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# CONJUGATE OPERATOR

The strategy for interpolation of missing traces is an inversion which determines the missing traces so as to minimize the high-dip-pass filtered output. For optimization, we use a conjugate-gradient algorithm. Conjugate-gradient inversion is an iterative method in which each iteration involves the application of a forward operator and its transpose.

The forward operator of a recursive dip filter can be represented as

 AqI> = BpI> orI> qI>=A-1BpI>, (5)

where A and B are matrices as follows:

 b1 a1 . . . . . . . . . . a1 b1 a1 . . . . . . . . . . a1 b1 a1 . . . . . . . . . . a1 b1 . . . . . . . . b2 a2 . . b1 a1 . . . . . . a2 b2 a2 . a1 b1 a1 . . . . . . a2 b2 a2 . a1 b1 a1 . . . . . . a2 b2 . . a1 b1 . . . . . . . . b2 a2 . . b1 a1 . . . . . . a2 b2 a2 . a1 b1 a1 . . . . . . a2 b2 a2 . a1 b1 a1 . . . . . . a2 b2 . . a1 b1
 q11 q12 q13 q14 q21 q22 q23 q24 q31 q32 q33 q34
=

 d1 c1 . . . . . . . . . . c1 d1 c1 . . . . . . . . . . c1 d1 c1 . . . . . . . . . . c1 d1 . . . . . . . . d2 c2 . . d1 c1 . . . . . . c2 d2 c2 . c1 d1 c1 . . . . . . c2 d2 c2 . c1 d1 c1 . . . . . . c2 d2 . . c1 d1 . . . . . . . . d2 c2 . . d1 c1 . . . . . . c2 d2 c2 . c1 d1 c1 . . . . . . c2 d2 c2 . c1 d1 c1 . . . . . . c2 d2 . . c1 d1
 p11 p12 p13 p14 p21 p22 p23 p24 p31 p32 p33 p34

where and the subscript and the superscript in pji and qji represent offset-axis and time-axis indices respectively.

The transpose operator of equation (5) is defined by

 p = BT (AT)-1q. (6)

In actual calculations, instead of inverting the matrix, we can use a tridiagonal solver, as follows: In the appendix, we list the program that applies the operator and its conjugate operator, along with the result of the dot-product test for demonstrating conjugacy (Claerbout, 1991).

Next: NMO CORRECTION AND SPECTRA Up: Ji and Claerbout: Trace Previous: REVIEW OF CLAERBOUT'S RECURSIVE
Stanford Exploration Project
12/18/1997