A multigrid methodology for assimilation of measurements into regional tidal models

Abstract

This paper presents a rigorous, yet practical, method of multigrid data assimilation into regional structured-grid tidal models. The new inverse tidal nesting scheme, with nesting across multiple grids, is designed to provide a fit of the tidal dynamics to data in areas with highly complex bathymetry and coastline geometry. In these areas, computational constraints make it impractical to fully resolve local topographic and coastal features around all of the observation sites in a stand-alone computation. The proposed strategy consists of increasing the model resolution in multiple limited area domains around the observation locations where a representativeness error is detected in order to improve the representation of the measurements with respect to the dynamics. Multiple high-resolution nested domains are set up and data assimilation is carried out using these embedded nested computations. Every nested domain is coupled to the outer domain through the open boundary conditions (OBCs). Data inversion is carried out in a control space of the outer domain model. A level of generality is retained throughout the presentation with respect to the choice of the control space; however, a specific example of using the outer domain OBCs as the control space is provided, with other sensible choices discussed. In the forward scheme, the computations in the nested domains do not affect the solution in the outer domain. The subsequent inverse computations utilize the observation-minus-model residuals of the forward computations across these multiple nested domains in order to obtain the optimal values of parameters in the control space of the outer domain model. The inversion is carried out by propagating the uncertainty from the control space to model tidal fields at observation locations in the outer and in the nested domains using efficient low-rank error covariance representations. Subsequently, an analysis increment in the control space of the outer domain model is computed and the multigrid system is steered optimally towards observations while preserving a perfect dynamical balance. The method is illustrated using a real-world application in the context of the Philippines Strait Dynamics experiment.

Appendix

1.1 Index notation

A ∈ ℂ^m×n complex m×n matrix, ${\bf{i}}_{K} \in \mathbb{N}^K$ a set ${\bf{i}}_{K} = \{ i_k \}_{k=1}^K$, i _k  ∈ { 1, 2, ... }, $({\bf{a}})_{{\bf{i}}_{K}} \in \mathbb{C}^{K}$ complex vector of length K containing the i _K th entries of a, $({\bf{a}})_{{\bf{i}}_{K}} = \big[ a_{i_1}, a_{i_2}, \ldots, a_{i_K} \big]^T$, $({\bf{A}})_{{\bf{i}}_{K}, {\bf{j}}_{L}} \in \mathbb{C}^{K \times L}$ complex K×L matrix containing the entries in the i _K th rows and j _L th columns of matrix A,

$$({\bf{A}})_{{\bf{i}}_{K}, {\bf{j}}_{L}} = \left[ \begin{array}{cccc} a_{i_1, j_1} & a_{i_1, j_2} & \ldots & a_{i_1, j_L} \\ a_{i_2, j_1} & a_{i_2, j_2} & \ldots & a_{i_2, j_L} \\ \vdots & \ddots & \ldots & \vdots \\ a_{i_K, j_1} & a_{i_K, j_2} & \ldots & a_{i_K, j_L} \\ \end{array} \right]_{K \times L}.$$