1. Seismic Reflection Data Processing and Interpretation A Workshop in Cairo 28 Oct. – 9 Nov. 2006 Cairo University, Egypt Dr. Sherif Mohamed Hanafy Lecturer Title: Migration (1)

2. What is the Migration Problem? For a horizontal reflector; Reflection point is midway between source and receiver.

3. What is the Migration Problem? For a dipping reflector; Reflection point shifted (migrated) upwards.

4. What is the Migration Problem? In seismic reflection processing, we usually assumes that the reflection point is mid way between source and receiver. Which is not true for dipping reflectors

5. What is the Migration Problem? What about reflectors dipping in two different directions?

6. What is the Migration Problem? Over a folded reflector;

7. What is the Migration Problem? In a more complex models;

8. Solving migration problem is simply, returning the reflection points to their original position and removing any duplications on the reflector images.

9. Solving migration problem. Mathematical point of view

10. Migration from Mathematical Point of View If we have LHS is the data space (recorded data) RHS is the model space G is Green’s Function m(x) is the model

11. xsxgSamp le i (0,0) 11 22 33 …. (0,0)(1,0)11601 (0,0)(1,0)21602 (0,0)(1,0)31603 …. (m,0)(0,0)1FWhere P is a vector contains 16 x 106 rows and one column (m,0)(1,0)2f + 1 …. (100,0) 160016 x 106 Solving Migration Problem To solve this equation, we discrediting it. Assume that we have a 100 source and a 100 receiver, then total number of ray paths=NoSor * NoRec = 100 * 100 = 10000. Assume that each ray path is recorded in 1600 samples, then we have a total of 16 x 10 6 recorded samples. Putting them in a table;

12. Solving Migration Problem Discrediting m(x) we get; We have a 100 point in x-direction and a 100 point in z-direction

13. xj (0,0)1 (1,0)2............ (0,1)fWhere m is a vector contains 10 4 rows and one column (1,1)f + 1................ (100,100)104 Solving Migration Problem Or in a table format, it gives

14. gsxij (0,0) 11 (1,0)(0,0)22........................ (1,0)(0,0)(0,10)fkWhere L ij is a matrix contains 16 x 10 6 rows and 10 4 columns (1,0) (0,10)f + 1k + 1................................ (100,0) (100,100)16 x 10 6 10 4 Solving Migration Problem Discrediting G(g|x)G(x|s) we get;

15. Solving Migration Problem The given equation can be solved as In matrix notation, we can write

16. Solving Migration Problem Solving this system of equations using least square, we have; Where ε is the error, in real world, ε never equal to zero because data has always noise and many approximation are made to reach this equation.

17. Solving Migration Problem Every trace in time domain is represented by one value in frequency domain, so working in frequency domain is much easier. In a P1, P2, and P3 dimension, the Lm and P of the last equation will be represented as vectors as shown in the figure

18. Solving Migration Problem One solution of equation (4) is the projection of Lm vector on P1-P2 plane

19. Solving Migration Problem So our solution will have the form The dot product will give zero because both vectors are perpendicular to each other.

20. Solving Migration Problem Knowing that

21. Solving Migration Problem the least square solution (LHS) = Migration (RHS).

22. Solving Migration Problem The least square solution of the migration problem shows that If we could !!!! Plot the error (in the vertical direction) and the model plane m1-m2 (in the horizontal plane), then each point on the resulted curve represent a solution to the problem, and the global minimum (m o ) represents the least square solution, i.e. the optimum solution to the problem

23. Solving Migration Problem But, practically, how can we solve the migration problem? Given: Data Required: Model Solution: ?????????????????????????????

24. End of this lecture Thank You for you attention All examples on this lecture is based on my work