## 1. Introduction

[2] Consider the 3-D acoustic one-way wave equation for upcoming wave in the frequency-space domain

where *U* is the wavefield, ω is the angular frequency, *x* is the lateral coordinate along the in-line direction, *y* is the lateral coordinate along the cross-line direction and *c* = *c*(*x*, *y*, *z*) is the velocity function.

[3] Now we consider the one-way propagator associated with equation (1), i.e., the solution of equation (1) with initial condition

For a sufficiently small vertical step Δ*z* = *z* − *z*′ (thin slab) and using the high-frequency approximation, the one-way thin slab propagator is given by *de Hoop et al.* [2000]:

where *k*_{x}, *k*_{y} are wave number and

For laterally homogeneous thin slab, i.e., *c*(, *x*, *y*) is independent of *x*, *y*, the propagator reduces to Gazdag's phase-shift operator [*Gazdag*, 1978]. In this case, the computation of equation (2) requires only one two-dimensional FFT. For inhomogeneous thin slab, however, the computation of equation (2) requires one two-dimensional FFT for each different velocity *c*(, *x*, *y*). This means a considerable computational effort. The split-step Fourier method introduced by *Stoffa et al.* [1990] requires much less computational cost by using a simple correction term applied in the ω, *x* domain to deal with lateral velocity variations. But this approach only works well for smooth velocity variations and near vertical propagation angles.

[4] *Le Rousseau and de Hoop* [2001] developed a scalar generalized-screen method which generalizes the phase-screen and the split-step Fourier methods to increase their accuracies with large and rapid lateral variations. Using two Taylor approximation and a perturbation hypothesis, this approach approximates the one-way wave operator by products of functions in space variables and functions in wave number variables. This approximation enables the dependency of equation (2) on *x*, *y* to be taken out of the integral thus resulting in a simplification of the computation. In spite of its great success, this method suffers a problem of branch points, and an integral contour deformation in the complex plane is needed.

[5] Recently, *Song* [2001] (an English translation of this paper can be obtained from chenjb@mail.igcas.ac.cn) suggested a theoretical method of expressing a multi-variable real function by products of single-variable functions. In fact, Song's method also works for complex functions. In this Letter, we will present a numerical implementation approach of Song's method for complex functions. Based on the numerical approach, we will obtain an approximation of the one-way wave operator which attains the goal of the generalized-screen method but suffers no problem of branch points.

[6] The new technique developed in this Letter can be used to construct fast 3-D wave-equation prestack depth migration algorithms. Now seismic imaging has been commonly applied to regions where geologic complexities are present. 3-D wave-equation prestack depth migration algorithms play a very important role in imaging regions with geologic complexities. However, because of the huge prestack data, the computational efficiency of the algorithms is in great demand. Therefore, the fast prestack depth migration algorithms based on this new technique will be of great significance in imaging complex geologic regions.