Variational Assimilation of Surface Wave Data for Bathymetry Reconstruction. Part II: Second Order Adjoint Sensitivity Analysis

Data assimilation methods have been proposed as a technique for reconstructing ocean bathymetry from observations of surface waves. To better understand this technique, we use second order adjoint (SOA) variational analysis to derive the sensitivity of the surface wave error to perturbations in the observations (such as their number, spacing and position relative to bathymetry profiles), given the reconstructed bathymetry. We apply SOA to the data assimilation scheme for the one-dimensional shallow water equations for bathymetry detection introduced in Khan and Kevlahan (2021). We derive the Hessian of a cost function J representing the error between forecast surface wave and the observations. We then use SOA to derive the sensitivity of the surface wave error given the reconstructed bathymetry to perturbations in the observations for both a compactly supported Gaussian bathymetry, and a sandbar profile bathymetry. We investigate the correlation between (i) low sensitivity of the surface wave given the reconstructed bathymetry, to the observations, and (ii) the error in the bathymetry reconstruction, as well as the sensitivity of the data assimilation scheme to perturbations of its parameters. Additionally, we determine whether the conclusions in Khan and Kevlahan (2021) for bathymetry reconstruction can be verified by the present sensitivity analysis. We observe that relatively large errors in the bathymetry reconstruction and large relative amplitudes of the Gaussian and sandbar bathymetry profiles are associated with higher sensitivity of the surface wave reconstruction error to the observations. However, sensitivity decreases when the observation network has a greater coverage of the bathymetry. Significantly, the sensitivity of the surface wave to the observations is orders of magnitude lower than the bathymetry reconstruction error itself. These results suggest optimal configurations of surface wave observations, help minimise costs for making observations, and could enhance the accuracy of tsunami models.