Inverse-Q filtered migration Yanghua Wang 1 ABSTRACT An inverse-Q filtered migration algorithm performs seis- mic migration and inverse-Q filtering simultaneously, in which the latter compensates for the amplitudes and corrects the phase distortions resulting from the earth attenuation ef- fect. However, the amplitudes of high-frequency compo- nents grow rapidly in the extrapolation procedure, so numeri- cal instability is a concern when including the inverse-Q filter in the migration. The instability for each frequency compo- nent is independent of data and is affected only by migration models. The stabilization problem may be treated separately from the wavefield-extrapolation scheme. The proposed strategy is to construct supersedent of attenuation coeffi- cients, based on given velocity and Q models, before per- forming wavefield extrapolation in the space-frequency do- main. This stabilized algorithm for inverse-Q filtered migra- tion is applicable to subsurface media with vertical and later- al variations in velocity and Q functions. It produces a seis- mic image with enhanced resolution and corrected timing, comparable to an ideal image without the earth attenuation effect. INTRODUCTION In seismic migration, surface-recorded seismic waves are back- propagated through the subsurface medium. This is an inverse pro- cess of wave propagation, as is inverse-Q filtering, in which the earth attenuation effects including energy absorption and velocity disper- sion are compensated for. Thus, a migration process also should ac- count for subsurface attenuation effects. By performing inverse-Q filtering and migration processes simultaneously, one can restore the high frequencies to balance the spectrum of the seismic image and correct the phases and the associated timings of reflections. This pro- cedure is referred to as inverse-Q filtered migration. In a viscoacoustic medium, seismic wave amplitudes and arrival times are changed because of energy absorption and velocity disper- sion, respectively. Consequently, migration by wavefield extrapola- tion without compensating for these earth attenuation effects pro- duces a result with a diffused image and incorrectly positioned re- flectors. Including an inverse-Q filter in migration, however, raises the problem of numerical instability. Because the amplitudes of high-frequency components grow rapidly in wavefield extrapola- tion, numerical round-off errors tend to be amplified drastically with increasing depth Dai and West, 1994; Mittet et al., 1995; Cui and He, 2004. This causes a huge amount of undesirable noise in the re- sult, even if the input data set is free of observed noise Wang, 2002, 2006. The improvement of an inverse-Q filter over seismic resolution is equal to three times the bandwidth increment plus two times the sig- nal-to-noise ratio S/N increment Wang, 2003. Only when the sum of these two factors is positive can the seismic resolution be en- hanced. Therefore, one of the focus points in an inverse-Q filtered migration procedure is stabilization to ensure that the resultant S/N is maximized. Zhang and Wapenaar 2002 suggest limiting the number of ex- trapolation steps and limiting the maximum angle of migration dip to derive a conditionally stable extrapolation operator. However, that does not solve the problem because of the limitation on migration depth and dip angle. The first stabilized algorithm for migration with inverse-Q filtering, presented by Wang and Guo 2004a, is imple- mented in the wavenumber-frequency domain. It applies only to a subsurface medium with a vertically variable velocity function and vertically variable Q model. Nevertheless, this preliminary investi- gation may offer some insight into stabilization issues in the extrapo- lation operators of wavefield downward continuation. This paper proposes a strategy to implement inverse-Q filtered migration for subsurface media with both vertical and lateral varia- tions in the velocity and Q models. A 1D wave equation, variable in the z-direction, is used at each lateral location of the model to derive a “supersedent” to supersede attenuation coefficients to stabilize the migration process. This 1D equation-based extrapolation operator has a maximum value of compensation among a series of compo- Manuscript received by the Editor 2 March 2007; revised manuscript received 21 August 2007; published online 29 November 2007. 1 Imperial College London, Centre for Reservoir Geophysics, Department of Earth Science and Engineering, London SW7 2AZ, UK. E-mail: yanghua.wang@ imperial.ac.uk. © 2008 Society of Exploration Geophysicists. All rights reserved. GEOPHYSICS, VOL. 73, NO. 1 JANUARY-FEBRUARY 2008; P. S1–S6, 4 FIGS. 10.1190/1.2806924 S1 Downloaded 28 Apr 2009 to 155.198.98.115. Redistribution subject to SEG license or copyright; see Terms of Use at http://segdl.org/