Multi-parameter waveform inversion; exploiting the structure of penalty-methods

In this talk I consider the problem of inverting waveforms for multiple medium parameters. The governing PDE is chosen to be the Helmholtz equation with compressibility and buoyancy as the unknowns. Both unknowns occur in the same equation and practice has shown it is very hard to estimate both equally accurate; the buoyancy estimate (or density if a slightly different parametrization is used) is typically much smoother than the compressibility (or velocity). Here I introduce a new waveform inversion algorithm: a full Newton-type method based on a penalty method which adds the PDE constraint as a quadratic penalty term. This method updates both the 'wavefields' and medium parameters, without explicitly solving PDE's. One of the main advantages is the availability of a sparse Hessian and exact gradient which are not the result of any PDE solves. We asses if the availability of the Hessian, which includes information about the coupling between the two medium parameters, can help reconstruct both compressibility and buoyancy.