On the data assimilation of initial distribution for 2-dimensional shallow-water equation model

The utilization of data assimilation (DA) techniques is prevalent in marine meteorology for the purpose of estimating the complete state of the system. This is done to address the practical limitations associated with measurement data, which can only be specified at finite number of discrete points within a limited domain. We develop an efficient DA algorithm to reconstruct the initial state of the shallow-water equations (SWE) within a 2-dimensional rectangular domain using sparse spatial measurement data. Our algorithm takes into account both the complete Coriolis force and the ocean bottom topography in the SWE model, resulting in accurate recovery of the initial status. After establishing the uniqueness of the solution to the nonlinear SWE with appropriate boundary conditions, we proceed to establish the conservation laws for the suitably defined energy quantity for this traveling wave system. This generalization of the known conservation laws for the simplified SWE system which ignores the Coriolis force and topography, allows us to reveal the influence of nonconstant sea floor topography on wave propagation. In order to restore the initial state through the minimization of a cost functional using DA techniques, we proceed by deriving the adjoint problem for our iteration process. Additionally, we establish a discrete scheme for the governing equations in the Arakawa C-grid framework, from which we rigorously derive the error associated with energy conservation in discrete form. The numerical implementations are also provided to validate our proposed scheme through the verification of energy conservation and the reconstruction effect of the initial state for various configurations.