Optimal determination of nudging coefficients using the adjoint equations

Abstract

The adjoint equations of a numerical model can be used for model-parameter estimation. In this study, a general computational procedure is developed to determine the size and distribution of any internal model parameter. The procedure is then applied to a one-dimensional shallow-water model in the context of analysis-nudging four-dimensional data assimilation (FDDA): the weighting coefficients used by the Newtonian nudging technique are determined such that the model error during the assimilation period is optimally reduced subject to some constraints. The sensitivity of these nudging coefficients to the optimal objectives and constraints is investigated using this simple grid-point model in an Observing Systems Simulation Experiments (OSSE) mode. The results show that in principle, it is feasible to determine a set of nudging weights which minimize the model error over the period covered by the observations. It is demonstrated, however, that the magnitude and distribution of these “optimal” nudging weights are sensitive to the prescribed estimate of the nudging weights and the corresponding coefficient matrix which define a penalty term in the cost function. The penalty term is a weak constraint on the size and distribution of the optimal nudging weights while the model is the strong constraint. The fit of the model to the data is greater when this constraint on the nudging weights is weaker, but then the nudging weights may be too large or even negative. Thus the “optimal” solution for this model parameter is not unique because specification of this penalty term in the cost function introduces a new uncertainty into the nudging FDDA framework. Nevertheless, this optimal-nudging approach does show promise, but the sensitivity of the technique to the penalty term requires further investigation under more realistic conditions. DOI: 10.1034/j.1600-0870.1993.t01-4-00003.x