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The shallow water equations (SWE) are a widely used model for the propagationof surface waves on the oceans. We consider the problem of optimallydetermining the initial conditions for the one-dimensional SWE in an unboundeddomain from a small set of observations of the sea surface height. In thelinear case we prove a theorem that gives sufficient conditions for convergenceto the true initial conditions. At least two observation points must be usedand at least one pair of observation points must be spaced more closely thanhalf the effective minimum wavelength of the energy spectrum of the initialconditions. This result also applies to the linear wave equation. Our analysisis confirmed by numerical experiments for both the linear and nonlinear SWEdata assimilation problems. These results show that convergence rates improvewith increasing numbers of observation points and that at least threeobservation points are required for the practically useful results. Betterresults are obtained for the nonlinear equations provided more than twoobservation points are used. This paper is a first step in understanding theconditions for observability of the SWE for small numbers of observation pointsin more physically realistic settings.

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