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

$$b_m = \sum_{k=1}^n z_k(\alpha_k^{2m-3}+\beta_k^{2m-3}),$$ (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  

$$c_1 = \left ( \sum_{k=1}^n z_k \left ( \frac{1}{\alpha_k} + \frac{1}{\beta_k} \right ) \right )^2 = t_0^2,$$ (3)

and  

$$c_2 = \frac{\sum_{k=1}^n z_k \left ( \frac{1}{\alpha_k} + \f... ... z_k \left (\alpha_k + \beta_k \right )} = \frac{1}{v_{rms}^2},$$ (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):  

$$c_3 = \frac{\left (\sum_{k=1}^n z_k (\alpha_k + \beta_k) \ri... ...)}{4 \left ( \sum_{k=1}^n z_k (\alpha_k + \beta_k) \right )^4}.$$ (5)

For one layer, equation (5) simplifies to

$$c_3 = \frac{z^2 \left [ (\alpha+\beta)^2 - (\frac{1}{\alpha}... ...{\beta})(\alpha^3 + \beta^3) \right ]}{4 z^4 (\alpha+\beta)^4},$$ (6)

which reduces to

$$c_3 = \frac{2 \alpha \beta - \frac{\alpha^3}{\beta} - \frac{\beta^3}{\alpha}}{4z^2 (\alpha+\beta)^4}.$$ (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

$$c_3 = \frac{2-\left ( \gamma^2 + \gamma^{-2} \right )}{4 t_0^2 v_{rms}^4 \left (\gamma^{1/2} + \gamma^{-1/2} \right )^4}.$$ (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 (v_rms), 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.

 

$$t^2 = t_0^2 + \frac{x^2}{v_{rms}^2} + \frac{x^4}{t_0^2 v_{rm... ... )}{4 \left( \gamma^{1/2} + \gamma^{-1/2} \right ) ^4} \right ]$$ (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.