On the sensitivity of a least-squares fit of discretized linear hyperbolic equations to data

Difficulties are investigated which occur when trying to specify a noise-free initial model state as the solution of a variational data assimilation problem. A linear shallow water model is used to investigate the existence and physical basis of the model fit to data. As in this context the shape of the cost function is of crucial importance, the interrelations between the cost function’s Hessian and specific model-data configurations are investigated. Special emphasis is put on the influence of the temporal/spatial data distribution and the choice of the scheme used for numerical model integration. It is illustrated how such details may cause intolerable uncertainties for those aspects of the recovered solution that are related to very small eigenvalues of the curvature operator. Due to the shortcomings of descent algorithms, uncontrolled large-amplitude error modes may remain invisible if a limited number of minimization cycles is applied. However, to render the retrieved smooth fields stable with respect to further iterations, prior knowledege has to be taken into account in the cost function definition.