Adjoint symmetry for Inverse ADCIRC

ADCIRC, a finite element circulation model for shelves, coasts and estuaries, is used for weak constraint variational data assimilation. That is, data will be smoothed in space and time using ADCIRC and boundary conditions as a weak constraints; the smoother is defined in terms of a quadratic penalty functional which is composed of dynamic, boundary condition and data error terms. The assimilation will be effected by the Inverse Ocean Modeling system (IOM). This system solves the nonlinear Euler-Lagrange (EL) problem using the iterated representer algorithm, which makes large, nonlinear but functionally smooth optimization problems feasible through Picard iterations on linear approximations of the nonlinear problem and by making preconditioned searches in the "data subspace" at each iterate. Much of the IOM is modular; the significant components required by the modeler are an iteration scheme and an adjoint operator. The iteration scheme has already been established; the purpose of this paper is to discuss the derivation and verification of the adjoint. Owing to the scientific purposes of data assimilation, we define the penalty functional in terms of the primitive formulation of the shallow water equations, rather than the wave formulation used by ADCIRC. This fundamental decision leads to a number of difficulties in the derivation of the adjoint and consequential loss of adjoint symmetry, and thus to a suboptimal solution. We show that this cannot be avoided without loss of the scientific objective of the project. We also identify and quantify the specific characteristics of the ADCIRC wave equation formulation which preclude computing THE optimal solution, and show that the suboptimal solution we find is still of scientific value.