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Theory: Non-hyperbolic moveout

The main characteristic of converted-wave data is their non-hyperbolic moveout. However, for certain offset/depth ratios, it is possible to approximate the non-hyperbolic 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 second-order approximation to the moveout equation to converted-wave data. In such cases the stacking velocity corresponds to the product of both P and S velocities known as converted-wave velocity.

Castle (1988) presents the third-order-approximation coefficient terms for the converted-wave 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):

 t2 = c1 + c2 x2 + c3 x4, (1)

where x represents full offset, c1 = b12, , 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 c3 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 c1 and c2, 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 c1, c2 and c3 into equation (1), I obtain an equation to perform non-hyperbolic 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 (vrms), 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 converted-waves 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 offset-depth 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