IDENTIFICATION OF BOTTOM DEFORMATIONS OF THE OCEAN FROM SURFACE MEASUREMENTS

In this paper, we consider a general scheme to solve two different inverse problems related to oceanography, that is retrieving either a tsunami or the shape of the seabed from the measurement of the free surface perturbation. We consider two-dimensional geometries and linear potential models in the frequency regime. Such general scheme consists first of recovering the potential in the whole domain, and second computing the seeked parameter at the bottom of the ocean, which in the two inverse problems is a function involved in a more or less complicated boundary condition. The first step amounts to solving an ill-posed Cauchy problem for the Laplace or Helmholtz equation, which we regularize by using a mixed formulation of the Tikhonov regularization and the Morozov principle to compute the regularization parameter. The computation of such Tikhonov-Morozov solution is based on an iterative method consisting of solving a sequence of weak formulations which are discretized with the help of a simple Lagrange type finite element method. In the particular case of the acoustic model, we need to solve a Laplace-type equation associated with the noisy Neumann boundary data and compute the noise amplitude of its solution. A probabilistic method is proposed to obtain such amplitude of noise. Some numerical experiments show the feasibility of our strategy.