2-D Seismic Data Processing Exercise

Post-stack migration

In this part of the workflow, we will experiment with seismic migration applied to the previously generated DMO stack of the Nankai dataset. The particular migration algorithm that we will use is Stolt migration based on the Fourier transform (Stolt, 1985).

1. The input to migration is the DMO stacked section (Figure 13.)
2. Stolt migration amounts to a 2D Fourier transform from the time-space $\{t,y\}$ coordinates of the stack to the frequency-wavenumber coordinates $\{\omega,k\}$ followed by mapping from $\omega$ to $\omega_0$ according to
   

   $$\omega = \sqrt{\omega_0^2 + \frac{v^2}{4}\,k^2}\;,$$ (2)

   
   
   where $v$ is the migration velocity, and the inverse Fourier transform from $\{\omega_0,k\}$ to $\{t_0,x_0\}$ coordinates of the time-migrated image.

   To compute and display the map from equation 2, run

   scons map2.view
   

   To display the 2-D Fourier transform (actually 2-D cosine transform) of the Nankai data before and after Stolt mapping with $v=2$ km/s (Figure 15), run

   scons cosft.view
   scons cosft2.view
   

   

   cosft,cosft2
   Figure 15. Nankai stack in the Cosine transform domain before (a) and after (b) Stolt migration with velocity 1500 m/s.

   

   
   

   

   mig2
   Figure 16. Nankai stack migrated with velocity of 1500 m/s.

   

   
   

3. Run

   scons mig2.view
   

   to display a seismic image migrated with the velocity of 1500 m/s. To look at the change brought by migration (Figure 16), run

   sfpen Fig/stack.vpl Fig/mig2.vpl
   

   What changes do you notice?

   

4. Of course, the real velocity is not constant and is likely to change in space. To try a more realistic velocity distribution, we will start with a velocity that increases with vertical time (Figure 17.) The velocity starts with 1500 m/s (water velocity) at the surface and remains constant until the water bottom reflection around 6 s. The velocity then increases to 1750 m/s at the vertical time of 8.5 s. To perform migration with a variable velocity using the Stolt method, we will migrate with a number of constant velocities in the range from 1500 to 1750 m/s and then slice through this ensemble of migrations to create an image (Mikulich and Hale, 1992). Thanks to the speed of the FFT algorithm, the whole operation is reasonably fast.

   

   vmig
   Figure 17. Variable migration velocity.

   

   
   

   To generate an image using an ensemble of Stolt migrations (Figure 18), run

   scons mig.vpl
   scons mig.view
   

   To look at the change brought by a variable velocity, run

   sfpen Fig/mig2.vpl Fig/mig.vpl
   

   Do you notice any interesting changes? Your task is to edit the SConstruct file to modify the migration velocity function used in order to improve the migration results.

   

   mig
   Figure 18. DMO stack migrated with variable velocity.

   

   
   

5. Let us select a small region of the image where a particularly interesting change occurs. One of such regions is shown in Figure 19. To display it on your screen, run

   scons zoom.view
   

   

   zoom
   Figure 19. Zoomed comparison: (a) unmigrated DMO stack (b) migrated with 1500 m/s, and (c) migrated with variable velocity.

   

   
   

   Edit and modify the SConstruct file to select a different region that you find interesting and modify the figure.

2-D Seismic Data Processing Exercise