Azimuth moveout for 3-D prestack imaging

Abstract

We introduce a new partial prestack-migration operator called "azimuth moveout" (AMO) that rotates the azimuth and modifies the offset of 3-D prestack data. Followed by partial stacking, AMO can reduce the computational cost of 3-D prestack imaging. We have successfully applied AMO to the partial stacking of a 3-D marine data set over a range of offsets and azimuths. When AMO is included in the partial-stacking procedure, highfrequency steeply dipping energy is better preserved than when conventional partial-stacking methodologies are used. Because the test data set requires 3-D prestack depth migration to handle strong lateral variations in velocity, the results of our tests support the applicability of AMO to prestack depth-imaging problems. AMO is a partial prestack-migration operator defined by chaining a 3-D prestack imaging operator with a 3-D prestack modeling operator. The analytical expression for the AMO impulse response is derived by chaining constant-velocity DMO with its inverse. Equivalently, it can be derived by chaining constant-velocity prestack migration and modeling. Because 3-D prestack data are typically irregularly sampled in the surface coordinates, AMO is naturally applied as an integral operator in the time-space domain. The AMO impulse response is a skewed saddle surface in the time-midpoint space. Its shape depends on the amount of azimuth rotation and offset continuation to be applied to the data. The shape of the AMO saddle is velocity independent, whereas its spatial aperture is dependent on the minimum velocity. When the azimuth rotation is small (≤20 •), the AMO impulse response is compact, and its application as an integral operator is inexpensive. Implementing AMO as an integral operator is not straightforward because the AMO saddle may have a strong curvature when it is expressed in the midpoint coordinates. An appropriate transformation of the midpoint axes to regularize the AMO saddle leads to an effective implementation.

Microlocal analysis of seismic inverse scattering in anisotropic elastic media

“…Because B prin M is homogeneous in ξ and Euler's relation, ξ, ∂ ξ B prin M = B prin M = ∓τ it follows directly that the group velocity is orthogonal to the slowness surface. Solving (2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14) reveals the formation of caustics. Caustics may form progressively in the presence of heterogeneities, or instantaneously in the presence of anisotropy even in the absence of heterogeneity.…”

“…Suppose the roots τ of (2-9) have constant multiplicity and Assumption 6 is valid microlocally on some neighborhood in T * (Z × R) \ 0. Let u in N (ν) be microlocal constituents of a solution describing the "incoming" modes, and suppose G M (µ) refers to an "outgoing" Green's function (2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18)(19). Microlocally, the single reflected/transmitted constituent of the solution is given by…”

“…Here (see (2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18) for the construction of φ M , φ N ),…”

“…In these equations we used the solution representation of the Hamilton system in (2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14). So far, the composition has not been subjected to any assumptions.…”

“…The result of this procedure is a composition of Fourier integral operators yielding seismic wavefield continuation, be it in the single scattering approximation. Relevant examples of seismic wavefield continuation are the transformation to zero offset [1] and the transformation to common (prescribed) azimuth [13]. The distribution kernel of transformation to zero offset is called dip moveout; the distribution kernel of transformation to common azimuth is called azimuth moveout.…”