Using S&M group theory to add fractures to the chalk matrix, we obtain the phase velocity curves in Figure as polar plots. The fractures are vertical and have constant azimuth along the inline direction. So we have a preferred inline-fracture orientation with a horizontal axis of symmetry. Examine just the resulting P-wave velocity. The solid line is the unfractured chalk velocity and it shows an isotropic response. Looking at the neighboring dashed lines, the fractured P velocities (4-5 km/s), the inline and crossline velocities coincide for purely horizontal propagation and up to 45 degrees from the horizontal. The discrepancy increases from there on and for purely vertical propagation until the inline and crossline differences reach a magnitude of 0.8 km/s. The shear waves exhibit similar behaviour, but the magnitude of those effects are generally smaller. Reflection amplitude behaviour at the interface will depend on both the stiffnesses above and below. So even if we see a strong anisotropy in velocity in one medium, there is still no guarantee that we will get a strong reflection amplitude anisotropy, since the other medium could vary in the same manner and the differences could be small.
Figures and show inline and crossline reflection amplitude comparisons. Very small differences exist for the PP reflection angles less than the critical angle. Approaching the critical angle, the reflection amplitudes show distinct changes between inline and crossline behaviour. The same is true for the PS reflection with the differences showing up at even smaller propagation angles. The most important effect, however, is the azimuthal change in critical angle. If we had designed an experiment to record out to the PS critical angle, that would give us the clearer information about anisotropy. However, for targets of interest one usually has precritical amplitudes available. Figures and tell us that the mode converted PS wave has a better chance in estimating the magnitude of anisotropy. Figure shows again phase velocity curves as a scalar plot. These plots differ from Figure in that the cracks are now fluid filled, while before they were unfilled voids; they show a decrease in the difference of the velocity curves.
Figures and are computed by incorporating not just a bulk excess compliance, but Hudson's approximation expressed in the S&M group domain Nichols (1989). These two pictures show for the PP azimuthal amplitude only small differences close to the critical angle, but the PS difference is now more pronounced, also influencing the position of the critical angle slightly.
Hudson's theory allows the variation of three parameters: the aspect ratio, the stiffness of the filling material, and the volume density of cracks. In the following examples, the crack filling material stiffness (a fluid) is maintained constant. First we consider a large aspect ratio and medium volume density. This model is an extreme example, and pushes the validity of our crack model. The P-wave sees a different bulk modulus, especially when compared to Figure . Figure shows the corresponding polar plots of phase velocity curves. The solid lines denote inline velocity curves. Comparing the P-wave velocities inline and crossline, we see no difference for horizontal propagation but some differences at other angles. The difference reverses sign along the propagation curve and is largest for near vertical propagation. The S-wave propagation exhibits somewhat different behaviour, but the effects are of similar magnitude. PP amplitude differences mainly show up near the PS critical angle. In the PS mode converted wave, we see the inline-crossline amplitude difference changing sign going from near vertical to PS critical angle propagation. We would expect this since the phase velocity curves crossing over in Figure . Again the PS critical angle is shifted with respect to the inline angle.
Next we consider in Figure a chalk and go to the other extreme, compared to Figure and crack it with a small aspect ratio, but giving a large volume density of fluid-filled cracks. We have many long, thin fluid-filled cracks (Fig. ). The large crack volume density lets us expect that the P-wave can traverse the chalk matrix relatively unobstructed in any direction (less azimuthal dependence). The shear wave however sees more displacement discontinuities due to the fluid inclusion over a much greater range of angles. For Figure , the P-waves show only minor azimuthal differences at medium propagation angles. The PP reflection amplitude in Figure shows hardly any differences inline and crossline. The PS converted mode (Figure ), however, shows more variability, with the differences now exhibiting a constant sign for all angles. Most importantly we recognize that the critical angle stays the same for inline and crossline experiments. In that case the PS critical angle does not give us any information about anisotropy.
The final example is a chalk matrix with a medium aspect ratio and a small volume density of cracks (Figure ). This is well within the validity limits of the theory. For fluid inclusion the reflection amplitudes in Figure and show hardly any sign of azimuthal amplitude variation. With a gas filling, the difference between inline and crossline appears again for PP as well as for converted mode PS reflections. The PS critical angle is shifted from inline to crossline. In this sense, discrimination between gas and fluid-filled cracks may be possible, but the ability to distinguish anisotropic from an isotropic behaviour is not guaranteed by looking only at inline and crossline experiments.
As shown by the above examples, we can give some estimation of the crack content and distribution by looking at azimuthal variation of reflection amplitudes. This analysis is different from just looking at constant azimuth reflection amplitudes, as is done in ``conventional'' 2-D AVO studies. In such constant azimuth studies, the search for anisotropy does not make much sense. We can always model the anisotropic AVO beautifully, but the problem lies in the inverse process, namely to estimate anisotropy from constant azimuth AVO. In that case an isotropic equivalent medium can give the same answer for a surface seismic experiment as an anisotropic medium. In this paper we have looked only at azimuthal variations and not at the ``absolute'' AVO behaviour. All our measurements are with respect to a fixed azimuth, that serves as a reference. Variations from that reference can indicate anisotropy due to fracture orientation and fracture content. The numerical modeling examples we showed above should be taken as a lower crack density bound, since they are based on a small volume density of cracks. Austin chalk may exhibit stronger fracturing in reality.