An evaluation of the nonlinear/non-gaussian filters for the sequential data assimilation

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1 Available online at Remote Sensing of Environment 112 (2008) An evaluation of the nonlinear/non-gaussian filters for the sequential data assimilation Xujun Han, Xin Li Cold and Arid Regions Environmental and Engineering Research Institute, Chinese Academy of Sciences, Lanzhou, Gansu , P.R. China Received 4 September 2006; received in revised form 13 June 2007; accepted 8 July 2007 Abstract This paper aims to investigate several new nonlinear/non-gaussian filters in the context of the sequential data assimilation. The unscented Kalman filter (UKF), the ensemble Kalman filter (EnKF), the sampling importance resampling particle filter (SIR-PF) and the unscented particle filter (UPF) are described in the state-space model framework in the Bayesian filtering background. We first evaluated those methods with a simple highly nonlinear Lorenz model and a scalar nonlinear non-gaussian model to investigate the filter stability and the error sensitivity, and then their abilities in the one-dimensional estimation of the soil moisture content with the synthetic microwave brightness temperature assimilation experiment in the land surface model VIC-3L. All the results are compared with the EnKF. The advantages and disadvantages of each filter are discussed. The results in the Lorenz model showed that the particle filters are suitable for the large measurement interval assimilation and that the Kalman filters were suitable for the frequent measurement assimilation as well as small measurement uncertainties. The EnKF also showed its feasibility for the non-gaussian noise. The performance of the SIR-PF was actually not as good as that of the UKF or the EnKF regarding a very small observation noise level compared with the uncertainties in the system. In the one-dimensional brightness temperature assimilation experiment, the UKF, the EnKF and the SIR-PF all proved to be flexible and reliable nonlinear filter algorithms for the low dimensional sequential land data assimilation application. For the high dimensional land surface system that takes the horizontal error correlations into account, the UKF is restricted by its computational demand in the covariance propagation; we must use the EnKF, the SIR-PF and other covariance reduction algorithms. The large computational cost prevents the UPF from being applied in practice Elsevier Inc. All rights reserved. Keywords: Bayesian filtering; Nonlinear/non-Gaussian; Sequential data assimilation; Kalman filter; Particle filter; Lorenz model; Monte Carlo methods; Land surface model; Microwave remote sensing 1. Introduction Data assimilation requires the estimation of the states of a dynamic system as a sequence of noisy observations becomes available. Bayesian methods provide a rigorous general framework for the dynamic state estimation problems (Gordon et al., 1993). There is a great deal of research in sequential data assimilation to employ various filtering methods. In recent years, several Bayesian filtering algorithms have been proposed and successfully applied to the nonlinear and non-gaussian processes. The purpose of this paper is to evaluate these new methods in the sequential data assimilation applications. Corresponding author. address: hanxj@lzb.ac.cn (X. Han). Bayesian filtering is optimal in a sense that it seeks the posterior distribution which integrates and uses all of the available information expressed by the probabilities. The Kalman filter and its variants have dominated this field for decades, but they are limited by the linear Gaussian assumption. The best-known algorithm to solve the problem of the nonlinear filtering is the extended Kalman filter (EKF) (Anderson & Moore, 1979). Because the calculation of the error covariance in the EKF for large dimensional systems is computationally demanding, some computational efficient forms of the EKF for complex models, such as the Reduced Rank Square Root (RRSQRT) Kalman filter (Verlaan, 1998; Verlaan & Heemink, 2001) and the Singular EvolutiveExtendedKalman(SEEK)filter(Pham et al., 1998), are available. The RRSQRT and the SEEK filters are based on an eigendecomposition of the error covariance matrix. They simplify the inversion of the matrices by operating with a lower /$ - see front matter 2007 Elsevier Inc. All rights reserved. doi: /j.rse

2 X. Han, X. Li / Remote Sensing of Environment 112 (2008) triangular square root of the covariance matrix. The Singular Evolutive Interpolated Kalman (SEIK) filter (Pham, 2001) which has been proposed as an improved version of the SEEK, and the ensemble Kalman filter (EnKF) (Burgers et al., 1998; Evensen, 1994) are ensemble based methods of the Kalman filter family. Verlaan and Heemink (2001) have compared the performances of those filters and pointed out that the EnKF is the most accurate one. For weakly nonlinear models the RRSQRT Kalman filter is to be preferred; for strongly nonlinear models the EnKF is a better choice. All the above filters are the approximation of the optimal Bayesian solution, including the particle filters that will be introduced below. The EnKF is often used as an alternative option to the ensemble based methods of the Kalman filters when the model is nonlinear. The Kalman filters hold the assumption that the Probability Density Function (PDF) is Gaussian. Although they work well for some types of the nonlinear systems, they may provide a poor performance in some cases when the true posterior is non-gaussian (Evensen, 1997), because in highly nonlinear systems, the initial Gaussian PDF can lead to a potential for non-gaussian PDF even during a short timescale of the state evolution (Bengtsson et al., 2003; Evensen & van Leeuwen, 2000; Miller et al., 1999). The EKF uses the first order terms of the Taylor series expansion of the nonlinear functions only. If the models are highly nonlinear, EKF will introduce large errors in the estimated statistics of the posterior distributions of the states and usually it is hard to find the linearization of the nonlinear model. Recently, the unscented Kalman filter (UKF) (Julier & Uhlmann, 1997), particle filter also known as the sequential Monte Carlo methods (Gordon et al., 1993), and their