Modelling and inversion of seismic data in anisotropic elastic media

Seismic data is modeled in the high frequency limit We consider general anisotropic media and our method is also valid in the case of multipathing caustics The data is modeled in two ways First using the Kirchho approximation where the medium is assumed to be piecewise smooth and reection and transmission occurs at the interface Secondly the data is modeled using the Born approximation in other words by a linearization in the medium parameters The main result is a characterisation of seismic data We construct a Fourier integral operator and a reectivity function which is a function of subsurface position and scattering angle and azimuth such that the data is given by the invertible ourier integral operator acting on the reectivity function Using this new transformation of seismic data to subsurface position angle coor dinates we obtain the following results on the problem of reconstructing the medium coecients Given the medium above the interface in the Kirchho approximation one can reconstruct the position of the interface and the angular dependent reection coecients on the interface We also obtain a criterium to determine whether the medium above the interface the background medium in the Born approximation is correctly chosen These results are new in medium with caustics In the Born approximation the singular medium perturbation can be reconstructe

Generalized Fourier Integral Operators on spaces of Colombeau type

Generalized Fourier integral operators (FIOs) acting on Colombeau algebras are defined. This is based on a theory of generalized oscillatory integrals (OIs) whose phase functions as well as amplitudes may be generalized functions of Colombeau type. The mapping properties of these FIOs are studied as the composition with a generalized pseudodifferential operator. Finally, the microlocal Colombeau regularity for OIs and the influence of the FIO action on generalized wave front sets are investigated. This theory of generalized FIOs is motivated by the need of a general framework for partial differential operators with non-smooth coefficients and distributional data

Directional Decomposition of Transient Acoustic Wave Fields

Electrical Engineering, Mathematics and Computer Scienc

Seismic inverse scattering in the downward continuation approach

Seismic data are commonly modeled by a linearization around a smooth background medium in combination with a high frequency approximation. The perturbation of the medium coefficient is assumed to contain the discontinuities. This leads to two inverse problems, first the linearized inverse problem for the perturbation, and second the estimation of the background, which is a priori unknown (velocity estimation). Here we give a reconstruction formula for the linearized problem using the downward continuation approach. The reconstruction is done microlocally, up to an explicitly given pseudodifferential factor that depends on the aperture. Our main result is a characterization of the wave-equation angle transform, derived from downward continuation, that generates the common image point gathers as an invertible Fourier integral operator, microlocally. We show that the common image point gathers obtained with this particular angle transform are free of so called kinematic artifacts, even in the presence of caustics. The assumption is that the rays in the background that are associated with the reflections due to the medium perturbation are nowhere horizontal. Finally, making use of the mentioned angle transform, pseudodifferential annihilators of the data are constructed. These annihilators detect whether the data are contained in the range of the modeling operator, which is the precise criterion in migration velocity analysis to determine whether a background medium is acceptable, even in the presence of caustics

Curvilinear wave-equation angle transform: Caustics, turning rays, absence of kinematic artifacts

Migration of seismic reflection data to common image-point gathers is an integral part of both migration velocity analysis (MVA) and amplitude (AVA) analysis. Its applicability in complex geology depends on whether these gathers will be artifact free, and is related to the formation of caustics and turning `rays' due to the heterogeneity of the velocity model used. Here, we discuss an angle transform — which by methods of (survey-sinking or shot-geophone) wave-equation migration maps data into image gathers — in special curvilinear coordinates that remains artifact free in the presence of caustics while allowing turning `rays' to illuminate steeply dipping reflectors

Kinematics of shot-geophone migration

Recent analysis and synthetic examples have shown that many prestack depth migration methods produce nonflat image gathers containing spurious events, even when provided with a kinematically correct migration velocity field, if this velocity field is highly refractive. This pathology occurs in all migration methods that produce partial images as independent migrations of data bins. Shot-geophone prestack depth migration is an exception to this pattern: each point in the prestack image volume depends explicitly on all traces within the migration aperture. Using a ray-theoretical analysis, we have found that shot-geophone migration produces focused (subsurface-offset domain) or flat (scattering-angle domain) image gathers, provided there is a curvilinear coordinate system defining pseudodepth with respect to which the rays carrying significant energy do not turn, and that the acquisition coverage is sufficient to determine all such rays. Although the analysis is theoretical and idealized, a synthetic example suggests that its implications remain valid for practical implementations, and that shot-geophone prestack depth migration could be a particularly appropriate tool for velocity analysis in a complex structure