Iteration and SUT-based Variational Filter

Transcription

1 Iteration and SUT-ased Variational Filter Ming Lei 1, Zhongliang Jing 1, and Christophe Baehr 2 1. School of Aeronautics and Astronautics, Shanghai Jiaotong University, Shanghai 2. Météo-France/CNRS, Toulouse mlei@sjtu.edu.cn, zljing@sjtu.edu.cn, christophe.aehr@math.univ-toulouse.fr Astract An iterative method ased on the concept of variational optimality and Ensemle Transform ET) as well as Scaled Unscented Transform SUT), called Iteration and SUT-ased Variational Filter ISVF), is introduced for nonlinear and high dimensionality dynamics. Using the SUT Kalman filter SUKF) [1], the ISVF suggests a novel correction scheme for estimation of ensemle mean and corresponding ensemle covariance, which incorporates a variational minimization as well as a ET-lie covariance update into the ordinary correction. Moreover for dealing with the dynamics with high dimensionality, the Truncated Singular Value Decomposition TSVD) is applied to generate a size-reduced set of sigma points. Finally numerical experiments are performed on Lorenz-95 model for efficiency validating. I. INTRODUCTION Data assimilation or equivalently estimation) practices have to face 3 prolems [2]: a) The nonlinearities of the dynamics; ) The high dimensions of the systems computational cost/speed or complexity); c) The non-gaussianity of the states proaility distriutions. It is well nown that the standard Kalman filter KF) is a linear optimal estimator and derived from a linear uniased minimum variance LUMV) criterion, suits for linear dynamics with Gaussian assumption oth for the dynamical and measurement noises. So far, variants of KF such as the extended Kalman filter EKF), the scaled unscented transform Kalman filter SUKF) [1], and the ensemle Kalman filter EnKF) [3], [5], were proposed to deal with nonlinear prolems. They explore different approaches to implement the moment propagation of the nonlinear dynamics. However there is almost no attempt to improve the update correction. Essentially the linear correction proposed originally y R.E. Kalman, is still preserved in the previous nonlinear filters. The SUKF as well as the EnKF is a data assimilation method which attempts to tacle some aspects of the aove prolems. Essentially, this is Monte Carlo implementation of the KF [4], with a set of ensemle memers as representation of the dynamical system. With a typically designed ensemle size, the filter can run efficiently. Moreover, y propagating forward the ensemle memers y the governing equations of a dynamical system and evaluating the statistics of state ased on the propagated ensemle memers, they can also ypass the prolem of nonlinearity requiring no linearization of the nonlinear system lie does for the EKF. It is usual to implicitly) assume that, oth the dynamical and oservation noise, and system states approximately) follow some symmetric distriutions not necessarily Gaussian). In practice, this assumption may not e always true. Due to the simplicity of their implementation and their acceptale accuracy, the KF-lie filterskf, EKF, EnKF and SUKF) are popular in the data assimilation community. The correction step of the KF-lie filters is a linear regression see II-A for details). This regression may e critical for strong nonlinear systems and could e disastrous for complex and high dimensional prolems. Therefore we are oliged to developed a new assimilation algorithm with a correction ased on variational minimization instead of the Kalman linear regression. Consequently the linear regression is replaced y quadratic term of oservation. Moreover, the SUT in SUKF is applied for the memers generation and the statistics mapping, an Ensemle Transform ET) scheme inspired y the Ensemle Transform KF ETKF) for the error covariance update. To deal with high dimensional systems, we incorporate the technique of Truncated Singular Value Decomposition TSVD) eside the SUT to have a variale-size ensemle fit to the most important directions. Thining aout a nonlinear dynamics with a smooth/perfect-model, in term of Taylor expansion, we want to transfer the variational optimal oject into a local KF-lie iteration. This can e initialized y the propagated acground, and generally compared with KF-lie variants a good computational accuracy is achieved. The most contents aove are reflected in [11], [12], however as a explicit form for optimisation implementation, we suggest here an iterative approach called Iteration and SUTased Variational Filter ISVF), which is validated y a nonlinear/highe dimensional system Lorenz-95 [13], the simulation results presented in section V are compared with a similar method [2], a modified scheme characterized y the Ensemle Kalman Filter EnKF) and the unscented transform. The wor is divided into 6 parts: In section II we riefly review the conceptions of several classical statistical corrections including the one used in our filter; Then the ISVF is depicted in section III and y the way outline the SUKF; In section IV some implementation issues involved in ISVF are discussed; Finally numerical experiments are descried with a discussion and conclusions given respectively in section V and VI. II. BACKGROUND The prolem of data assimilation, equivalently a filtering prolem, is to estimate the proaility law, η = pdfx y 1: ). Since we have the transition model M 1, in 5), the estimation could e sequential and therefore can e considered as a Marov process conditioned to the oservations.

