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# Second-order reflection traveltime derivatives

In this appendix, I derive equations connecting second-order partial
derivatives of the reflection traveltime with the geometric properties
of the reflector in a constant velocity medium. These equations are
used in the main text of Chapter for the amplitude
behavior description. Let be the reflection traveltime
from the source *s* to the receiver *r*. Consider a formal equality
| |
(266) |

where *x* is the reflection point parameter, corresponds to the
incident ray, and corresponds to the reflected ray.
Differentiating () with respect to *s* and *r* yields
| |
(267) |

| (268) |

According to Fermat's principle, the two-point reflection ray path must
correspond to the traveltime stationary point. Therefore
| |
(269) |

for any *s* and *r*. Taking into account () while
differentiating () and (), we get
| |
(270) |

| (271) |

| (272) |

where
Differentiating equation () gives us the additional
pair of equations
| |
(273) |

| (274) |

where
Solving the system () - () for and and substituting
the result into () - () produces the
following set of expressions:
| |
(275) |

| (276) |

| (277) |

In the case of a constant velocity medium, expressions () to
() can be applied directly to the explicit
equation for the two-point eikonal
| |
(278) |

Differentiating () and taking into account the trigonometric
relationships for the incident and reflected rays (Figure
), one can
evaluate all the quantities in () to () explicitly.
After some heavy algebra, the resultant expressions for the traveltime
derivatives take the form
| |
(279) |

| (280) |

| |
(281) |

| (282) |

| |
(283) |

| |
(284) |

| |
(285) |

Here *D* is the length of the normal (central) ray, is its dip angle
(, ),
is the reflection angle
, *K* is the reflector
curvature at the reflection point , and
*a* is the dimensionless function of and defined in ().
The equations derived in this appendix were used to get the equation

| |
(286) |

which coincides with () in the main text.

** Next:** Solving the Cauchy problem
** Up:** Three-dimensional seismic data regularization
** Previous:** Conclusions
Stanford Exploration Project

12/28/2000