3D generalized nonhyperboloidal moveout approximation

Introduction

Reflection moveout approximation is an important ingredient for velocity analysis and other time-domain processing techniques (Yilmaz, 2001). As a function of source-receiver offset, the two-way reflection traveltime has the well-known hyperbolic expression, which is exact for plane reflectors in homogeneous isotropic or elliptically anisotropic overburden and approximately valid for small offsets in other cases. This behavior is generally valid for any pure-mode reflections thanks to the source-receiver reciprocity (Thomsen, 2014). At larger offsets, moveout may deviate from hyperbola and behave nonhyperbolically due to the effects of either anisotropy or heterogeneity (Fomel and Grechka, 2001).

In 2D, many extended moveout approximations have been proposed and designed to work with large-offset seismic data. They have led to better stacked sections and successful inversions for anisotropic parameters (e.g. Fomel, 2004; Castle, 1994; Alkhalifah, 1998; Tsvankin and Thomsen, 1994; Alkhalifah and Tsvankin, 1995; Taner et al., 2005; Ursin and Stovas, 2006; Hake et al., 1984; Blias, 2009; Golikov and Stovas, 2012; Pech et al., 2003; Blias, 2013; Aleixo and Schleicher, 2010). Fomel and Stovas (2010) proposed an approximation, which includes five independent parameters that can be defined from traveltime derivatives at the zero-offset ray and one far-offset ray. This approximation was named generalized moveout approximation (GMA) because its functional form reduces to several other known approximation forms with particular choices of parameters and thus, provides a systematic view on the effect of various choices of parameters on the approximation accuracy. Its application to homogeneous TI media was studied by Stovas (2010) and the GMA analog in $\tau$ -$p$ domain was developed by Stovas and Fomel (2012). The case of P-SV waves in horizontally layered VTI media was investigated by Hao and Stovas (2015).

The most basic expression for a 3D moveout approximation that works in arbitrary anisotropic heterogeneous media with small offsets can be expressed as the NMO ellipse and originates in the second-order Taylor polynomial of traveltime squared around zero offset (Tsvankin and Grechka, 2011; Grechka and Tsvankin, 1998). Several large-offset 3D moveout approximations have also been proposed and applied to seismic velocity analysis in azimuthally anisotropic media (Farra et al., 2016; Pech and Tsvankin, 2004; Al-Dajani and Tsvankin, 1998; Grechka and Pech, 2006; Xu et al., 2005; Vasconcelos and Tsvankin, 2006; Al-Dajani et al., 1998).The general expression for the quartic coefficients was studied by Fomel (1994) and Pech et al. (2003) based on an extension of normal-incident-point theorem.

In this paper, we revisit the 2D generalized nonhyperbolic moveout approximation and develop its natural extension to 3D. We subsequently show that the proposed approximation can be reduced to other known forms with different choices of parameters. Using numerical tests, we show that the 3D GMA can be several orders of magnitude more accurate than previously proposed 3D moveout approximations, at the expense of increasing the number of adjustable parameters. The accuracy and analytical properties of the proposed approximation make it an appropriate choice for 3D moveout approximation in the case of long-offset seismic data.

3D generalized nonhyperboloidal moveout approximation