Amplitude corrections of the mapping into angle domain

Tisserant and Biondi (2003) showed that in 3-D the transformation to angle domain is accomplished by mapping the image-domain offset wavenumber plane $\left(k_{x_h},k_{y_h}\right)$ into the plane defined by the reflection aperture angle $\gamma$and the reflection azimuth $\phi$.This mapping is performed at fixed depth wavenumber k_z and midpoint wavenumbers k_xm and k_ym.

Appendix A provides the analytical form of this transformation. The analytical form is fairly intricate and not easy to interpret because the reflection azimuth $\phi$enters only indirectly as a parameter for rotating the midpoint and offset wavenumbers. Figure  shows a graphical representation of the mapping to angle domain, and thus it illustrates the problem that I am addressing in this paper more intuitively than the formulas. The figure shows a uniformly sampled Cartesian grid in the $\left(\gamma,\phi\right)$ domain mapped into the $\left(k_{x_h},k_{y_h}\right)$ plane. Each dot corresponds to one value of $\gamma$ and $\phi$,for fixed k_z, k_xm and k_ym. The ranges for $\gamma$ and $\phi$are: $-80^\circ\leq \gamma\leq 80^\circ$and $-30^\circ\leq \phi\leq 30^\circ$.The dots are densely clustered close to the origin (corresponding to small values of $\gamma$), and become sparse away from the origin (corresponding to large values of $\gamma$). If this strong variability in the mapping density were not taken into account when a summation is performed in the angle domain, the summation would result in a strongly distorted image.

In this paper I consider the effects of averaging the image over reflection azimuth because it is the most challenging and interesting situation, as we will see briefly. However, similar considerations are needed when summing over aperture angles (for example when computing the ``stacked'' image after the application of a residual moveout correction to improve reflectors' coherency).

 

cig-1-data6 Figure 2 ADCIGs for a synthetic data set. Left: Image for only one azimuth ($\phi=16^\circ$). Right: Average of ADCIGs for all the azimuth within the range of $-60^\circ\leq \phi\leq 60^\circ$.

The effects of ignoring the variability in the mapping density are demonstrated in Figure . This figure shows two ADCIGs obtained by imaging a synthetic data set with a full source-receiver 3-D prestack migration. The data set contains 5 dipping planes, from zero dip to 60 degrees dip. The azimuth of the planes is 45 degrees with respect to the direction of the acquisition. The velocity was $V\left(z\right)= 1.5 + .5 z$ km/s, which corresponds to the upper limit among the typical gradients found in the Gulf of Mexico. The acquisition geometry had one single azimuth and the source-receiver offset range was -1.6-3 km. More detailed description of this data set can be found in Biondi (2001, 2003); Vaillant and Biondi (2000).

Figure a shows an ADCIG computed at fixed reflection azimuth $\phi$ of 16 degrees. This azimuth corresponds - only approximately, because the reflection azimuth changes with the aperture angle - to the reflection azimuth for the deepest reflector. Correspondingly, the deepest reflector has a flat moveout along the aperture angles, but the shallower ones are frowning downward. Figure b shows the result of averaging over azimuths all the ADCIGS within the range of $-60^\circ\leq \phi\leq 60^\circ$.In this case the moveouts are flat for all the reflection angles larger than 10 degrees, but the amplitude of the image is strongly attenuated for all these angles. This distortion of the image amplitudes is caused by the variable density of the mapping from the the $\left(k_{x_h},k_{y_h}\right)$ plane into the $\left(\gamma,\phi\right)$ plane illustrated in Figure .

The solution to this problem seems straightforward. We can correct the amplitudes by applying the jacobian of the transformation from the $\left(k_{x_h},k_{y_h}\right)$ plane into the $\left(\gamma,\phi\right)$ plane (Appendix A presents the formulas to evaluate the jacobian). However, while this correction yields a much improved result, it is not sufficient. Figure a shows the effect of including the jacobian while transforming the image into angle domain. The amplitudes of the ADCIG are now distorted in the opposite direction of the previous result (Figure b). Now the wide aperture angles have a good amplitude response, but the narrow angles have been too strongly attenuated and the $\gamma=0$ trace has been zeroed. This behavior is simply explained by the fact that the jacobian is zero at $\gamma=0$.This singularity of the jacobian is graphically represented in Figure  by the fact that all the dots corresponding to $\gamma=0$ fall into the origin of the plane, where the dot density becomes effectively infinite.

A simple solution to this problem is suggested when we examine the inverse mapping, as it is graphically illustrated in the sketch in Figure . In this case an integration segment along the line defined by constant $\gamma$ and close to the origin of the $\left(k_{x_h},k_{y_h}\right)$ plane would expand into a long segment extending well beyond the usual range of $-180^\circ\leq \phi\leq 180^\circ$.However, because of the periodicity of the mapping, the segment is actually folded into the the $-180^\circ\leq \phi\leq 180^\circ$,and its effective length is limited to 360 degrees ($2\pi$). Taking into account of this fact, we can correct the jacobian weighting and recover the image amplitudes close to normal incidence. The details of the correction and its derivation are presented in Appendix A.

The effects of taking into account the folding of the azimuth axis in the mapping are demonstrated in Figure b. Now the angles close to near incidence have been properly imaged. Figure  shows windows of the ADCIGs shown in Figure  with narrower aperture angle ranges. The comparison of Figure a and Figure b demonstrates the improvements achieved by taking into account the folding of the azimuth axis.

 

cig-2-data6 Figure 3 ADCIGs for a synthetic data set. Left: Image obtained when the simple jacobian weighting is applied before averaging over azimuths. Right: Image obtained when the jacobian weighting takes into account the folding of the azimuth axis.

 

fold
Figure 4 Graphical representation of the stretching involved in the mapping from the $\left(k_{x_h},k_{y_h}\right)$ plane into the $\left(\gamma,\phi\right)$ plane.

 

cig-2-data6-win Figure 5 Zoom into the ADCIGS shown in Figure  to examine the differences between the two panels close to normal incidence.

Figures - illustrate the effect of the amplitude correction from another point of view. They show depth slices taken at the depth of 1,140 meters (corresponding to the reflector dipping at 45 degrees) before the stacking over azimuths. The reflection amplitudes are thus shown as function of both the aperture angle $\gamma$ and the azimuth $\phi$.Because of the poor azimuthal resolution close to normal incidence, the azimuthal range is wide for small $\gamma$;it narrows as $\gamma$ increases. Comparing Figure  with Figure  it is evident that the amplitude correction boost up the relative amplitudes of the image at large $\gamma$.

 

zaz-60-60-dense-all-v3-data6 Figure 6 Depth slice taken at the depth of 1,140 meters (corresponding to the reflector dipping at 45 degrees) before the stacking over azimuths. Notice that the azimuthal resolution is strongly dependent on the aperture angle.

 

zaz-60-60-dense-all-jac-v3-data6 Figure 7 Depth slice taken at the same depth as the slice shown in Figure  (z=1,1140 meters) after application of the proposed angular dependent weighting. Notice that the amplitudes close to normal incidence have been attenuated, but not zeroed, and the ones at large aperture angle have been boosted up.

 

• Relative phase shift across aperture angles ($\gamma$)
• Determining the azimuthal range