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Elastic Wave Scattering in Laterally Inhomogeneous Mediaby Submitted to the Department of Earth, Atmospheric, and Planetary Sciences on January 28, 1991 in partial fulfillment of the requirements for the degree of Doctor of Philosophy ABSTRACT
The earth is often modeled as a series of simple homogeneous layers. Such
an approach can lead to synthetic seismograms which match the dominant arrivals
in the field data very well, but lack the random travel time and amplitude
fluctuations and signal generated noise commonly observed on seismic recordings.
These secondary features are often due to scattering from small-scale variations
in the earth. The small-scale variations are too numerous and distributed
too irregularly to allow deterministic characterization, so these features
are often characterized by their statistical distribution. This thesis is
concerned with modeling elastic waves in randomly heterogeneous media.
We first explore the general principles and assumptions concerning statistical
characterization and introduce several commonly used statistical models. Both
analytical and numerical techniques have been applied to this problem. Most
analytical techniques assume scattering is weak and use the Born or Rytov
approximation to generate relatively simple closed form solutions. These solutions
can be limiting in some applications because they neglect the effects of multiple
scattering, and assume the incident wave travels through a smooth background
medium. In the random media studied here, it is shown that these assumptions
can cause serious errors in the amplitude and phase of the scattered wavefield.
In order to investigate these errors, a new numerical technique is developed.
The technique starts with the elastodynamic equation of motion. Using the
Born approximation and perturbation analysis, the elastic wave is reduced
to a single scattering wave equation which can be solved with finite differences.
The utility of the new technique is that both the single and multiple scattering
(as calculated by conventional finite difference techniques) solutions can
be generated for the same complex velocity model. In Chapter 3, this is done
for two different random media. The first is an impedance scattering medium;
a medium which has impedance variations, but no velocity variations. In such
a medium, the dominant scattering mechanism is back scattering and the efficiency
which energy is scattered varies inversely with the size of the heterogeneity.
In this medium, the two solutions (single and multiple scattering) agreed
well, except around the first arrival. Near the first arrival, the amplitude
of the single scattering solution is consistently greater than the multiple
scattering solution. This is a consequence of the Born approximation, which
does not account for the removal of energy in the incident wave due to scattering.
The general shape and arrival time of the scattered field is consistent with
the multiple scattering solution.
In the second model, the material properties were chosen so that the medium
contained significant velocity anomalies, but almost no impedance anomalies.
Because scattering is stronger in this medium, agreement between the two solutions
is not as good as the previous case. Again, the single scattering solution
had too much energy in the first arrival, which in turn led to an overestimated
scattered field. Unlike the previous example, the velocity anomalies also
created significant travel time differences between the two solutions. These
errors were present in both the scattered and incident waves and occurred
because the Born approximation assumes the incident wave travels in the background
field (which is often assumed to be homogenous).
It is generally agreed that the Earth's crust and lithosphere have heterogeneities.
However, the distribution and exact nature of these heterogeneities has not
yet been resolved. Using the techniques presented in this thesis and data
from NORSAR and NORESS arrays we develop a model for the statistical heterogeneities
present under Fennoscandia. In the course of choosing the final model, we
investigated many randomly heterogeneous models. We began with a simple, single
layered model with a Gaussian autocorrelation function. We also considered
other single layered models with more roughness, like that proposed by Frankel
and Clayton (1986), as well as multi-layered models like that proposed by
Flatté and Wu (1988). Based on coherency measurements and travel time
and amplitude fluctuation, we propose that the random velocity variations
in the lithosphere can be modeled as a three layered random medium. Satisfactory
results were obtained when the power spectrum of the fluctuations in the uppermost
layer (0 - 3 km) was a bandlimited white spectrum (0.5 km-1<
| k | > 1.1 km-1, where k
is the wavenumber vector) and the rms velocity variation was 2%. The middle
layer was meant to simulate the remaining portion of the crust (3 - 35 km)
and the fluctuations in this layer were described by the 0th order von Kármán
function. The correlation length of the von Kármán function was
10 km and there was 3% rms variation in velocity. The third layer extended
from the base of the crust to a depth of 250 km and was characterized by an
anisotropic Gaussian correlation function. The horizontal and vertical correlation
lengths in this region were 20 km and 5 km, respectively, and there was 2%
rms variation in velocity.
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