A 4D-Var Method with Flow-Dependent Background Covariances for the Shallow-Water Equations

Abstract

The 4D-Var method for filtering partially observed nonlinear chaotic dynamical systems consists of finding the maximum a-posteriori (MAP) estimator of the initial condition of the system given observations over a time window, and propagating it forward to the current time via the model dynamics. This method forms the basis of most currently operational weather forecasting systems. In practice the optimization becomes infeasible if the time window is too long due to the non-convexity of the cost function, the effect of model errors, and the limited precision of the ODE solvers. Hence the window has to be kept sufficiently short, and the observations in the previous windows can be taken into account via a Gaussian background (prior) distribution. The choice of the background covariance matrix is an important question that has received much attention in the literature. In this paper, we define the background covariances in a principled manner, based on observations in the previous b assimilation windows, for a parameter b ≥ 1. The method is at most b times more computationally expensive than using fixed background covariances, requires little tuning, and greatly improves the accuracy of 4D-Var. As a concrete example, we focus on the shallow-water equations. The proposed method is compared against state-of-the-art approaches in data assimilation and is shown to perform favourably on simulated data. We also illustrate our approach on data from the recent tsunami of 2011 in Fukushima, Japan.

Optimization Based Methods for Partially Observed Chaotic Systems

“…However, we have found in simulations on Figure 2 that even for relatively large values of σ Z and h, for large dimensions, and not very short assimilation windows, these approximations seem to be working reasonably well. Besides theoretical importance, the Gaussian approximation of the smoother can be also used to construct the prior (background) distributions for the subsequent intervals in a flow-dependent way, as we have shown in Paulin et al [2017] for the non-linear shallow-water equations, even for realistic values of σ Z and h. These flow-dependent prior distributions can considerably improve filtering accuracy. Going beyond the approximately Gaussian case (for example when σ Z , h, and T are large, or the system is highly non-linear) in a computationally efficient way is a challenging problem for future research (see Bocquet, Pires, and Wu [2010] for some examples where this situation arises).…”

“…Instead, we have computed the Newton's method iterations described in (2.28) based on preconditioned conjugate gradient solver, with the gradient and the product of the Hessian with a vector were evaluated based on adjoint methods as described by equations (3.5)-(3.7) and Section 3.2.1 of Paulin, Jasra, Beskos, and Crisan [2017] (see also Le Dimet, Navon, and Daescu [2002]). This means that the Hessians were approximated using products of Jacobian matrices that were stored in sparse format due to the local dependency of the equations (3.1).…”

Model and Data Reduction for Data Assimilation: Particle Filters Employing Projected Forecasts and Data with Application to a Shallow Water Model

“…For SWE we consider both complete observations with all model variables observed (p = 100%) or sparse observations with every 100th variable observed (p = 1%). We additionally consider three basic observation scenarios for experiments (see [56]):…”

“…With the projection matrices computed, we assimilate using Proj-OP-PF starting at t = 48h and continuing for the next 24h, with observations performed and assimilated with the timestep of τ D = 60min (as compared to a one minute observational time scale in [56]). This effectively discards the first day as transient between the initial condition and the development of the coherent structures.…”