variants as well, have been introduced into other areas besides electric engineering. The unscented Kalman filter founds on the intuition that it is easier to approximate a probability distribution than to approximate an arbitrary nonlinear function or transformation. Compared with the first-order accuracy of EKF, the estimation accuracy of UKF is improved to the third-order for Gaussian data, and at least second-order for non-gaussian data (Julier & Uhlmann, 2004). The UKF can be considered as a generalization of the RRSQRT Kalman filter. The RRSQRT Kalman filter just uses the leading eigenvectors of the covariance matrix, where it is less than or equal to the dimension of the state; while the UKF uses all the eigenvectors. This reduction may speed up the calculations. Gove and Hollinger (2006) used the UKF in a simple physiological model for the simultaneous state and parameter estimation. Frolov et al. (2006) has applied the UKF to the coastal data assimilation problems successfully. Take that into consideration, we intend to evaluate the UKF as an alternative assimilation method of the EKF. The particle filters use a number of particles to represent the PDF of the system states. So it is an efficient solution to deal with the non-gaussian PDF because it tracks the full posterior state space. With the help of the newly emerging nonlinear/non-gaussian filters, including the UKF and the variants of the particle filter, this paper focuses on the nonlinear filtering of the system states. The UKF and the particle filter methods find extensive applications in nonlinear systems. Several works have paid close attention to the application of the particle filter in the nonlinear system data assimilation (Kivman, 2003; Pham, 2001; Van Leeuwen, 2003; Xiong et al., 2006). We will evaluate the performances of those new filters in two kinds of experiment in different respects within the nonlinear model under the different model, observation uncertainty assumptions and the assimilation frequency in the hope of finding their potential use in the land data assimilation problems. This paper is organized as follows. In the next section, we introduce the state-space model in order to formulate a dynamic system. Section 3 briefly outlines the Bayesian filtering algorithms. Those approaches include the unscented Kalman filter, the ensemble Kalman filter, the sampling importance resampling particle filter and the unscented particle filter; in Section 4 we evaluate those methods with a simple highly nonlinear Lorenz model and a scalar nonlinear non-gaussian model to investigate the filter stability and the error sensitivity. Then in Section 5 we examine those filters' performances in the one-dimensional land surface data assimilation applications with the microwave radiative transfer model. Section 6 presents the discussions. We summarize the paper, conclude the discussions and specify the future research directions in the final section. 2. Dynamic state-space model The state-space approach provides a general framework for describing the dynamic state estimation problems. Thus we use the state-space formulation to introduce the filtering algorithms. The evolution of discrete-time dynamic systems can be expressed as follows: x k ¼ f ðx k 1 ; v k 1 Þ y k ¼ hðx k ; n k Þ: Where x k a R n x and x k 1 a R n x denote the state vector of the system at time k and k 1 with initial PDF p(x 0 ) respectively, which evolves over time as a first order Markov process according to the conditional PDF p(x k x k 1 ). The observation y k a R n y is conditionally independent given x k and the observation Eq. (2) is represented by the PDF p(y k x k ) which is often named as likelihood, n x,n y are the dimensions of the state and the observation vectors respectively. The vectors v k 1 a R n v, is an i.i.d (independent identically distributed) system noise sequence and n k a R n n is an i.i.d observation noise sequence, n v,n n are the dimensions of the system noise and the observation noise vectors respectively. The mappings f : R n x R n v YR n x and h : R n x R n n YR n y represent the system and the observation models respectively. In this paper, we only consider the system and the observation models with the additive noise. 3. Bayesian filtering Before we proceed, we briefly introduce the notation used in this paper: the Gaussian distribution is denoted by N(u,Σ) with mean u and covariance matrix Σ, xˆ k and xˆ k k 1 represent the filtered state and the predicted state of x k respectively; P xk and ð1þ ð2þ

3 1436 X. Han, X. Li / Remote Sensing of Environment 112 (2008) P xk k 1 are the filtered forecast error covariance and the predicted forecast error covariance respectively; P yk is the error covariance matrix of the observation predictions and P xyk is the cross covariance matrix between the state and the observation predictions; K k is the Kalman gain matrix; R v and R n are the error covariance matrixes of the system and the observation respectively. The essence of the state estimation problem in the Bayesian filtering is to construct the posterior PDF p(x k y k ) of the state based on all the available information (Gordon et al., 1993). The posterior PDF can be calculated in two steps theoretically: prediction and update. In the prediction step, we integrate the state PDF from the precious state using the system model. The update operation modifies the prediction PDF making use of the latest observation. The Bayesian filtering is aimed to apply the Bayesian statistics and the Bayes rule to the probabilistic problems. In a Bayesian framework, the posterior filtering density p(x k y k ) of the state given the observations y k constitutes the complete solution to the sequential data assimilation problem, and allows us to calculate any optimal" (depend on the specific definition of the optimality) estimate of the state, such as the conditional mean: Z ˆx k ¼ E½x k jy k Š¼ x k pðx k jy k Þdx k : ð3þ The optimal method to recursively update the posterior density as new observations arrive is given by the recursive Bayesian estimation algorithm. Given the initial PDF p(x 0 ), transition PDF p(x k x k 1 ), and likelihood PDF p(y k x k ), the objective of the filtering is to estimate the optimal state at time k given the observations up to time k. From Bayesian