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In this Appendix I review the derivation of the expression for upward continuation of a scalar wavefield given by Berryhill 1979.

The Kirchhoff integral equation Schneider (1978) may be written
for an up-going scalar wavefield *U*(*x*,*y*,*z*,*t*) as

(3) |

If the wavefield is
only known along a line in the plane and is assumed to be independent of the
coordinate perpendicular to this line, the integration can be carried out
over the perpendicular coordinate. The geometry of this situation is
illustrated in Figure A-A-1 where the
wave field is recorded along the
*x* direction and is assumed to be constant in the *y* direction.
Following Berryhill 1979 the integral over *dA* is performed
for an observation point *U*(0,*z*,*t*) directly above the origin of the
coordinate axes:

(4) |

datumfig1 Geometry for the upward continuation of a wavefield.
Figure 12 |

In order to do the integration in the *y* direction the following substitutions
can be made according to Figure A-A-1:

(5) |

Equation (A-A-3) is of the form of a convolution in time. For reference I write the general form of the convolution integral as:

where corresponds to . Defining Equation (A-A-3) can now be written as (looking only at the time part): noting that: andEquation (A-A-3) can now be written as:

(6) |

(7) |

The integrand in equation (A-A-4) can be written as

(8) |

Finally the expression for the upward continued wavefield is written as

(9) |

For a wavefield transformation from one datum *U*(*x*,*z*=*z _{1}* ,

(10) |

The effect of the filtering operation implied by equation (A-A-6) is to perform a phase-shift of the traces input to the datuming algorithm. This is illustrated in Figure A-2. The input trace is convolved with the second derivative of to produce the output trace which goes into the summation of equation (A-8). The filter is tapered to eliminate undesirable truncation effects.

For downward continuation, the input traces are crosscorrelated with the filter after the summation of equation (A-8) has been performed on the unfiltered traces.

quedemo The wavelet (upper left) is convolved with the second derivative of (upper right) to yield the phase shifted trace (bottom).
Figure 13 |

11/17/1997