2 Therefore there are two proaility pdfs for the system description: one is the prediction pdf, η = pdfx xa 1 ), which uses the Marov transition spanned y the dynamics; the other is the update pdf, η a = pdfxa x, y 1:), which implements the correction via the latest availale oservation. Recursively finding these two pdfs solves the assimilation/filtering prolem. Different approaches were explored to fulfil the prediction pdf η to nonlinear prolems, for instance the EKF, the EnKF and the SUKF as well as their variants [1]. However there is almost no attempt to improve the update pdf η. A linear regression is always acting in the previous nonlinear filters to deal with nonlinear systems, unfortunately it is not always true especially for the case of strong nonlinear complex systems. To introduce the ISVF, first we review three update schemes, then we propose an iterative variational version of the SUKF, with an attempt to reduce the update error of the SUKF for nonlinear/high-dimension systems. A. LUMV-ased linear moment update The Kalman Filter-lie Estimator KFE) is a linear optimal algorithm and suits for linear/gaussian dynamics, its update implementation is a linear regression due to the structure. Some variants of the KFE such as the EKF, the SUKF, as well as the EnKF, proposed to deal nonlinear prolems, use the same linear regression formulas for updating a acground statistics to analysis ones. Particularly, assuming that there exists matrices A R n and B R n m, the analysis x a of true state x R n can e represented as a linear regression of the oservation y R m and generally m n. We call the x a a linear analysis of true state x; the analysis is said to e a linear uniased minimum variance LUMV) analysis if the estimate x a is uniased and its linear estimation mean square error is minimized. By the criterion of LUMV, one can determine the regression coefficients A and B and therefore determine the analysis and corresponding covariance of x a y x a = A + By, = x P xy,p yy,) 1 y ) + P xy, P yy,) 1) y, 1a) P a xx, = P xx, P xy,p yy,) 1 P xy,) T, 1) where x = E[x z], y = E[y z], P xx = E[x x )x x ) T z], P xy = E[x x )y y ) T z], P yy = E[y y )y y ) T z], and assumed that P yy is nonsingular. x and y denote state and oservation with unnown distriution and conditioned on oservations z. B. ET-ased linear moment update As a suoptimal KF-lie estimator, the Ensemle Transform KF ETKF) was suggested for oservation assimilation in [6]. It uses the ensemle transform and normalization to quicly otain the acground error covariance so as to reduce the forecast error variance. In particular, suppose that there are N ensemles sampled in a assimilation cycle using the numerical decomposition algorithm. A n N square root matrix S x, can e otained from the acground error covariance P xx, and such that P xx, = S x, S x, )T. Similarly let S a x, e an n N square root matrix of the analysis error covariance P a xx,. Then the S a x, can e updated from the S x, y a product with a N N transform matrix T [6]: S a x, = S x,t 2) where T is designed to e the orthogonal eigenvector matrix V of H S x, )T R 1 H S x, ) postmultiplied y the inverse square root of the sum of the N N identity matrix I N N and the eigenvalue matrix D, i.e., T = V D + I N N ) 1/2. H is the linearization of oservation function H with respect to x, i.e., H = H / x. More explicitly, T is related to a singular value decomposition as elow H S x,) T R 1 H S x,) = V D V T 3) where D is a diagonal matrix containing the eigenvalues of H S x, )T R 1 H S x, ), and V containing the orthogonal eigenvectors. Thus the analysis error covariance P a xx, is determined y P a xx, = S a x,s a x,) T = S x,t T T S x,) T. 4) Moreover, once x a and Sa x, are determined, the analyzed ensemle {x a +1,i }N i=1 used y the next cycle of assimilation can e generated via x a +1,i = x a + N 1S a x, ) i, i = 1,, N, the S a x, ) i denotes the i-th column of S a x,. After the analyzed ensemle eing generated at the eginning of a new assimilation cycle, it will e propagated forward the dynamics to produce the acground ensemles. C. Variational nonlinear moment update The variational moment update was suggested to apply for nonlinear assimilation [7]. Firstly we formulate the state estimation prolem in elow scenario for a state process x and its oservation process y at time : x = M, 1 x 1 ) + u, y = H x ) + v, 5a) 5) where the transition M, 1 and the oservation H are possily oth nonlinear. The dynamical and oservation noises, denoted y u and v, are uncorrelated and supposed to e zero-mean Gaussian with nown covariance Q for u, and R for v respectively. Based on Eq.5), the total Proaility Density Function PDF) can e formulated as a product of the marginal PDF of the acground P and the oservation P o, i.e., P = P P o = explnp +lnp o ). So a gloal optimal analysis turns to maximize the total PDF, equivalently one formulates the costfunction J y min J = min lnp lnp o ) = minj + J o ). To express the cost-function J at time, we denote the acground and the oservation errors y ε = x x and ε o = y H x ) and the acground and oservation error covariance y B and R. If one supposes that x, y, B and R are nown at any time, J can e regarded as a function of state x and show that J x ) = 1 2 x x ) T B 1 x x )+ 1 2 y H x ) ) T R 1 y H x ) ). 6)

