Reduced-rank sigma-point Kalman filter for geophysical data assimilation

Abstract Performance of an advanced, derivativeless, sigma-point Kalman filter (SPKF) data assimilation scheme in a strongly nonlinear dynamical model is investigated. The SPKF data assimilation scheme is compared against standard Kalman filters such as the extended Kalman filter (EKF) and ensemble Kalman filter (EnKF) schemes. Three particular cases—namely, the state, parameter, and joint estimation of states and parameters from a set of discontinuous noisy observations—are studied. The problems associated with the use of tangent linear model (TLM) or Jacobian when using standard Kalman filters are eliminated when using SPKF data assimilation algorithms. Further, the constraints and issues of SPKF data assimilation in real ocean or atmospheric models are emphasized. A reduced sigma-point subspace model is proposed and investigated for higher-dimensional systems. A low-dimensional Lorenz 1963 model and a higher-dimensional Lorenz 1995 model are used as the test beds for data assimilation experiments. The results of SPKF data assimilation schemes are compared with those of standard EKF and EnKF, in which a highly nonlinear chaotic case is studied. It is shown that the SPKF is capable of estimating the model state and parameters with better accuracy than EKF and EnKF. Numerical experiments showed that in all cases the SPKF can give consistent results with better assimilation skills than EnKF and EKF and can overcome the drawbacks associated with the use of EKF and EnKF.

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Singular-Vector-Based Covariance Propagation in a Quasigeostrophic Assimilation System

Monthly Weather Review ◽

10.1175/mwr2909.1 ◽

2005 ◽

Vol 133 (5) ◽

pp. 1295-1310 ◽

Cited By ~ 13

Author(s):

Alexander Beck ◽

Martin Ehrendorfer

Keyword(s):

Kalman Filter ◽

Data Assimilation ◽

Variational Data Assimilation ◽

Dynamic Information ◽

Dynamic Background ◽

Background Error ◽

Singular Vectors ◽

Reduced Rank ◽

The Impact ◽

Static Background

Abstract Variational data assimilation systems require the specification of the covariances of background and observation errors. Although the specification of the background-error covariances has been the subject of intense research, current operational data assimilation systems still rely on essentially static and thus flow-independent background-error covariances. At least theoretically, it is possible to use flow-dependent background-error covariances in four-dimensional variational data assimilation (4DVAR) through exploiting the connection between variational data assimilation and estimation theory. This paper reports on investigations concerning the impact of flow-dependent background-error covariances in an idealized 4DVAR system that, based on quasigeostrophic dynamics, assimilates artificial observations. The main emphasis is placed on quantifying the improvement in analysis quality that is achievable in 4DVAR through the use of flow-dependent background-error covariances. Flow dependence is achieved through dynamical error-covariance evolution based on singular vectors in a reduced-rank approach, referred to as reduced-rank Kalman filter (RRKF). The RRKF yields partly dynamic background-error covariances through blending static and dynamic information, where the dynamic information is obtained from error evolution in a subspace of dimension k (defined here through the singular vectors) that may be small compared to the dimension of the model’s phase space n, which is equal to 1449 in the system investigated here. The results show that the use of flow-dependent background-error covariances based on the RRKF leads to improved analyses compared to a system using static background-error statistics. That latter system uses static background-error covariances that are carefully tuned given the model dynamics and the observational information available. It is also shown that the performance of the RRKF approaches the performance of the extended Kalman filter, as k approaches n. Results therefore support the hypothesis that significant analysis improvement is possible through the use of flow-dependent background-error covariances given that a sufficiently large number (here on the order of n/10) of singular vectors is used.

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Comments on “Sigma-Point Kalman Filter Data Assimilation Methods for Strongly Nonlinear Systems”

Journal of the Atmospheric Sciences ◽

10.1175/2009jas3245.1 ◽

2009 ◽

Vol 66 (11) ◽

pp. 3498-3500 ◽

Cited By ~ 17

Author(s):

Thomas M. Hamill ◽

Jeffrey S. Whitaker ◽

Jeffrey L. Anderson ◽

Chris Snyder

Keyword(s):

Kalman Filter ◽

Nonlinear Systems ◽

Data Assimilation ◽

Sigma Point ◽

Strongly Nonlinear

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Sigma-point Kalman filter data assimilation methods for strongly nonlinear dynamical models.

