Improved Boussinesq-Type Equations for Highly-Variable Depth ∗
Boussinesq type models | highly variable coefficients | refocusing
Intermediate depth, Boussinesq-type modeling is used to generalize previously known results for surface water waves propagating over arbitrarly shaped topographies. The improved reduced wave model is obtained after studying how small changes in the linear dispersion relation (over a ﬂat bottom) can become dramatically important in the presence of a highly-ﬂuctuating topography. Numerical validation of the dispersive properties, regarding several possible truncations for the reduced models, are compared with the complete (non-truncated) linear potential theory model. Moreover, linear L2 estimates are extended from the analysis of KdV-type models to include the improved Boussinesq systems in contrast with potential theory. Discrepancies observed among the diﬀerent possible reduced models become even more important in the waveform inversion problem. The time reversal technique is used for recompressing a long ﬂuctuating signal, representing a highly scattered wave that has propagated for very long distances. When properly backpropagated (through a numerical model), the scattered signal refocuses into a smooth proﬁle representing the onset of the ocean’s surface disturbance. Previous Boussinesq models underestimate the original disturbance’s amplitude. The improved Boussinesq system agrees very well with the full potential theory predictions.