Theory of interval traveltime parameter estimation in layered anisotropic media

Introduction

Moveout approximations play an important role in conventional seismic data processing (Yilmaz, 2001). Bolshykh (1956) and Taner and Koehler (1969) laid the groundwork for studies on moveout approximations by proposing to employ the Taylor expansion of reflection traveltimes around zero offset. This approach has led to many developments in traveltime approximations for isotropic and anisotropic media (Malovichko, 1978; Alkhalifah, 1998; Sena, 1991; Ursin and Stovas, 2006; Hake et al., 1984; Grechka and Tsvankin, 1998; Fomel and Stovas, 2010; Stovas, 2015; Al-Dajani et al., 1998; Al-Dajani and Tsvankin, 1998; Alkhalifah and Tsvankin, 1995; Tsvankin and Thomsen, 1994; Taner et al., 2005; Golikov and Stovas, 2012; Blias, 2009; Aleixo and Schleicher, 2010). Some of the early developments are summarized by Castle (1994). In the small range of offsets, the reflection traveltime has the well-known hyperbolic form and its processing involves only one controlling parameter, namely the NMO (normal moveout) velocity. In the case of horizontally stacked isotropic layers, the effective NMO velocity can be related to the interval velocity through Dix inversion (Dix, 1955). Its 3D counterpart was described as generalized Dix equation (Grechka and Tsvankin, 1999; Tsvankin and Grechka, 2011; Grechka and Tsvankin, 1998)

In the case of long-offset seismic data and more complex media, both the hyperbolic moveout approximation and the Dix inversion formula need to be modified due to the moveout nonhyperbolicity (Fomel and Grechka, 2001). Hake et al. (1984) studied this problem in VTI (vertically transversely isotropic) media and proposed a 2D averaging formula for the quartic coefficient of the traveltime-squared expansion. Tsvankin and Thomsen (1994) introduced a different functional form for nonhyperbolic moveout approximation with correct asymptotic behavior at large offsets and propose a Dix-type formula for inversion of the quartic coefficient for interval anisotropic parameters. The moveout approximations by Tsvankin and Thomsen (1994) and its 3D extensions have led to subsequent developments on this topic (Al-Dajani and Tsvankin, 1998; Pech and Tsvankin, 2004; Xu et al., 2005; Alkhalifah and Tsvankin, 1995; Al-Dajani et al., 1998; Pech et al., 2003). Various alternative nonhyperbolic moveout approximations have been investigated in the literature. Several of them can be related to the generalized moveout approximation (Fomel and Stovas, 2010; Sripanich and Fomel, 2015a; Sripanich et al., 2016).

In the case of 3D orthorhombic media, several well-known moveout approximations make use of the rational approximation (Al-Dajani et al., 1998) in combination with the weak anisotropy assumption (Farra et al., 2016; Pech and Tsvankin, 2004; Grechka and Pech, 2006; Xu et al., 2005; Vasconcelos and Tsvankin, 2006). This kind of approximation is readily applicable for a single homogeneous orthorhombic layer. For a horizontally layered model, the rational approximation suggests averaging azimuthally dependent interval quartic coefficients using expressions for VTI media (Tsvankin and Thomsen, 1994; Tsvankin, 2012; Hake et al., 1984). However, this is justifiable only when the azimuthal anisotropy is mild (Vasconcelos and Tsvankin, 2006; Al-Dajani et al., 1998). In this paper, we derive exact expressions for averaging interval quartic coefficients in a 3D horizontal stack of general anisotropic layers. Next, we specify these expressions explicitly for two particular settings: a horizontal stack of aligned orthorhombic layers and a horizontal stack of azimuthally rotated orthorhombic layers. These expressions lead to exact Dix-like layer-stripping formulas for interval parameter estimation in layered anisotropic media.

Theory of interval traveltime parameter estimation in layered anisotropic media