Ensemble Kalman filter versus particle filter for a physically-based coupled surface–subsurface model
Highlights
► We study uncertainty propagation for a model of surface/subsurface flow interactions. ► We compare Monte Carlo Data Assimilation schemes for our nonlinear/non-Gaussian model. ► SIR-particle filter and EnKF improve accuracy of posterior PDF using small ensembles. ► SIR outperforms EnKF when measurements of streamflow alone are assimilated. ► As opposed to EnKF, SIR updates maintain physical consistency of model dynamics.
Introduction
It is widely recognized that problems in catchment hydrology and water resource management involve strong interactions between surface water and groundwater [1]. A number of physically based hydrological models that incorporate some representation of groundwater-surface water interactions have been recently developed (e.g., [2], [3], [4], [5], [6], [7], [8], [9], [10], [11]). The objective of these models is to accurately reproduce a number of hydrological processes, such as rainfall partitioning between runoff and infiltration, soil moisture redistribution, and groundwater recharge. This is achieved by solving various equations for the description of surface runoff and variably saturated flow in porous media. The uncertainties associated to calibrated model parameters, initial conditions, and boundary conditions induce model errors that can propagate in time, yielding final predictions drastically different from reality. For this reason, a suitable approach to the problem of simulating hydrological processes is to consider them as realizations of a stochastic process, searching for the most probable one or even for the full probability distribution of the physical state of the system. Field observations provide important but noisy information of the true system state. These measurements are nowadays often available, at least in the form of discharge at the outlet. In this framework, data assimilation (DA) methods are techniques that try to combine these measurements with the simulation model of the system in order to improve accuracy and, most importantly, to quantify uncertainties (i.e., the probability distribution of the errors) of model predictions [12], [13].
Most of the DA techniques are based on the filtering theory [14], in which the problem unknown is the evolution in time of the probability density function (pdf) of the system state conditioned to all the previous observations. This pdf is called filtering pdf and is the solution of a recursive formula that consists of two steps: (i) the forecast step, in which the pdf is propagated in time using the model dynamic equations; (ii) the analysis step, in which the forecast is updated by Bayes theorem using the newly acquired observations. This approach can be addressed analytically only in few simple cases. For example, if the model is linear with Gaussian additive noise, filtering pdfs are Gaussian and the first two moments (mean and covariance) completely characterize the density function. The widely known Kalman filter [15] provides in this case the full solution of the filtering problem. Several approaches are available for dealing with nonlinear dynamics or observation relationships and we refer the reader to [16] for a general review.
One of the most common nonlinear filters is the ensemble Kalman filter (EnKF), first introduced by Evensen [17] in the context of ocean modeling. Nowadays EnKF is widely used in different fields of application, including hydrological modeling, mainly because of its ease of implementation and computational efficiency. EnKF uses a Monte Carlo approach to evaluate the state mean and covariance matrix and performs an update step based on the Kalman filter. This method has been applied extensively to hydrological models, including, e.g., the one-dimensional Richards equation [18], three-dimensional saturated groundwater flow [19] and transport [20], integral-balance saturated–unsaturated subsurface models [21], and conceptual rainfall-runoff models [22], [23], [24], [25].
Although EnKF has given accurate results in many applications, a number of issues remain unresolved. An interesting question to ask is what is the influence of the Gaussian approximation, inherent in any KF-based method, when the filtering density function is not Gaussian. Another interesting and partially unanswered question is related to the optimality of the scheme when using KF-based algorithms for nonlinear processes [26], [27]. Several modifications to the EnKF and alternative DA techniques have been presented in the literature to handle non-Gaussian distributions. For example, Sun et al. [28] compares four different deterministic ensemble filters for the estimation of the hydraulic conductivity, while Zhou et al. [29] and Schöniger et al. [30] propose two versions of the EnKF with transformed data, in which ensemble realizations are transformed into a Gaussian distribution before the update. Instead of looking at variations of EnKF that address non-Gaussianity, in this paper we focus on the sequential importance resampling (SIR) particle filter, a DA method developed specifically to avoid the Gaussian assumption [31], [32]. Particle filters (PF) are very flexible and easy to implement, an appealing characteristic that fostered their use in many research areas, such as object recognition, target tracking, signal processing, financial analysis, and robotics. For a comprehensive description of the principal PF methods see, for instance, Arulampalam et al. [33] and Doucet et al. [34].
