Migration / Image Gather Extraction

The migration program used in this project was developed by Biondi. It uses a variation of the split-step method Stoffa et al. (1990) with the Double Square Root (DSR) equation Claerbout (1985). So for each depth step wave-fields at different reference velocities are generated, then an interpolation in the space domain is used as needed. For this study, three reference velocities were used. By using the split-step with the DSR, Biondi's algorithm is very effective in positioning reflectors correctly, even in regions of sharp velocity contrast, such as sediment in contact with salt Claerbout (1985).

The extraction of angle gathers using this migration algorithm is relatively straight forward. After the wave-field has been downward continued, a slant stack is applied before imaging Biondi (2000):

$$D(\omega, {\bf m}, x_h; z=0) \; \stackrel{{DSR \it}}{\Longrightarrow} \; D(\omega, {\bf m}, x_h; z) \;$$ (4)

$$D(\omega, {\bf m},x_h;z) \; \stackrel{{Slant Stack \it}}{\Longrightarrow} \; D(\omega, {\bf m}, P_h; z) \;$$ (5)

$$D(\omega, {\bf m}, P_h; z) \; \stackrel{{Imaging \it}}{\Longrightarrow} \; D(t=0, {\bf m}, P_h; z) \;.$$ (6)

In order to ensure true amplitudes in the image, migration was done as in my other paper in this report Gratwick (2001). Amplitude weighting was applied according to Sava and Biondi (2001). Angle gathers are subsets of $D(t=0, {\bf m}, P_h; z)$ with the midpoint constant Biondi (2000). The angle gathers are actually not exactly a function of angle. Instead they are a function of offset ray parameter (P_h). Offset ray parameter and angle are related by equation (7):

$$\frac{\delta t}{\delta h} \; = \; P_h \; = \; \frac{2\sin \theta \cos \phi}{V(z,{\bf m})}.$$ (7)

In equation (7), $\theta$ is the incident angle, $\phi$ is the geologic dip, and $V(z,{\bf m})$ is the velocity function. Thus, if the geology is relatively flat and the velocity function can be well approximated, incidence angle is a simple function of offset ray parameter.