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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 $\gamma$ 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, $c_2 = \frac{b_1}{b_2}$, and $c_3 = \frac{b_2^2 - b_1 b_3}{4b_2^4}$, with
b_m = \sum_{k=1}^n z_k(\alpha_k^{2m-3}+\beta_k^{2m-3}),\end{displaymath} (2)
where k indicates the stratigraphic layers present in the model. Here and hereafter, $\alpha_k$ and $\beta_k$ respectively denote the P velocity and the S velocity for the $k^{\mbox {th}}$-layer. Tessmer and Behle (1988) show that  
c_1 = \left ( \sum_{k=1}^n z_k \left ( \frac{1}{\alpha_k} + \frac{1}{\beta_k} \right ) \right )^2 = t_0^2,\end{displaymath} (3)
c_2 = \frac{\sum_{k=1}^n z_k \left ( \frac{1}{\alpha_k} + \f...
 ... z_k \left (\alpha_k + \beta_k \right )} = \frac{1}{v_{rms}^2},\end{displaymath} (4)
where $v_{rms}^2=\alpha_{rms} \cdot \beta_{rms}$, this is only true when $\gamma$ is constant. The formal definition for c3 is as follows Castle (1988):  
c_3 = \frac{\left (\sum_{k=1}^n z_k (\alpha_k + \beta_k) \ri...
 ...)}{4 \left ( \sum_{k=1}^n z_k (\alpha_k + \beta_k) \right )^4}.\end{displaymath} (5)
For one layer, equation (5) simplifies to
c_3 = \frac{z^2 \left [ (\alpha+\beta)^2 - (\frac{1}{\alpha}...
 ...{\beta})(\alpha^3 + \beta^3) \right ]}{4 z^4 (\alpha+\beta)^4},\end{displaymath} (6)
which reduces to
c_3 = \frac{2 \alpha \beta - \frac{\alpha^3}{\beta} - \frac{\beta^3}{\alpha}}{4z^2 (\alpha+\beta)^4}.\end{displaymath} (7)
Now, the simple trick I use to make an educated guess for $\gamma$ with PS data alone is to consider [from the results of c1 and c2, equations (3) and (4)] that $\alpha_{rms}^2= v_{rms}^2 \gamma$ and $\beta_{rms}^2 = v_{rms}^2 \gamma^{-1}$, remember that $\gamma$ is approximately constant in all layers. With these assumptions, equation (5) simplifies to
c_3 = \frac{2-\left ( \gamma^2 + \gamma^{-2} \right )}{4 t_0^2 v_{rms}^4 \left (\gamma^{1/2} + \gamma^{-1/2} \right )^4}.\end{displaymath} (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 ($\gamma$). 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 $\gamma$ using PS data alone.

t^2 = t_0^2 + \frac{x^2}{v_{rms}^2} + \frac{x^4}{t_0^2 v_{rm...
 ... )}{4 \left( \gamma^{1/2} + \gamma^{-1/2} \right ) ^4} \right ]\end{displaymath} (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 $\gamma$. It is also important to note that the sensitivity of equation (9) to $\gamma$ 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 $\alpha=\beta$ (this never happens in practice), i.e., no converted waves, the value of $\gamma$ equals 1, and equation (9) reduces to the conventional normal moveout equation. This is also a result of the one layer assumption.

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