perspective, when an observation y k becomes available at time k, we can obtain this posterior PDF p(x k y k ) via Bayes rule: pðx k jy k Þ¼ pðy kjx k Þpðx k jy k 1 Þ pðy k jy k 1 Þ where: Z pðx k jy k 1 Þ¼ pðx k jx k 1 Þpðx k 1 jy k 1 Þdx k 1 Z pðy k jy k 1 Þ¼ pðy k jx k Þpðx k jy k 1 Þdx k : The posterior PDF provides a complete solution to the filtering problems and the Kalman filter provides an optimal solution to the linear Gaussian system. However, the real-world is nonlinear in general; the multi-dimensional integrals are intractable and some approximate methods must be used. We will describe how, when the analytic solution is intractable, these approximations of the optimal Bayesian solution can be derived in the nonlinear filtering algorithms. In this paper, we just want to evaluate the available practical algorithms. For this reason no new algorithms are developed; we use the algorithms described by Van der Merwe et al. (2000), Van der Merwe (2004) and Arulampalam et al. (2002) for describing the unscented Kalman filter and the particle filters. ð4þ ð5þ ð6þ 3.1. Unscented Kalman filter (UKF) The unscented Kalman filter was developed to address the deficiencies of linearization of the EKF by providing a more direct and explicit mechanism for transforming the mean and covariance information (Julier & Uhlmann, 2004). The UKF is founded on the intuition that it is easier to approximate a probability distribution than it is to approximate an arbitrary nonlinear function or transformation (Uhlmann, 1994). Van der Merwe (2004) gives a detailed description as follows: We start with introducing the deterministic sampling approach named unscented transformation. The unscented transformation (UT) is a method for calculating the statistics of a random variable which undergoes a nonlinear transformation. The UKF is implemented by a scaled version of UT. The scaled UT is a generalizing extension of the UT. Let n x be the dimension of state vector x with mean x and covariance P x ;we consider the propagation of x through an arbitrary nonlinear function: y ¼ gðxþ: For calculating the mean and the covariance of y using scaled UT, the operation process is as follows: a set of 2n x +1 points χ and associated weights S={χ i,ω i ;i=0,..., 2n x } are deterministically drawn so that they completely capture the true mean and covariance of the state vector x. The selection scheme is: v 0 ¼ x pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi v i ¼ x þð ðn x þ kþp x i ¼ 1; N ; n x pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi v i ¼ x ð ðn x þ kþp x i ¼ n x þ 1; N ; 2n x Þ i Þ i x ðmþ 0 ¼ k n x þ k i ¼ 0 x ðcþ 0 ¼ k n x þ k þð1 a2 þ bþ i ¼ 0 ðmþ xi ¼ x ðcþ i ð7þ 1 ¼ 2ðn x þ kþ i ¼ 1; N ; 2n x where λ=α 2 (n x +κ) n x is the scaling parameter. Those points are called the sigma-points. The different superscripts (m) and (c) of the weights are used to calculate the mean and covariance respectively. The complete scaled unscented transformation is given as the following: 1. Choose the parameters k, α and β. Choose k(k 0) guarantees positive semi-definiteness of the covariance matrix, a common default is k=0. The α (0 α 1) controls the spread of the sigma points and should ideally be a small number to avoid non-local effects when the nonlinearities are strong. β is a non-negative weight which can be used to incorporate knowledge of the higher order moments of the distribution. For a Gaussian prior the optimal choice is β=2. 2. Calculate the set of 2n x +1 sigma-points and weights. 3. Propagate each sigma point through the nonlinear transformation: ð8þ Y i ¼ gðv i Þ i ¼ 0; N ; 2n x ð9þ

4 X. Han, X. Li / Remote Sensing of Environment 112 (2008) Calculate the mean, the covariance of y and the crosscovariance: ȳ X2n x i¼0 P y X2n x i¼0 P xy X2n x i¼0 x ðmþ i Y i x ðcþ i ðy i ȳþðy i ȳþ T x ðcþ i ðv i xþðy i ȳþ T ð10þ Base on the scaled UT, the so-called unscented Kalman filter with the additive noise has been derived as follows (Julier & Uhlmann, 2004; Van der Merwe, 2004): While calculating the sigma-points according to the above selection scheme, the new state vector χ is augmented with these sigma-points: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi v k 1 ¼½ˆx k 1 ˆx k 1 þ ðn x þ kþp xk 1 ˆx k 1 ðn x þ kþp xk 1 Š a. Prediction step: v kjk 1 ¼ f ðv k 1; v k 1 Þ ˆv kjk 1 ¼ X2n x x ðmþ i v i;kjk 1 i¼0 P xkjk 1 ¼ X2n x i¼0 x ðcþ i ðv i;kjk 1 ˆx kjk 1 Þðv i;kjk 1 ˆx kjk 1 Þ T þ R v b. Update step: q v kjk 1 ¼½ ˆx kjk 1 ˆx kjk 1 þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðn x þ kþp xkjk 1 Y kjk 1 ¼ hðv kjk 1 ; n k Þ ŷ kjk 1 ¼ X2n x P yk ¼ X2n x i¼0 P xyk ¼ X2n x i¼0 i¼0 x ðmþ i Y i;kjk 1 q ˆx kjk 1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðn x þ kþp xkjk 1 x ðcþ i ðy i;kjk 1 ŷ kjk 1 ÞðY i;kjk 1 ŷ kjk 1 Þ T K k ¼ P xyk ½P yk þ R n Š 1 x ðcþ i ðv i;kjk 1 ˆx kjk 1 ÞðY i;kjk 1 ŷ kjk 1 Þ T ˆx k ¼ ˆx kjk 1 þ K k ðy k ŷ kjk 1 Þ P xk ¼ P xkjk 1 þ K k P yk K T k The advantage of the UKF over the EKF is its derivative-free nonlinear estimation; there is no need of calculation of Jacobians and Hessians (Julier & Uhlmann, 2004), and it is as efficient as the EKF. Unlike the linearization operation of EKF, the UKF uses the nonlinear models directly; it captures the posterior mean and covariance accurately to the third-order for the Gaussian distribution and at least second-order for the non-gaussian distribution, but both the EKF and the UKF use the Gaussian approximation. In our paper, for the numerical stability we use the square-root version of the UKF (SR-UKF) introduced by Van Š der Merwe (2004); it propagates and updates the square-root of the state covariance directly in Cholesky factored form, using the following linear algebra techniques, QR decomposition, Cholesky factor updating and the efficient pivot-based least squares. The SR-UKF has equal estimation accuracy and efficiency compared to UKF, but more robust Ensemble Kalman filter (EnKF) The native formulation of the ensemble Kalman filter was first introduced by Evensen (1994) and Burgers et al. (1998). Numerous applications involved in this ensemble filter method and its other versions have demonstrated that it is an efficient and computationally acceptable strategy for the high dimensional sequential data assimilation. In the prediction step, an ensemble