A Study On The Application Of Adijoint Method For Numerical Models Of Tide And Sea Temperature

Recent development in the field of variational data assimilation has demonstrated that the adjoint approach is one of potential methods for fitting numerical model to observations. In this thesis the theoretical aspects and applications of adjoint method to the oceanographic models are studied. Two methods are applied in deriving the adjoint equations. One is the Lagrange multiplier method and another method is based upon the Gateaux differential of functional (called Geaux differential method for short). Two ways to formulate the discretized adjoint equations and gradients of cost functional are discussed. In the first (inite difference of adjoint, one derives the continuous adjoint equations from the linearized continuous forward model equations and then formulates the continuous adjoint equations and gradient of cost functional. In the second (djoint of finite difference, one derives the finite-difference adjoint equations and gradient directly from the finite difference of the forward model. The latter has been widely accepted as a valid way to obtain the discretized adjoint equations and gradients of cost functional. Furthermore the relation of the two adjoint systems is investigated by using a simple model in the thesis. Moreover the problems concerning the first guess and multiplicity for the optimal adjoint system are examined in the thesis. The variational adjoint method is applied to the assimilation of the observed data into the two dimensional non-linear tidal model, in which horizontal kinematic nonlinearities, nonlinear bottom friction and horizontal eddy diffusion are included. The adjoint model is developed from the discretized tidal model formulated on Arakawa C-grid with an ADI finite difference scheme which can speed up the calculation and decrease the need of disk space. The open boundary conditions are optimized by incorporating the data from tide stations with/without TOPEX/POSEIDON altimeter data with the tidal model. It is shown that improved numerical results are obtained after the open boundary conditions are optimized. The variational adjoint method is also used for assimilating the observed data into the sea surface temperature (SST) numerical models. The frame of SST model adopted here is based on Wang and Su (1992). The initial field of SST is estimated by assimilating the SST ship report data into the model. It shows that the adjoint calculations have successfully improved the accuracy of SST hindcasting. It is emphasized at the end of the thesis that future studies in relation with the application of second adjoint model should be carried out in order to solve the problems arised from the assimilation procedure of the adjoint systems (herein called first order adjoint).