Acoustic Modeling of Near Borehole Anomalies via the Generalised Random Transform

In this thesis we develop from first principles an analytical solution of the scattering wave equation in a 3D acoustic medium. The emerging inversion formula is analogous to a Generalized Radon Transform of the velocity structure of the medium over surfaces of constant travel time. If we assume that the scatterers are a composite of localized isolated perturbations of a constant velocity background medium, our inversion formula can be simplified to be analogous to a Radon transform of the velocity structure of the medium of interest. An inverse Radon transform is readily available and we apply this to obtain a simple expression for the scattering potential (a measure of the velocity perturbations) of the medium. We address the special data acquisition configuration of the Full Waveform Acoustic Logging (FWAL) tool and convert the inverse scattering equations into a form directly applicable to data collected by this seismic tool.

An algorithm based on our theory is applied to six synthetic 2D models. We ignore the effects of the borehole and any fluids they contain. For real data a re-scaling of the magnitude of the scattered data will have to be applied for our inversion technique to give satisfactory results. We address 2D models in this thesis, since data is cheaper to generate these than their 3D counterparts. We argue that the very acoustic nature of our acquisition tool prohibits us from discerning the direction around the borehole from which the scattering occurs, and therefore any real 3D medium will appear to be the FWAL tool as an infinite number of 2D slices along an axis of symmetry.

Five of the six models analyzed provide favorable results, which demonstrate the feasibility of our algorithm in reconstructing point scatterers in very complex formations, dipping layers and beds, pinched-out layers often prevalent in fault zones, fractured regions and metamorphosised rocks. Our algorithm did not satisfactorily image inclusions and regions with velocity gradients.

In future work, we will apply the algorithm to real data from well logging experiments. We also hope to extend the theory presented in this thesis to an elastic medium.