of the state vectors replicates are generated and propagated through Eq. (1) in parallel, each state vector representing a particular realization of the model states. Then the forecast error covariance can be computed from those ensembles implicitly. During the update step, the EnKF updates each ensemble replicate separately, using the observation and the state forecast error covariance matrix. Finally the EnKF state estimation is given by the means of the ensemble replicates. We also present the schematic representation of the EnKF by two steps: a. Prediction step: w x i ¼ f ðx i k 1 ; vi k 1 Þ kjk 1 P xkjk 1 ¼ AðAÞT N 1 A ¼½x w kjk 1 x kjk 1; N ; w N xkjk 1 x k Š x kjk 1 ¼ 1 N X N b. Update step: i w x i ŷk i ¼ i hðxw kjk 1; n k Þ ȳ k ¼ 1 N P yk X N i ¼ SðSÞT N 1 ŷ i k kjk 1 S ¼½ŷ 1 k ȳ k ; N ; ŷ N k ȳ k Š P xyk ¼ AðSÞT N 1 K k ¼ P xyk ½P yk þ R n Š 1 x i k ¼ xw i kjk 1 þ K K ðy i k ŷi k Þ P xk ¼ EðEÞT N 1 E ¼½x 1 k ˆx k ; N ; x N k ˆx k Š ˆx k ¼ 1 N X N x i k i¼1

5 1438 X. Han, X. Li / Remote Sensing of Environment 112 (2008) where N is the number of ensembles, ˆx i kjk 1 is the ith ensemble i member forecasted at time k and x k 1 is the ith updated ensemble member at time k 1; ŷ i k is the ith ensemble member of observation predictions. Researches have revealed that the Gaussian assumption didn't prevent the EnKF from giving the reasonable results for non-gaussian ensemble (Zhou et al., 2006). The dependence of its performance on the Monte Carlo sampling techniques can make the EnKF computationally expensive and it requires a number of ensemble members for higher estimate accuracy. The basis of the UKF is deterministic sampling; it utilizes a small set of sample points to get hold of the higher order information of the mean and the covariance. That is the essential difference between the UKF and the EnKF. But the UKF is hindered outside of the large dimensional applications because of the high storage and high computational cost in covariance propagation Particle filter (SIR-PF) The Particle filter, also known as the sequential Monte Carlo method is a kind of recursive Bayesian filter based on the Monte Carlo simulation. It is based on the sequential importance sampling (SIS) algorithm; the key idea is to represent the posterior PDF by a set of random drawn samples with associated weights. The estimates are computed based on those samples and their weights. With large numbers of samples, this Monte Carlo characterization becomes an equivalent representation to the usual functional description of the posterior PDF, and the SIS filter approaches the optimal Bayesian estimation. See Arulampalam et al. (2002) for overall explanations. Compared with the Kalman filters, the particle filter has no matrixes operations in itself; it is explicitly derived from the Bayesian theory. To avoid intractable integrations in the Bayesian filtering, the posterior PDF at time can be approximated as: pðx k jy k Þ XN i¼1 x i k dðx k x i k Þ ð11þ where x i k is assumed to be an i.i.d state sample drawn from the posterior PDF p(x k y k )withassociatedweightsx i k,andδ denotes the P Dirac delta function. The weights are normalized so that N i¼1 xi k ¼ 1 and are chosen by using the principle of the importance sampling. This principle relies on the following process: since it is usually impossible to sample from the true posterior PDF, an alternative is to sample from a proposal distribution (also called importance density) denoted by q(x k y k ). Then the weights can be defined as: x i k ~ pðxi k jy kþ qðx i k jy kþ ð12þ A common problem with the SIS filter is the weight degeneracy or the sample impoverishment, where after several iterations, only very few particles have non-zero importance weights. This degeneracy implies that large computational effort is devoted to updating particles whose contributions to the approximation of the p(x k y k ) are practically negligible. An operative measure for the degeneracy is the effective sample size N eff introduced by Bergman (1999) and defined as: N eff ¼ N 1 þ Varðx i k Þ ð14þ where x i k ¼ pðxi k jy kþ=qðx i k jxi k 1 ; y kþ. But in practice, the true N eff is difficult to evaluate, and an approximate N w eff of N eff can be obtained by: N w eff ¼ 1 P N i¼1 ð f x i k Þ2 ð15þ where f x i k is the normalized weight. The small Nw eff indicates a severe degeneracy problem. This effective sample size definition is used in the next step to judge whether to do resampling or not. A straightforward approach to reduce the degeneracy phenomenon is to increase the number of particles. That is often impractical for the corresponding computational demand and the variance increasing. There are two practical methods to alleviate this problem: (1) a good choice of proposal distribution and (2) resampling the particles. The optimal proposal distribution is pðx k jx i k 1 ; y kþ (Arulampalam et al., 2002). In general, it is not straightforward to incorporate the observation information into this proposal. The basic idea of resampling is to duplicate the particles with high normalized weights and discard the particles with a low normalized weight while keeping the number of the particles not changing. There are many resampling algorithms, such as the residual resampling (Liu & Chen, 1998), the stratified resampling (Kitagawa, 1996), the deterministic resampling (Kitagawa, 1996) and the systematic resampling (Carpenter et al., 1999); the resampling method we applied is the residual resampling which is one computationally cheaper algorithm. The residual resampling procedure involves the following steps: (a) set ˆN i ¼ tn f x i kb. N denotes the original number of particles, and the operator takes the integer part of its argument; (b) perform a sampling importance resampling procedure to select the remaining N P k ¼ N P N i¼1 tnf x i kb samples with new weights f x i k ¼ N P 1 k ð f x i k N ˆN i Þ. f x i k denotes the normalized weight. For convenience, the transition prior PDF is the most popular choice of the proposal distribution: In a sequential case, we could define the recursive weight update equation: x i k ¼ xi k 1 pðy k jx i k Þpðxi k jxi k 1 Þ qðx i k jxi k 1 ; y : ð13þ kþ qðx k jx i k 1 ; y kþ¼pðx k jx i k 1 Þ By substitute Eq. (13) into Eq. (12), we obtain: x i k ¼ xi k 1 pðy kjx i k Þ: ð16þ ð17þ