3 Note that due to the Gaussian assumption of ε and εo, there should includes two normalized terms 1 2 ln 2πB ) and 1 2 ln 2πR ) in the right hand side of 6), consider that they will e constants and have no effect on the minimization of 6), so e omitted for simplicity. Finally the original prolem, maximizing total proaility of system at time, is turned to minimize the cost-function J x ) iteratively from a point x x with a nonnegative gradient constraint J x ) = J x )/ x : x a = arg min x J x ), suject to J x ) 0, 7) Explicitly the original prolem of nonlinear update reduces to minimize a quadratic cost-function J x ). There are a numer of efficient nonlinear minimization algorithms, for instance the conjugate-gradient method [9] can e one choice for minimization implementation. Alternatively we can derive an iterative correction formula for introducing the optimization as follows. The cost-function J x ) conditioned on the zero gradient, that is J x ) = 0, implies B 1 x x ) x H x ) ) T R 1 y H x ) ) = 0. 8) Suppose that B, R and x are nown at time. Let xi e an approximation of the analysis x a for the ith iteration and can e initialized y the acground x, moreover the propagation and the oservation operators are smooth enough, the difference etween system states ecomes small during the successive iterations, show that H x i+1 ) = H x i ) + H i x i+1 x i ) + O x i+1 x i 2 ), 9) where H i = x H x =x i, and denotes a norm. By sustituting 9) into 8) and discarding the quadratic terms, we have B 1 xi+1 x )+H i ) T R 1 = H i ) T R 1 Hi x i+1 x i ) y H x i ) ), 10) Then replacing the term x i+1 x y xi+1 x i + xi x in LHS of 10) and after some simple rearrangement yields x i+1 ) 1 Hi = x i + B 1 + H i ) T R 1 H i ) T R 1 + B 1 + H i ) T R 1 Hi y H x i ) ) ) 1 B 1 x x i ). 11) Thus Eq.11) can e rewritten as KF-lie gain and innovation, one can compare elow expression with the Eq.1), x i+1 = x i + K i y H x i ) R H i ) T B 1 xi x ) ), 12) where for simplification of K i and using the argument in [17], if the matrices A and C are positive definite, there exists the identity A 1 + B T C 1 B) 1 B T C 1 = AB T BAB T + C) 1. Thus considering the properties of covariances B and R, we get K i = B H i ) T H i B H i ) T + R ) 1. 13) The Eq.12) is similar to the correction proposed in [18], which adopted an increment to the state x i generated y the previous iteration. A numer of simulations presented in section V show that, for nonlinear dynamics with smooth/perfect-model, the Eq.12) is suitale for eing used directly in local KF-lie iteration. Qualitatively it rings a considerale improvement to the computational accuracy compared with the ordinary variants of KF-lie estimators. To oserve the Eq.6), the quadratic cost-function is completely characterized y the acground and oservation error covariances, B and R. The R is fixed and mostly constant for a deterministic oservation equation. Thus B plays a ey role in the variational filtering. Different strategies to determine B were explored, however as an implementale choice, we set B PX a, for numerical simulations in section III-B. III. ITERATION AND SUT-BASED VARIATIONAL FILTER In this section, we propose the Iteration and SUT-ased Variational Filter ISVF) ased on the concept of SUT [1] and an iterative variation update scheme in Eq.12). 1 To serve as a support of the ISVF explanations, the traditional SUT-ased Kalman filter SUKF) ased on the dynamics 5) is outlined firstly. A. SUT-ased Kalman filter SUKF) Suppose that an analyzed mean x a 1 and covariance P a xx, 1 at time 1 are nown, it is possile to generate a sigma points set {X 1,i a }2L with their associated weights {W 1,i } 2L [10]. Then we propagate the sigma points through the dynamical operator M,+1 and otain an another set of acground ensemles {X,i : X,i = M,+1X 1,i a ), i = 1,, 2L}. Consequently the ensemle acground mean X and covariance PX, are estimated y X = P X, = W 1,i X,i, W 1,i X,i X )X,i X ) T 14a) β α 2 )X,0 X )X,0 X ) T + Q. 14) where the second term in RHS of Eq.14) is introduced further to reduce the approximation error, β = 2 is an optimal choice when x 1 follows a Gaussian distriution [10]. It is customary to first compute the cross covariance PX Y, and the projection covariance P Y, as well as the projection 1 Note that all discussion aout the state estimation prolem in this paper, is confined to nonlinear and symmetric systems. Here y nonlinear and symmetric, we mean that in Eq.5), there are not only the dynamical and oservation noises, u and v respectively, with some symmetric distriutions not necessarily Gaussian), ut also the system state x.

