ω-dependent smoothing in GPSPI Frequency-dependent velocity smoothing in GPSPI migration Chad M. Hogan and Gary F. Margrave ABSTRACT The GPSPI algorithm uses a phase shift that is a high-frequency approximation to the exact solution: the square-root Helmholtz operator symbol. The square-root Helmholtz operator symbol dramatically changes character between high- and low-frequency limits, while the GPSPI phase shift does not. A frequency-dependent smoothing of the velocity model (and therefore the phase shift) is used to approximate the character change of the square-root Helmholtz symbol, and is implemented via the spatial resampling within the FOCI algorithm. This results in heavy smoothing of the phase shift at low frequencies, and virtually no smoothing at higher frequencies. The frequency-dependent smoothing results in an image of much higher quality than the image generated using the usual GPSPI phase shift. INTRODUCTION The GPSPI algorithm (Margrave and Ferguson, 1999) provides a highly-accurate prestack depth migration algorithm. This algorithm depends upon a numerical implementation of the “infinitesimal extrapolator”, Ψ(x, z +Δz,ω)= T α(z:z+Δz) Ψ(x,z,ω) ≈  R φ(k x ,z,ω)α(x, k x ,ω,z : z +Δz )e ikxx dk x (1) where α (x, k x ,ω,z : z +Δz )= ⎧ ⎨ ⎩ exp  iΔz  ω 2 v(x) 2 − k 2 x  , |k x |≤ |ω| v(x) exp  −   Δz  ω 2 v(x) 2 − k 2 x     , |k x | > |ω| v(x) . (2) Here Ψ represents the wavefield as a function of horizontal position x, vertical depth z , depth-step Δz , and temporal frequency ω. T is the infinitesimal extrapolation operator characterized by symbol α, where α is a function of x, horizontal wavenumber k x , ω, and z . φ is the wavefield Ψ after Fourier-transforming from x to k x . In this case, T extrapolates the wavefield from depth z to depth z +Δz . For a full description of this algorithm and its efficient implementation, see Margrave and Ferguson (1999) and Margrave et al. (2006). THE SQUARE-ROOT HELMHOLTZ OPERATOR SYMBOL Fishman (2002) identifies the term  ω 2 /v(x) 2 − k 2 x in equation 2 as the limiting form of a high-frequency approximation to the square-root Helmholtz operator symbol (i.e. an “infinite-frequency symbol”). Figure 1 shows the real part of the infinite-frequency sym- bol as used in GPSPI. This symbol is calculated for a velocity model consisting of three blocks of constant velocity. On the left, velocity v 1 is relatively high; in the middle, ve- locity v 2 is relatively low; and on the right, velocity v 3 is moderate. Figure 2 shows an ensemble of exact square-root Helmholtz operator symbols for high, moderate, low, and CREWES Research Report — Volume 18 (2006) 1