6 X. Han, X. Li / Remote Sensing of Environment 112 (2008) For this choice of the proposal distribution, the generic particle filter is also known as the sampling importance resampling (SIR) particle filter (SIR-PF). A schematic representation is: a. Prediction step: (1) Initialization: Draw sample of x i 0 from the initial PDF p(x 0) (2) Prediction: Draw sample of x i k from the transition PDF pðx k j x i k 1 Þ b. Update step: (3) Evaluate the importance weights x i k ¼ xi k 1 pðy k j x i k Þ (4) Normalize the importance weights f x i k ¼ xi k = PN x j k j¼1 (5) Calculate the effective particle set size N w eff with Eq. (15) (6) Resampling: if N w eff is smaller than the resample threshold, resample the particles according to their weights f x i k, to obtain N samples approximately distributed according to p(x k y k ). (7) For i=1,...,n, reset x i k ¼ f x i k ¼ 1=N (8) Calculate the filtered state estimation: ˆx k 1 X N x i k N. i¼1 There is no specific assumption about the model characteristic (e.g. linear or nonlinear) and the probability distribution (Gaussian or non-gaussian) of the noise in the particle filter, so it can handle any form of the PDF even if it is multimodal in practice. The prediction step for the SIR-PF is to propagate each particle through the system model; this step is the same as the EnKF. The calculation of associated weights for predicted particles is the core of the update step Unscented particle filter (UPF) The transition prior distribution was used as the proposal distribution in the SIR-PF because of its simplicity. But this kind of proposal distribution sometimes results in a high uncertainty because the most recent observation y k is neglected in the proposal distribution pðx k j x i k 1Þ. When the likelihood is peaked by comparing with the prior distribution, this problem will become more serious. We will demonstrate it in the experiment. Van der Merwe (2004) proposed a new particle filter named the unscented particle filter. In this new version particle filter, the UKF is utilized for the proposal generation that not only incorporates the latest observation, but also generates proposals which tend to overlap with the true posterior more consistently. Employing the UKF to approximate the mean and the covariance of the proposal distribution for each particle is the central idea of the UPF. The new proposal distribution involves the information of the latest observation and then we can draw new particles from this distribution to represent the proposal distribution density. Those particles fall into the overlap field of the prior and the likelihood as much as possible. The UPF has been successfully applied in the object tracking field. The intrinsic difference between the SIR-PF and the UPF is the proposal distribution. a. Prediction step: (1) Initialization: Draw sample of x i 0 from p(x 0) (2) Update each particle with the UKF: Draw sample of x i k from the proposal distribution q N ðx i k j y kþ¼nð P xk i; Pi k Þ, where P xk i; Pi k are calculated from the UKF for each particle with the help of algorithms introduced in Section 3.1 b. Update step: (3) Evaluate the importance weights x i k ¼ pðy k j x i xi k Þpðxi k j xi k 1 Þ k 1 q N ðx i k j y kþ (4) Normalize the importance weights f x i k ¼ xi k = PN x j k j¼1 (5) Calculate the effective particle set size N w eff with Eq. (15) (6) Resampling: if N w eff is smaller than the resample threshold, resample the particles according to their weights f x i k,to obtain N samples approximately distributed according to p(x k y k ). (7) For i=1,...,n, reset x i k ¼ f x i k ¼ 1=N (8) Calculate the filtered state estimation: ˆx k 1 X N x i k N. i¼1 In this section we have introduced all algorithms that we used in this paper; their performances will be demonstrated in the following sections. Firstly, the filter stability and the error sensitivity are examined in the simple Lorenz model. Secondly, we compare them in a scalar nonlinear non-gaussian state estimation problem, and lastly we establish a synthetic microwave brightness temperature assimilation experiment with the land surface model and the microwave radiative transfer model. 4. Numerical experiments with Lorenz model and a simple scalar model For easy comparison, we first evaluate those filters in experiments A and B with the simple Lorenz model which is an extensively used testbed for the nonlinear data assimilation algorithms because of its high nonlinear characteristics (e.g. initial value sensitivity); this feature is very helpful to show the necessity of importing sequential data assimilation for model predictions. After this we carry out an extra experiment C in a nonlinear non-gaussian problem to recognize the differences among the algorithms more clearly Experiment setup in Lorenz model The well known Lorenz model defined by three differential equations (Lorenz, 1963): dx ¼ rðy xþ dt dy ¼ qx y xz dt dz ¼ xy bz dt ð18þ where σ is the Prandtl number, ρ is a normalized Rayleigh number and β is a non-dimensional wavenumber. We chose the commonly used values for the parameters in the equation:σ =10, ρ=28,β=8/3. This system of equations is integrated by the fourth order Runge Kutta method, with an integration step We constructed a reference solution for t [0,40] with the initial conditions (x 0,y 0,z 0 )=( , , ). So there

7 1440 X. Han, X. Li / Remote Sensing of Environment 112 (2008) are 4000 steps in the solution space. The background solution was computed by adding a noise with the distribution N(0,2) to the initial conditions of the reference solution. The covariance of the initial conditions was assumed to be the same as the observation error covariance. The observations were generated by adding normal distributed noise to the reference solution. The model error covariance is assumed to be (2.0,12.13,12.31) for this three state variables system which is the same as what Evensen and pffiffiffiffiffi van Leeuwen (2000) used. The white model noise is scaled by Dt to decrease the aggregation of the state error variance (Evensen & van Leeuwen, 2000; Evensen, 2003). The experiment design is very similar to that of Miller et al. (1999) and Evensen and van Leeuwen (2000), but we used many different observation and model error covariances to examine the filter stability and error sensitivities in the next