G. H. F. Gardner

A three‐dimensional perspective on two‐dimensional dip moveout

Prestack migration in a constant‐velocity medium spreads an impulse on any trace over an ellipsoidal surface with foci at the source and receiver positions for that trace. The same ellipsoid can be obtained by migrating a family of zero‐offset traces placed along the line segment from the source to the receiver. The spheres generated by migrating the zero‐offset impulses are arranged to be tangent to the ellipsoid. The resulting nonstandard moveout equation is equivalent to two consecutive moveouts, the first requiring no knowledge of velocity and the second being standard normal moveout (NMO). The first of these is referred to as dip moveout (DMO). Because this DMO-NMO algorithm converts any trace to an equivalent set of zero‐offset traces, it can be applied to any ensemble of traces no matter what the variations in azimuth and offset may be. In particular, this three‐dimensional perspective on DMO can be used with multifold inline data. Then it becomes clear that velocity‐independent DMO operates on rad...

Hyperbolic traveltime analysis of first arrivals in an azimuthally anisotropic medium; a physical modeling study

Wave propagation in a fractured medium is modeled physically using layers of Plexiglas with thin films of water, held under moderate uniaxial confining pressure. The system exhibits anisotropy comparable to that of measured earth materials; i.e., shear‐wave splitting to waves with 3 percent velocity differences and P-wave directional anisotropy of at least 20 percent. SV polarizations demonstrate the concept of the shear‐wave window with the conversion of an SV body wave to an internal head wave with P-wave velocity, a head wave which is present in both the fractured medium and the control solid (unfractured) medium. For an azimuthally anisotropic medium, moveout curves are hyperbolic for a surface line oriented parallel to the fractures but are nonhyperbolic for a line oriented perpendicular to the fractures. Q anisotropy is observed in the system, with strongest attenuation on propagation paths perpendicular to the fractures.

Normal moveout in anisotropic media

Elliptical anisotropy may be considered a more general case of isotropy, in which the medium has been stretched in one direction (Helbig, 1983; Dellinger and Muir, 1988; Verwest, 1989). As a consequence, certain principles and relationships hold for both isotropy, where wavefronts are circular, and elliptical anisotropy, where the wavefronts are circles which have been stretched in one direction (i.e., ellipses); while they may not hold for the general case of anisotropy, where wavefronts have more complex shapes. Examples are the dip moveout (DMO) relation (Uren et al., 1990) and the method of images (Dellinger and Muir, 1988).

Some effects of velocity variation on AVO and its interpretation

In recent years interest has increased in the interpretation of the amplitude variation of reflected signals as a function of offset (AVO). A more meaningful relationship for interpreting reflection coefficients at the target horizon is amplitude variation with incident angle (AVA). The challenge is to convert from AVO to AVA. The effects of velocity variation in the overburden on amplitude variation with offset (AVO) and on the final inversion of AVO data into velocity, density, and Poisson’s ratio can be significant. Examples are given here for subsurface medium with a vertical velocity gradient range of -0.2s-1 to 0.8s-1. When the medium is treated as homogeneous in the conversion from AVO to AVA, this velocity variation causes significant errors (about 10 percent) in both the gradient of AVA and in the normal incident reflection coefficient. Such errors produce errors of similar magnitude in the inversion of AVA data into the elastic parameters of velocity, Poisson’s ratio, and density. The errors dep...

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 fun...

Effects of irregular sampling on 3-D prestack migration

In a direct application of Kirchhoff migration each trace is added to the migrated volume by spreading the data along impulse response curves with suitable change in the amplitude and shape of the wavelets. Overlapping impulse responses form the correct answer where they form an envelope and are supposed to cancel elsewhere. For uniform data (constant offset, constant azimuth, constant midpoint spacing and constant velocity) the cancellation is excellent and the reflectors have the correct amplitude. This paper shows how failure to meet any of these constancy criteria results in noise. Modified summation procedures can reduce the noise at the expense of a loss of resolution.

Another look at the question of azimuth

The question of azimuth seems to be an emotional issue in 3-D seismic exploration. Some people believe that 3-D surveys should contain a wide range of source‐receiver azimuths so that the subsurface is sampled from all directions and the acquisition is closer to a “true” 3-D survey. Others, on the other hand, argue the case for narrow azimuth acquisition. A few years ago, Malcolm Lansley, then with Halliburton Geophysical, gave an excellent presentation regarding the question of azimuth at an SEG summer workshop. In this talk (which was recently presented to the Houston Geophysical Society), Lansley provided an extensive summary of the advantages and disadvantages of wide azimuth surveying, and listed many aspects that one should consider when deciding between narrow and wide azimuth acquisition.

A physical model of shear‐wave propagation in a transversely isotropic solid

It is now understood that seismic anisotropy is a comparatively common phenomenon in sedimentary layers. The elastic properties of most sedimentary rocks have been shown to be anisotropic. (Anisotropy means that the physical property of the material is a function of the measuring direction). Seismologists are generally concerned with velocity variation with the direction of propagation.

Fracture detection using P‐wave AVO

A 3-D modeling program for general anisotropy was used to obtain the reflection coefficients of the P-wave reflected from two anisotropic, vertically fractured media beneath an isotropic medium. Two lines were recorded, perpendicular and parallel to the fractures. Although the AVO response of a media containing gas-filled fractures increases with offset when the recorded line was parallel to the fractures, we observed a decreasing AVO response when the offset orientation was perpendicular to the direction of the fractures. Media containing fractures filled with water present no such differences; the AVO response increases with offset, perpendicular and parallel to the fractures. The results were compared to a generalized Zoeppritz solution for reflections from a layer with vertical fractures. The case study suggests that AVO response for different source-receiver azimuths may contain information about the internal structure of the subsurface.

Experimental observation of surface wave propagation for a transversely isotropic medium

Velocity anisotropy of surface‐wave propagation in a transversely isotropic solid has been observed in a laboratory study. In this study, Phenolite™, an electrical insulation material, was used as the transversely isotropic media (TIM), and a vertical seismic profiling (VSP) geometry was used to record seismic arrivals and to separate surface waves from shear waves. Results show that surface waves that propagate with different velocities exist at certain directions.