Abstract
Scattering of surface waves in a three dimensional layered elastic medium with
embedded heterogeneities is described in this thesis with the Born approximation. The
dyadic decomposition of the surface wave Green's function provides the crucial element
for an efficient application of Born theory to surface wave scattering. This is
because the dyadic Green's function
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allows for an efficient bookkeeping of the different
processes that contribute to the scattered surface wave: excitation, propagation, scattering
(conversion), and oscillation. One can argue that the most crucial (and surprisingly
also the simplest) expression in this thesis is equation (3) of chapter 2. The resulting
surface wave scattering theory for buried heterogeneities in a flat geometry (chapter 2),
can easily be extended to incorporate the effects of surface topography (chapter 3), and
a spherical geometry (chapters 6 and 7).
In practice, the Born approximation imposes a lower limit on the periods that can
be analyzed. This limit depends both on the properties of the heterogeneity and on the
source receiver separation. An analysis of the surface wave coda recorded in stations of
the NARS array shows that the surface wave coda level differs substantially for different
regions. For paths through eastern and middle Europe, the Born approximation
breakS down for periods shorter than 30 s., while for paths through the western Mediterranean
periods as short as 20 s. can be analyzed with linear theory (chapter 8).
In exploration seismics, linear theory is usually used to establish a relation
between the heterogeneity and the reflected waves, as well as for the inversion of these
reflection data. It is therefore not surprising that the surface wave coda can in principle
be used to map the heterogeneity in the Earth, with an inversion scheme which is reminiscent
to Kirchoff migration as used in exploration seismics (chapter 2). In a simple
field experiment the feasibility of such an inversion scheme is established (chapter 4).
It is also possible to formulate the waveform inversion of surface wave data as a (huge)
matrix problem. The least squares solution of these matrix equations can iteratively be
constructed. These reconstructed models have the same characteristics as the models
found with a simple holographic inversion (chapter 8).
Inversion of the surface wave coda recorded in stations of the NARS array produce
chaotic models of scatterers which are difficult to interpret unambiguously. Apart
from a lack of enough data to perform a good imaging, this inversion is hampered by an
appreciable noise component in the surface wave coda. This noise level might be
acceptable if the data set were redundant, so that this noise component can be averaged
out. However, the 42 available seismograms lead to an underdetermined system of
linear equations, which make it likely that the noise in the surface wave coda introduces artifacts in the reconstructed model (chapter 9).
Born theory for surface waves describes the distortion of the wavefield due to the
heterogeneity of the medium. This distortion consists of true surface wave scattering
due to abrupt lateral inhomogeneities, as well as a distortion of the direct surface wave
due to smooth variations of the heterogeneity. Up to first order, ray geometrical effects
follow from linear scattering theory (chapter 5). Furthermore, the scattering coefficient
for forward scattering of unconverted waves is proportional to the phase velocity perturbation
of these waves (chapter 3). This makes it possible to reconstruct phase velocity
fields for surface waves using a large scale linear waveform inversion of the direct
surface wave (chapter 8).
This inversion is applied to the direct surface wave train recorded in stations of
the NARS array. This results in detailed reconstructions of the phase velocity of the
fundamental Rayleigh mode. In this inversion, a variance reduction of approximately
40% is achieved. By combining this information for different frequencies, detailed
models of the S-velocity under Europe and the Mediterranean are reconstructed
(chapter 9). With the present data set, the resolution of this model differs considerably
from region to region. The only way to overcome this restriction is to use more data,
which can be realized by employing dense networks of digital seismic stations.
There is still a considerable amount of research to be performed on scattering
theory of elastic waves. Apart from the restriction of linearity, the theory presented in
this thesis is only valid in the far field. This means that the inhomogeneity should be
several wavelengths removed from the source and the receiver (and their antipodes). In
practice, this is a troublesome limitation, because seismic stations are often located on
top of heterogeneities, and earthquakes usually occur in heterogeneous areas such as
subduction zones. The interaction terms are valid both in the far field and in the near
field (chapter 7), so that in order to resolve the far field restriction, the propagator terms
need to be investigated. Future theoretical research should also address the problem of
conversions between surface waves and body waves. This issue is related to the near
field problem, because in the near field the concepts of "surface waves" and "body
waves" are poorly defined.
It would be interesting to use portable seismic stations for local investigations by
recording scattered surface waves in the vicinity of strong lateral variations in the crust
and upper mantle. In this way, it should be possible to probe tectonic features such as
subduction zones using scattered surface waves. The waveform inversions of the direct
surface waves, as presented in this thesis, can be applied to other regions of the Earth
with a good coverage with digital seismic stations (e.g. Japan, the continental US), and
possibly for lower frequencies on a global scale. In this way, large scale waveform
inversions for both the phase and amplitude of surface wave data may dramatically
increase our knowledge of the Earth's interior
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