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
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the dyadic Green's function 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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