Limited temporal frequency bandwidth

Another major source of inaccuracy is the limited temporal frequency band of the recorded data. Figure 5 schematically explains this phenomenon. We begin with an image gather in the offset domain, perfectly described in its limited k_z-k_h band (a). From the image we model data described in $\omega-k_h$ coordinates (b). In the case of wide temporal frequency bands, we record the entire data (b, black shape). After migration, we return to the image domain (c, black shape) described by k_z-k_h coordinates, from which we compute angle-gathers by RTT (d, black shape). Inverse Fourier transform along k_z brings us back in the angle domain, with correct amplitudes for all angles smaller than $\gamma_{max}$.

Alternatively, we can record a narrower frequency band (b, gray shape). After migration, we obtain the image as described by the DSR equation (c, gray shape). Conversion to angle follows the same style as in the full-band case (d, gray shape). The major distinction between the two cases is that we do not recover correct amplitudes even for the ($-\gamma_{max},+\gamma_{max}$) range.

Again, this second cause of inaccuracy is related to data sampling and cannot be corrected during processing. One possible way to recover consistent amplitudes is to low-pass-filter the angle-domain gathers to appropriate maximum depth wavenumbers (k_z).

The same logic applies to offset ray-parameter gathers, as depicted in Figure 5 (e,f).

 

adconv
Figure 5 Schematic simulation of migration, modeling and conversion to the angle domain. For angle-domain gathers, start with the image in (a), model to obtain the data in (b), migrate (c), and convert to angle (d). For offset ray-parameter gathers, start with the image in (a), model to obtain the data in (b), convert to offset ray-parameter in the data space (e), and image (f). The black shapes correspond to a wide temporal frequency band, and the gray shapes correspond to band-limited data.