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Retarded Muir recurrence

The kz square root may be computed with the square root function in your computer or by Muir's expansion. For Fourier domain calculations incorporating causality, you must use a complex square root function. This will also take care of the evanescent region automatically--you no longer have the discontinuity between evanescent and nonevanescent regions. The square root of a complex number is multivalued, so you better first check that your computer chooses the phase as described in Figure . Mine did. But I found that limited numerical accuracy prevented me from achieving strict positivity of the real part of the impedance until I replaced the expression by its algebraic equivalent .

For finite differencing we will need the Muir recurrence. Let r0 define the cosine of the angle that starts the Muir recurrence, often 0 or 45.This is another free parameter for optimization. This angle is also an angle of exact fit for all orders of the recurrence.
 (52)
Starting from the Muir recursion is equation (53):
 (53)

For a diffraction program we will be evaluating .Since R can be proven to have a positive real part, the exponential should never grow. Finite difference calculations are normally done with retarded time. To retard time, is expressed as
 (54)
As discussed earlier in this chapter, you probably don't want the time shift of retardation to be associated with viscous effects. So you will probably want to downward continue instead with
 (55)
Notice the signs and distinction of from .

From equation (53) we see that R-s should have a positive real part. I found that numerical roundoff sometimes prevented it. So the Muir recurrence was reorganized to incorporate the retardation. Let
 (56)
Equation (53) becomes
 (57)
From Muir's rules, you can see that R ' will always have a positive real part if we start it that way, so we start it from
 (58)
(Combining (58) and (56) gives the same 15 equation as does (53).) Mathematically (55) is identical to
 (59)
but numerically the exponential in (59) is assured to decay in z.

Next: Stepping in depth Up: ACCURACY THE CONTRACTOR'S VIEW Previous: Viscosity and causality
Stanford Exploration Project
10/31/1997