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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
| |
(1) |
where x is the reflection point parameter, corresponds to the
incident ray, and corresponds to the reflected ray.
Differentiating (A-1) with respect to s and r yields
| |
(2) |
| (3) |
According to Fermat's principle, the two-point reflection ray path must
correspond to the traveltime stationary point. Therefore
| |
(4) |
for any s and r. Taking into account (A-4) while
differentiating (A-2) and (A-3), we get
| |
(5) |
| (6) |
| (7) |
where
Differentiating equation (A-4) gives us the additional
pair of equations
| |
(8) |
| (9) |
where
Solving the system (A-8) - (A-9) for and and substituting
the result into (A-5) - (A-7) produces the
following set of expressions:
| |
(10) |
| (11) |
| (12) |
In the case of a constant velocity medium, expressions (A-10) to
(A-12) can be applied directly to the explicit
equation for the two-point eikonal
| |
(13) |
Differentiating (A-13) and taking into account the trigonometric
relationships for the incident and reflected rays (Figure
), one can
evaluate all the quantities in (A-10) to (A-12) explicitly.
After some heavy algebra, the resultant expressions for the traveltime
derivatives take the form
| |
(14) |
| (15) |
| |
(16) |
| (17) |
| |
(18) |
| |
(19) |
| |
(20) |
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
| |
(21) |
which coincides with () in the main text.
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Stanford Exploration Project
12/30/2000