Keywords: Boussinesq, bathymetry reconstruction, inverse problem, finite elements, dispersive waves, nonlinear waves.

Knowledge of the seabed configuration in coastal areas is of primary importance in many coastal engineering problems, ranging from beach erosion to the design of coastal structures. The marine systems are very dynamic. Erosion and deposition processes can easily lead to variations on submarine topography, and thus nautical charts might quickly become outdated.

The determination of bathymetry, where there are no surveys, can be carried out using observations of the elevation of the free surface and, or of the surface particle velocity conjugated with inverse modelling algorithms. The increasing development of remote sensing techniques has fostered the interest in developing numerical methods for reconstruction of the ocean bottom, from records of the free surface.

Two types of inversion methods can be used: frequency domain inversion and spatial domain inversion. Depth inversion in frequency or time domain essentially uses time stacks of surface information obtained through remote sensing methods for determining the characteristics of propagating waves: direction, frequency, wave number and, or wave velocity. On the other hand, the aim of depth inversion methods in the spatial domain is the bathymetry reconstruction when the surface information is sparse in time and dense in space, using a wave propagation model.

In this paper we present and apply a two dimensional depth inversion model in the spatial domain, based on the classical Boussinesq dispersive non-linear wave equations. The model is used for reconstruction of the bathymetry at the entrance of the Leixões Harbour on the Portuguese north-western Atlantic coast. It comprises two algorithms: a wave phase velocity inversion algorithm and a depth inversion algorithm. The wave phase velocity is computed from one or more pairs of time-lagged distributions of the free surface elevation, through a least squares method. The depth inversion algorithm computes the depth based on the knowledge of both the phase velocity and the free surface elevation at a given time instant. A finite element method with a Galerkin formulation and triangular elements is used for discretization of the differential equations of the model. Both algorithms use dense space information of the free surface elevation as input information.

This is the first part of a more general project. Part 1 consists of the algorithmic development, with numerical generation of free surface elevation profiles. Part 2 consists of the model calibration, with laboratory produced profiles. Part 3 is the field application with profiles obtained from the sea (e.g. radar, satellite). In the first part, the spatial distributions of the free surface elevation were numerically generated using the FUNWAVE computational code. Regular waves with height, period and direction of propagation typical of the Portuguese coast were used. Free surface elevation distributions in the neighbourhood of the Leixões harbour were extracted which allowed estimation of the wave velocity and the local depth distributions. An analysis of the local and global errors is produced.

The wave velocity inversion algorithm seems to be quite efficient when comparing the estimated wave velocity with the expected linear phase velocity for the same wave period. On the other hand, the seabed configuration was well recovered all over the domain, but near the boundaries the non-physical boundary conditions might have lead to some perturbation. The reconstructed seabed also displays an excessive smoothing. In future studies the filtering window size shall be reduced in order to investigate its possible smearing effect. An iterative procedure is used but the convergence criterion proved not to be the most convenient. A better convergence criterion must be defined for the model before we can proceed into the next stages of the model implementation.

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