The principle of causality: no response before a stimulus, is not considered
in the *R* definition [equation (2)]. The omission of causality
translates into improper behavior of the high frequencies.
In order to include causality into the definition of *R*, let be the causal,
positive, discrete representation of the differentiation operator,

(3) |

which is simplified by writting .

Claerbout (1985) proposes the use of the following Muir recursion starting
from *R _{0}*=

(4) |

This recursion produces a continuous fraction. Studying the limit for we obtain:

(5) |

If we let *X ^{2}* =

(6) |

The following change of variables:

(7) |

transforms the Muir recursion (4) into:

(8) |

Again, taking the limit for in this recursion we will obtain

(9) |

This quadratic expression yields to two square roots for ,

(10) |

We need to select the one that is able to handle the evanescent region, *i.e*,
the square root that goes to zero at *k*_{x} = 0, which corresponds to the positive
square root.

(11) |

(12) |

I already showed that the real part of is positive (Figure 1); therefore, , as defined in equation (12), also has a positive real part.

Finally, we downward continue the data in the Fourier domain by multiplying by:

(13) |

Expression (13) incorporates the causality and viscosity concepts in phase shift migration, controls the evanescent energy and will not allow discontinuity between evanescent and non-evanescent regions.

4/29/2001