Direct Nonlinear Wave-equation Tomography: An Initial Study

Abstract:

We consider the problem of wave equation tomography, formulated in the
frequency domain, including scattering. Via a resolvent estimate, we
can obtain a corresponding time-domain formulation using strictly
finite-frequency surface data. We propose a method and algorithm for
direct nonlinear reconstruction of the wavespeed, which we base on
so-called d-bar techniques. The key component is the introduction of a
scattering transform which nonlinearly connects the wavespeed with the
surface data and so-called complex geometrical optics (CGO) solutions.
These CGO solutions satisfy a Lippmann-Schwinger-type equation with a
Fadeev's Green's function.

We solve a boundary integral equation, while introducing a spectral
parameter in its kernel (which is directly related to Fadeev's Green's
function), and, using the data, obtain so-called (non-physical) CGO
solutions at the surface; with these solutions, and the data, we then
recover the scattering transform as a function of this spectral
parameter. The scattering transform appears in an integral equation
following a non-local Riemann-Hilbert problem, the solution of which
determines the wavespeed via an expansion in the spectral parameter.

To mitigate the ill-posedness of the inverse problem we introduce a
regularization approach based on truncating the scattering
transform. This affects the resolution. We then argue that one can
increase the resolution by applying a residual iterative
reconstruction with guaranteed convergence making use of iterative
regularization via hierarchical compression.