Numerical Linear Algebra in Data Assimilation

TL;DR

This paper reviews data assimilation methods combining observations with model outputs to improve system state estimates, emphasizing the role of numerical linear algebra in solving large-scale problems.

Contribution

It provides a comprehensive derivation of key data assimilation techniques and discusses recent advances and challenges from a numerical linear algebra perspective.

Findings

Comparison of variational and sequential approaches

Analysis of advantages and disadvantages of advanced methods

Discussion of numerical linear algebra challenges in large-scale problems

Abstract

Data assimilation is a method that combines observations (that is, real world data) of a state of a system with model output for that system in order to improve the estimate of the state of the system and thereby the model output. The model is usually represented by a discretised partial differential equation. The data assimilation problem can be formulated as a large scale Bayesian inverse problem. Based on this interpretation we will derive the most important variational and sequential data assimilation approaches, in particular three-dimensional and four-dimensional variational data assimilation (3D-Var and 4D-Var) and the Kalman filter. We will then consider more advanced methods which are extensions of the Kalman filter and variational data assimilation and pay particular attention to their advantages and disadvantages. The data assimilation problem usually results in a very large optimisation problem and/or a very large linear system to solve (due to inclusion of time and space dimensions). Therefore, the second part of this article aims to review advances and challenges, in particular from the numerical linear algebra perspective, within the various data assimilation approaches.