Residual Stolt migration

The main idea of Sava (1999b) is that we can do residual Stolt migration by defining the updated depth wave number k_z for various velocity ratios $\gamma$ through the equation

$$\begin{array} {r} k_z= \frac{1}{2}\sqrt{\frac{1}{\gamma^2} ... ...0^2}- \left\vert{\vec k_m}+{\vec k_h}\right\vert^2},\end{array}$$ (6)

where

• k_z0 and k_z are depth wavenumbers before and after residual migration,
• $\bf v_0$ the background velocity field,
• $\gamma$ is defined as the $\frac{\bf v_0}{\bf v_n}$,and
• ${\vec k_m}$ and ${\vec k_h}$ are the offset and midpoint wavenumbers.

To make sure that residual Stolt migration behaves as we anticipate, I perform a test on the Marmousi synthetic. The top panel of Figure 2 shows every 15th common reflection point (CRP) gather of the Marmousi synthetic migrated with the correct velocity. Note how the CRP gathers are generally flat. The middle panel shows the same CRP gathers after migrating the data with a velocity $3\%$lower. The focusing has degraded and moveout in the gathers has increased. The bottom panel shows the result of performing residual migration with a $\gamma$ value of .97. Note how, as expected, the image is better focused and the gathers are flatter.

 

marm1
Figure 2 The top panel is the Marmousi synthetic migrated with the correct velocity. The center panel is the result of migrating with a velocity 3% less. The bottom panel is the result of residual Stolt migration with $\gamma=97\%$. Note how the gathers are almost as flat as in the original migration.

Figure 3 shows an angle image gather for seven different values of $\gamma$ ranging from .8 to 2.0. Note how we see the events move from curving down (left) to curving (up) as we increase $\gamma$. Unfortunately, the events also move as we change $\gamma$. We can eliminate this movement for flat events, and reduce it significantly for dipping events, by modifying our equation for k_z,

 

moveme Figure 3 CRP gathers after performing residual Stolt migration with a $\gamma$value between .80 and 1.20. Note the significant reflector movement.

 

$$\begin{array} {r} k_z= \frac{1}{2}\sqrt{\v2vkernelthrp- \ga... ...mma^2\left\vert{\vec k_m}+{\vec k_h}\right\vert^2} .\end{array}$$ (7)

Figure 4 shows the same image gather using equation (7). The reflector movement is generally eliminated from the gather.

 

nomove Figure 4 CRP gathers after performing residual Stolt migration using Equation 7. Note the lack of movement compared to Figure 3.