two experiments. The parameters of the UKF were set to be α=1, β=2, and κ=0. The ensemble (or particle) numbers of the EnKF, the SIR-PF and the UPF were set to 200; for our experiments, we found that 200 particles were enough to obtain good results. Increasing the number of particles additionally hardly made any sense. In addition, the resample threshold for particle filters was chosen as 1. In the first experiment, we want to compare the performances of the four filters by assimilating the observations on the x coordinate in two different observation intervals and observe the effects of the data assimilation in other directions in the high nonlinear circumstances; in the second experiment, we analyze the filter stability and the error sensitivity with changed error statistics. We select two criterions for comparing the filter performances: the RMSE (Root Mean Square Error) and the error standard deviations (STD) of the state estimation, because the RMSE is very commonly used in literatures and it facilitates the quantitative comparison. In order to reduce the influences that the random error brings, 100 independent simulations with different random generator seeds were carried out to calculate the mean value of the RMSE and the temporal average of the error STD of the state estimation. This procedure is very necessary for comparing different Monte Carlo based filters' performances to lessen the random noise influences Experiment A: a vertical view of the filter performances in the sequential data assimilation In the data assimilation system, we are not only interested in the estimation accuracy of the assimilated state which can be Table 1 Table of the algorithms, the mean of the RMSE and the error standard deviations for experiment A calculated over 100 independent runs with Δt obs =0.1 Algorithm x y z RMSE S.D. RMSE S.D. RMSE S.D. Unscented Kalman filter (UKF) Ensemble Kalman filter (EnKF) Particle filter (SIR-PF) Unscented particle filter (UPF) Table 2 Table of the algorithms, the mean of the RMSE and the error standard deviations for experiment A calculated over 100 independent runs with Δt obs =0.25 Algorithm x y Z RMSE S.D. RMSE S.D. RMSE S.D. Unscented Kalman filter (UKF) Ensemble Kalman filter (EnKF) Particle filter (SIR-PF) Unscented particle filter (UPF) measured in reality, but also hope for the estimate improvements of the related state estimation which can't be measured in the same system. This can be done in the data assimilation framework with the help of the system error covariance and the system model intrinsic mechanisms. We take the observation error covariance as the same diagonal matrix (2, 2, 2). Firstly, we set the distance between the observations as Δt obs =0.1 (i.e., 1 observation every 10 model steps). Secondly, we change the observation interval into Δt obs =0.25 (i.e., 1 observation every 25 model steps) to study the performances of those filter with infrequent observations. Table 1 illustrates the results obtained from the experiments described above in the short observation interval case. From the view point of the RMSE and the STD deviation, all filters provide similar results in all three variables, and the EnKF is superior to the SIR-PF and the UKF in this condition. The UPF is better than all the others. The STD of the EnKF is the largest one in all three directions. The resampling step reduces the STD of the particle filters. Table 2 illustrates the results as the distance increases. It is clear that the performances of all filters decrease both in the RMSE and the STD. As the above results demonstrate, the SIR- PF and the UPF hold the better performances. The UKF is somewhat inferior to the EnKF in both the RMSE and the error STD, which is different from what we see in Table 1. However, the performance difference between the UPF and the SIR-PF is very small. The RMSE of the background solution is (11.419,13.122,11.368). The advantages of the data assimilation for the state estimation with those filters are proved. Those two tables merely show the mean RMSE value and the mean error STD; it is well known that the random sampling method makes the results of the filters fluctuated. So we have calculated the variance of the RMSE in 100 realizations from the above two cases; the results showed that the filters on the basis of the Monte Carlo sampling possessed big variances in the RMSE value generally Experiment B: a horizontal view of the filter stability and the error sensitivity The filter stability is an important aspect from the theory to the practice as well as the error sensitivity. We have indicated some results in the preceding experiments with the mean value of the RMSE, the error STD and the variance of the RMSE in all

8 X. Han, X. Li / Remote Sensing of Environment 112 (2008) Table 3 Table of the algorithms, the mean of the RMSE and the error standard deviations for experiment B with different measurement error covariances and Δt obs =0.1 calculated over 100 independent runs Algorithm RMSE S.D. RMSE S.D. Unscented Kalman filter (UKF) Ensemble Kalman filter (EnKF) Particle filter (SIR-PF) Unscented particle filter (UPF) Table 5 Table of the Algorithms, the mean of the RMSE and the error standard deviations for experiment B with different model error covariances and Δt obs =0.1 calculated over 100 independent runs Algorithm 4 8 RMSE S.D. RMSE S.D. Unscented Kalman filter (UKF) Ensemble Kalman filter (EnKF) Particle filter (SIR-PF) Unscented particle filter (UPF) three directions of the Lorenz system. The explorations of the relation between filter stability and the error statistics are considered as follows: Observation error sensitivity: In this section, we want to evaluate the filter performances in different model and observation error covariance matrix conditions, as the appropriate state estimation of these filters relies on the specific error statistics information. Firstly, we use two different small observation error variances of 0.2 and which are smaller than the system error variance 2.0. This will make the distribution of the observation noise very peaked; the likelihood of the state vectors may happen to lie in one of the tails of the prior distribution. Sampling from the prior does not generate enough particles in the overlapping region of the prior distribution and