Imaging reflections in elliptically anisotropic media

Joe Dellinger and Francis Muir

This paper appeared in pages 1616 to 1618 of the December, 1988, copy of ``GEOPHYSICS''.

© 1988 Joe Dellinger, Francis Muir, and the SEG

ABSTRACT

In an isotropic medium, waves reflected from a mirror form a virtual image of their source. This property of planar reflectors is generally not true in the presence of anisotropy. In their short note, Blair and Korringa (1987) show that for the special case of SH waves from a point source in a transversely isotropic medium an aberration free image is formed for any orientation of the mirror. While their proof is mathematical, we show the same result in an intuitive, pictorial fashion and in the process discover that although the image is indeed aberration free it is still distorted.

STRETCHING ISOTROPY

SH waves in a transversely isotropic medium display elliptical anisotropy. That is, wavefronts spreading away from a point source are ellipses rather than circles. To make an elliptically anisotropic wave equation, start with the regular isotropic wave equation and apply a linear coordinate transformation. Graphically, a linear transformation (in two dimensions) is just a combined stretch and rotation.

A snapshot of a wavefield shows the wavefield at one instant of time. Since stretching a snapshot of a wavefield is the same as linearly transforming the underlying wave equation, we can get a snapshot of an elliptically anisotropic wavefield by simply stretching the corresponding isotropic snapshot.

In Figure 1 we show how a mirror works in the standard isotropic case. The starburst at the top of the picture represents a point source, and the starburst at the bottom of the picture is its image. If we stretch Figure 1 at a 45-degree angle to the reflector, and then rotate it so that the reflector is again horizontal, we obtain Figure 2 . From the argument in the previous paragraph this figure shows how a mirror in an elliptically anisotropic medium works. Since the starbursts are supposed to represent point sources, they are not stretched.

We can easily see that Blair and Korringa were right. We do still have an image of the source, but it is mispositioned. The line connecting the source and its image no longer intersects the reflector at a right angle, but it does point out the position where the wavefront first encounters the reflector.

Stretching a point

Aberration free means that each point in the source is imaged as a point. We see that this is clearly satisfied in Figure 2 . What about something bigger than a point? Examining the two figures we can see that the mirror image of the square near the left edge in Figure 2 is a strongly sheared parallelogram.

Mirrors in an elliptically anisotropic media do make aberration-free images, but not distortion-free images.

Stretching and depth migration

We know that the amount of ellipticity cannot be determined from surface-to-surface data of this kind if the stretch is normal to the surface. What if the stretch is at an arbitrary angle? Such a stretch, like that in Figure 2 , can be thought of as a vertical stretch combined with a horizontal simple shear about the surface of recording. Since there is no change where we measure our data, CDP gathers must still look like CDP gathers. The true physical location of the depth point in the earth is unknown. It does not even have to be under the survey.

In principle, amplitude information could provide an answer, since the symmetry is broken when we do not stretch the antenna patterns of our source and receiver along with the coordinate system. The telltale amplitude variations might easily be masked or missed, however, especially if the anisotropy is not pronounced or the amplitudes are not handled carefully.

Where does this leave depth migration? Nearly where it was, but not quite. Conventional depth migration is still a valid focusing operation, but it incorporates a pseudo time-to-depth mapping based on an isotropic assumption. It should be followed by a vertical stretch and horizontal shear when this information is available from, say, well-logs or VSPs.

Multiple layers

The same stretching principle easily extends to the case of multiple layers. Each layer must be independently stretched to become isotropic, stretching must be consistent across layer boundaries, and the surface must be unchanged. Start at the surface and stretch all the layers beneath. Make the first layer isotropic but leave the surface unstretched. Proceed to the top of the second layer. Treat this as the new ``surface,'' and stretch all the layers beneath again to make the second layer isotropic. Continue down through all the layers. Each stretching step leaves the surface data unchanged and makes another layer isotropic. Figure 3 illustrates this stretching process for a two layer model. Helbig (1983) discusses these ideas in detail, but considers only stretching perpendicular to layer boundaries.

Just as for a single layer, an isotropic migration with the correct velocities will focus the data, but a second layer-by-layer stretching step is necessary to map the focussed image to true depth.

Does stretching have meaning?

It is possible to stretch one wavetype in isolation, such as SH or P in transverse isotropy. The standard elastic wave equation cannot be stretched, however, because the orthogonality of particle motion between wave types would be broken in the process. This does not mean that the stretching operation has no usefulness. Geophysicists normally assume hyperbolic moveout on all events, even though it is not strictly true. In practice it is usually a good approximation for reasonable offsets. Assuming a hyperbolic moveout, however, is the same as assuming an elliptical wavefield, so this simple stretch model applies in any case where the hyperbolic move-out approximation is appropriate.

REFERENCES

Blair, J. M. and Korringa, J., 1987, Aberration-free image for SH reflection in transversely isotropic media: Geophysics, 52, 1563-1565.

Helbig, K., 1983, Elliptical anisotropy - Its significance and meaning: Geophysics, 48, 825-832.