Naoshi Aoki's

Deblurring Migration Image Lab

Figure: (left) reference migration image from a grid model. (Right) deblurred migration image.

# Objective

Migration image is an approximation of reflectivity model and is blurred by many factors such as a complex subsurface structure and limited seismic acquisition geometry. In this lab, we deblur migration image with 2 different methods: 1) Deblurring Filter, and 2) Least Squares Migration.

# Why Migration Image is blurred?

Forward modeling of acoustic data is mathematically represented with the operator L such that:

where d is scattered data, m is a reflectivity model and L is the forward modeling operator associated with the survey geometry, the source wavelet and the velocity model.

The reflectivity model can be obtained by inversion. However  is very expensive to compute. Instead of inversion, we often use an adjoint modeling, where another name for this is migration, because it is much cheaper and still useful.

The equation clearly shows a difference between inversion and migration. The migration result is  blurred version of m.

# Deblurring Filter Theory

To partly deblur the migration image  by an approximation to , we can employ a local layered media assumption (Hu and Schuster, 1998; Hu et al., 2001) or a multi-channel matching filter (Guitton, 2004). For the matching filter, we first find a local filter  that honors the following equations:

where m' is a reference model we need to provide and d' is synthetic data calculated by forward modeling Lm'. After finding the operator F, apply the same F to the actual migration image  so that we obtain an estimate of m.

This non-stationary F can be computed inexpensively because its size is only several wavelength wide and tall (Figure).

# Least Square Migration Theory

Least squares migration (LSM) is a linearized inversion of seismic data for the reflectivity model. This employs a gradient optimization method such as the steepest descent or the conjugate gradient methods (Nemeth et al., 1999; Chavent and Plessix, 1999; Rickett, 2003; Symes, 2008). The steepest descent method updates the estimated reflectivity model in the following way:

where  and  are the n+1 and n-th reflectivity models;   is the model update for the n-th iterative step;  n is the step length that can be found by minimizing the misfit functional with respect to the scalar , and  is the misfit gradient vector that is parallel to the steepest descent direction of the misfit function. The spatial resolution of LSM images is expected to be better than those seen in standard migration images because the wavelet is deconvolved and the acquisition footprint noise is largely suppressed with the inverse Hessian. A drawback of this technique is the computational expense: LSM typically requires 10 or more iterations, which is about 20 times or more the CPU cost of conventional migration.

# Procedure

2D deblurring filter: DeblurGrid2D, mchef2D, convmat2D

Regularized LSM: RLSM.m.

Others:

2.   Data preparation

Open "main_deblurring_lab". This shows the workflow for this lab.

First, type "create_folders" to create folders. Next, open "Prepare_data" and read the script. This provides actual common shot gathers and prestack migration image, and reference migration image. After you convince the script's objectives, type "Prepare_data" to run the script. You will see the following figures:

Table: List of figures shown in "Prepare_data"

Actual Model

Actual CSG (Animated)

Migrated Actual CSG (Animated)

Migration Image

Reference Model Image

Synthetic CSG (Animated)

Migrated Synthetic CSG (Animated)

Reference Migration Image

3.   Deblurring Filter

is used in line 27-28. Run "run_deblurring_filter " so that you obtain a deblurred migration image. See parameters and the descriptions that you used. Read the "DeblurGrid2D.m" script and investigate how it works. Try other parameters so that you get the best deblurred image.

4.   LSM

Type "run_LSM". This script calculates LSM up to 10 iterations. You can monitor the processing progress in a figure. It takes about a half hour. You can reduce the number of iteration if you don't have time. After 10 iterations, you will see convergence curves and several plots. Compare the deblurring filter and LSM results (Figure).

# Suggested Tests and Future Developments

Change background velocity value for generating actual data (Line19 of "Prepare_data")

Use an erroneous source wavelet (Line 36 of "Prepare_data").

Test your actual model (Change "get_actual_model").

Test a different spacing of a grid model.

Use your deblurred image as a priori model for a regularized LSM (Line 4 of "RLSM").

Increase a number of LSM iterations up to 30 iterations (nit in Line 74 of "RLSM").

Use LSM with a deblurring filter (Lines 97 and 151 of RLSM).

Store computed F for multiple applications in LSM algorithm.

Create more efficient matching filter (Maybe FFT version?).

# References

[1] Aoki, N, 2008 , Fast Least Squares Migration with a deblurring filter, MS thesis.

[2] Chavent, G. and R.-E. Plessix, 1999, An optimal true-amplitude least-squares prestack depth-migration operator: Geophysics, 64, 508-515.

[3] Guitton, A., 2004, Amplitude and kinematic corrections of migrated images for

nonunitary imaging operators: Geophysics, 69, 1017-1024.

[4] Hu, J., and G. T. Schuster, 1998, Migration deconvolution,: Mathematical Methods in Geophysical Imaging V, 3453, 118-124.

[5] Hu, J., G. T. Schuster, and P. A. Valasek, 2001, Poststack migration deconvolution:

Geophysics, 66, 939-952.

[6] Kuhl, H., and M. D. Sacchi, 2003, Least-squares wave-equation migration for

AVP/AVA inversion, : Geophysics, 68, 262-273.

[7] Lecomte, I., 2008, Resolution and illumination analyses in PSDM: A ray-based approach: The Leading Edge, 27, 650-663.

[8] Nemeth, T., C. Wu, and G. T. Schuster, 1999, Least-squares migration of incomplete reection data: Geophysics, 64, 208-221.

[9] Rickett, J. E., 2003, Illumination-based normalization for wave-equation depth

migration: Geophysics, 68, 1371-1379.

[10] Symes, W. W., 2008, Approximate linearized inversion by optimal scaling of prestack depth migration: Geophysics, 73, R23-R35.

[11] Toxopeus, G., J. Thorbecke, K. Wapenaar, S. Petersen, E. Slob, and J. Fokkema, 2008, Simulating migrated and inverted seismic data by _ltering a geologic model:

Geophysics, 73, T1-T10.