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Sampling the posterior: an approach to non-Gaussian data assimilation
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Apte, A., Hairer, Martin, Stuart, A. M. and Voss, J. (2007) Sampling the posterior: an approach to non-Gaussian data assimilation. Physica D: Nonlinear Phenomena, Vol.230 (No.1-2). pp. 50-64. doi:10.1016/j.physd.2006.06.009 ISSN 0167-2789.
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Official URL: http://dx.doi.org/10.1016/j.physd.2006.06.009
Abstract
The viewpoint taken in this paper is that data assimilation is fundamentally a statistical problem and that this problem should be cast in a Bayesian framework. In the absence of model error, the correct solution to the data assimilation problem is to find the posterior distribution implied by this Bayesian setting. Methods for dealing with data assimilation should then be judged by their ability to probe this distribution. In this paper we propose a range of techniques for probing the posterior distribution, based around the Largevin equation; and we compare these new techniques with existing methods.
When the underlying dynamics is deterministic, the posterior distribution is on the space of initial conditions leading to a sampling problem over this space. When the underlying dynamics is stochastic the posterior distribution is on the space of continuous time paths. By writing down a. density, and conditioning on observations, it is possible to define a range of Markov Chain Monte Carlo (MCMC) methods which sample from the desired posterior distribution, and thereby solve the data assimilation problem. The basic building-blocks for the MCMC methods that we concentrate on in this paper are Langevin equations which are ergodic and whose invariant measures give the desired distribution; in the case of path space sampling these are stochastic partial differential equations (SPDEs).
Two examples are given to show how data assimilation can be formulated in a Bayesian fashion. The first is weather prediction, and the second is Lagrangian data assimilation for oceanic velocity fields. Furthermore the relationship between the Bayesian approach outlined here and the commonly used Kalman filter based techniques, prevalent in practice, is discussed. Two simple pedagogical examples are studied to illustrate the application of Bayesian sampling to data assimilation concretely. Finally a range of open mathematical and computational issues, arising from the Bayesian approach, are outlined. (c) 2006 Elsevier B.V All rights reserved.
Item Type: | Journal Article | ||||
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Subjects: | Q Science > QA Mathematics Q Science > QC Physics |
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Divisions: | Faculty of Science, Engineering and Medicine > Science > Mathematics | ||||
Journal or Publication Title: | Physica D: Nonlinear Phenomena | ||||
Publisher: | Elsevier BV | ||||
ISSN: | 0167-2789 | ||||
Official Date: | June 2007 | ||||
Dates: |
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Volume: | Vol.230 | ||||
Number: | No.1-2 | ||||
Number of Pages: | 15 | ||||
Page Range: | pp. 50-64 | ||||
DOI: | 10.1016/j.physd.2006.06.009 | ||||
Status: | Peer Reviewed | ||||
Publication Status: | Published | ||||
Access rights to Published version: | Restricted or Subscription Access | ||||
Title of Event: | Data Assimilation for Geophysical Systems Program | ||||
Type of Event: | Other | ||||
Location of Event: | Stat & Appl Math Sci Inst, Res Triangle Pk, NC | ||||
Date(s) of Event: | 2005 |
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