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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
zerooffset 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
constantvelocity medium, the expression for x_{p}, equation (),
simplifies to:
 
(43) 
Then xx_{p}, 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 (twoway) zero offset traveltime to the seabed and subsea
reflection at any midpoint as a function of the corresponding zerooffset
traveltimes at the midpoint location, y_{0}:
 

 (45) 
 
 (46) 
where the small angle approximation was employed as before. Substituting the
zerooffset 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 (). Locallyplanar reflectors are not
required to implement equation ().
Next: Derivation of Snell Resampling
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Previous: Conclusions on the 3D
Stanford Exploration Project
5/30/2004