Second Order Information in Data Assimilation

Francois X. Le Dimet, I. M. Navon, Dacian N. Daescu

In variational data assimilation (VDA) for meteorological and/or oceanic models, the assimilated fields are deduced by combining the model and the gradient of a cost functional measuring discrepancy between model solution and observation, via a first order optimality system. However existence and uniqueness of the VDA problem along with convergence of the algorithms for its implementation depend on the convexity of the cost function. Properties of local convexity can be deduced by studying the Hessian of the cost function in the vicinity of the optimum thus the necessity of second order information to ensure a unique solution of the VDA problem. In this paper we present a comprehensive review of issues related of second order analysis of the problem of VDA along with many important issues closely connected to it. In particular we study issues of existence, uniqueness and regularization through second order properties.We then focus on second order information related o statistical properties and on issues related to preconditioning and optimization methods and second order VDA analysis. Predictability and its relation of the structure of the Hessian of the cost functional is then discussed along with issues of sensitivity analysis in the presence of data being assimilated. Computational complexity issues are also addressed and discussed. Automatic differentiation issues related of second order information are also discussed along with the computational complexity of deriving the second order adjoint. Finally an application aimed at illustrating the use of automatic differentiation for deriving the second order adjoint as well as the Hessian/vector product applied to minimizing a cost functional of a meteorological problem using the Truncated-Newton method is presented. Results verifying numerically the computational cost of deriving the second order adjoint as well as results related to the spectrum of the Hessian of the cost functional are displayed and discussed.