10.24124/2009/bpgub578 ◽

2009 ◽

Author(s):

Jaison Thomas Ambadan

Keyword(s):

Kalman Filter ◽

Data Assimilation ◽

Dynamical Models ◽

Nonlinear Dynamical ◽

Sigma Point ◽

Strongly Nonlinear

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Reduced-Rank Sigma-Point Kalman Filter and Its Application in ENSO Model

Journal of Atmospheric and Oceanic Technology ◽

10.1175/jtech-d-13-00172.1 ◽

2014 ◽

Vol 31 (10) ◽

pp. 2350-2366 ◽

Cited By ~ 3

Author(s):

K. K. Manoj ◽

Youmin Tang ◽

Ziwang Deng ◽

Dake Chen ◽

Yanjie Cheng

Keyword(s):

Kalman Filter ◽

Covariance Matrix ◽

Unscented Kalman Filter ◽

Southern Oscillation ◽

Estimation Accuracy ◽

Dimensional System ◽

Square Root ◽

Reduced Rank ◽

Sigma Point ◽

Enso Events

Abstract The huge computational expense has been a main challenge while applying the sigma-point unscented Kalman filter (SPUKF) to a high-dimensional system. This study focuses on this issue and presents two methods to construct a reduced-rank sigma-point unscented Kalman filter (RRSPUKF). Both techniques employ the truncated singular value decomposition (TSVD) to factorize the covariance matrix and reduce its rank through truncation. The reduced-rank square root matrix is used to select the most important sigma points that can retain the main statistical features of the original sigma points. In the first technique, TSVD is applied on the covariance matrix constructed in the data space [RRSPUKF(D)], whereas in the second technique TSVD is applied on the covariance matrix constructed in the ensemble space [RRSPUKF(E)]. The two methods are applied to a realistic El Niño–Southern Oscillation (ENSO) prediction model [Lamont-Doherty Earth Observatory model, version 5 (LDEO5)] to assimilate the sea surface temperature (SST) anomalies. The results show that both the methods are more computationally efficient than the full-rank SPUKF, in spite of losing some estimation accuracy. When the truncation reaches a trade-off between cost expense and estimation accuracy, both methods are able to analyze the phase and intensity of all major ENSO events from 1971 to 2001 with comparable estimation accuracy. Furthermore, the RRSPUKF is compared against ensemble square root filter (EnSRF), showing that the overall analysis skill of RRSPUKF and EnSRF are comparable to each other, but the former is more robust than the latter.

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A Hybrid Kalman Filter Algorithm for Large-Scale Atmospheric Chemistry Data Assimilation

Monthly Weather Review ◽

10.1175/mwr3269.1 ◽

2007 ◽

Vol 135 (1) ◽

pp. 140-151 ◽

Cited By ~ 13

Author(s):

R. G. Hanea ◽

G. J. M. Velders ◽

A. J. Segers ◽

M. Verlaan ◽

A. W. Heemink

Keyword(s):

Kalman Filter ◽

Data Assimilation ◽

Atmospheric Chemistry ◽

Large Scale ◽

Transport Model ◽

Reduced Rank ◽

Nonlinear Properties ◽

Rank Approximation ◽

Kalman Filter Algorithm ◽

Better Than

Abstract In the past, a number of algorithms have been introduced to solve data assimilation problems for large-scale applications. Here, several Kalman filters, coupled to the European Operational Smog (EUROS) atmospheric chemistry transport model, are used to estimate the ozone concentrations in the boundary layer above Europe. Two Kalman filter algorithms, the reduced-rank square root (RRSQRT) and the ensemble Kalman filter (EnKF), were implemented in a prior study. To combine the best features of these two filters, a hybrid filter was constructed by making use of the reduced-rank approximation of the covariance matrix as a variance reducer for the EnKF. This hybrid algorithm, complementary orthogonal subspace filter for efficient ensembles (COFFEE), is coupled to the EUROS model. The performance of all algorithms is compared in terms of residual errors and number of EUROS model evaluations. The COFFEE results score somewhere between the EnKF and RRSQRT results for less than approximately 30 model evaluations. For more than approximately 30 model evaluations, the COFFEE results are, in all cases, better than the EnKF and RRSQRT results. The results of the COFFEE simulations with more than about 60 model evaluations proved to be significantly better than all the EnKF and RRSQRT simulations (even better than those with 100 and 200 modes or ensemble members). The performance of both the EnKF- and RRSQRT-type filters is affected by the nonlinear properties of the model and observation operator, because both rely on linearization to some extent. To further study this aspect, several measures of nonlinearity were calculated and linked with the performance of these algorithms.

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