The azimuthal resolution of 3-D Angle Domain Common Image Gathers (ADCIGs) strongly varies with the reflection aperture angle. This dependence may cause severe distortions in the image when 3-D ADCIGs are averaged over azimuths. To correct for these distortions, I derive an effective weighting method based on the jacobian of the transformation to angle domain. The proposed method avoids the underweighting of the reflections close to normal incidence by properly taking into account the ``folding'' of the azimuth axis. A simple scheme to limit the range of the azimuthal averaging as a function of the opening angle further attenuates noise in the image. A synthetic example illustrates the practical application of the proposed methodology .
Tisserant and Biondi (2003) presented a method to create Angle Domain Common Image Gathers (ADCIGs) in 3-D. In 3-D ADCIGs the image is decomposed at each physical location (x,y,z) depending on the aperture angle and the azimuth of the reflections. Given the the limited azimuthal range of many common acquisition geometries (e.g. marine streamer data), it is often useful to average the ADCIGs over azimuths and to limit the azimuthal average to a subrange of the possible azimuths. These procedures can attenuate coherent noise that was recorded in the data (e.g. multiples) and/or caused by computational shortcuts Biondi (2003). However, because of the variable resolution of the angle decomposition in the azimuthal direction, the averaging over azimuths may cause distortions in the amplitudes and the phases of the final image.
In this report I address the problems of 1) balancing the amplitudes across aperture angles while performing the stack over azimuths, 2) determining an ``optimal'' azimuthal subrange as a function of the aperture angle. Both of these problems are related to the strong dependence of the azimuthal resolution with aperture angle. The azimuthal resolution decreases as the aperture angle gets closer to normal incidence; at the limit, all azimuths are equally illuminated at normal incidence. To preserve the relative amplitudes between the whole range of aperture angles, we need to introduce a proper weighting factor when stacking the ADCIGs over azimuths. If no normalization factor is applied during the summation, the reflections with angles close to normal incidence would totally overshadow the reflections with wider aperture angles. This normalization factor must be obviously based on the jacobian of the transformation into angle domain, but a straight application of the jacobian would underweight the reflections close to normal incidence. To avoid such a problem, I define a weighting method that takes into account the ``wrapping'' of the azimuth axis close to normal incidence.
Figure 1 Graphical representation of the mapping from the offset wavenumber plane into the plane. Each dot corresponds to one value of and ,for fixed kz, kxm and kym.