Direct downward continuation from topography using explicit wavefield extrapolation Saleh M. Al-Saleh 1 , Gary F. Margrave 2 , and Sam H. Gray 3 ABSTRACT Downward-continuation migration algorithms are power- ful tools for imaging complicated subsurface structures. However, they usually assume that extrapolation proceeds from a flat surface, whereas most land surveys are acquired over irregular surfaces. Our method downward continues data directly from topography using a recursive space-fre- quency explicit wavefield-extrapolation method. The algo- rithm typically handles strong lateral velocity variations by using the velocity value at each spatial position to build the wavefield extrapolator in which the depth step usually is kept fixed. To accommodate topographic variations, we build space-frequency wavefield extrapolators with laterally vari- able depth steps LVDS. At each spatial location, the differ- ence between topography and extrapolation depth is used to determine the depth step. We use the velocity and topograph- ic values at each spatial lateral position to build extrapolators. The LVDS approach does not add more data nor does it re- quire preprocessing prior to extrapolation. We implemented the LVDS method and applied it to a source profile prestack migration technique. We also implemented the previously developed zero-velocity layer approach to use for compari- son. For both algorithms, we modeled the acoustic source as an approximate free-space Green’s function, not as a simple extrapolated spatial impulse. Tests on a synthetic data set modeled from rough topography and comparisons with the zero-velocity layer approach confirm the method’s effective- ness in imaging shallow and deep structures beneath rugged topography. INTRODUCTION Many recursive explicit space-frequency wavefield extrapolation methods are approximations to the generalized phase shift plus inter- polation GPSPI algorithm Margrave and Ferguson, 1999, the limiting form of the phase-shift plus interpolation PSPI method Gazdag and Squazerro, 1984. Also, GPSPI reduces to the phase- shift algorithm Gazdag, 1978, Claerbout 1985 when the velocity is constant. At each output point, the wavefield is computed explicitly using an operator specified by the velocity at the output location. Unlike ray-based algorithms, e.g., Kirchhoff methods, these downward-continuation approaches usually assume that extrapola- tion proceeds from one horizontal surface to the next, so they cannot directly handle data recorded from irregular topographic surfaces over which most land surveys are acquired. Further, they are often computationally more expensive than ray-based methods, and cur- rent approaches that deal with topography typically increase their computational cost. One of the oldest topographic approaches is to static shift data i.e., apply constant time shifts to individual traces to a horizontal datum followed by downward continuation. This approach is inac- curate for nonvertically traveling energy and can produce artifacts in the shallow section after migration Gray, 1997. Ji and Claerbout 1992 and Bevc 1997 use a more accurate approach by upward- continuing the data to the highest elevation using wave-equation da- tuming Berryhill, 1979 prior to migration, which needs more com- putational effort than merely static shifting the data. The zero-velocity layer approach is more efficient than wave- equation datuming Beasley and Lynn, 1992; Gray, 1997 but still re- quires some processing before migration. In that approach, data are static shifted to a datum above or equal to the highest elevation be- fore migration. The wavefield extrapolation then is carried out by as- suming a zero velocity for diffraction effects only between the da- tum and recording surface. The velocity used to calculate the static shifts is also used in the thin-lens term of the extrapolator. In a similar approach, Reshef 1991 proposes a variable depth- step approach, and Margrave and Yao 2000 use a method in the nonstationary phase-shift algorithm Margrave and Ferguson, 1999 to extrapolate zero-offset data directly from topography. In these ref- erences, the zero-velocity layer method and the variable depth step Manuscript received by the Editor 2 December 2008; revised manuscript received 5 June 2009; published online 15 December 2009. 1 Saudi Aramco, Dhahran, Saudi Arabia. E-mail: saleh.saleh.6@aramco.com; alsalehsaleh@yahoo.com. 2 University of Calgary, Calgary,Alberta, Canada. E-mail: margrave@ucalgary.ca. 3 CGG Veritas, Calgary,Alberta, Canada. E-mail: sam.gray@cggveritas.com. © 2009 Society of Exploration Geophysicists. All rights reserved. GEOPHYSICS, VOL. 74, NO. 6 NOVEMBER-DECEMBER 2009; P. S105–S112, 9 FIGS. 10.1190/1.3263914 S105 Downloaded 10 Feb 2010 to 24.149.197.34. Redistribution subject to SEG license or copyright; see Terms of Use at http://segdl.org/