In person (MC 5417) and online talk (for Zoom Link please contact ddelreyfernandez@uwaterloo.ca)

## Speaker

Professor Nicholas Kevlahan, McMaster University Department of Mathematics and Statistics

## Title

Two problems in data assimilation for the shallow water equations

## Abstract

The shallow water equations (SWE) are a widely used model for the propagation of surface waves in oceans, lakes and rivers. Common applications include modelling the propagation of tsunami waves, storm surges and flooding. In this talk we describe two data assimilation problems for the SWE, both based on sparse observations of the free surface height.

The goal of the first problem is to reconstruct the initial conditions for a surface wave. In the case of the relatively simple linear one-dimensional problem we prove a theorem that gives sufficient conditions on the number and spacing of the observations that ensure convergence to the true initial conditions. These results are confirmed numerically for the nonlinear equations. We then consider the associated two-dimensional nonlinear problem. We compare observations arranged in straight lines, in a grid, and along concentric circles, and determine the optimal number and configuration of observation points such that convergence to the true initial conditions is achieved.

In the second (ill posed) problem our goal is to determine under which conditions observations of the free surface are sufficient to reconstruct the bathymetry to a given accuracy (e.g. sufficient for modelling wave propagation). We use density-based global sensitivity analysis (GSA) to assess the sensitivity of the surface wave and reconstruction error to model parameters and second order adjoint analysis (SOA) to derive the sensitivity of the surface wave error, given the reconstructed bathymetry, to perturbations in the observations.

*This is joint work with Bartek Protas (McMaster) and Ramsha Khan (University of Stockholm)*

**References**

Kevlahan, N.K.-R. Khan, R. & Protas, B. 2019 On the convergence of data assimilation for the one-dimensional shallow water equations with sparse observations. Adv Comput Math 45(5), 3195-3216 .

Khan, R. and Kevlahan, N.K-R. 2022 Data assimilation for the two-dimensional shallow water equations: Optimal initial conditions for tsunami modelling. Ocean Model. 174, 102009.

Khan, R. and Kevlahan, N.K.-R. 2021 Variational assimilation of surface wave data for bathymetry reconstruction. Part I: algorithm and test cases, Tellus A: Dynamic Meteorology and Oceanography, 73:1, 1-25.

Khan, R. and Kevlahan, N.K-R. 2022 Variational Assimilation of Surface Wave Data for Bathymetry Reconstruction. Part II: Second Order Adjoint Sensitivity Analysis. Tellus A: Dynamic Meteorology and Oceanography, 74, 187-203.