the likelihood distribution, i.e. samples drawn from the prior distribution do not cover the likelihood region. This will cause the particle degeneracy in the standard particle filter SIR-PF which uses the prior as the proposal PDF. The performance of the SIR-PF will be very poor since the contributions of the most particles are insignificant. It is necessary to choose a better proposal distribution which incorporates the latest observation when the narrow likelihood appears. This requirement results in the development of the UPF, in which the new proposal has the ability to move the particles towards the highly likelihood regions. We can see this effect from the following results clearly. The results are shown in Tables 3 and 4. These changes have improved all the filters estimation results, especially as far as the UKF and the EnKF are concerned, because of the more precise observations; and the results demonstrate the degeneracy phenomenon in the SIR-PF and the advantage of the UPF over the SIR-PF. When the observation noise variance is 0.2, the behaviors of particle filters in opposition to Kalman filters will be similar to experiment A. The particle filters behave well in the long observation interval and the Kalman filters do well in another place. But when the observation uncertainty is reduced about one hundred times in the case of variance 0.002, it is evident that the SIR-PF with the prior proposal performs very poorly. By contrast, the UPF with a better proposal (i.e. UKF) obtained better results. The Kalman filters retain stable performances. The most worthy being paid attention to is that the EnKF is the best one in 10 model steps frequency assimilation, while we can find that the UPF works better than the SIR-PF and the Kalman filter in the bigger assimilation interval where the UPF holds the first-rate result. Nevertheless, the particle filters are unable to assimilate the dense accurate observation information to improve the state estimation, as that is shown in Table 3. Model error sensitivity: We have noticed the observation error covariance impacts on the filter performance significantly; we will examine the impact of the different model error covariances next. Tables 5 and 6 summarize the results of the changed model error variance value of 4 and 8 on the x coordinate respectively. These results showed that the impacts on the performances of uncertain model errors are not distinguishable when being compared with the original small model error variance 2. The EnKF and the UPF hold the best results of the short observation interval and the big observation interval respectively, and the variations in the RMSE and the error STD are the same as the preceding results Experiment C: a nonlinear non-gaussian state estimation application In this section we compare these new filters in a simple scalar nonlinear system with the gamma process noise and the Gaussian observation noise. Van der Merwe (2004) has used this model to validate the UKF, the SIR-PF and the UPF. Now we add the EnKF comparison. This additional test could be helpful to improve our understanding of the limitations of the Table 4 Table of the algorithms, the mean of the RMSE and the error standard deviations for experiment B with different measurement error covariances and Δt obs =0.25 calculated over 100 independent runs Algorithm RMSE S.D. RMSE S.D. Unscented Kalman filter (UKF) Ensemble Kalman filter (EnKF) Particle filter (SIR-PF) Unscented particle filter (UPF) Table 6 Table of the algorithms, the mean of the RMSE and the error standard deviations for experiment B with different model error covariances and Δt obs =0.25 calculated over 100 independent runs Algorithm 4 8 RMSE S.D. RMSE S.D. Unscented Kalman filter (UKF) Ensemble Kalman filter (EnKF) Particle filter (SIR-PF) Unscented particle filter (UPF)

9 1442 X. Han, X. Li / Remote Sensing of Environment 112 (2008) Table 7 Table of the algorithms, the mean of the RMSE and the error standard deviations for experiment C calculated over 100 independent runs Algorithm RMSE S.D. Unscented Kalman filter (UKF) Ensemble Kalman filter (EnKF) Particle filter (SIR-PF) Unscented particle filter (UPF) SIR-PF and show the feasibility of the EnKF under the non- Gaussian noise. The system model is characterized as the following equation: x k ¼ / 1 x k 1 þ 1 þ sinðxkðk 1ÞÞ þ v k ð19þ where v k is a gamma random variable with mean 1.5 and variance 0.75, and ω=4e 2 and ϕ 1 =0.5. The observation model is: y k ¼ / 2x 2 k þ n k kv30 / 3 x k 2 þ n k kn30 ð20þ where ϕ 2 =0.2 and ϕ 3 =0.5. n k is a Gaussian distribution random variable with mean 0.0 and variance 1e 5. We want to estimate the state sequence x k from assimilating the noisy observations y k in t [0,100]. The initial value of x k was set to 1.0. The parameters values for the UKF were set to be α=1, β = 1, and κ = 2. All of the filters used 200 ensembles (or particles). Table 7 summarized the performances of the different filters. It shows the mean values of the RMSE and the temporal averaged error standard deviation values of the state estimations in 100 independent runs. Because of the small observation noise variance, the likelihood of the observations will be very narrow, which in turn caused the sample impoverishment problem in the SIR-PF. The performance of the UPF is very evident. The EnKF also shows its applicability for this non-gaussian state estimation because of its ensemble representation of the posterior distribution. At present, we have evaluated the performances of all the four filters in the simple high nonlinear Lorenz model and the scalar state estimation problem. The different behaviors are shown. It should be noted that these results may not be very general indications. Different system models have totally different driving factors that will affect the whole performances of the filters. The Lorenz model may not be an ideal carrier to identify the filters' performances in the operational sequential data assimilation. So it will be very useful to examine these filters in a practical data assimilation system; therefore we include these algorithms into a hydrologic data assimilation system. 