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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