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

<I>AqI> = <I>BpI> <I>orI> <I>qI>=<I>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

=

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

where and the subscript and the superscript in p_jⁱ and q_jⁱ represent offset-axis and time-axis indices respectively.

The transpose operator of equation (5) is defined by

p = B^T (A^T)^-1q.

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).