4 mean Y [10] in order to compute the KF gain K, such that Y = P X Y, = P Y, = W 1,i H X,i), W 1,i X,i X) H X,i) Y ) T 15a) β α 2 ) X,0 X) H X,0) Y ) T, 15) W 1,i H X,i) Y) H X,i) Y ) T β α 2 ) H X,0) Y ) H x,0) Y ) T. 15c) Once the latest oservation y is availale, one can update the acground mean and covariance to the analysis ones in the spirit of Eq.1) y K = PX ) Y, P 1 Y, + R, 16a) x a = X + K y H X) ), 16) P a xx, = P X, K P X Y,) T. B. Iteration and SUT-ased variational filter ISVF) 16c) We suppose the analysis mean x a 1 and the square root matrix S a x, 1 of covariance Pa xx, 1 are nown, the sigma points set {X 1,i a }2L can e generated [10] y { x a 1, x a 1 ± α } L + λs a x, 1) i, i = 1, 2,, L, 17) where S a x, 1 ) i denotes the i-th column of a square root matrix S a x, 1 of Pa xx, 1. A set of weights {W 1,i} 2L associated to the previous sigma-points is also allocated y λ W 1,0 = α 2 L + λ) α 2, 18) 1 W 1,i = 2α 2 L + λ), i = 1, 2,, 2L, The choice of parameters α, λ and L is independent of the holding of the aove identities. After generation of the sigma points at time 1, to propagate them pass through the system Eq.5a) and denote the propagated ensemles y { X,i : X,i = M,+1 X a 1,i), i = 0, 1,, 2L }, 19) which can e regarded as an analogy to the acground ensemle in the framewor of the EnKF. We split the ISVF procedures into propagation and filtering steps. 1) Propagation step: The ensemle acground mean X and covariance PX, can e computed similarly with E- qs.14a)14). Since we propose to use the iterative update formulated in Eq.12) to compute the gain K and analyzed estimate x a, explicitly it ecomes unnecessary to compute the projection ensemle mean Y, cross covariance P X Y, and the projection covariance PY, in filtering step. To present the covariance in terms of square root matrice for numerical efficiency, the matrices S x, = P X, )1/2 are introduced and the acground covariance is estimated y X = W 1,i X,i, 20a) PX, =S x,s x,) T + Q. 20) [ S x, = W αβ 1,0 X,0 X) W 1,1 X,1 X),, ] W 1,2L X,2L X), 20c) W αβ 1,0 =W 1, β α 2. 20d) 2) Filtering step: To initialize y the sample mean X, the covariances P X, in terms of square root matrices S x, ) and the oservation noise covariance R, then incorporated the latest oservation y, a variational correction in Eq.12) can e implemented iteratively until the convergence condition Eq.21c) eing satisfied, show that K i =P X,H i ) T[ H i P X,H i ) T + R ] 1, 21a) x i+1 =x i + K i [ y H x i ) R H i ) T P X,) 1 x i X ) ], 21) x i+1 x i C, 21c) S x, =PX,) 1/2, 21d) V D V T =H i+1 S x,) T R 1 Hi+1 S a x, =S x,v [ D + I L L] 1/2, S x,), 21e) 21f) where we denote H i = x H x =x i and similarly for Hi+1. C is a specified threshold for iterative accuracy control. D is a L L matrix defined y a full diagonal matrix formed with all eigenvalues sorted in descending order, i.e., D = diagσ,1 2,, σ2,l ). The L L-matrix V = [e,1,, e,l ] contains the corresponding eigenvectors e,i. L is the dimension of the augmented state [x T, ut ]T. The procedure of ith iteration is stopped once the condition in Eq.21c) is satisfied, then the analysis and its covariance at time are otained y x a = xi+1 and P a xx, = Sa x, Sa x, )T respectively. Then there exist at least 2 questions: 1) How is convergence in terms of inequality 21c)) guaranteed? 2) How many iterations were needed to reach the threshold for example C=0.001 in implementation? that is, the prolem of convergence condition and convergence rate of Eqs.21), the answer involves a lot of derivation and discussion, thus is omitted here. IV. IMPLEMENTATION OF THE ISVF A. Truncated Covariance-ased sigma-points sampling Considering some large-scale prolems such as in meteorological or geophysical sciences, to deal high dimensionality of state such as 10 8 dimensions is one of great challenge. A rute-force algorithm is clearly impractical. Similarly to the wors in [2], [20], we employ a truncated singular value decomposition TSVD) to generate a size-reduced perturation for sigma points reduction. A ran-reduction of the covariance matrix is employed to approximate the original one y using the TSVD [2].