5. Hydrologic data assimilation The soil moisture plays an important role in the water resources management. The microwave remote sensing provides a practicable tool to retrieve the soil moisture information from the space. As a result of the uncertainties in the model and Table 8 Table of the algorithms, the mean of the RMSE and the error standard deviations for the brightness temperature assimilation at 12 h assimilation frequency calculated over 10 independent runs Algorithm 1st layer 2nd layer 3rd layer RMSE S.D. RMSE S.D. RMSE S.D. Unscented Kalman filter (UKF) Ensemble Kalman filter (EnKF) Particle filter (SIR-PF) the uncertainties in the remote sensing data, it will be useful to combine those two data sources for improving the understanding of the soil moisture spatial distributions. To assess the performances of those filters more practically, we conducted a one-dimensional direct radiance assimilation experiment with the synthetically generated microwave brightness temperature observation in order to improve the near surface soil moisture estimation from the contaminated" model output VIC-3L land surface model and synthetic experiments The model we used is the VIC-3L model, which is a macroscale land surface model (Liang et al., 1994) and is used as the substitute for the system model Eq. (1). It represents the surface and the subsurface hydrologic processes on a grid cell basis, and jointly solves the energy and water budgets at the land surface. Typically, the grid cell ranges from 1/8 to 2. VIC- 3L characterizes the sub-grid scale variation approximately by partitioning the grid cell areas into different vegetation classes. There are three soil layers in the soil column which we denote as layers 1, 2, and 3. The true depth and the composition of the soil column is usually imperfectly known; we set the first layer to a depth of 5 cm for this study to match the microwave penetration depth in the near surface soil moisture estimation. The study area is the basin of the Model Parameter Estimation Experiment (MOPEX) project (Duan et al., 2006); we just regard this basin as one model grid to carry out this onedimensional experiment. The forcing data collected to drive the land surface model are the precipitation, the air temperature, the incoming shortwave radiation, the incoming long-wave radiation, the atmospheric density, the atmospheric pressure, the vapor pressure and the wind speed. The soil type is the loam and Table 9 Table of the algorithms, the mean of the RMSE and the error standard deviations for the brightness temperature assimilation at 24 h assimilation frequency calculated over 10 independent runs Algorithm 1st layer 2nd layer 3rd layer RMSE S.D. RMSE S.D. RMSE S.D. Unscented Kalman filter (UKF) Ensemble Kalman filter (EnKF) Particle filter (SIR-PF)

10 X. Han, X. Li / Remote Sensing of Environment 112 (2008) Fig. 1. Assimilation results of the soil moisture at the second layer (upper) and the third layer (lower) with an assimilation frequency of Dt obs =12 hours. the vegetation type is the mixed forest of 6 types in this study area. All the data of the soil and the vegetation are derived from MOPEX project. The calibrated VIC-3L parameters originate from Wang (2006). The synthetic experiment covers a four-month period from 1st May 1998 to 31st August The model simulation was performed at 1 h time step, and the spin-up period was chosen as the time section from 1st May 1998 to 30th June We relate the brightness temperature observation to the land surface soil moisture content with a microwave radiative transfer model named Q/H model (Choudhury et al., 1979; Wang & Choudhury, 1981), which is the substitute for the observation model Eq. (2). It is a commonly used semiempirical model that describes the rough surface emission as a function of the surface roughness and the dielectric properties as: R e H ¼ 1 e H ¼½Q r V þð1 QÞ r H Š H R e V ¼ 1 e V ¼½Q r H þð1 QÞ r V Š H : ð21þ Where H and V denote the horizontal polarization and the vertical polarization respectively. R e H and Re v are the reflectivities of the rough surface. ε H and ε V are the emissivities of the rough surface. r H and r V are the Fresnel reflectivities. Q and H are the surface roughness parameters, and are often with the values between 0 and 1 to define the surface roughness effect on the reflectivity (Shi et al., 2005). In our experiment, Q and H had the values of 0.1 and 0.95 respectively. The clay and the sand percentage were and 41 respectively; these values are derived from the North American Land Data Assimilation Systems (NLDAS) project. The initial soil moisture state values of all three layers were set to 0.2; it can be shown that a two month spin-up period would dispel the influences of the misspecification of the initial conditions. We first simulate the VIC-3L model with the original forcing to acquire the truth of the three layers soil moisture content and the surface temperature data. The synthetic brightness temperature observations of 6.9 GHz and 10.7 GHz, both vertical and horizontal polarizations, are generated through the first layer soil moisture content and the surface temperature with the same soil parameters, vegetation parameters, forcing data and initial states. There are four observations at each assimilation step consequently, and a zero mean Gaussian distributed random error is added to the brightness temperature values with the following assumed observation error covariance. Because the soil moisture is mostly sensitive to the precipitation, we attributed the input uncertainties to the precipitation and perturbed it by adding 50% magnitude. Then the background solution is defined by the soil moisture content from running the VIC-3L with this perturbed precipitation. Those perturbed precipitation data are also used as the forcing in the assimilation step. The model error and the observation error statistics would determine the performance of the data assimilation algorithm, but they are very hard to know explicitly in reality. Because of the evaluation objective, we assume that we have known those errors in advance. The model error covariance matrix is assumed to be a diagonal matrix (0.0001, , ), according to the higher variability of the model prediction of the soil moisture content in the upper layers. The observation error covariance matrix used in assimilating the brightness temperature is assumed as: Varð6:9VÞ Varð6:9HÞ Varð10:7VÞ Varð10:7HÞ ¼