Isotropic Migration Research Articles

Anisotropic reverse‐time migration for tilted TI media

ABSTRACTSeismic anisotropy in dipping shales results in imaging and positioning problems for underlying structures. We develop an anisotropic reverse‐time depth migration approach for P‐wave and SV‐wave seismic data in transversely isotropic (TI) media with a tilted axis of symmetry normal to bedding. Based on an accurate phase velocity formula and dispersion relationships for weak anisotropy, we derive the wave equation for P‐wave and SV‐wave propagation in tilted transversely isotropic (TTI) media. The accuracy of the P‐wave equation and the SV‐wave equation is analyzed and compared with other acoustic wave equations for TTI media. Using this analysis and the pseudo‐spectral method, we apply reverse‐time migration to numerical and physical‐model data. According to the comparison between the isotropic and anisotropic migration results, the anisotropic reverse‐time depth migration offers significant improvements in positioning and reflector continuity over those obtained using isotropic algorithms.

VTI anisotropic corrections and effective parameter estimation after isotropic prestack depth migration

In the method for applying anisotropic corrections after isotropic prestack depth migration (PSDM), the correction, which is calculated and implemented in the depth domain, is defined as a time difference between isotropic and anisotropic traveltimes, under the assumption that the vertical velocity is known. The definition of this correction uses a special postmigration common-image-gather (CIG) ordering, which collects the migrated data according to the input-trace’s source and receiver distance from the surface CIG location. In this postmigration domain, the dip of the events can be directly related to their horizontal position in the CIG, called the imaging offset, and the separation of flat and dipping reflectors becomes easy to perform. The dependency of the seismic anisotropic effect on the subsurface dip angle is well pronounced in these CIGs. After application of an isotropic PSDM, effective anisotropic-parameter estimation is performed at selected CIG locations by using a simple two-parameter scan procedure. The optimal anisotropic parameters can be used to perform a final anisotropic PSDM or to apply a residual correction to the isotropically migrated data. We demonstrate the method for P-wave data in 2D media with vertical transverse isotropy (VTI) symmetry by using both synthetic and real data. We also present a strategy for handling the ambiguity between the vertical velocity and the anisotropic parameters.

Lateral displacement by transient mobility in chemisorption of CO on Pt(997).

We investigated adsorbed states of CO on Pt(997) at 11 K using infrared reflection absorption spectroscopy. At 11 K, thermal migration is suppressed and thus the initial chemisorption at terrace sites and step sites is controlled by the transient mobility of the adsorbing molecule. The initial occupation ratio between atop CO on the terrace and atop CO at the step is directly determined to be 3.6:1. With a simple isotropic migration model, we estimated the mean lateral displacement from the first impact point to the initial chemisorption site to be 6.8 A. We also discuss the origin of transient mobility of CO on metal surfaces.

A Semilinear Parabolic System for Migration and Selection in Population Genetics

The semilinear parabolic system that describes the evolution of the gene frequencies in the diffusion approximation for migration and selection at a multiallelic locus is investigated. The population occupies a finite habitat of arbitrary dimensionality and shape. The selection coefficients depend on position; the drift and diffusion coefficients may do so. For two alleles (the scalar case), the global analysis of D. Henry (1981, “Geometric Theory of Semilinear Parabolic Equations,” Lecture Notes in Mathematics, Vol. 840, Springer-Verlag, Berlin) is extended from homogeneous, isotropic migration (corresponding to the Laplacian) to arbitrary migration (corresponding to an arbitrary elliptic operator). For multiple alleles, sufficient conditions are given for the global loss of an allele that is nowhere the fittest. In the case of no dominance with at least one change in the direction of selection, sufficient conditions are established for global convergence to a stable equilibrium with all the intermediate alleles absent and one or two extreme alleles present. Sufficient conditions on the migration pattern for casting the elliptic operator into variational form are proved; in this case, the above results become more explicit.

Finite Element Prestack Reverse‐Time Migration for Anisotropic Elastic Waves

AbstractBased on the finite element method and the interface point method of shortest path ray tracing, reverse‐time migration for anisotropic elastic waves is carried out. The migrated sections of synthetic data are clear and accurate. This demonstrates the reliability of the migration system. Furthermore, we migrate anisotropic wave field data with isotropic migration method to find out the effect of ignoring anisotropy. The results show that there are considerable errors on isotropically migrated sections where the positions of geological bodies are inconsistent with those in the real models.

Ray-Based 2d Anisotropic Pre-Stack Depth Migration on Synthetic and Real Data

Pre-stack depth migration (PSDM), while costly, is a powerful tool for imaging and velocity analysis. Seismic velocity anisotropy is now thought to be widespread in sedimentary rocks, particularly shales. We have written a 2D pre-stack depth migration code for use in transversely isotropic media with a vertical symmetry axis (TIV). We use it to investigate the effects of anisotropy on isotropic pre-stack migration by means of a simple synthetic example. Results suggest that it is difficult to find plausible isotropic models which give good images of reflectors at all dips. We further apply our PSDM code to a 2D line of real data from offshore West Africa, where there is a thick shale sequence, using isotropic and anisotropic velocity models. A dipping fault plane is imaged more clearly with the anisotropic model, but some of the near horizontal reflectors are less well imaged suggesting that this model could be improved and confirming the sensitivity of pre-stack depth migration to all aspects of the velocity model.

