Since multi-component modelling provides an improved image of a target horizon, I present a 9C-3D numerical modelling approach for comparison with real seismic data, and synthetic data derived from fully elastic numerical methods. This modelling is posed in the slowness domain due to its advantages over the time domain. To accomplish this modelling, I extrapolate the known source wave.eld from the surface to the reflector, multiply it by the reflection coeffcient, and then extrapolate back to the surface. As the phase shift operator is prerequisite for performing this modelling, I compute the effective ray parameter that is used to obtain vertical slowness, an essential parameter in order to compute the phase shift operator. The presented modelling takes two data acquisition geometries into account. The first corresponds to micro-seismic and walk-away vertical seismic profiling (VSP) and the second corresponds to surface seismic methods. In order to obtain multi-component data, I build a rotation matrix based on azimuth and dip between the grid point and P-wave polarization vector. The implementation of this matrix on the extrapolated wavefield transforms the source polarization into the orientation of multi-component geophone and yields the multi-component data that can be used for analysis. Considering the first data acquisition geometry, the execution of the rotation matrix takes place before the refection coefficient is multiplied. To execute the proposed modelling for the second data acquisition geometry, it is necessary to obtain the reflection (R) and transmission (T ) coefficients in the plane wave domain. To do so, a normal for each individual plane wave based on the local velocity and vector cross product of this normal with the normal to the reflector are computed. This cross product yields a ray parameter that presently is used to compute corresponding R and T coefficients for a given plane wave.
For the sake of simplicity, I first consider isotropic media and follow the procedure described in above paragraph for a known SH wavefield. By considering this case, I reveal a problem associated with the data acquisition geometry. Further, as seismic anisotropy plays an important role in exploration field, I perform the proposed modelling for transverse isotropic media (VTI: when axis of symmetry is vertical, HTI:when axis of symmetry is horizontal) in behalf of the generally occurrence of these media in geophysical exploration field. To continue a 9C-3D modelling for anisotropic media, I obtain the R and T coefficients in plane wave domain for VTI and HTI media. Additionally, the influence of anisotropy on amplitude versus offset (AVO) analysis is also examined. For HTI media, I present two more approaches for avoiding the problems associated with the previously proposed modelling. According to the first approach, I use a relationship between the cosine of any angle with the horizontal axis and the angle of incidence considered with respect to vertical, and azimuth. Further, this relationship leads to the effective ray parameter for HTI media that is an essential component for the implementation of the proposed modelling. In the second approach, I solve the Christoffel equation for obtaining vertical slowness that is used to obtain the phase shift operator for the given media. Authentication of modelling is demonstrated in light of the physical modelling presented by another student of CREWES (Consortium for Research in Elastic Wave Exploration Seismology) for orthorhombic media.
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