Prestack phase‐shift migration of separate offsets

Abstract

Prestack phase‐shift migration is implemented by evaluating the offset‐wavenumber ([Formula: see text]) integral using the stationary‐phase method. Thus, the stationary point along [Formula: see text] must be calculated prior to applying the phase shift. This type of implementation allows for migration of separate offsets, as opposed to migration of the whole prestack data when using the original formulas. For zero‐offset data, the stationary point ([Formula: see text]) is known in advance, and, therefore, the phase‐shift migration can be implemented directly. For nonzero‐offset data, we first evaluate the [Formula: see text] that corresponds to the stationary point solution either numerically or through analytical approximations. The insensitivity of the phase to [Formula: see text] around the stationary point solution (its maximum) implies that even an imperfect [Formula: see text] obtained analytically can go a long way to getting an accurate image. In transversely isotropic media, the analytical solutions of the stationary point ([Formula: see text]) include more approximations than those corresponding to isotropic media (i.e., approximations corresponding to weaker anisotropy). Nevertheless, the resultant equations, obtained using Shanks transform and perturbation theory, produce accurate migration signatures for strong anisotropy (η ≈ 0.3) and even large offset‐to‐depth ratios (>2). The analytical solutions are particularly accurate in predicting the nonhyperbolic moveout behavior associated with anisotropic media, a key ingredient to performing an accurate nonhyperbolic moveout inversion for strongly anisotropic media. Although the prestack correction achieved using the phase‐shift method can also be obtained using a cascade of NMO correction, dip‐moveout (DMO) correction, and zero‐offset time migration, the prestack approach can handle sharper velocity models more efficiently. In addition, the resulting operator is sharper than that obtained from the DMO method. Synthetic, including the Marmousi, data applications of the proposed prestack migration demonstrate its usefulness.

The offset-midpoint traveltime pyramid of P-waves in homogeneous orthorhombic media

“…We adopt the special Fourier transform convention used in Yilmaz and Claerbout (1980), Alkhalifah (2000a), Yilmaz (2001, p. 156), and neglecting the factor 1∕ð2πÞ in front of the inverse of Fourier transform for convenience. The stationary phase method for oscillatory integrals is discussed by Bleistein and Handelsman (1986) in detail.…”

“…In the oscillatory function, the P-wave traveltime at the stationary point is described by an offset-midpoint traveltime equation, which is also called the offset-midpoint traveltime pyramid or Cheops' pyramid (Claerbout [1985], pp. 164-166; Alkhalifah, 2000a) because of the shape of the offset-midpoint traveltime surface. This treatment of the phase-shift migration leads to the Kirchhoff prestack migration using straight rays, which is extremely efficient for time-domain migration of prestack seismic data, and subsequent migration-based velocity analysis.…”

“…The multiple integrals in phaseshift migration can be treated by the stationary phase method. Alkhalifah (2000aAlkhalifah ( , 2000b uses the stationary phase method to obtain an asymptotic solution of the multiple integral of an oscillatory function in the prestack phase-shift offset-midpoint migration for transversely isotropic media with a vertical symmetry axis (VTI). In the oscillatory function, the P-wave traveltime at the stationary point is described by an offset-midpoint traveltime equation, which is also called the offset-midpoint traveltime pyramid or Cheops' pyramid (Claerbout [1985], pp.…”

Double‐square‐root one‐way wave equation prestack tau migration in heterogeneous media

“…The table-driven method or analytical approximation to determine the point of the stationary phase introduced by Alkhalifah (2000) can be used for reference here. …”

“…Theoretically, however, almost all existing prestack time migration methods based on the double-square-root equation (Deregowski and Rocca 1981;Alkhalifah 2000;Pestana et al 2001) have limitations in laterally variant media. In this paper, we present a prestack tau migration technology that truly honours the lateral inhomogeneity under complex overburdens based on the modified double-square-root one-way wave equation.…”