5 A singular value decomposition is applied to P a xx, = V D V ) T, where D = diagσ,1 2,, σ2,l ) is the eigenvalues σ,i 2 s of Pa xx, sorted in descending order. L is the dimension of [x T, ut ]T and V = [e,1,, e,l ] the matrix of eigenvectors. Then an approximation can e made y P a xx, = L σ,ie 2,i e T,i i=1 l i=1 σ 2,ie,i e T,i = V D V ) T, 22) where the truncation size l is a time-variale integer and l < L. The V and D are uild with a reduced dimension, L l and l l, respectively. An efficient scheme for l choice such that l l l l u is suggested in [2], here l l and l u denote a lower and an upper specified ound. Thus aiming to compromise the computational cost and accuracy, when the truncation size l is determined at time, the L L-sized S a x, 1 in 17) should e replaced y L l -sized {Ŝ a x, 1 ) i σ 1,i e 1,i, i = 1,, l } and the constant L in 18) should e replaced y l, respectively. B. The choice of λ When the ISVF is implemented, an issue aout the positive semi-definiteness of the covariance matrices deserves a special attention. Normally, we require l +λ > 0, the weights W 1,i are positive for i > 0. Nevertheless, the weight W 1,0 can e positive if λ l 1 α 2 )/α 2. If it is lie this, when computing the acground covariances according to Eqs.15)15c), the positive semi-definiteness may not e guaranteed. However, one may notice that, in Eq.20c), the effective weight of the term X,0 X αβ is actually W 1,0 = W 1, β α 2 β > 0). By this way, in order to guarantee the positive semi-definiteness, we choose the parameters λ and β properly to satisfy that W αβ 1,0 > 0 and l + λ > 0. Given W 1,0 = λ/[α 2 l + λ)] + 1 1, we get at the end α2 λ = 1 α4 + α β) α α 2 β) l. 23) Therefore since l is ounded within [l l, l u ], setting λ > l l [ 1 α 4 +α 2 2+β)]/[α 2 2+α 2 β)], one can guarantee the positive semi-definiteness of the covariance. C. Computation of the acground error matrix B We see from Eq.6) that the quadratic cost-function is completely characterized y the acground error matrix B and oservation error covariance R. If R is nearly fixed and mostly a constant, B plays a ey role. Different strategies for the B s determination [16] : For random sampling approaches, B is a time-variant statistic and can e empirically computed as a conditional acground covariance online. Particularly, if B comes from a random memers set {X,i } with size N, one may show that B N P xx, = 1 N N 1 =1 X,i x )X,i x )T, where x = N i=1 X,i /N. We call EnKF an ensemle Kalman Filter using the empirical B N matrix. This scheme is simple and direct, we will use it for the numerical comparisons. For variational filtering methods, B is seen as a function of nonlinear evolution M 1, and supposed to e nown and time-variant. It can e computed with the empirical statistics of a one-step-ahead prediction of an ensemle. In the proposed ISVF we simply choose to put B P X, in Eqs.21a)21). V. SIMULATION AND DISCUSSION We adopt a relative root mean square error relative RMSE for short) for performance validation, its definition is E = 1 M j=1 M xa,j xtrue,j 2 / x true,j 2, = 1,, T, where x true,j and x a,j denote the truth and analysis of the vectorvalued state at time for jth run of Monte Carlo simulation. M and T are total runs of Monte Carlo and simulation steps. 2 means the L 2 norm. We attempt to implement a comparison etween the ISVF and the Ensemle Unscented Kalman Filter ) [2]. The is a modified scheme characterized y the Ensemle Kalman Filter EnKF) and the unscented transform, and has more accuracy improvement compared with the ordinary EnKF [2]. Particularly the comparison is due to several reasons: 1) Both of the methods use the unscented transform for sigma points generation and the ensemle mean and covariance estimation; 2) Both confine to the dynamics of nonlinear/symmetric ; 3) Only the update scheme is different: performs the LUMV criterion in II-A while ISVF implements the variational correction in II-C. The Lorenz-95 is chosen for performance examination, which originally introduced y Lorenz in 1995 [13] and descries the propagation of atmospheric wave along a meridian circle on the earth. A simplified version that circle state is divided into n-dimension at time, is formulated y x i, t = x i+1, x i 2, )x i 1, x i, + F, 24) where the dimension index i = 1,, n. The cyclic oundary conditions, x 0, = x n, and x 1, = x n 1, as well as x n+1, = x 1,, are adopted to determine state components x i. To set the constant F = 8. To solve the system y numerical integration with a fourth-order Runge-Kutta with integration step t = 0.05 unit, which corresponds to aout a 6-hour interval in practice [14]. The perturation u of dynamics is set to e a zero-mean Gaussian with nown n n diagonal Q. We choose the oserver H simply to e a timeinvariant identity operator. Specifically, given a state x = [x,1,, x,n ] T at the th assimilation cycle, the oservations are otained y y = H x ) + v = x + v 25) where v follows the n-dimensional Gaussian distriution N 0, R ) with a nown covariance R. Note that in this case the mean update in Eq.7) is reduced to x a { = arg min y x) T R 1 x y x) +x X) T PX,) 1 x X) }, 26)

