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Scattering of Elastic Waves using Non-Orthogonal Expansionsby 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. Return to Theses Return to ERL Home Updated: May, 1999
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