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The main characteristic of convertedwave data is their nonhyperbolic moveout.
However, for certain offset/depth
ratios, it is possible to approximate the nonhyperbolic
moveout as a hyperbola Tessmer and Behle (1988).
Tessmer and Behle (1988) extend the work of Taner and Koehler (1969)
for converted waves. They apply a secondorder
approximation to the moveout equation to convertedwave
data. In such cases the stacking velocity corresponds to the product of
both P and S velocities known as convertedwave
velocity.
Castle (1988) presents the thirdorderapproximation
coefficient terms for the convertedwave moveout equation.
In this note, I simplify this term and present it as a function
of alone.
Equation (1) is the expanded traveltime function
of reflected PP or SS data presented by Taner and Koehler (1969):

t^{2} = c_{1} + c_{2} x^{2} + c_{3} x^{4},

(1) 
where x represents full offset, c_{1} = b_{1}^{2}, , and
, with
 
(2) 
where k indicates the stratigraphic layers present in the model.
Here and hereafter, and respectively denote the P velocity and
the S velocity for the layer.
Tessmer and Behle (1988) show that
 
(3) 
and
 
(4) 
where , this is only true when is constant.
The formal definition for c_{3} is as follows Castle (1988):
 
(5) 
For one layer, equation (5) simplifies to
 
(6) 
which reduces to
 
(7) 
Now, the simple trick I use to make an educated guess for with PS data alone
is to consider [from the results of c_{1} and c_{2}, equations (3) and (4)]
that and
, remember that is approximately constant in all layers.
With these assumptions, equation (5) simplifies to
 
(8) 
Introducing the final results for c_{1}, c_{2} and c_{3} into equation (1),
I obtain an equation to perform nonhyperbolic moveout for PS data that is
dependent on only two parameters: 1) the multiplication of the P and S velocities, or the
converted wave rms velocity (v_{rms}),
and 2) the Vp/Vs ratio (). It is also important to note that this equation is valid for a
constant Poisson's ratio in all layers. With these simplifications and equations, it is possible to
obtain an approximate value of using PS data alone.
 
(9) 

Equation (9) is the central result of this paper. It is possible to note
that the moveout equation is more than a hyperbolic relation, since it involves a
third term. Another important characteristic of equation (9)
is that it depends only on two parameters: 1) the convertedwaves rms velocity, and 2) the
Vp/Vs ratio. This important characteristic will allow us to invert for
a value of . It is also important to note that the
sensitivity of equation (9) to probably is not too high, since the third term
of the equation also depends on the offsetdepth ratio.
It is important to note that for the specific case of (this never happens in practice),
i.e., no converted waves,
the value of equals 1, and equation (9) reduces to the conventional normal moveout
equation. This is also a result of the one layer assumption.
Next: Numerical examples
Up: Rosales: NHNMO
Previous: Introduction
Stanford Exploration Project
10/23/2004