TY  - JOUR
A1  - Bergemann, Kay
A1  - Gottwald, Georg
A1  - Reich, Sebastian
T1  - Ensemble propagation and continuous matrix factorization algorithms
N2  - We consider the problem of propagating an ensemble of solutions and its characterization in terms of its mean and covariance matrix. We propose differential equations that lead to a continuous matrix factorization of the ensemble into a generalized singular value decomposition (SVD). The continuous factorization is applied to ensemble propagation under periodic rescaling (ensemble breeding) and under periodic Kalman analysis steps (ensemble Kalman filter). We also use the continuous matrix factorization to perform a re-orthogonalization of the ensemble after each time-step and apply the resulting modified ensemble propagation algorithm to the ensemble Kalman filter. Results from the Lorenz-96 model indicate that the re-orthogonalization of the ensembles leads to improved filter performance.
Y1  - 2009
UR  - http://onlinelibrary.wiley.com/journal/10.1002/%28ISSN%291477-870X
U6  - https://doi.org/10.1002/qj.457
SN  - 0035-9009
ER  - 
TY  - JOUR
A1  - Bergemann, Kay
A1  - Reich, Sebastian
T1  - A localization technique for ensemble Kalman filters
N2  - Ensemble Kalman filter techniques are widely used to assimilate observations into dynamical models. The phase- space dimension is typically much larger than the number of ensemble members, which leads to inaccurate results in the computed covariance matrices. These inaccuracies can lead, among other things, to spurious long-range correlations, which can be eliminated by Schur-product-based localization techniques. In this article, we propose a new technique for implementing such localization techniques within the class of ensemble transform/square-root Kalman filters. Our approach relies on a continuous embedding of the Kalman filter update for the ensemble members, i.e. we state an ordinary differential equation (ODE) with solutions that, over a unit time interval, are equivalent to the Kalman filter update. The ODE formulation forms a gradient system with the observations as a cost functional. Besides localization, the new ODE ensemble formulation should also find useful application in the context of nonlinear observation operators and observations that arrive continuously in time.
Y1  - 2010
UR  - http://onlinelibrary.wiley.com/journal/10.1002/(ISSN)1477-870X
U6  - https://doi.org/10.1002/Qj.591
SN  - 0035-9009
ER  - 
TY  - JOUR
A1  - Bergemann, Kay
A1  - Reich, Sebastian
T1  - A mollified ensemble Kalman filter
N2  - It is well recognized that discontinuous analysis increments of sequential data assimilation systems, such as ensemble Kalman filters, might lead to spurious high-frequency adjustment processes in the model dynamics. Various methods have been devised to spread out the analysis increments continuously over a fixed time interval centred about the analysis time. Among these techniques are nudging and incremental analysis updates (IAU). Here we propose another alternative, which may be viewed as a hybrid of nudging and IAU and which arises naturally from a recently proposed continuous formulation of the ensemble Kalman analysis step. A new slow-fast extension of the popular Lorenz-96 model is introduced to demonstrate the properties of the proposed mollified ensemble Kalman filter.
Y1  - 2010
UR  - http://onlinelibrary.wiley.com/journal/10.1002/(ISSN)1477-870X
U6  - https://doi.org/10.1002/Qj.672
SN  - 0035-9009
ER  - 
TY  - JOUR
A1  - Bergemann, Kay
A1  - Reich, Sebastian
T1  - An ensemble Kalman-Bucy filter for continuous data assimilation
JF  - Meteorologische Zeitschrift
N2  - The ensemble Kalman filter has emerged as a promising filter algorithm for nonlinear differential equations subject to intermittent observations. In this paper, we extend the well-known Kalman-Bucy filter for linear differential equations subject to continous observations to the ensemble setting and nonlinear differential equations. The proposed filter is called the ensemble Kalman-Bucy filter and its performance is demonstrated for a simple mechanical model (Langevin dynamics) subject to incremental observations of its velocity.
Y1  - 2012
U6  - https://doi.org/10.1127/0941-2948/2012/0307
SN  - 0941-2948
VL  - 21
IS  - 3
SP  - 213
EP  - 219
PB  - Schweizerbart
CY  - Stuttgart
ER  -