Theory for 1D GPR data inversion for a dissipative layered medium

We present a data driven method of full waveform inversion in one dimension. This means that the inversion is carried out as a sequence of processing steps. The first step is known as Marchenko redatuming. In this step we retrieve focusing functions from the measured data. In the second step we isolate the last event in the focusing function to obtain the local reflection coefficient of a particular reflecting boundary. This is done for the dissipative and equivalent effectual model. An effectual medium amplifies a propagating wave in the same way as a dissipative medium attenuates it. From these two models the reflection coefficient of the corresponding lossless medium can be computed. This is then inverted for the electric permittivity. Once the permittivity is found, the individual layer thicknesses are obtained from the travel times. The ratio of the reflection coefficient in the physical and effectual medium provides an estimate of the attenuation in each layer from which the conductivity in each layer can be found. We show that in this case the full waveform inversion is a linear problem. We need reflection and transmission data measured at two sides of the medium. We use an unconditionally convergent iterative technique to compute the focusing functions. The method only needs the up-and downgoing parts of the electric field at the receiver levels. A 1D numerical example with a lossy model shows that the proposed GPR inversion method is effective on modeled data.