6 and the cost-function is a comination of two quadratics. The following experiments adopt α = 1, β = 2, the convergence threshold C = To explore the different effects on the choices of state and oservation noise level, as well as state dimensionality n, we consider 4 cases for comparison : Case 1: Lower level of noise Parameters choice: state dimensionality n = 40, the lower ound l l = 20 and upper ound l u = 30, the initial ound l 1 = 25, the process noise Q = 5. Id40, 40), the oservation noise R = 0.1 Id40, 40). 50 runs of Monte Carlo, corresponding relative RMS errors E and the estimated trajectories are plotted in Fig.1a). To oserve Fig.1a), the ISVF exhiits a etter accuracy and almost has no transient stage, also exhiits a etter staility than the. Similarly the has a tight approximation comparing to the ISVF, however much ias is presented in transient stage, there exists a explicit gap during filtering procedure. To see the averaged consumption time for calculating in Tale I, the ISVF is less than the y up to 3s. Case 2: Higher level of noise Similarly the last case, we eep the same parameters such as the n = 40, l l = 20 and l u = 30, we only increase the noise level to Q = 10. Id40, 40) and R = 0.5 Id40, 40). After 50 runs of Monte Carlo cycles, the relative RMS error is on Fig.1). To oserve the Fig.1), the curve of the ISVF eeps oscillating with an explicit amplitude compared to the first case, and maintains a certain level and lasts to the end of simulation. The quantities of oth curves in axis y are increased compared with those in case 1. However the gap etween them are reduced to a small quantity, it is the result of simultaneously increasing the noises of dynamical and oservation. Moreover the effect rought y the noise increasing is more distinct to ISVF, logically the results meet also theoretical prediction. To chec the averaged elapsed time in Tale I, the time consumption of two methods is slightly increased compared with case 1. Case 3: Higher state dimension and lower noise level To increase the state dimensionality n from 40 to 80, associatively the scope [l l, l u ] is increased from [20, 30] to [20, 40] and set the initial value l l = 30, while specify a lower level of noise y Q = 5. Id80, 80) and R = 0.1 Id80, 80). 50 runs of Monte Carlo cycles, the relative RMS error and trajectory comparisons are plotted in Fig.2a) and 2) respectively. Fig.2a) shows that, with the state dimension n eing increased from 40 to 80 while eeping lower level of noise, the is sensile to the dimensionality increasing and its relative RMS error tends to a larger increased amplitude compared to the case 2. Moreover the higher amplitude maintains to the end, while the ISVF shows eing impacted y a relative small quantity of increasing compared with the, and eeps the amplitude staly in a lower level until the end of simulation. On the other hand, to oserve the effect of dimensional increasing on the time computation in Tale.I: for the, the average time consumption is more than 18 time the used in case 1, while the ISVF is more than 13 time. Intuitively dimensionality-increasing indicates size-increased covariance matrices and thus means the increased time consumption y the methods with a more algera operation on matrices. TABLE I. A COMPARISON OF CONSUMPTION TIME BETWEEN THE ISVF AND THE ENUKF Parameters Choice ISVF Case Case Case Case Case 4: Higher state dimension and noise level To e different from the case 3, we eep no changes on other parameters ut respecify a higher level noise, Q = 10. Id80, 80) and R = 0.5 Id80, 80), the results with 50 runs of Monte Carlo are plotted in Fig.3a) and 3). We focus on the coupled effects rought y the stronger perturation from noise and dimensionality simultaneously. To see from Fig.3a), the relative RMS error produced y the has large higher gap compared with the ISVF s: the ISVF maintains an increased quantity of RMS error and seems almost