Introduction

Implicit 2-D finite-difference wavefield extrapolation Claerbout (1985) has proved itself as a robust, accurate migration method. It naturally and efficiently deals with lateral variations in velocity without the need for asymptotic approximations, such as ray-tracing. The implicit formulation can also be bullet-proofed to ensure unconditional stability Godfrey et al. (1979).

Unfortunately, despite the widespread popularity of 3-D seismology, implicit 3-D wavefield extrapolation has yet to find wide-spread application. Whereas 2-D extrapolation requires the inversion of a tridiagonal system, the simple extension from 2-D to 3-D leads to a blocked tridiagonal system, which is prohibitively expensive to solve.

Typically, the matrix inversion problem is avoided by an explicit finite-difference approach Holberg (1988). Explicit extrapolation has proved itself effective for practical 3-D problems; since stable explicit filters can be designed Hale (1990b), and McClellan filters provide an efficient implementation Hale (1990a). However, unlike implicit methods, stability can never be guaranteed if there are lateral variations in velocity Etgen (1994). Additionally, accuracy at steep dips requires long explicit filters, which can conflict with rapid lateral velocity variations, and can be expensive to apply.

The problem can also be avoided by splitting the operator to act sequentially along the x and y axes. Unfortunately this leads to azimuthal operator anisotropy, and requires an additional phase correction operator Graves and Clayton (1990); Li (1991). Zhou and McMechan (1997) have presented an alternative to the traditional 45$^\circ$ equation, with form similar to the 15$^\circ$ equation plus an additional correction term. Although splitting their equations results in less azimuthal anisotropy than with the standard 45$^\circ$ equation, the splitting approximation is still needed to solve the equations.

Mixed domain methods such as Fourier finite-differences Ristow and Ruhl (1994), which shuttle between the $(\omega,x)$ and $(\omega,k)$ domains, often depend on an implicit extrapolation step in the $(\omega,x)$ domain. Although these implicit operators have only a residual effect, splitting errors may still cause unacceptable reflector mispositioning. The Fourier finite-difference plus interpolation method Biondi (2000) reduces operator anisotropy by extrapolating once from a reference velocity field above the medium velocity, once from a reference velocity below the medium velocity, and interpolating between the two. Although very accurate, this algorithm is also significantly more expensive than conventional $(\omega,x)$ migration algorithms, since it requires two $(\omega,x)$ extrapolations as well as multiple $(\omega,k)$ reference fields.

In this chapter, I apply helical boundary conditions to the implicit operators at the heart of $(\omega,x)$ migration, showing how this can lead to azimuthally isotropic migration impulse responses without the need for either additional phase-correction operators or multiple passes of the finite-difference operator.