On the observability of bottom topography from measurements of tidal sea surface height

The question of whether features of the ocean bottom topography can be identified from measurements of water level is investigated using a simplified one-dimensional barotropic model. Because of the nonlinear dependence of the sea surface height on the water depth, a linearized analysis is performed concerning the identification of a Gaussian bump within two specific depth profiles, (1) a constant depth domain, and, (2) a constant depth domain adjoining a near-resonant continental shelf. Observability is quantified by examining the estimation error in a series of identical-twin experiments varying data density, tide wavelength, assumed (versus actual) topographic correlation scale, and friction. For measurements of sea surface height that resolve the scale of the topographic perturbation, the fractional error in the bottom topography is approximately a factor of 10 larger than the fractional error of the sea surface height. Domain-scale and shelf-scale resonances may lead to inaccurate topography estimates due to a reduction in the effective number of degrees of freedom in the dynamics, and the amplification of nonlinearity. A realizability condition for the variance of the topography error in the limit of zero bottom depth is proposed which is interpreted as a bound on the fractional error of the topography. Appropriately designed spatial covariance models partly ameliorate the negative impact of shelf-scale near-resonance, and highlight the importance of spatial covariance modeling for bottom topography estimation.