not eing affected y the coupled perturations comparing to the case 3. To oserve the time consumption in Tale I, the average increased time consumption of the is roughly as same as that in the case 3 actually 1.02 times the consumption increased), and the ISVF is also almost unchanged. Additionally we should note that, in some further surveys with the dimensionality scale more than that in this case, the ISVF occasionally reports numerical prolem and sequentially the output may e inaccurate due to the searching procedure eing interrupted, while the relative RMS error of the has a slight tendency of divergence. Interestingly y comparing the case 1-2 and 3-4, we notice that the accuracies of the oth methods are sensitive to the increasing of state dimensionality and noise level. The gap of the RMS error curves exhiited y two methods explicitly ecomes larger with the dimensionality increasing from 40 to 80, then under the coupled wor of two factors, RMS error curves of oth methods will ecome worse than ever. Moreover the computational time only sensitive to the dimensionality increasing, e.g., to oserves Tale I, the consummated time e increased roughly y 2.6 and 3 times when dimensionality increased 2 times, i.e. from 40 to 80, respectively. Summarily y comparing the relative RMS error and time consumption using 4 cases, we show that the proposed ISVF exhiits a etter state estimation accuracy than the s, meanwhile the time consumption of the ISVF grows slower than the s. The RMS error of the ISVF shows a relative staility throughout the procedure of simulations and almost have no transient stage compared with the. VI. CONCLUSION In this wor, aimed to strongly nonlinear and large-scale assimilation prolems, an iterative scheme called ISVF is suggested. Summarily the improvements achieved y the ISVF hold in three points: First, we attempt to rea the linear assumption in KF-lie estimators concerning the update scheme. This linear regression with respect to oservation governs all KFlie nonlinear filters explored previously. Considering the practical dynamics characterized y stronger nonlinearity and

7 Relative RMS erroe E Relative RMS erroe E Time step ) Time step ) a) Effects of the low level of state and oservation noise covariance Q = 5. Id40, 40) and R = 0.1 Id40, 40) on the estimate accuracy. ) Effects of the high level of state and oservation noise covariance Q = 10. Id40, 40) and R = 0.5 Id40, 40) on the estimate accuracy. Fig. 1. Comparison of the relative RMS error E etween the ISVF and with State dimensionality n = 40 and 50 runs of Monte Carlo. 2.4 Truth Relative RMS erroe E 10 0 Trajectories Time step ) 2.25 Time step ) a) Comparison of the relative RMS error E etween the ISVF and ) Comparison of the trajectory etween the truth and the estimates conducted y the ISVF and Fig. 2. Effects of the low level of state and oservation noise covariance Q = 5. Id80, 80) and R = 0.1 Id80, 80) on the estimate accuracy and the trajectory. State dimensionality n = 80 and 50 runs of Monte Carlo. high dimensionality, the relation etween estimate and oservation does not e necessarily linear. Meanwhile the linear assumption may e introduces more error and ottlenecs all efforts of performance enhancement. Therefore we suggest in the ISVF that the scheme of KF-lie mean update could eing replaced y a variational minimization. The implementation promises a gloal and optimal solution in possile solution space. Second, an iterative update scheme is derived for the correction step, it is shown to e simple and stale for implementation. Third, the scheme of KF-lie covariance update is replaced y the ET scheme descried in II-B, there the maintaining of the cross and project covariance ecomes unnecessary. Therefore a loc of computational load is cut and naturally the estimation efficiency is improved. Fourth, y introducing the TSVD formulated in Eq.22), the sampling size is efficiently reduced especially for largescale prolems. In the introduced experiments, the accuracy of estimate with respect to the noise level and the system dimensionality, was demonstrated via a Lorenz-95 system. We compared the proposed ISVF with the in terms of the relative RMS error and time consumption y a numer of Monte Carlo runs, the results valid the outperforming and efficiency of ISVF. Finally we should note that, the prolem of convergence condition and convergence rate of Eqs.21) is not presented here, however considering its importance for the proposed ISVF, we will give a extensive discussion in the future.

