Convergent least-squares optimisation methods for variational data assimilation

TL;DR

This paper investigates the convergence issues of the Gauss-Newton method in 4D-Var data assimilation and proposes convergent variants that improve initial state estimates for better weather forecasts.

Contribution

It introduces and compares two convergent Gauss-Newton variants, line search and regularisation, for long time-window 4D-Var data assimilation.

Findings

Conventional Gauss-Newton may not converge with high background uncertainty or long time-windows.

Convergent variants can achieve more accurate initial state estimates.

Improved initial states can lead to more accurate weather forecasts.

Abstract

Data assimilation combines prior (or background) information with observations to estimate the initial state of a dynamical system over a given time-window. A common application is in numerical weather prediction where a previous forecast and atmospheric observations are used to obtain the initial conditions for a numerical weather forecast. In four-dimensional variational data assimilation (4D-Var), the problem is formulated as a nonlinear least-squares problem, usually solved using a variant of the classical Gauss-Newton (GN) method. However, we show that GN may not converge if poorly initialised. In particular, we show that this may occur when there is greater uncertainty in the background information compared to the observations, or when a long time-window is used in 4D-Var allowing more observations. The difficulties GN encounters may lead to inaccurate initial state conditions for subsequent forecasts. To overcome this, we apply two convergent GN variants (line search and regularisation) to the long time-window 4D-Var problem and investigate the cases where they locate a more accurate estimate compared to GN within a given budget of computational time and cost. We show that these methods are able to improve the estimate of the initial state, which may lead to a more accurate forecast.