Theory for 1D full waveform inversion of surface GPR data

In one dimension, full waveform inversion is shown to be a linear problem under several conditions. I show that if the magnetic permeability can be assumed constant and electric conductivity to be zero, measuring the magnetic field at the surface or in the air suffices as input data. I present the theory using integral equations that describe the electric field inside the medium in terms of contrast sources. The electric field inside the medium can be computed from the measured magnetic field by solving a Marchenko equation. Once this field is known only the contrast function is unknown and can be found by matrix inversion. If the electric field is also measured the inverse problem can be solved recursively. In one dimension depth is intrinsically unknown and I use recording time as a replacing coordinate. After the electric permittivity is known as a function of one-way travel time from surface to a depth level inside the medium, the depth level can be found by an integral. This produces electric permittivity as a function of depth and full waveform inversion is complete. A simple numerical example demonstrates the method.