[p2] Mathematical aspect
[p231] 
Accounting for Model Error: WeakConstraint 4DVariational Data Assimilation

N. K. Nichols (University of Reading), A. ElSaid (University of Reading), A. S. Lawless (University of Reading) 
Abstract 
The behaviour of the weakconstraint fourdimensional variational data assimilation problem (wc4DVAR) and its sensitivity to the input data governing the problem are investigated here. The aim of wc4DVAR is to provide a statistically optimal estimate of the state of a dynamical system given an imperfect model of the system, a model forecast, and observations of the states over time. It differs from conventional strong constraint 4DVAR by providing estimates for the model errors as well as the states. We consider two formulations of the wc4DVAR problem  the model error and state estimation formulations  and examine these as a function of observation accuracy and configuration; forecast accuracy and error covariance structure; model error variance and correlation length scales; and assimilation window length. The variational problems are solved by a gradient iteration procedure. We investigate the behaviour of the convergence of the two formulations via numerical examples. Results are shown using a linear advection model with periodic boundary conditions. Although the optimization problems are mathematically equivalent, we show that the two formulations exhibit very different performance characteristics when the data are varied. 