The independent set perturbation method for efficient computation of sensitivities with applications to data assimilation and a finite element shallow water model

An adjoint model for a 2D Galerkin/Petrov-Galerkin finite element (FE) shallow water (S-W) model is developed using the Independent Set Perturbation (ISP, [40]) sensitivity analysis. Its performance in a full 4-D Var setup with a limited area shallow water equations model is assessed by comparing with the adjoint model derived by the automatic di erentiation approach (TAMC), where it is used for optimising the initial conditions. It is shown that the ISP sensitivity analysis provides a very simple approach of forming the adjoint code/gradients/di erentiation of discrete forward models (even complex governing equations, discretization methods and non-linear parameterizations) and is realised using a graph colouring approach combined with a perturbation method. Importantly, the adjoint is automatically updated as the forward code continues to be developed. In the test cases, it is shown that the adjoint model using the ISP sensitivity analysis can achieve the accuracy of traditional adjoint models derived by the automatic di erentiation method (TAMC). Further comparison shows that the CPU time required for running the adjoint model using the ISP sensitivity analysis is much less than that required for the automatic di erentiation derived adjoint model since the ISP derived adjoint CPU time scales linearly with the problem size. TheISPsensitivityanalysisisfurtherappliedtoahighlynon-linearPetrov-Galerkin FE model. The perturbation size used in deriving the tangent linear model with the ISP sensitivity analysis method is then optimised and the resulting approach used to assimilate both sparse (more realistic) and dense observational data for optimising the initial conditions. A simple first order formula is developed to calculate the perturbation size for each variable, at each node and time level. By applying the ISP sensitivity method