (1–4), 4–26.
Smith, P.J., Beven, K.J., Tawn, J.A., 2008. Detection of structural inadequacy inprocess-based hydrological models: a particle filtering approach. Water Resour.Res. 44, W01410. doi:10.1029/2006WR005205.
Sorooshian, S., Dracup, J.A., 1980. Stochastic parameter estimation procedures forhydrologic rainfall–runoff models: correlated and heteroscedastic error cases.Water Resour. Res. 16, 430–442.
Sorooshian, S., Duan, Q., Gupta, V.K., 1993. Calibration of rainfall–runoff models:application of global optimization to the Sacramento soil moisture accountingmodel. Water Resour. Res. 29, 1185–1194.
Thiemann, M., Trosset, M., Gupta, H., Sorooshian, S., 2001. Bayesian recursiveparameter estimation for hydrologic models. Water Resour. Res. 37 (10), 2521–2535.
Thyer, M., Renard, B., Kavetski, D., Kuczera, G., Franks, S.W., Srikanthan, S., 2009.Critical evaluation of parameter consistency and predictive uncertainty inhydrological modelling: a case study using Bayesian total error analysis. WaterResour. Res. 45, W00B14. doi:10.1029/2008WR006825.
Van der Knijff, J.M., Younis, J., de Roo, A.D.P., 2008. LISFLOOD: a GIS-baseddistributed model for river-basin scale water balance and flood simulation. Int.J. Geogr. Inf. Sci.. doi:10.1080/13658810802549154.
Vrugt, J.A., Bouten, W., Sorooshian, S., 2003. A shuffled complex evolutionmetropolis algorithm for optimization and uncertainty assessment ofhydrologic model parameters. Water Resour. Res. 39 (8), 1201. doi:10.1029/2002WR001642.
Vrugt, J.A., Diks, C.G.H., Gupta, H.V., Bouten, W., Verstraten, J.M., 2005. Improvedtreatment of uncertainty in hydrologic modeling: combining the strengths ofglobal optimization and data assimilation. Water Resour. Res. 41, W01017.doi:10.1029/2004WR003059.
Vrugt, J.A., Gupta, H.V., Nuallain, B.O., Bouten, W., 2006. Real-time data assimilationfor operational ensemble streamflow forecasting. J. Hydrometeorol. 7 (3), 548–565. doi:10.1175/JHM504.1.
Vrugt, J.A., ter Braak, C.J.F., Clark, M.P., Hyman, J.M., Robinson, B.A., 2008. Treatmentof input uncertainty in hydrologic modeling: doing hydrology backwards withMarkov Chain Monte Carlo simulation. Water Resour. Res. 44, W00B09.doi:10.1029/2007WR006720.
Weerts, A.H., El Serafy, G.Y.H., 2006. Particle filtering and ensemble Kalman filteringfor state updating with hydrological conceptual rainfall–runoff models. WaterResour. Res. 42, W09403. doi:10.1029/2005WR004093.
Wösten, J.H.M., Lilly, A., Nemes, A., Le Bas, C., 1999. Development and use of adatabase of hydraulic properties of European soils. Geoderma 90 (3–4), 169–185.
Zhao, R.J., Liu, X.R., 1995. The Xinanjiang model. In: Singh, V.P. (Ed.), ComputerModels of Watershed Hydrology. Water Resources Publications, Littleton,Colorado, pp. 215–232.
