Phase shift migration uses the dispersion relation of the scalar wave equation Gazdag (1978) in order to perform wavefield downward continuation. This relation presents computational implementation problems when facing evanescent waves due to negative number in the square root. The solution is to redefine the Fourier kernel for the downward continuation by adding a damping factor, ,Claerbout (1999a). However, this redefinition faces stability problems related to the choice of .
Concepts like causality Claerbout (1999b) and viscosity Claerbout (1995); van Trier and Symes (1990) can be incorporated in the redefinition of the Fourier kernel in order to perform phase shift migration. This modification does not discard the evanescent energy in the computational implementation of phase shift migration; it is steady with respect to the damping factor, and it incorporates cosmetic features to the final image.
This work presents a review of the theory for the damping factor in the dispersion relation. Numerical examples show how the omission of causality and viscosity in the redefinition of the dispersion relation translates into a loss of high frequency information in the final migration result. Finally, I will show examples with the redefinition of the dispersion relation that incorporates causality and viscosity in phase-shift migration.