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## Lateral velocity variation

When the velocity varies laterally, the matrix M in the extrapolation operator at equation (7) has coefficients which varies along the diagonal, but we can still derive a symmetric matrix (Godfrey et al., 1979). The symmetry property, which will guarantee the unconditional stability, can be obtained by putting the velocity term on both sides as follows

 (20)

Now the small block matrices which are located along the diagonal in split matrices Me and Mo will have a symmetric form

where

and

The eigenvalues of are given by

and lie on the unit circle since a and b are imaginary, and .It follows that the matrix norms and .To prove the unconditional stability of the algorithm we need only show that . This follows immediately since , each of which is unity according to the preceding argument.

Next: WIDE-ANGLE DEPTH MIGRATION Up: 15-DEGREE DEPTH MIGRATION Previous: Accuracy
Stanford Exploration Project
12/18/1997