Herman Jaramillo

The link of Kirchhoff migration and demigration to Kirchhoff and Born modeling

The Kirchhoff approximation provides a representation of seismic data as a summation of imaged data along isochron surfaces (demigration). The asymptotic inversion of this representation provides a migration as a summation of seismic data along diffraction surfaces. We replace Born inversion techniques with Kirchhoff inversion techniques and further show the link between the Kirchhoff and Born representations after the Born linearized reflection coefficient is replaced by the Kirchhoff reflection coefficient.

Comparing common‐offset Maslov, Gaussian Beam, and coherent state migrations

Several techniques to circumvent limitations in produc tion Kirchho migration have recently been proposed Kirchho migration can be extended to handle phases through caustics as well as true amplitude inversion For large scale production work however where Green s function information is often precomputed storage of traveltimes phases and amplitudes for multiple arrivals can become prohibitive Wave eld extrapolation tech niques are also being used although these techniques can be computationally expensive

True-amplitude transformation to zero offset of data from curved reflectors

Transformation to zero offset (TZO), alternatively known as migration to zero offset (MZO), or the combination of normal moveout and dip moveout (NMO/DMO), is a process that transforms data collected at finite offset between source and receiver to a pseudozero offset trace. The kinematic validity of NMO/DMO processing has been well established. The TZO integral operators proposed here differ from their NMO/DMO counterparts by a simple amplitude factor. (The form of the operator depends on how the input and output variables are chosen from among the combinations of midpoint or wavenumber with time or frequency.) With this modification in place, the dynamical validity for planar reflectors of the proposed TZO operators of this paper have been established in earlier studies. This means that the traveltime and geometrical spreading terms of the finite offset data are transformed to their counterparts for zero offset data, while the finite offset reflection coefficient is preserved. The main purpose of this st...

Seismic data mapping

This work describes Seismic Data Mapping (SDM), its definition, properties, applications, limitations and goals. A variety of problems such as offset-continuation, azimuthal-continuation (AMO), layer replacement and datuming can be cast as special cases of SDM. In general, this process is a cascade of prestack migration and inverse migration (demigration). We find a simplification of the operator that requires less computational effort than either the individual prestack migration and demigration processes that generated it.

To: “A unified approach to 3-D seismic reflection imaging, Part II: Theory” (M. Tygel, J. Schleicher, and P. Hubral, GEOPHYSICS, 61, 759–775)

It came to our attention that in the paper of Tygel et al. (1996), formula (A-5) is incorrect. First, it is different from the cited equation (17) of Tygel et al. (1995). Second, the derivation of formula (17) in Appendix A of Tygel et al. (1995) is invalid for the purposes of Tygel et al., (1996) because in the 1995 paper, a rotation is treated, whereas in Tygel et al. (1996), a projection matrix is needed. Further investigation of the subject was needed to resolve these inconsistencies.

Prestack migration error in transversely isotropic media

Previous studies have shown the dependence of migration error on reflector dip when poststack migration is done with an algorithm that ignores the presence of anisotropy. Here we do a numerical study of the offset dependence of migration error that can be expected when common-offset data from factorized transversely isotropic media are imaged by an isotropic prestack migration algorithm. Anisotropic ray tracing, velocity analysis and prestack migration in the common-offset domain are the basic tools for this analysis, which we apply to models with constant vertical gradient in velocity that are characterized by a particular combination of Thomsens anisotropy parameters: TJ= (€6)/ (1+ 26). The results show that the offset dependence of error in imaged position, and therefore the quality of stacked, imaged data, depends largely, but not completely, on the anisotropy parameter TJ.Generally, the larger the value of TJ,the larger the problem of mis-stacking. Over a wide range of reflector dip, time-misalignment of imaged features on common-reflection-point gathers is considerably less than the error in imaged position on the zero-offset data. For all the model parameters studied, we expect stacking quality to be worst for reflector dip around 50 degrees. Reflections from horizontal reflectors and those with dip close to 90 degrees should stack well in all cases, and mistacking is not severe for overturned reflectors.

Demigration and migration in isotropic inhomogeneous media

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.

Improving Performance And Accuracy of 3D Kirchhoff Migration

Since 3D Prestack Kirchhoff Depth Migration (KPSDM) has become one of the leading imaging tools for hydrocarbon exploration, its accurate and precise handling of the kinematical and dynamical aspects of the wavefield have become center stage to the R&D efforts worldwide. In a separate paper in this proceeding by the same author, describe a modified antialiasing filter weight that corrects for amplitude artifacts observable in earlier designs. Here we continue the efforts of developing an efficient true amplitude migration algorithm by suggesting a simplification of the traditional filtering done during this process that will improve the performance and precision of the results.

Multimode reverse time VSP imaging over complex structures at Yucca Mountain, Nye County, Nevada

Structural interpretation of vertical seismic profiles (VSP’s) is always a challenge because conventional plots of the reflected wavefield itself, observed along a borehole, do not bear much resemblance to the reflectors responsible for the reflected events. Wyatt and Wyatt (1981) and Dillon and Thomson (1983) pioneered this problem by mapping or migrating the reflected events from space-time into spacespace domains. A more elaborate migration of an extensive multiple offset VSP was demostrated by Mons, et al (1985).