Introduction

Moveout analysis is arguably one of the most important steps in seismic processing, which leads to an expeditious construction of subsurface models. The classical work of Taner and Koehler (1969) represents the foundation for modern moveout analysis, where the reflection traveltimes of pure P waves are approximated by a Taylor expansion around zero offset. The degree of matching between the data and the computed traveltime can be judged based upon the semblance criterion (Neidell and Taner, 1971). Velocities that correspond to a higher degree of matching (semblance) are picked, and an accurate subsurface model can be inferred. In a 2D isotropic subsurface, only one parameter — the normal-moveout (NMO) velocity — needs to be estimated but more parameters are required when 3D data and effects from anisotropy are under consideration (Yilmaz, 2001; Oh and Alkhalifah, 2016; Tsvankin, 2012; Alkhalifah and Tsvankin, 1995; Grechka and Tsvankin, 1998a; Masmoudi et al., 2017; Thomsen, 2014). Aiming to improve the process, many researchers have proposed strategies such as refining the semblance criterion for higher resolution semblance peaks (Wilson and Gross, 2017; Luo and Hale, 2012; Chen et al., 2015; Fomel, 2009; Kirlin, 1992; Kirlin and Done, 2001; Abbad et al., 2009), taking into account amplitude variations with offset (AVO) (Sarkar et al., 2001; Yan and Tsvankin, 2008; Sarkar et al., 2002), and implementing inversions by maximizing semblance-based objective function to avoid too many costly parameter scans (Grechka and Tsvankin, 1999; Vasconcelos and Tsvankin, 2006). However, it can generally be said that the higher the number of parameters to scan, the lower the computational efficiency of semblance-based techniques.

Fomel (2007) proposes an alternative approach to this problem and formulated an automatic NMO correction of CMP gathers based on local slopes, which can directly and automatically be estimated from the gathers themselves (Fomel, 2002). This concept evolves from the original velocity-independent imaging strategies of Ottolini (1983) and early results of Wolf et al. (2004), which rely on the connection between the slopes of reflection traveltime surfaces and the time-domain processing parameters. As a result, the process of subsurface model building simply amounts to mapping from the automatically measured slopes to the corresponding NMO velocity. In the simple case of 2D hyperbolic reflection traveltimes, the NMO velocity $v_{nmo}$ can be found from the slope $p$ as follows (Fomel, 2007):

$$\frac{1}{v^2_{nmo} (t_0,x)} = \bar{W}(t_0,x) = p(t_0,x)\frac{t}{x}~,$$ (1)

where $t$ is the hyperbolic reflection traveltime, $t_0$ denotes the zero-offset traveltime, $x$ is offset, and $p = dt/dx$. The slowness squared $\bar{W}(t_0,x)$ is obtained as an attribute volume in terms of $t_0$ and $x$. In conventional time processing workflow (Yilmaz, 2001), a profile of slowness squared $W(t_0)$ at each CMP location as opposed to a volume of $\bar{W}(t_0,x)$ is generally preferred, which leads to a research question on how one can best obtain $W(t_0)$ from $\bar{W}(t_0,x)$. Moreover, reflection traveltime approximations are often nonhyperbolic with more independent parameters needed to honor the effects from anisotropy and heterogeneity. Stovas and Fomel (2016) present several relationships between local slopes/curvatures ($d^2t/dx^2$) and moveout parameters for different nonhyperbolic traveltime approximations. Nonetheless, the problem of using local slopes to achieve both moveout correction (event flattening) and parameter estimation (velocity analysis) automatically in this regime remains a challenging task.

Alternatively, Burnett and Fomel (2009) and Burnett (2011) propose to accomplish both processes by a combination of non-physical flattening and moveout inversion. The former is achieved by using predictive painting (Fomel, 2010), which automatically traces the CMP events according to the local structural information from slopes without relying on any prior assumption regarding their shape. This is notably different from directly mapping local slopes to moveout parameter as in equation 1. The traced events are subsequently warped (numerical nonstationary shifts) until they are flat and a seismic image can be obtained from stacking these flattened gathers. The term `non-physical' is used here to emphasize the difference between the conventional physical flattening based on an assumption of some particular shape of traveltime surfaces (e.g., hyperbolic in equation 1 or any other nonhyperbolic forms extracted from the physics of wave propagation) and the warping approach, which flattens the gathers numerically. In view of subsurface model building, the byproduct of this process, the nonstationary shifts, can be stored and applied to a time attribute volume — thereby the name time-warping — which yields reflection traveltimes of the corresponding flattened CMP events. We can then invert for the best-fit parameters of any preferred moveout approximation from this knowledge on reflection traveltimes of all CMP events and obtain properties of the subsurface.

Many moveout approximations suitable for various settings exist in the literature and choosing which one to use depends largely on the data coverage, the types of subsurface media, the desired traveltime accuracy, and the computational efficiency — more parameters to scan is often more computationally inefficient. Conventionally, moveout approximations have the well-known hyperbolic expression, which is exact for homogeneous isotropic or elliptically anisotropic media, but is only approximately applicable to small-offset data in other cases. For long-offset data, the moveout become nonhyperbolic and require more parameters to capture the effects from possible anisotropy (Castle, 1994; Alkhalifah, 2011; Alkhalifah and Tsvankin, 1995; Blias, 2009; Aleixo and Schleicher, 2010; Ursin and Stovas, 2006; Farra and Pšenčík, 2013; Tsvankin and Thomsen, 1994; Al-Dajani et al., 1998; Hake et al., 1984; Golikov and Stovas, 2012; Farra et al., 2016). Fomel and Stovas (2010) propose the 2D highly accurate generalized moveout approximation (GMA), which requires five independent parameters. This approximation is named `generalized' due to the fact that its functional form can reduce to several other known moveout approximations with different choices of parameters, and therefore provides a systematic view on the effects of various parameter choices on the approximation accuracy. Its 3D extension (3D GMA) is proposed by Sripanich et al. (2017) and requires a total of seventeen parameters to accurately describe reflection traveltimes in 3D layered anisotropic media.

In this study, we first show that using the time-warping workflow, we can construct a highly overdetermined inverse problem for moveout parameters. We improve upon the work of Burnett and Fomel (2009) and Burnett (2011) that rely on second-order moveout approximations (NMO ellipse), and utilize in our approach instead the more accurate 3D GMA appropriate for 3D layered anisotropic models. Furthermore, we acknowledge the cheap forward traveltime computations with 3D GMA and propose to employ a global inversion scheme within a transdimensional (hierarchical Bayesian) framework (Malinverno and Briggs, 2004; Sambridge et al., 2006,2013), instead of the previous linearized least-squares inversion by Burnett (2011). This allows us to obtain valuable statistical posterior probability distributions of all estimated moveout parameters and data uncertainty as opposed to only one `best-fit' solution according to the least-squares minimization criterion. We provide synthetic and real-data examples including one from the SEAM Phase II unconventional reservoir model to demonstrate the performance and discuss important implications from using the proposed traveltime-only inversion approach for subsurface model building.