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Prior to the dip filtering, however, it is necessary to compute
the PRT in the Fourier domain, rather than in the time domain.
The equations that follow show a method for computing the PRT
in the frequency domain. In the time domain, the equation for
a parabola is
| |
(1) |

where is the zero offset time, *x* the offset, and *q* the
curvature of the parabola. The modeling equation, which
superposes on parabolas, from the radon domain to the CMP domain
is
| |
(2) |

and the adjoint transform, which sums along parabolas, is
| |
(3) |

We can easily transform these equations in the Fourier domain as follows:
| |
(4) |

| (5) |

With constant, equations (4) and (5) describe a simple matrix
multiplication with the operator :
| |
(6) |

Hence, we can write equations (4) and (5) in the following more
familiar way for each frequency:
| |
(7) |

| (8) |

where is the adjoint of . I have computed the
radon and offset domains in Figure 2 using equations
(7) and (8). The following is a possible algorithm for
computing the radon domain:
- Fourier transform the data.
- For each frequency, starting from the lowest to the
highest, solve equation (8).
- Inverse Fourier transform .

Because we can compute a *pseudo*-inverse for **L** noniteratively in the
Fourier domain Kostov (1989), the PRT is generally not
computed in the time domain.

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Stanford Exploration Project

4/29/2001