Data assimilation for the two-dimensional shallow water equations: Optimal initial conditions for tsunami modelling

Accurate modelling of tsunami waves requires complete boundary and initial data, coupled with the appropriate mathematical model. However, necessary data is often missing or inaccurate, and may not have sufficient resolution to capture the dynamics of such nonlinear waves accurately. We demonstrate that variational data assimilation for the continuous shallow water equations (SWE) is a feasible approach for recovering initial conditions. We showed that the necessary conditions for reconstructing one-dimensional initial conditions in Kevlahan et al. (2019) can be extended to the maximum Euclidean distance between pairwise observations to two-dimensions. We use Sadourny finite-difference finite volume simulations to verify convergence to the true initial conditions can be achieved for observations arranged in multiple configurations, for both isotropic and anisotropic initial conditions, and with realistic bathymetry data in two dimensions. We compare observations arranged in straight lines, in a grid, and along concentric circles, and assess the optimal number and configuration of observation points such that convergence to the true initial conditions is achieved. These idealised results with simplified two-dimensional geometry are a first step towards more physically realistic settings. Recent advances in altimetry observation data now permit much denser measurements of sea surface height than is possible with a fixed buoy network. This provides the opportunity to use the method developed here for more accurate tsunami forecasts in realistic settings.