Role of the Observability Gramian in Parameter Estimation: Application to Nonchaotic and Chaotic Systems via the Forward Sensitivity Method

Abstract

Data assimilation in chaotic regimes is challenging, and among the challenging aspects is placement of observations to induce convexity of the cost function in the space of control. This problem is examined by using Saltzman’s spectral model of convection that admits both chaotic and nonchaotic regimes and is controlled by two parameters—Rayleigh and Prandtl numbers. The problem is simplified by stripping the seven-variable constraint to a three-variable constraint. Since emphasis is placed on observation positioning to avoid cost-function flatness, forecast sensitivity to controls is needed. Four-dimensional variational assimilation (4D-Var) is silent on this issue of observation placement while Forecast Sensitivity Method (FSM) delivers sensitivities used in placement. With knowledge of the temporal forecast sensitivity matrix V, derivatives of the forecast variables to controls, the cost function can be expressed as a function of the observability Gramian VTV using first-order Taylor series expansion. The goal is to locate observations at places that force the Gramian positive definite. Further, locations are chosen such that the condition number of VTV is small and this guarantees convexity in the vicinity of the cost function minimum. Four numerical experiments are executed, and results are compared with the structure of the cost function independently determined though arduous computation over a wide range of the two nondimensional numbers. The results are especially good based on reduction in cost function value and comparison with cost function structure.