3D generalized nonhyperboloidal moveout approximation

Using similar notations as before, we express the reflection traveltime of each CMP event in terms of offset $x$ and $y$ in a given acquisition coordinate frame as $t(x,y)$. The 3D generalized nonhyperboloidal moveout approximation (3D GMA) can be expressed as (Sripanich et al., 2017):

$$t^2(x,y) \approx t^2_0 + W(x,y) + \frac{A(x,y)}{t^2_0+B(x,y)+\sqrt{t^4_0+2t^2_0B(x,y)+C(x,y)}} ~,$$ (4)

where

$$W(x,y) = W_1x^2+ W_2xy+ W_3y^2~,$$

$$A(x,y) = A_1x^4+A_2x^3y+A_3x^2y^2+A_4xy^3+A_5y^4~,$$

$$B(x,y) = B_1x^2+B_2xy+B_3y^2~,$$

$$C(x,y) = C_1x^4+C_2x^3y+C_3x^2y^2+C_4xy^3+C_5y^4~,$$ (5)

and $t_0$ again denotes the reflection traveltime at zero offset. The total number of independent parameters in equation 4 is seventeen including $t_0$, $W_i$, $A_i$, $B_i$, and $C_i$. In the case of $x=0$ or $y=0$, equation 4 reduces to the generalized nonhyperbolic moveout approximation of Fomel and Stovas (2010). Sripanich et al. (2017) shows that the 3D GMA (equation 4) can predict reflection traveltimes in 3D homogeneous or complex horizontally layered anisotropic models with a nearly exact level of accuracy for all practical purposes.

In equation 4, the parameters $t_0$, $W_i$, and $A_i$ are defined with respect to the zero-offset ray, whereas $B_i$ and $C_i$ are obtained from finite-offset rays (Sripanich et al., 2017). With these definitions, the parameters $t_0$, $W_i$, and $A_i$ are related to previously known time-processing parameters: $W_i$ simply denote the reciprocals of NMO velocities and govern the so-called NMO ellipse (Grechka and Tsvankin, 1998a). $A_i$ represent the quartic coefficients that govern the nonhyperbolicity of reflection traveltimes. A common approach to simplify and relate the quartic coefficients ($A_i$) to anisotropic parameters of the subsurface medium is done using the pseudoacoustic approximation, where $A_i$ can then be expressed in terms of $\eta_i$ (Alkhalifah, 2003; Sripanich and Fomel, 2016; Alkhalifah and Tsvankin, 1995; Sripanich et al., 2017; Stovas, 2015). With regard to this study, we keep the $A_i$ notation to maintain the generality of the proposed moveout inversion approach.

On the other hand, the $B_i$ and $C_i$ are designed to be dynamic and can vary with respect to different choices of finite-offset rays, which lead to better flexibility and accuracy of traveltime fitting. As a result, they cannot be directly related to medium parameters and one may expect that there are many possibilities of $B_i$ and $C_i$ that can lead to equally good approximation accuracy. Therefore, we expect that the non-uniqueness of $B_i$ and $C_i$ can lead also to the non-uniqueness in the final inverted models.