Isotropic case

If the reflector has the shape of a dipping plane, and the medium is homogeneous and isotropic, the normal moveout curve is a hyperbola of the form Levin (1971)  

$$t^2(h) = t_0^2 + {h^2 \over V_n^2}\;,$$ (60)

where

$$\begin{eqnarray} t_0 & = & {{2\,L} \over V_z}\;, \ V_n & = & {{V_z} \over {\cos{\alpha}}}\;,\end{eqnarray}$$ (61) (62)

L is the length of the zero-offset ray, and $\alpha$ is the reflector dip angle. Formula (60) is not accurate if the reflector is both dipping and curved. The Taylor series expansion of the reflection moveout in this case has the form of equation (26) with the coefficients Fomel (1994)

$$\begin{eqnarray} a_2 & = & {{\cos^2{\alpha}\,\sin^2{\alpha}\,G} \over {4\,V_z^2... ...2 \alpha} + \sin{2 \alpha}\, {{G\,K_3} \over {K_2^2\,L}}\right)\;,\end{eqnarray}$$ (63) (64)

where  

$$G={{K_2\,L} \over {1 + K_2\,L}}\;,$$ (65)

$\alpha$ and K₂ are the dip angle and curvature of the reflector at the reflection point of the central (zero-offset) ray, and K₃ is the third-order curvature. If the reflector has an explicit representation of the form z=z(x), then the parameters in formulas (63) and (64) have the expressions

$$\begin{eqnarray} \tan{\alpha} & = & {{d z} \over {d x}}\;, \ L & = & {z \over {... ...d^3 z} \over {d x^3}}\,\cos^4{\alpha} - 3\,K_2^2\,\tan{\alpha}\;.\end{eqnarray}$$ (66) (67) (68) (69)

Leaving only three terms in the Taylor series leads to the approximation  

$$t^2(h) = t_0^2 + {h^2 \over V_n^2} + {{\tan^2{\alpha}\,G\,h^4} \over {V_n^2\,\left(V_n^2 t_0^2 + G\,h^2\right)}}\;,$$ (70)

where we have included the denominator in the third term to stabilize the traveltime behavior at large offsets according to the obvious limit  

$$\lim_{h \rightarrow \infty} t^2(h) = {h^2 \over V_z^2}\;.$$ (71)

As indicated by formula (68), the sign of the curvature K₂ is positive if the reflector surface is locally convex. The sign is negative if the reflector is concave. Therefore, the coefficient G expressed by formula (65) and, likewise, the nonhyperbolic term in (70) can take both positive and negative values. This means that only for concave reflectors in homogeneous media do nonhyperbolic moveouts resemble those in VTI and vertically heterogeneous media. Convex surfaces produce nonhyperbolic effects with the opposite sign. For obvious reasons, formula (70) is not accurate for strong negative curvatures $K_2 \approx 1 / L$, which cause focusing of the reflected rays and triplications of the reflection traveltimes.

In order to evaluate the accuracy of approximation (70), we can compare it with the exact expression for the case of a point diffractor. A point diffractor is formally a convex reflector with an infinite curvature. The exact expression for normal moveout is written in the present notation as  

$$t(h) = {{\sqrt{z^2 + {(z\,\tan{\alpha} - h)^2}} + \sqrt{z^2 + {(z\,\tan{\alpha} + h)^2}}} \over {2\,V_z}}\;,$$ (72)

where z is the depth of the diffractor, and $\alpha$ is the central ray angle. Figure () shows the relative error of approximation (70) as a function of the ray angle for the half-offset h equal to the depth z. We can see that the maximum error occurs at $\alpha \approx 50^{\circ}$ and is about 1%. We can expect formula (70) to be even more accurate for reflectors with smaller curvatures.

 

nmoerr Figure 4 Relative error of the nonhyperbolic moveout approximation for a curved reflector in the case of a point diffractor. The relative error corresponds to the half-offset h equal to the diffractor depth z and is plotted against the central ray angle.