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Even after application of the waterbottom reflection coefficient, the AVO response of the
pseudoprimary section created by equation (1) does not match that of the
corresponding NMOcorrected primary section. Refer to Figure 2 and note that
for constantAVO waterbottom reflection (and a free surface reflection coefficient of
1), the amplitude of the waterbottom multiple at
offset h_{p}+h_{m} is simply the amplitude of the primary at offset h_{p}, scaled by the
negative waterbottom reflection coefficient. Still, the question remains: what are h_{m}
and h_{p}? For the case of constant velocity, we can use trigonometry to derive h_{m}
and h_{p} as a function of the zero offset traveltimes of the primary reflection and
water bottom ( and , respectively), and the sourcereceiver offset x.
In constant velocity, the multiple and primary legs of the raypath are similar triangles:
 
(3) 
Also, for a firstorder waterbottom multiple,
h_{p} + h_{m} = x.
These two independent equations can be solved and simplified to give expressions for
h_{p} and h_{m}:
 
(4) 
I omit the general form of the expression for orders of multiple higher than one, although
it is straightforward to derive.
avo
Figure 2 Assuming a constant AVO waterbottom reflection
and constant velocity, we can write the AVO of waterbottom multiples with offset h_{p}+h_{m}
as a function of the AVO of the primary recorded at a shorter offset, h_{p}.

 
To obtain an estimate of the waterbottom reflection coefficient, I solve a simple least squares
problem to estimate a function of location, , which when applied to a small window of
dimension around the NMOcorrected waterbottom reflection, , optimally resembles the NMOcorrected
[equation (1)] firstorder waterbottom multiple reflection, . To achieve
this, is perturbed to minimize the following quadratic functional.
 
(5) 
may not be reliable at far offsets, due to either NMO stretch or nonhyperbolicity,
so in practice, an estimate of the single bestfitting waterbottom reflection coefficient
is made using the from ``useful'' offsets only.
Next: Leastsquares imaging of multiples
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Stanford Exploration Project
6/10/2002