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## Two conjugate transpose datuming algorithms

We can epitomize the datuming algorithm in a depth varying velocity by writing the successive operators in matrix form. For simplicity I will take only a datum with three depth levels similar to the one in Figure . The algorithm can be generalized for any number of depth levels and any datum geometry. For each value of we have
 (6)
where represents the zero-offset data after Fourier transformation along the time axis. The matrix

performs the Fourier Transform matrix of the horizontal space variable. Therefore the matrix

is composed of inverse Fourier Transform block matrices, each being the conjugate transpose to the direct transformation. The matrix

contains three blocks corresponding to the product of diagonal matrices of the form
 (7)
On each diagonal are the phase-shifting exponentials necessary to upward extrapolate the wavefield to a depth level. The value of (kz)ij is given by the dispersion relation:

 Datumatrix Figure 2 Datuming for only three depth levels. Data extrapolated to the first depth level is extracted by matrix A, to the second level by matrix B and to the third by matrix C.

The matrices A,B,C are matrices corresponding to the shape of the datum. For the case shown in Figure the matrices A,B,C are

The conjugate transpose algorithm can be obtained by reversing the order of matrix multiplication in equation (6) and transposing-conjugating each matrix. The matrices A,B,C are diagonal matrices and therefore equal to the transpose. For each value of the frequency the conjugate datuming is
 (8)
where represents the wavefield recorded on the topographic datum. The matrices W*i are conjugate transposed of the diagonal matrices Wi which are explicitly represented in equation (7).

Next: Wave-equation Datuming for laterally Up: WAVE-EQUATION DATUMING ALGORITHM Previous: WAVE-EQUATION DATUMING ALGORITHM
Stanford Exploration Project
11/17/1997