The spectral conjugate gradient method in variational adjoint assimilation for model terrain correction I: Theoretical frame

Abstract

A spectral conjugate gradient (SCG) method is proposed within the mathematical framework of a variational adjoint assimilation system to correct the bottom terrain of a shallow-water equations model. The formulation of this method is described from the mathematical point of view with determination of the descent direction by using Andrei’s limited-memory form and the step length by solving the tangent linear model. It is proved to benefit from (i) the iterative regularization strategy, (ii) the inverse Hessian approximation involving the second-order information from Broyden-Fletcher-Goldfarb-Shanno class (BFGS), and (iii) the optimal step length. The regularization term introduced to the cost function will allow the incorporation of the known information about the desired bottom terrain and guarantee the uniqueness of the optimal solution.