Earth Resources Laboratory :: Abstracts

Scattering of Elastic Waves using Non-Orthogonal Expansions

Matthias Georg Imhof

Submitted to the Department of Earth, Atmospheric, and Planetary Sciences on June 29, 1996 in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Geophysics

Abstract

This thesis is concerned with scattering of acoustic and elastic waves from scatterers embedded in a homogeneous background. The scatterers and the background can be a mixture of fluid and solid domains, e.g. solid scatterers submerged in water. The background will always be a homogeneous half- or fullspace.

Commonly, wavefields are expanded into an orthogonal set of basis functions, e.g. planar or cylindrical waves. Unfortunately, these expansions converge rather slowly for complex geometries. The new approach enhances convergence by summing multipole solutions to the wave equation with different centers of expansions. For this reason, the method is also called Multiple MultiPole expansions (MMP) or Generalized Matching Technique (GMT). The non-orthogonal expansion functions allow irregularities of the wavefields (e.g. due to a rough boundary) to be resolved locally from a nearby center of expansion. This means that the wavefields are expanded into a non-orthogonal set of basis functions. The incident wavefield and the fields induced by the scatterers are matched by evaluating the boundary conditions at discrete matching points along the domain boundaries. Due to the non-orthogonal expansions, no unique answer can be found. Instead, more matching points than actually needed are used. The resulting overdetermined system is solved in the least-squares sense.

Since there are free parameters such as location and number of expansion centers as well as kind and orders of expansion functions used, numerical experiments are performed to measure the performance of different discretizations. An empirical set of rules governing the choice of these parameters is found from these numerical experiments. The resulting scheme is thoroughly tested against numerical experiments performed by finite differences and physical experiments in an ultrasonic watertank. As an application, the method is used to study the effects of shallow-subsurface cavities on reflection seismic data.

To account for heterogeneous scatterers, a hybrid scheme with finite elements is devised. Multiple multipole expansions are used to expand the scattered fields in homogeneous scatterers and in the background. Contrarily, the wavefields inside heterogeneous scatterers are modelled by the finite element method. By condensation, the finite element regions are then collapsed into superelements directly coupling the MMP expansions.