An investigation of incremental 4D‐Var using non‐tangent linear models

Abstract

SUMMARYWe investigate the convergence of incremental four-dimensional variational data assimilation (4D-Var) when an approximation to the tangent linear model is used within the inner loop. Using a semi-implicit semi-Lagrangian model of the one-dimensional shallow water equations, we perform data assimilation experiments using an exact tangent linear model and using an inexact linear model (a perturbation forecast model). We find that the two assimilations converge at a similar rate and the analyses are also similar, with the difference between them dependent on the amount of noise in the observations. To understand the numerical results, we present the incremental 4D-Var algorithm as a Gauss-Newton iteration for solving a least-squares problem and consider its fixed points.

Parameter space Kalman smoothers for multi-decadal climate analysis in high resolution coupled Global Circulation Models

“…The locally iterated (extended) Kalman filter (IKF) is a Gauss-Newton method for approximating a maximumlikelihood estimate Bell and Cathey (1993), and actually it is algebraically equivalent to nonlinear three dimensional variational (3D-Var) analysis algorithms (Cohn, 1997). Later, Bell (1994) showed that the iterated Kalman smoother (IKS) represents a Gauss-Newton method to obtain an approximate maximum likelihood, as was shown later for incremental 4D-Var (Lawless et al, 2005) and has been summarised in section 3.1 above. Thus, the IKF (as the IKS) circumvents the need for choosing a step size, which is sometimes a source of difficulty in descent methods.…”

“…Incremental 4D-Var has been shown to be an inexact GaussNewton method applied to the original nonlinear cost function (Lawless et al, 2005). As we assume GaussianR and P b 0 and a perfect-model framework except for the sources of model uncertainty in z 0 , the conditional mode given by the linear least 5 squares minimization (12) is the same that the conditional mean (also called the minimum variance estimate) given by the explicit solution…”

A note on the analysis error associated with 3D-FGAT

“…The method is equivalent to solving the full nonlinear 4D-Var problem using a Gauss-Newton method (Lawless et al, 2005). The algorithm for 3D-FGAT is very similar to incremental 4D-Var, except that the tangent linear model is approximated by the identity, so that in the linearized cost function (1) we have δx i = δx 0 for all times t i .…”

“…The system is discretized using a semiimplicit semi-Lagrangian integration scheme, as described in Lawless et al (2003). We define the problem on a periodic domain of 1000 grid points, with a spacing x = 0.01 m between them, so that x ∈ [0 m, 10 m] and assume a model time step t = 0.0092 s. Other parameters for the problem are as defined in Lawless et al (2005). A 3D-FGAT scheme for this system is set up over a time window [−T, T], with observations of u and φ at every spatial point at times −T, 0, T. The true state at time −T is defined using the initial conditions from Case I of Lawless et al (2005) and we take T = 0.23, so that there are 50 time steps in the assimilation interval.…”

“…We define the problem on a periodic domain of 1000 grid points, with a spacing x = 0.01 m between them, so that x ∈ [0 m, 10 m] and assume a model time step t = 0.0092 s. Other parameters for the problem are as defined in Lawless et al (2005). A 3D-FGAT scheme for this system is set up over a time window [−T, T], with observations of u and φ at every spatial point at times −T, 0, T. The true state at time −T is defined using the initial conditions from Case I of Lawless et al (2005) and we take T = 0.23, so that there are 50 time steps in the assimilation interval. The solution to this problem is given by a stationary field over the orography and two outgoing gravity waves moving away from the orography in opposite diretcions.…”