On the convergence of data assimilation for the one-dimensional shallow water equations with sparse observations

The shallow water equations (SWE) are a widely used model for the propagation of surface waves on the oceans. In particular, the SWE are used to model the propagation of tsunami waves in the open ocean. We consider the associated data assimilation problem of optimally determining the initial conditions for the one-dimensional SWE in an unbounded domain from a small set of observations of the sea surface height and focus on how the structure of the observation operator affects the convergence of the gradient approach employed to solve the data assimilation problem computationally. In the linear case we prove a theorem that gives sufficient conditions for convergence to the true initial conditions. It asserts that at least two observation points must be used and at least one pair of observation points must be spaced more closely than half the effective minimum wavelength of the energy spectrum of the initial conditions. Our analysis is confirmed by numerical experiments for both the linear and nonlinear SWE data assimilation problems which reveal a relation between the convergence rate of gradient iterations and the number and spacing of the observation points. More specifically, these results show that convergence rates improve with increasing numbers of observation points and that at least three observation points are required for the practically useful results. Better results are obtained for the nonlinear equations provided more than two observation points are used. This paper is a first step in understanding the conditions for observability of the SWE for small numbers of observation points in more physically realistic settings.