The basic idea of PF is to use directly the Bayesian formula for the computation of the filtering pdf at each analysis step. The resulting update is similar to the static GLUE (generalized likelihood uncertainty estimation [35]) and Bootstrap filters [36]. In SIR the state pdf is approximated via Monte Carlo by associating to each realization (also called particle) appropriate weights, which are updated using the likelihood function of the assimilated observations. A resampling step is used to propagate through the model dynamic equations only the particles that correspond to non-negligible weights. The weights can be thought of as a measure of the “nearness” of each particle to the true system state. Theoretically, it can be shown that, without any assumption on the relevant pdfs, the empirical probability distribution associated to SIR converges to the filtering density when the ensemble size tends to infinity [37].
PF methods were used to address parameter uncertainty and state estimation in conceptual hydrological models [38], [39] and in a coupled hydrogeophysical model applied to an infiltration experiment [40]. Noh et al. [41] introduced a lagged regularized PF for a process-based distributed hydrological model, while van Delft et al. [42] combined PF and EnKF to solve a flood forecasting problem. PF had been also used to account for initial condition uncertainties in the framework of ensemble streamflow prediction [43], to assimilate remote sensing derived water levels in a one-dimensional hydraulic model [44], and to assimilate satellite soil moisture measurements in a one-dimensional mechanistic soil water model [45].
The purpose of this work is to present an experimental comparison between EnKF and SIR in the context of a detailed process-based hydrological modeling framework. Comparison between these two methods are already present in the literature related to hydrological applications, but are mostly restricted to conceptual models. Weerts and El Serafy [23] compared the EnKF and PF in a conceptual rainfall-runoff model, finding that, for their application, EnKF outperforms SIR for low flows. A comparison on a more complex land surface model was done by Zhou et al. [27], demonstrating that EnKF can well approximate the optimal filtering pdf also in case of non-normal moisture behavior. Dechant and Moradkhani [46] concluded that, using a large number of realizations, SIR is better than EnKF when assimilating microwave radiance data, improving predictions of the snow water equivalent and operational streamflow forecast.
Following the study presented in Camporese et al. [47], we want to assess the relative performance of EnKF and SIR as implemented in a coupled model of surface–subsurface water interactions. A further objective of the paper is to focus on implementation issues related to the assessment of the physical, statistical, and numerical consistency of the state variables within the model dynamics during the entire assimilation process, topics that rarely have been considered in previous data assimilation studies, at least in hydrological models. Finally, we want to discuss the advantages and drawbacks of EnKF and SIR in hydrological applications.
The paper is organized as follows. The main aspects of the filtering theory are first summarized to highlight the theoretical differences between SIR and EnKF. Then we describe how these methods are combined with our hydrological model, emphasizing the most correct way to initialize and propagate the ensemble of state vectors in order to obtain physically consistent realizations and to improve the accuracy of the system state estimation. The behavior and performance of the filters are then analyzed in a series of numerical experiments involving a three-dimensional synthetic v-shaped catchment for which a rainfall-drainage simulation is carried out. The two methods are evaluated in terms of their ability to retrieve the correct watershed response and compared for different assimilation frequencies, different scenarios of state variable observations (streamflow only, pressure head only, both), and for biased initial conditions or biased atmospheric forcing.
Section snippets
Problem statement
Let and represent the vectors of the true system state variables and of the observations at a given assimilation time t, respectively. Given an initial state , the state and the observations satisfy two generic equations of the form:where is the discrete transient function arising from the numerical discretization of the physical model and is the observation function relating the true state to the
Model setup
A series of numerical simulations were conducted on a synthetic three-dimensional test case. We considered a tilted 3 m deep v-catchment with a surface area of 1.62 km2, characterized by a homogeneous and isotropic saturated hydraulic conductivity field, with no-flow conditions at the bottom and at the lateral boundaries. Table 1 summarizes the model discretization and parameter values used for the numerical experiments.
We would like to stress that the main objective of this research is to
Summary and conclusions
We applied the ensemble Kalman filter (EnKF) and sequential importance resampling (SIR) data assimilation methods to a distributed physically based hydrological model that couples surface and subsurface flow. Both schemes allow to sequentially incorporate field measurements, such as pressure head, soil moisture, and/or streamflow, into the nonlinear model dynamics. As EnKF and SIR are based on the Monte Carlo approach, they are suitable for considering different sources of errors in the model,
Acknowledgements
We would like to thank the anonymous reviewers for their careful revisions and useful suggestions that helped improving the quality of the manuscript. The support of the European Commission (Grant FP7-ENV-2009-1-244151), the CARIPARO Foundation (Grant “NPDE-Nonlinear Partial Differential Equations: models, analysis, and control-theoretic problems”), the “Ing. Aldo Gini” Foundation and the University of Padua (Grant STPD08RWBY) is gratefully acknowledged.
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