Wavefield extrapolation by nonstationary phase shift

The phase shift method of wavefield extrapolation applies a phase shift in the Fourier domain to deduce a scalar wavefield at one depth level given its value at another. The phase shift operator varies with frequency and wavenumber and assumes constant velocity across the extrapolation step. We use nonstationary filter theory to generalize this method to nonstationary phase shift (NSPS) which allows the phase shift to vary laterally depending upon the local propagation velocity. For comparison, we derive the popular PSPI (phase shift plus interpolation) method in the limit of an exhaustive set of reference velocities. NSPS and this limiting form of PSPI can be written as generalized Fourier integrals which reduce to ordinary phase shift in the constant velocity limit. However, only NSPS has the physical interpretation of a laterally varying phase shift which forms the scaled, linear superposition of impulse responses (i.e. Huygen’s wavelets). The difference between NSPS and PSPI is clear when they are compared in the case of a piecewise constant velocity variation. Define a set of windows such that the j window is unity when the propagation velocity is the j distinct velocity and is zero otherwise. NSPS can be computed by applying the window set to the input data to create a set of windowed wavefields, each of which is phase shift extrapolated with the corresponding constant velocity, and the extrapolated set is superimposed. PSPI proceeds by phase shift extrapolating the input data for each distinct velocity, applying the j window to the j extrapolation, and superimposing. PSPI has the unphysical limit that discontinuities in the lateral velocity variation cause discontinuities in the wavefield while NSPS shows the expected wavefront “healing”. We then formulate a finite aperture compensation for NSPS which has the practical result of absorbing lateral boundaries for all incidence angles. Wavefield extrapolation can be regarded as the crosscorrelation of the wavefield with the expected response of a point diffractor at the new depth level. Aperture compensation simply applies a laterally varying window to the infinite, theoretical diffraction response. The crosscorrelation becomes spatially variant, even for constant velocity, and hence is a nonstationary filter. The nonstationary effects of aperture compensation can be simultaneously applied with the NSPS extrapolation through a laterally variable velocity field. INTRODUCTION In a general context, wavefield extrapolation refers to the mathematical technique of advancing a wavefield through space or time. Such techniques can be used in both seismic migration and seismic modeling. In this paper, we will restrict the scope of wavefield extrapolation to the problem of deducing a scalar wavefield at one depth level in the earth given knowledge of its properties at another level. We also assume that the wave propagation velocity, v, depends only on the lateral spatial coordinates, (x,y), and not on the depth, z. Consequently, our technique is intended for use in a recursive scheme in which vertical velocity variations are handled, in the usual manner, through Margrave and Ferguson 30-2 CREWES Research Report — Volume 9 (1997) an appropriate choice of depth levels and only lateral velocity variations are directly addressed by our theory. Wavefield extrapolation by phase shift (Gazdag, 1978) has many desirable properties and one overriding difficulty. On the positive side, the phase shift operator is theoretically exact for constant velocity, unconditionally stable, shows no grid dispersion, and is accurate for all scattering angles. (We prefer the term scattering angle to the more commonly used dip because the latter is often confused with the geologic dip of reflectors.) The major difficulty is that it is not immediately apparent how lateral velocity variations can be incorporated into a phase shift method because the space coordinate has been Fourier transformed. As a result, extrapolation techniques for v(x) (we use v(x) as synonymous with the phrase “a laterally variable velocity field”) are usually formulated in the space-frequency domain (Gazdag 1980, Berkhout 1984, Holberg 1988, Hale 1991, and others) as a dip-limited approximation to the inverse Fourier transform of the phase shift operator. The velocity dependence of such a local space domain extrapolator is then varied with the local velocity of the computation grid. However, since the multidimensional Fourier transform is a complete description of a wavefield, it follows that it must be possible to extrapolate a wavefield through lateral velocity variations with a Fourier domain technique. We present such a technique here and illustrate its relation to established methods. Black et al. (1984) and Wapenaar (1992) have presented similar Fourier methods (see also Wapenaar and Dessing, 1995, and Grimbergen et al. ,1995). We present our work in the context of nonstationary filter theory (Margrave, 1997) and show its direct link to the popular PSPI (phase shift plus interpolation) method of Gazdag and Squazzero (1984). NSPS (nonstationary phase shift) is presented as an explicit closed-form expression for one-way wavefield extrapolation through v(x) which has the physical interpretation of a laterally varying, or nonstationary, phase shift. Next, we give a detailed comparison between NSPS and PSPI for the case of a step velocity model. Both analytic and numerical results show that NSPS gives more physically plausible results. As a further demonstration of the utility of our approach we conclude with a modification of NSPS which has perfectly absorbing (that is, reflections are surpressed at all dips) lateral boundaries. This is achieved through the compensation of the NSPS operator for finite recording aperture. THEORETICAL DEVELOPMENT We begin with a summary of PSPI and show how to formulate the most accurate, limiting form of PSPI as a generalized Fourier integral. Then, using results from the theory of nonstationary linear filters, we show that the PSPI limiting form is a type of nonstationary filter called a combination filter. Such filters are linear and have definable properties; however, they do not form the linear superposition of impulse responses which Huygen’s principle suggests is desirable in wave propagation. This motivates the use of a nonstationary convolution filter which does form the desired linear superposition and is the basis for our NSPS algorithm. We give expressions for NSPS and PSPI in the dual (space-wavenumber) domain and in the full Fourier domain. Wavefield extrapolation by nonstationary phase shift CREWES Research Report — Volume 9 (1997) 30-3 The PSPI method PSPI (phase shift plus interpolation, Gazdag and Squazzero, 1984) is a rational attempt to build an approximate extrapolation through v(x) from a set of constant velocity phase shift extrapolations using a suitable set of reference velocities, {vj}. For simplicity, we present the theory in 2D as the extrapolation of a wavefield from z=0 to z=∆z. (A summary of our mathematical notation appears in Appendix A.) After an initial Fourier transform over time, we denote the wavefield at z=0 as Ψ(x,0,ω), (ω is temporal frequency) and the desired extrapolated wavefield at z=∆z as Ψv(x)(x,∆z,ω), where the subscript provides information about the velocity field. Phase shift extrapolation with each vj produces a reference wavefield, Ψvj(x,∆z,ω), given by Ψvj x,∆z,ω = φ k x,0,ω αvj k x,ω e ik xdk x – ∞ ∞ (1)