Prestack Least Squares Migration Lab

OBJECTIVE: Least Squares migrate synthetic data with a MATLAB script. Determine how migration section is affected by accuracy of migration velocity, sampling interval, and aperture width.

PROCEDURE:

• Load into your working directory MATLAB scripts for the main program (testforw.m), diffraction stack modeling (forw.m), Ricker wavelet generator (ricker.m). migration program (mig.m), and veltime model generator (TIME.m). Name each file by their names given above.
• In MATLAB, type "testforw" to generate and migrate the shot gather synthetic data for a point scatterer model.
• Turn code into steepest descent least squares migration code. That is, the steepest descent formula is given by m'=m-alpha L^Tdelta d, where m' is the new migration image, m is the old image, delta d is the residual data d^predicted - d^observed, L^T is the migration operator and alpha is the step length. In practice the step length is an empirically derived fraction of the analytic step length. Here we can try with the analytical optimal step length (steplength.m) by adding this step length calculation into the iterations. Also turn the code into conjugate gradient least square migration code and apply optimal step length in the iterations.
  1. What are the effects of migration velocity error on estimates of depth?
  2. What are the effects of migration velocity error on migration noise?
  3. What are the effects of migration velocity error on reflector shape?
  4. Just before executing the mig.m program zero out every 2nd trace so that the spatial sampling interval is 50 m rather than 25 m. Migrate these data, and comment about aliasing artifacts.
• Trace Sampling Effects: Then retain every 4th trace so that the spatial sampling interval is 100 m rather than 15 m, and migrate these data. Answer the following questions.
  1. What are the effects of a coarse spatial sampling interval on the migration and LSM sections?
  2. What parts of the model generate the strongest migration artifacts, the steep or shallow dipping structures?
• Dot product test. In order for the conjugate gradient algorithm to work you must insure that the the migration subroutine truly produces the adjoint L^T of the forward modeling operator L. You can test for adjointness by performing the dot product test by numerically checking for equality of the following equation (Lm,d) = (m,L^Td). Perform the dot product test on forw.m and mig.m.
• Option: Migrate radar lab.