A study on the theory of second order adjoint model

The Hessian matrix, which is formed by the second order partial derivatives of the cost function with respect to control variables, plays an important role in the procedure of variational data assimilation(VDA), sensitivity analysis, etc.,and it can be obtained by establishing the first order adjoint(FOA) and second order adjoint (SOA) models for direct model.The derivations of the FOA and SOA models of shallow water equations model are given in detail, which is based upon the Gateaux differential of functional and the concepts of the adjoint operators in Hilbert space.We obtain the result for SOA model of the shallow water equations model, which improves the theory established in the paper of Zhi Wang et al.(1992).