Deconvolution-type imaging condition effects on shot-profile migration amplitudes
Amplitude preservation in Pre-Stack Depth Migration (PSDM) processes that use wavefield extrapolation must be ensured – first, in the operators used to continue the wavefield in time or depth, and second, in the imaging condition used to estimate the reflectivity function. In the later point, the conventional correlation-type imaging condition must be replaced by a deconvolution-type imaging condition. Migration performed in common-shot profile domain obtains the final migrated image as the superposition of images resulting of migrate each shot separately. The amplitude obtained in a point of the migrated image corresponds to the sum of the reflectivities for each shot which has illuminated such point, along the angles determined by the velocity model and the positions of the source and the receiver. The deeper the reflector, the lower the amplitude of the illumination field will be. As result, the correlation-type imaging condition produces images with an unbalanced amplitude decrease with depth. A deconvolution-type imaging condition scales the amplitudes through a correlation, using the weighting function dependent on the spectral density or the illumination of the downgoing wave field. In this article, two possible scaling functions have been used in the case of a single shot. In the case of data with multiple shots, five scaling possibilities are presented with the spectral density or the illumination function. The results of applying these imaging conditions to synthetic data with multiple shots show that the values of the amplitude in the migrated images are influenced by the coverage of the common midpoint, compensating this effect only in one of the imaging conditions described. Numerical experiments with synthetic data generated using Seismic Unix and the Sigsbee2a data are presented, highlighting that in velocity fields with strong vertical and lateral velocity variations, the balance of the amplitudes of the deep reflectors relative to the shallow reflectors is strongly influenced by the imaging condition applied.
Cazzola, L., Arienti, M.T., Bonomi, E. & Cardone, G. (2002). Amplitude preserving Monte Carlo 3D pre-stack migration. 64th EAGE Conference & Exhibition, Pre-stack Time Imaging.
Cohen, J. & Stockwell, Jr. (2006). Seismic Un*x: a free package for seismic research and processing. Toulsa: Center for Wave Phenomena, Colorado School of Mines. Release No. 39- 2006.
Chattopadhyay, S. & McMechan, A. (2008). Imaging conditions for prestack reserve time migration. Geophysics, 73(3), 581- 589.
Claerbout, J. (1971). Toward a unified theory of reflector mapping. Geophysics, 36(3), 467-481.
Claerbout, J. (1985). Imaging the earth’s interior. New York: Blackwell Scientific Publications.
Gazdag, J. & Sguazzero, P. (1984). Migration of seismic data by phase shift plus interpolation. Geophysics, 49(2), 124-131.
Godin, O.A. (1999). Reciprocity and energy conservation within the parabolic approximation. Wave Motion, 29(2),175-194.
Guitton, A., Valenciano, A., Bevc, D. & Claerbout, J. (2007) Smoothing imaging condition for shot-profile migration. Geophysics, 72(3), S149-S154.
Paffenholz, J., Stefani, J., McLain, B. & Bishop, K. (2002). SIGSBEE2a Synthetic subsalt dataset: image quality as function of migration algorithm and velocity model error. 64th EAGE Conference, Extended abstracts.
Schleicher, J., Costa, J. & Novais, A. (2008). A comparison of imaging conditions for wave-equation shot-profile migration. Geophysics, 73(6), S219- S227.
Shin, C., Jang, S. & Min, D. J. (2001). Improved amplitude preservation for pre-stack depth migration by inverse scattering theory. Geophysical Prospecting, 49(5), 592–606.
Valenciano, A. & Biondi, B. (2003). 2D Deconvolution imaging condition for shot-profile migration. SEG Technical Program, 22(1).
Vivas, F. & Pestana, R. (2010). True amplitude one-way wave equation migration in the mixed domain. Geophysics, 75(5), S199-S209.
Vivas, F., Pestana, R. C. & Bjorn, U. (2009). A new stabilized least-squares imaging condition. J. of Geophysics and Eng., 6(3), 264-268.
Wapenaar, C. (1990). Representation of seismic sources in the one-way wave equations. Geophysics, 55(6), 786-790.
Zhang, G. (1993). System of coupled equations for up-going and down-going waves. Acta Math. Appl. Sinica, 16(2), 251-263.
Zhang, Y., Zhang, G. & Bleistein, N. (2003). True amplitude wave equation migration arising from true amplitude oneway wave equations. Inverse Problems, 19(5), 1113-1138.