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The theory developed here is similar to the traditional survey-sinking
migration approach where the source and receiver wavefields are downward
continued into the earth Claerbout (2001).
However, in the traditional application of zero-offset survey sinking
downward continued source wavefields do not have an intrinsic
initial horizontal velocity.
In the teleseismic case, planar teleseismic sources nearly always
have non-zero horizontal velocities.
This discrepancy is reconciled by deriving modified survey-sinking equations.
To help conceptualize the equations required for this context,
consider a reference frame moving with a horizontal velocity
equal to that of the source wavefield.
In this reference frame a dipping plane wave propagates vertically.
Hence, the downward continuation operators applied to
recorded teleseismic data must transform to the double square root
equations in a reference frame moving with a horizontal velocity
equal to that of the source.
To enable source horizontal propagation, wavefields downward
continued to the next depth step are now dependent on horizontal
position (*x*) in addition to depth (*z*).
That is,

| |
(1) |

where subscripts *s* and *g* correspond to source and receiver
wavefields, respectively.
To begin the derivation the differential is expressed explicitly,
| |
(2) |

where partial derivatives with respect to the *x* and *z* coordinates may be readily
associated with wavefield parameters.
In a *v*(*z*) medium the constant horizontal slowness of the source wavefield, *p*, is defined by
| |
(3) |

the vertical slowness of source wavefield by
| |
(4) |

the variable horizontal slowness of the scattered receiver wavefield,
*p*_{scat}, by
| |
(5) |

and the vertical slowness of receiver wavefield by
| |
(6) |

where *v*_{s} and *v*_{g} represent the source and receiver wavefield velocities,
respectively. Differential pairs and , and
and are linked geometrically by
| |
(7) |

and
| |
(8) |

where and represent angles of the source and receiver propagation direction with
respect to the z-axis, respectively.
Replacing trigonometric expressions in equations (7) and (8) with
ray parameter equivalents, and by downward continuing
the source and receiver wavefields by equal depth steps (i.e. ) one may rewrite equation (2) as
| |
(9) |

Equation (9) demonstrates that the effects of the propagating
source wavefield are transferred to the receiver wavefield. Accordingly,
using the chain rule and algebraic manipulation, the partial
derivative of the receiver wavefield, *U*, with respect to depth is
| |
(10) |

Relating Fourier domain variables to the angular components of the
individual plane waves
| |
(11) |

and applying a Fourier Transform over the temporal coordinate in equation (10) yields
| |
(12) |

The solution to differential equation (12) expressed in a
discrete sense is,
| |
(13) |

With this expression, the receiver wavefield is downward continued
and the image of the reflector at each model point (x,z)
constructed through the usual imaging condition Claerbout (1971),
| |
(14) |

Although the method derived here is strictly for 2-D geometry, it may
be extended to account for out-of-plane
propagation by assuming an underlying 2-D medium
and accounting for the effects of 2.5-D propagation through
incorporation of the invariant cross-line horizontal
slowness. This method is also potentially applicable in 3-D, but
this is not explored here.

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Stanford Exploration Project

7/8/2003