Radar data collected on two sides of a horizontally dissipative layered medium are required to invert for the medium parameters. The two-sided reflection and transmission responses are reduced to two single-sided reflection responses. One is the measured dissipative medium response, and the other is the reflection response of the corresponding effectual medium, which has negative dissipation. Marchenko-type equations are solved using these two reflection responses. The obtained focusing functions in the dissipative and effectual media are used to invert for the permittivity and the permeability under the assumption of weak dissipation in reflection. Once these parameters are known, the travel times are used to estimate the layer thicknesses. Finally, the focusing functions are used to estimate the conductivity in each layer. The method does not require any model information and runs as a fully automated process. A numerical example shows that the method works well for a horizontally dissipative layered medium. Statistical analysis for several noise models shows that the method is robust at least up to 40 dB additive and multiplicative white noise. ... Read more

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. ... Read more