8 Truth Relative RMS erroe E 10 0 Trajectories Time step ) 2.25 Time step ) a) Comparison of the relative RMS error E etween the ISVF and ) Comparison of the trajectory etween the truth and the estimates conducted y the ISVF and Fig. 3. Effects of the higher level of state and oservation noise covariance Q = 10. Id80, 80) and R = 0.5 Id80, 80) on the estimate accuracy and the trajectory. State dimensionality n = 80 and 50 runs of Monte Carlo. ACKNOWLEDGMENT The authors would lie to than the support from the Chinese Natural Science Foundation granted No and No ), and the ANR PREVASSEMBLE project. REFERENCES [1] S. J. Julier, J. K. Uhlmann, Unscented filtering and nonlinear estimation, Proc. IEEE ) [2] Luo X., Moroz I.M., Ensemle Kalman filter with the unscented transform Physica D: Nonlinear Phenomena, 238 5), 2009) [3] G. Evensen, The ensemle Kalman filter: theoretical formulation and practical implementation, Ocean Dyn ) [4] M. D. Butala, J. Yun, Y. Chen, R. A. Frazin, F. Kamalaadi, Asymptotic convergence of the ensemle Kalman filter, in: IEEE International Conference on Image Processing, [5] G. Evensen, Data Assimilation: The Ensemle Kalman Filter, Springer, [6] C. H. Bishop, B. J. Etherton, S. J. Majumdar, Adaptive sampling with ensemle transform alman filter. part I: theoretical aspects, Mon. Wea. Rev ) [7] Rihan, F. A., Collier, C. G., Ballard, S. P. and Swarric, S. J., Assimilation of Doppler radial winds into a 3D-Var system: Errors and impact of radial velocities on the variational analysis and model forecasts. Quarterly Journal of the Royal Meteorological Society, 134,2008) [8] B. D. O. Anderson, J. B. Moore, Optimal Filtering, Prentice-Hall, [9] J. G. Gilert, J. Nocedal, Gloal Convergence Properties of Conjugate Gradient Methods for Optimization. SIAM Journal on Optimization, Vol.2, No.1,1992) [10] S. J. Julier, The scaled unscented transformation, in: The Proceedings of the American Control Conference, Anchorage, AK, [11] Ming Lei, Zhongliang Jing and Shiqiang Hu, Scaled Unscented Transform-ased Variational Optimality Filter, 15th International Conference on Information Fusion. Singapore, 9-12th, July [12] Ming Lei, Christophe Baehr, Unscented/Ensemle Transform-ased Variational Filter, Physica D: Nonlinear Phenomena, Vol.246, No.1, March [13] E. N. Lorenz, 1995: Predictaility: A prolem partly solved. In Seminar on Predictaility, volume Vol. I, ECMWF, Reading, UK, Availale at pdf/seminar/1995. [14] E. N. Lorenz, K. A. Emanuel, sites for supplementary weather oservations: Simulation with a small model, J. Atmos. Sci ): [15] P. Weigel, A. Liniger, and C. Appenzeller, The Discrete Brier and Raned Proaility Sill Scores, Mon. Wea. Rev., 135,2007) [16] P. Saov, G. Evensen and L. Bertino, Asynchronous data assimilation with the EnKF, Tellus Ser A., [17] K.B.Petersen, M.S.Pedersen, The Matrix Coooo. [18] A. Tarantola, Inverse prolem theory and methods for parameter estimation. SIAM, 2005, pp.342. [19] Gu, Y. and D. S. Oliver, An iterative ensemle Kalman filter for multiphase fluid flow data assimilation. SPE Journal,2007, Vol.12, pp: [20] Hansen P. C., The truncated svd as a method for regularization, BIT, 27 4), 1987,