Next: Derivation of Snell Resampling Up: Least-squares joint imaging of Previous: Conclusions on the 3-D

# HEMNO Equivalence with Levin and Shah's Equations

In this appendix, I prove that the HEMNO equation is equivalent to Levin and Shah's traveltime equation Levin and Shah (1977) in the limit of small dip angle. They show that in a constant velocity medium with dipping target reflector and multiple generator, the moveout equation of the S102G'' pegleg multiple (see Figure ) is:
 (39)
where and are the dip angle (in radians) of the multiple generator and target reflector, respectively. and are the zero-offset traveltimes to the two reflectors, x is offset, and V is the medium velocity. For small dip angles (i.e., less than 5 degrees), we can make the small angle approximation for angles , , and to update equation () accordingly:
 (40)
Multiplying out the squares in equation () and collecting terms gives:
 (41)
The and terms are negligible for small angles, so we can ignore these terms and further simplify equation ():
 (42)
I will now show that the HEMNO equation () is equivalent to the Levin/Shah equation () under the constant velocity and small dip angle assumptions. First I make some preliminary definitions. In a constant-velocity medium, the expression for xp, equation (), simplifies to:
 (43)
Then x-xp, which will be needed later, simplifies to:
 (44)
Since the reflectors in this derivation are assumed planar and the velocity is assumed constant, using equations () and (), we can directly write the (two-way) zero offset traveltime to the seabed and subsea reflection at any midpoint as a function of the corresponding zero-offset traveltimes at the midpoint location, y0:
 (45) (46)
where the small angle approximation was employed as before. Substituting the zero-offset traveltimes () and () into the HEMNO equation () yields:
 (47) (48)
Equation () is equivalent to equation (). Therefore, we have proven the equivalence of the moveout equations of the true and approximate raypaths shown in Figure , subject to the small dip angle approximation. As before, and terms were dropped in going from equation () to equation (). Although explicit seabed and subsea reflector dip angles, and , are contained in equation (), they were introduced only to show equivalence to equation (). Locally-planar reflectors are not required to implement equation ().

Next: Derivation of Snell Resampling Up: Least-squares joint imaging of Previous: Conclusions on the 3-D
Stanford Exploration Project
5/30/2004