A Caltech Library Service

Data assimilation: Mathematical and statistical perspectives

Apte, A. and Jones, C. K. R. T. and Stuart, A. M. and Voss, J. (2008) Data assimilation: Mathematical and statistical perspectives. International Journal for Numerical Methods in Fluids, 56 (8). pp. 1033-1046. ISSN 0271-2091. doi:10.1002/fld.1698.

Full text is not posted in this repository. Consult Related URLs below.

Use this Persistent URL to link to this item:


The bulk of this paper contains a concise mathematical overview of the subject of data assimilation, highlighting three primary ideas: (i) the standard optimization approaches of 3DVAR, 4DVAR and weak constraint 4DVAR are described and their interrelations explained; (ii) statistical analogues of these approaches are then introduced, leading to filtering (generalizing 3DVAR) and a form of smoothing (generalizing 4DVAR and weak constraint 4DVAR) and the optimization methods are shown to be maximum a posteriori estimators for the probability distributions implied by these statistical approaches; and (iii) by taking a general dynamical systems perspective on the subject it is shown that the incorporation of Lagrangian data can be handled by a straightforward extension of the preceding concepts. We argue that the smoothing approach to data assimilation, based on statistical analogues of 4DVAR and weak constraint 4DVAR, provides the optimal solution to the assimilation of space–time distributed data into a model. The optimal solution obtained is a probability distribution on the relevant class of functions (initial conditions or time-dependent solutions). The approach is a useful one in the first instance because it clarifies the notion of what is the optimal solution, thereby providing a benchmark against which existing approaches can be evaluated. In the longer term it also provides the potential for new methods to create ensembles of solutions to the model, incorporating the available data in an optimal fashion. Two examples are given illustrating this approach to data assimilation, both in the context of Lagrangian data, one based on statistical 4DVAR and the other on weak constraint statistical 4DVAR. The former is compared with the ensemble Kalman filter, which is thereby shown to be inaccurate in a variety of scenarios.

Item Type:Article
Related URLs:
URLURL TypeDescription
Voss, J.0000-0001-7740-8811
Additional Information:© 2007 John Wiley & Sons. Issue online: 18 February 2008, Version of record online: 27 December 2007, Manuscript Accepted: 25 October 2007, Manuscript Revised: 24 October 2007, Manuscript Received: 25 April 2007. Funded by ONR. Grant Number: N00014-05-1-0791.
Funding AgencyGrant Number
Office of Naval Research (ONR)N00014-05-1-0791
Subject Keywords:data assimilation; Bayesian statistics; 3DVAR; 4DVAR; filtering; smoothing; stochastic PDEs
Other Numbering System:
Other Numbering System NameOther Numbering System ID
Andrew StuartJ73
Issue or Number:8
Record Number:CaltechAUTHORS:20160805-165730529
Persistent URL:
Official Citation:Apte, A., Jones, C. K. R. T., Stuart, A. M. and Voss, J. (2008), Data assimilation: Mathematical and statistical perspectives. Int. J. Numer. Meth. Fluids, 56: 1033–1046. doi:10.1002/fld.1698
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:69497
Deposited By: Linda Taddeo
Deposited On:08 Aug 2016 23:11
Last Modified:11 Nov 2021 04:15

Repository Staff Only: item control page