On Controlling the Shape of the Cost Functional in Dynamic Data Assimilation: Guidelines for Placement of Observations and Application to Saltzman’s Model of Convection

Abstract

Over the decades the role of observations in building and/or improving the fidelity of a model to a phenomenon is well documented in the meteorological literature. More recently adaptive/targeted observations have been routinely used to improve the quality of the analysis resulting from the fusion of data with models in a data assimilation scheme and the subsequent forecast. In this paper our goal is to develop an offline (preprocessing) diagnostic strategy for placing observations with a singular view to reduce the forecast error/innovation in the context of the classical 4D-Var. It is well known that the shape of the cost functional as measured by its gradient (also called adjoint gradient or sensitivity) in the control (initial condition and model parameters) space determines the marching of the control iterates toward a local minimum. These iterates can become marooned in regions of control space where the gradient is small. An open question is how to avoid these “flat” regions by bounding the norm of the gradient away from zero. We answer this question in two steps. We, for the first time, derive a linear transformation defined by a symmetric positive semidefinite (SPSD) Gramian G=F¯TF¯ that directly relates the control error to the adjoint gradient. It is then shown that by placing observations where the square of the Frobenius norm of F¯ (which is also the sum of the eigenvalues of G) is a maximum, we can indeed bound the norm of the adjoint gradient away from zero.