Variational Data Assimilation with a Variable Resolution Finite-Element shallow-water Equations Model

Keyun Zhu Supercomputer Computations Research Institute, The Florida State University, Tallahassee, Florida

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I. Michael Navon Department of Mathematics and Supercomputer Computations Research Institute, The Florida State University, Tallahassee, Florida

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Xiaolei Zou Supercomputer Computations Research Institute, The Florida State University, Tallahassee, Florida

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Abstract

The adjoint model of a finite-element shallow-water equations model was obtained with a view to calculate the gradient of a cost functional in the framework of using this model to carry out variational data assimilation (VDA) experiments using optimal control of partial differential equations.

The finite-element model employs a triangular finite-element Galerkin scheme and serves as a prototype of 2D shallow-water equation models with a view of tackling problems related to VIDA with finite-element numerical weather prediction models. The derivation of the adjoint of this finite-element model involves overcoming specific computational problems related to obtaining the adjoint of iterative procedures for solving systems of nonsymmetric linear equations arising from the finite-element discretization and dealing with irregularly ordered discrete variables at each time step.

The correctness of the adjoint model was verified at the subroutine, level and was followed by a gradient cheek conducted once the full adjoint model was assembled. VDA experiments wore performed using model-generated observations. In our experiments, assimilation was carried out assuming that observations consisting of a full-model-state vector are available at every time step in the window of assimilation. Successful retrieval was obtained using the initial conditions as control variables, involving the minimization of a cost function consisting of the weighted sum of difference between model solution and model-generated observations.

An additional set of experiments was carried out aiming at evaluating the impact of carrying out VDA involving variable mesh resolution in the finite-element model over the entire assimilation period. Several conclusions are drawn related to the efficiency of VDA with variable horizontal mesh resolution finite-element discretization and the transfer of information between coarse and fine meshes.

Abstract

The adjoint model of a finite-element shallow-water equations model was obtained with a view to calculate the gradient of a cost functional in the framework of using this model to carry out variational data assimilation (VDA) experiments using optimal control of partial differential equations.

The finite-element model employs a triangular finite-element Galerkin scheme and serves as a prototype of 2D shallow-water equation models with a view of tackling problems related to VIDA with finite-element numerical weather prediction models. The derivation of the adjoint of this finite-element model involves overcoming specific computational problems related to obtaining the adjoint of iterative procedures for solving systems of nonsymmetric linear equations arising from the finite-element discretization and dealing with irregularly ordered discrete variables at each time step.

The correctness of the adjoint model was verified at the subroutine, level and was followed by a gradient cheek conducted once the full adjoint model was assembled. VDA experiments wore performed using model-generated observations. In our experiments, assimilation was carried out assuming that observations consisting of a full-model-state vector are available at every time step in the window of assimilation. Successful retrieval was obtained using the initial conditions as control variables, involving the minimization of a cost function consisting of the weighted sum of difference between model solution and model-generated observations.

An additional set of experiments was carried out aiming at evaluating the impact of carrying out VDA involving variable mesh resolution in the finite-element model over the entire assimilation period. Several conclusions are drawn related to the efficiency of VDA with variable horizontal mesh resolution finite-element discretization and the transfer of information between coarse and fine meshes.

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