Migration-Velocity Analysis for Ti and Orthorhombic Background Media

A knowledge of the background velocity model is crucial to achieve the accurate reservoir description now expected from imaging and inversion. Conventional methods for reconstructing the background velocity model, like migration-velocity methods, often assume an isotropic subsurface and can yield inaccurate reservoir descriptions when the subsurface contains anisotropic rock formations. Here, we generalize the migration-velocity concept by permitting the background velocity to be transversely isotropic (TI) with respect to the vertical axis or othorhombic. The scheme consists of scanning different anisotropic velocity models using a phase-shift migration and of picking anisotropic parameters based on amplitude variations of migrated results (focusing analysis). As the anisotropic background velocity model is generally described by several elastic coefficients, it is important to adopt an efficient scanning procedure. We have chosen to work with common azimuthal sections. For a given common azimuthal section, we sequentially scan two parameters : normal move-out velocity and the anisotropic parameter known as anellipticity. These two scans allow us to reconstruct an azimuthally isotropic velocity model. The procedure is then repeated for different common azimuthal sections; each common azimuthal section leads to a new azimuthally isotropic velocity model if the medium is azimuthally anisotropic. The number of common azimuthal sections, and therefore the number of azimuthally isotropic velocity models, needed to reconstruct an azimuthally anisotropic velocity model is dependent on the type of symmetries. For example, only three common azimuthal sections are needed for an orthorhombic medium. As the contribution of isotropic component of most rock formations is generally more important than the anisotropic one, we have proposed to base the picking of values of anellipticity on the subtraction of the result of isotropic migration from that of TI migration instead of using directly.

Estimation of anisotropy and anisotropic 3-D prestack depth migration, offshore Zaire

Chevron (operator), Teikoku Oil, and Unocal hold the offshore Zaire concession, West Africa. Recently, Unocal performed an anisotropic 3-D prestack depth migration in an attempt to optimally image tilted fault blocks in the presalt section. Depth migration was required because large‐displacement faults juxtaposed a high‐velocity massive carbonate and a low‐velocity marl. Large compressional‐wave anisotropy caused by thin layering was inferred for most of the depth section. Evidence on the nature and magnitude of the anisotropy came from comparing stacking velocities with well velocities, analyzing a multioffset vertical seismic profile (VSP), thin‐layer modeling in the massive carbonate using well log data, and rock physics laboratory velocity measurements on core from the marl. The subsurface was characterized as being locally transversely isotropic and the Thomsen parameters were used to describe the anisotropy. This required that five parameters be specified at each subsurface location to represent the compressional‐wave 3-D velocity field. The observed anisotropy deviated far from the elliptical condition, and modeling indicated that isotropic migration would exhibit misfocusing and mispositioning of events, and its quality would be dip‐dependent. The anisotropic migration was performed using Kirchhoff summation for which the traveltimes were determined using a finite‐difference scheme. Interpreted horizons in the migrated depth section accurately tied well depths.

Kirchhoff migration and velocity analysis for converted and nonconverted waves in anisotropic media

We develop asymptotic expressions and outline a procedure to perform Kirchhoff migration in anisotropic media. This technique is based on a new Green’s tensor representation for azimuthally isotropic media obtained by using analytical forms for the ray amplitudes and traveltimes. Since in real applications the usage of general anisotropy in a migration scheme will be limited by the availability and reliability of the velocity model considered, we also develop a new anisotropic velocity analysis scheme to generate realistic anisotropic models for migration in azimuthally isotropic media for nonconverted and converted qP-qSV waves. This velocity analysis technique is based on nonhyperbolic traveltime‐offset formulas explicitly given in terms of the five elastic constants of azimuthal isotropy. The imaging technique is applied to nonconverted‐as well as converted‐wave surface seismic data. In both cases the method provides accurate images of the subsurface. Even with a weak to moderate percentage of anisotropy, we show that an isotropic migration algorithm cannot properly image the subsurface. This paper provides new nonconventional techniques for the velocity analysis and migration in anisotropic media and shows the feasibility of exploiting converted and nonconverted waves.

The migrator’s equation for anisotropic media

The migrator’s equation, which gives the relationship between real and apparent dips on a reflector in zero‐offset reflection seismic sections, may be readily implemented in one step with a frequency‐domain migration algorithm for homogeneous media. Huygens’ principle is used to derive a similar relationship for anisotropic media where velocities are directionally dependent. The anisotropic form of the migrator’s equation is applicable to both elliptically and nonelliptically anisotropic media. Transversely isotropic media are used to demonstrate the performance of an f-k implementation of the migrator’s equation for anisotropic media. In such a medium SH-waves are elliptically anisotropic, while P-waves are nonelliptically anisotropic. Numerical model data and physical model data demonstrate the performance of the algorithm, in each case recovering the original structure. Isotropic and anisotropic migration of anisotropic physical model data are compared experimentally, where the anisotropic velocity function of the medium has a vertical axis of symmetry. Only when anisotropic migration is used is the original structure recovered.