# Adam El-Said, The University of Reading

## Conditioning of Weak-Constraint 4DVAR

##### Coauthors

Prof. N.K. Nichols, Dr A.S. Lawless#### Abstract:

Data assimilation merges observations with a dynamical model to find the optimal state estimate of a system given a set of observations. It is cyclic in that it is applied at fixed time intervals and the beginning of each cycle incorporates a forecast from the previous cycle known as the background or apriori estimate. Variational data assimilation aims to minimise a non-linear, least-squares objective function that is constrained by the flow of the (perfect) dynamical model (4DVAR). Relaxing the perfect model assumption gives rise to weak-constraint 4DVAR, which has two formulations of interest. We call these the `model error forcing' and `state estimation' formulations.

Gradient-based iterative solvers are used to solve the wc4DVAR problem. We gain insight into accuracy and convergence by studying the condition number of the Hessian of both formulations. We have derived new theoretical bounds on the condition number of the Hessians of both formulations. The theoretical bounds give us insight into the sensitivities of both formulations to changes in the assimilation parameters. We demonstrate the bounds using the linear advection equation and the non-linear chaotic Lorenz-95 model and discuss the results.