Abstract
Small-scale heterogeneity alters the arrival time of waves in a way that cannot be explained by ray theory. It is
because ray theory is a high-frequency approximation that does not take the finite-frequency of wavefields into
account. A theory based on the first Rytov approximation is
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developed wherein the effect of finite-frequency
waves is included. It is found that the developed scattering theory predicts well the arrival time of waves
propagating in media with small-scale inhomogeneity that has a characteristic length smaller in size than the width
of Fresnel zones. In the regime for which scattering theory is relevant, it is found that caustics are easily
generated in wavefields, but this does not influence the good prediction of finite-frequency arrival times of waves
by scattering theory. The regime of scattering theory is relevant when the characteristic length of inhomogeneity is
smaller than the width of Fresnel zones.
By using ray perturbation theory, a condition for the development of triplications is defined for plane wave
sources and for point sources in 2-D and 3-D media. This theory is applied to two cases of slowness media:
1-D slowness perturbation models and 2-D Gaussian random media. The focus position in 1-D slowness models
is proportional to the inverse of the square root of the relative slowness fluctuations. For Gaussian random
media, the distance at which caustics generate is dependent on the relative slowness perturbation to the power of
minus two thirds.
The developed scattering theory is tested in a 2-D numerical finite-differences experiment using models with
small-scale heterogeneity and in a 3-D physical laboratory experiment where ultrasonic waves propagate through
samples of granite with small grain-sizes of the minerals. For both kind of modelling experiments, the length-scale
of inhomogeneity of the slowness models is smaller than the width of the Fresnel zones. The results of the 2-D
and 3- D modelling experiment confirm that the developed scattering theory is better than ray theory in predicting
the arrival time of waves propagating in complex media for which the conditions for ray theory are not valid. In
general, ray theory overestimates the observed timeshifts of waves or even worse the ray theoretical timeshifts
are anti-correlated with the observed ones for waves propagating in media where the effect of scattering is
significant. By comparing the characteristic value of the Fresnel zone with the characteristic length of
heterogeneity in different experiments from exploration seismics and seismology, it is shown that the resolution in
present-day seismological tomography is that the limits of the validity of ray theory. With an emphasis on surface
wave tomography, I show that it is important to implement the effect of finite-frequency waves in tomographic
imaging techniques in order to retrieve small-scale structured Earth models with the correct theory.
The theory for the scattering of waves is applied in global surface wave tomography for Love waves at 40 s and
150s. The estimated tomographic surface wave models derived from ray theory and scattering theory are similar,
because a restrictive regularisation condition is incorperated in the inversion so that structures with length-scales
smaller than the Fresnel zones are mostly suppressed.
I treat a special case of inverse problem theory, namely the spectral leakage problem. The term spectral leakage
indicates that observed data affected by structures with a length-scale that is not account for in a given inverse
can leak into the long-wavelength structures that are part of the estimated model. In the case of global surface
wave tomography, surface wave scattering theory is used in an inversion including the spectral leakage correction
of phase velocity measurements for Love waves at periods of 40 s and 150 s. The global phase velocity models
from the spectral leakage inversion are obtained without applying any damping, but they show, however, a good
correlation with tectonic features such as plate boundaries, ridges and trenches.
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