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Normal moveout

A seismic trace represents a signal d(t) recorded at a constant location x. The normal moveout operator transforms a trace into a ``vertical propagation'' signal, $m(\tau) = d(t)$, by stretching t into $\tau$ Claerbout (1999). For conventional PP seismic processing the NMO transformation is generally described as a hyperbola. For PS waves the traveltime moveout does not approximate a hyperbola for large values of offset-to-depth ratio even for an Earth's model consisting on horizontal layers embedded in a constant P and S velocities.

One of the main characteristic of converted-wave data is their non-hyperbolic moveout in CMP gathers. However, for certain values of offset-to-depth ratio, it is possible to approximate the non-hyperbolic moveout in a CMP gather to a hyperbola Tessmer and Behle (1988).

Castle (1988) developed a non-hyperbolic moveout equation for converted-wave data. This equation consists on three coefficients instead of the two coefficients for the hyperbolic equation. Castle's 1988 non-hyperbolic moveout equation is a function of both the P-velocity and the S-velocity. Next, I present an expression for the non-hyperbolic moveout equation in terms of $\gamma$.

Equation [*] is the third-order approximation for the total traveltime function of reflected PP or SS data presented by Taner and Koehler (1969):

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

where x represents surface-offset, the first coefficient c1 = b12, the second coefficient $c_2 = \frac{b_1}{b_2}$, and the third coefficient $c_3 = \frac{b_2^2 - b_1 b_3}{4b_2^4}$, where
b_m = \sum_{k=1}^n z_k \left ( v_{p_k}^{2m-3}+v_{s_k}^{2m-3} \right ).\end{displaymath} (2)
The summation index k indicates the stratigraphic layers in the model, vp is the vertical P-velocity, and vs is the S-velocity.

For the second order approximation of the total traveltime function, Tessmer and Behle (1988) show that the first coefficient, c1, is simplified as  
c_1 = \left ( \sum_{k=1}^n z_k \left ( \frac{1}{v_{p_k}} + \frac{1}{v_{s_k}} \right ) \right )^2 = t_0^2,\end{displaymath} (3)
and the second coefficient, c2, reduces to  
c_2 = \frac{\sum_{k=1}^n z_k \left ( \frac{1}{v_{p_k}} + \fr...
 ...k \left (v_{p_k} + v_{s_k} \right )} = \frac{1}{v_{\rm eff}^2},\end{displaymath} (4)
where $v_{\rm eff}^2=v_p \cdot v_s$. This simplification for the second coefficient (c2) is valid only for constant values of $\gamma$.

The third coefficient, as presented by Castle (1988), is  
c_3 = \frac{\left (\sum_{k=1}^n z_k (v_{p_k} + v_{s_k}) \rig...
 ...3)}{4 \left ( \sum_{k=1}^n z_k (v_{p_k} + v_{s_k}) \right )^4}.\end{displaymath} (5)
For one layer, equation [*] simplifies to

c_3 = \frac{z^2 \left [ (v_p+v_s)^2 - (\frac{1}{v_p} + \frac{1}{v_s})(v_p^3 + v_s^3) \right ]}{4 z^4 (v_p+v_s)^4},\end{displaymath} (6)
which reduces to

c_3 = \frac{2 v_p v_s - \frac{v_p^3}{v_s} - \frac{v_s^3}{v_p}}{4z^2 (v_p+v_s)^4}.\end{displaymath} (7)
Equation [*] represents the simplification for the third coefficient (c3) as a function of both the P-velocity and the S-velocity. However, equation [*] can be rewritten using the results of equation [*] and [*], that is, $v_{p_{\rm rms}}^2= v_{\rm eff}^2 \gamma$ and $v_{s_{\rm rms}}^2 = v_{\rm eff}^2 \gamma^{-1}$.Remember, the velocity ratio $\gamma$ is approximately constant in all layers. With these assumptions, the new expression for the third coefficient (c3) in the total traveltime function is  
c_3 = \frac{2-\left ( \gamma^2 + \gamma^{-2} \right )}{4 t_0^2 v_{\rm eff}^4 \left (\gamma^{1/2} + \gamma^{-1/2} \right )^4}.\end{displaymath} (8)

Figure 1
Non-hyperbolic term as a function of the offset-to-depth ratio.
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The coefficient c3 is the term in the total traveltime function that controls the non-hyperbolicity characteristic for PS reflections. Figure [*] plots equation [*], as a function of the offset-to-depth ratio. Notice that the non-hyperbolicity is primarily observed for large values of the offset-to-depth ratio, this conclusion validates the assumption of Tessmer and Behle (1988), that is the non-hyperbolic moveout can be approximated with a hyperbola for small values of the offset-to-depth ratio. Also, note that the correction is always negative and ignoring it will result in overestimates values for the velocities.

Substituting equations [*], [*] and [*] into the total traveltime equation [*], I obtain the non-hyperbolic moveout equation for PS data  
t^2 = t_0^2 + \frac{x^2}{v_{\rm eff}^2} + \frac{x^4}{t_0^2 v...
 ...)}{4 \left( \gamma^{1/2} + \gamma^{-1/2} \right ) ^4} \right ].\end{displaymath} (9)
Equation [*] depends only on two parameters: 1) the effective velocity ($v_{\rm eff}$), and 2) the P-to-S velocity ratio ($\gamma$). It is also important to note that this equation assumes a constant value of $\gamma$ in all layers. For the non-physical case of vp=vs, i.e. no converted waves, $\gamma$ equals 1, and equation ([*]) reduces to the conventional hyperbolic normal moveout equation. This equation also assumes a single layer model.

Figure [*] shows the traveltime computed with equation [*], with a constant P-velocity of 2 km/s, and S-velocity of 0.6 km/s, a maximum absolute offset of 7 km, and four horizontal reflectors at depths of 0.8, 1.5, 2.2, 2.9 km. The dotted curve represents non-hyperbolic moveout, equation ([*]), and the solid curve represents the hyperbolic moveout equation, that is omitting the third term in equation ([*]). Observe that for deeper reflectors and small offset both curves match reasonably well.

Figure 2
Non-hyperbolic traveltime comparison with the hyperbolic approximation.
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