Specular reflectivity estimation

In this section we present the results of inverting the synthetic shot gathers for specular reflectivity. A total of 141 shot gathers is migrated to produce the estimates, which is identical to the field data requirement. The migration model used is identical to the elastic model of Table 1 for depths above the gas reflector, but continues through the reservoir zone as a homogeneous halfspace with the properties of the shale layer just above the gas sand. Figure displays the $R_{pp}(z,x_h;{\bf x_m})$ elastic $\grave{P}\!\acute{P}$ estimate, and Figure the $\theta_{pp}(z,x_h;{\bf x_m})$ estimate, for a single fixed surface position.

 

Rpp
Figure 3 $R_{pp}(x_h,z;{\bf x_m})$; migration of marine synthetics.

 

theta
Figure 4 $\theta_{pp}(x_h,z;{\bf x_m})$; marine field data migration. The contours represent specular reflection angle in 5$^{\circ}$ increments, starting with a 5$^{\circ}$ contour at the nearest offset.

Figures and are true common depth point images, as a function of source-receiver offset, as opposed to migrated CMP gathers. Since each gather corresponds to a constant x slice perpendicular to the (x,z) plane (the third dimension being the offset coordinate), reflected events in the true CDP are always flat (provided the velocity model is correct), independent of reflector structure in the (x,z) plane. The near offset traces in Figure have the correct relative $R_{pp}(\theta\approx 0^{\circ})$ (peak) amplitudes to within a few percent, as well as the correct Zoeppritz AVO response. The shallow event at 1960 m is a little weak since it is within a Fresnel zone of the mute pattern in the shot gathers. The two reflections beneath the reservoir reflection at 2960 m are somewhat attenuated due to the low Q values in the gas zone which are not present in the migration model. Figure is a contoured version of an identical gather to Figure , except it displays the calculated $\theta_{pp}$ specular angles as a function of depth and source-receiver offset. (This is actually the $\theta_{pp}$ estimate from the raw field data, since the synthetic $\theta_{pp}$ is more difficult to interpret due to large zero-valued areas where no reflections exist). At the reservoir depth, we see that the specular angles of reflection range from about $3^{\circ}$ to $33^{\circ}$, which is correct to within $1^{\circ}$ compared to raytraced values. The kinks in the contours at 1970 m and 2610 m are accurate evidence of Snell's Law effects on $\theta_{pp}$ at abrupt changes in the migration velocity model.

Figure compares the migration estimate of $R_{pp}(\theta)$ at the gas reflection (2960 m) to the true Zoeppritz $\grave{P}\!\acute{P}$ response as a function of specular angle. The plotted open circles are the correct Zoeppritz values. The filled circles are the migrated $R_{pp}(\theta)$ estimates taken from combining the information in Figures and , after applying a constant scale factor normalization (loss of absolute amplitude in migration: only relative amplitudes are recoverable).

 

Rtheta
Figure 5 $R_{pp}(\theta)$ true vs. $R_{pp}(\theta)$ estimated from migrated data.

The extra curve is a quadratic fit to the estimated $R_{pp}(\theta)$. One can immediately see the good agreement between the correct and estimated Zoeppritz specular reflectivity. This level of accuracy in the migration estimate of R_pp on the gas event is representative of the R_pp recovery on the other reflection events in the gather. In particular, it is evident that the finite aperture amplitude correction inherent in our method has compensated well for the marine end-on recording geometry in this variable background model. It is important to note that the accurate estimation of R_pp, $\theta_{pp}$ and the aperture correction are valid for (laterally) heterogeneous media in general.

Finally, some preliminary stability issues in the $R_{pp}(\theta)$ estimation have been investigated. The migration has proved extremely stable in the presence of random noise, certainly up to a s/n level of 2:1. The migration amplitudes are less stable in the presence of reasonable velocity-depth model errors (10%), but acceptable nonetheless. As the velocity-depth errors increase, the kinematic error tends to destroy the amplitude accuracy faster than the error due to geometric divergence (as a function of offset in the R_pp gather). This underlines the probable need for good residual migration velocity analysis before inverting $R_{pp}(\theta)$ for elastic parameter variations.

Considering amplitude loss effects, using an acoustic transmission loss model instead of an elastic model produces no error on the normal incidence R_pp values (no shear conversions at $\theta\approx 0^{\circ}$), and less than 10% decrease in amplitude at the far offsets at the target zone. Considering anelastic Q attenuation losses, prescaling the shot gathers with a time-variant rms Q gain function before migration resulted in correct normal incidence amplitudes, but an increase in far offset amplitudes in the target zone of about 100%. Thus, it seems that pre-migration Q amplitude compensation may not be appropriate for preserving migrated $R_{pp}(\theta)$ amplitudes, but rather, the compensation needs to be done properly in the depth model domain during depth migration. Finally, approximations to the WKBJ geometrical divergence amplitude within the migration by simple factors such as 1/r or $1/\tau$ produced accurate amplitudes at normal incidence, and were about 10% too weak at far offsets along the gas reflection. Although this error will increase for targets shallower than 3000 m, the approximation seems acceptable for this earth model and acquisition geometry. However, performing a time-variant divergence correction as a function of the rms velocity ($V_{rms}^2\tau$) greatly overemphasized the far offset amplitudes. Hence, as with Q compensation, it seems most reasonable to make approximate compensation for geometric divergence in the spatial domain (depth migration), and not in the time domain (time migration).