Demigration and migration in isotropic inhomogeneous media

Abstract

It is a generally accepted principle that seismic migration can be carried out by stacking amplitudes along a diffraction curve and mapping its result into the corresponding diffractor location. See, for example, Hagedoorn (1954), Lindsey and Hermann (1985), Rockwell (1971), Schneider (1978), and Schleicher et al. (1993). This problem has been studied in a more formal way from the inverse scattering theory point of view by using the Generalized Radon Transform (GRT), starting with Norton and Linzer (1981), who treated ultrasonic experiments in medical imaging and followed by Miller et al. (1984), Beylkin (1985) and others. Bleistein (1987) introduces the inverse scattering solution by using the Fourier transform instead of the GRT. An extension to general anisotropic media is obtained by Burridge et al. (1995). While those migration techniques using the GRT and Fourier transfoms are based on the Born approximation, here we base our derivations on the Kirchhoff approximation. While the former representation is linear in the perturbat ion of the medium parameters, the latter is linear in the reflection coefficient that we seek, that coefficient, in turn, being a nonlinear function of the medium perturbations. In analogy with the diffraction-stack formula in Bleistein (1987), we interpret the Kirchhoff modeling formula as an isochron stack and derive from it demigration and migration operators. Using the superposition principle, we derive two alternative (migration and demigration) operators. Seismic true amplitude migration maps the recorded data into imaged data weighted by the oblique-incidence specular reflection coefficient. Seismic inverse migration (or demigration) maps true amplitude migrated data into data that would be recorded for a given set of earth model parameters and a defined measurement configuration along the recording surface. From the above, the output section of a modeling algorithm and an demigration algorithm should be the same; however the input section for the demigration algorithm consists of output data traces with format identical to that of the recorded data while the input of a modeling algorithm is an earth model. From the seismic data processing point of view, the use of cascade migration and demigration algorithms to solve practical problems such as DMO or offset continuation is preferred to the use of cascade inversion and modeling algorithms. The reason for this is that the output of a migration program is ready to be used as input for a demigration program, while the output traces of an inversion program have to be pre-processed (travel times and amplitudes should be picked) before going into the modeling program. This preprocessing is not only tedious but a source of human error. Mathematically there is no fundamental difference between cascading modeling and inversion operators or cascading demigration and migration operators for solution of imaging problems. The intermediate data differ from one another but these data are used only while deriving the final composed operator. They will not be of any use after that.