Introduction

Moveout analysis plays an important role in seismic processing and subsurface parameter estimation (Yilmaz, 2001). With regard to pure-mode reflections, the two-way moveout traveltimes are commonly expressed as a Taylor expansion around zero offset with only even powers in the offset direction (Taner and Koehler, 1969). The absence of odd powers is due to the source–receiver reciprocity of pure-mode reflections (Tsvankin, 2012; Pech et al., 2003). The second-order and fourth-order traveltime derivatives in the expansion are related to normal moveout (NMO) velocities and quartic coefficients. Their general expressions have been studied previously (Pech and Tsvankin, 2004; Fomel, 1994; Pech et al., 2003) and are usually written in terms of the one-way traveltime derivatives of a normal-incidence ray traveling from the reflector to the surface according to the normal-incidence-point (NIP) theorem (Krey, 1976; Hubral and Krey, 1980; Fomel and Grechka, 2001; Hubral, 1983; Gritsenko, 1984; Chernjak and Gritsenko, 1979).

A common assumption for evaluating the aforementioned one-way traveltime derivatives and relating them to subsurface medium parameters is to consider a 1-D horizontally-layered anisotropic medium. This assumption generally leads to traveltime and offset being functions of only horizontal phase slownesses (ray parmeters) and allows the traveltime derivatives to be evaluated explicitly, providing a direct bridge between both quantities (e.g, Sripanich and Fomel, 2016; Koren and Ravve, 2017). Despite being strictly applicable to 1-D lateral homogeneous media, this simple connection is normally used to test the accuracy of new moveout approximations in multi-layered media and to relate the estimated (effective) parameters in practice to contributions from different layer (interval) parameters based on Dix-type inversions (e.g, Fomel and Stovas, 2010; Thomsen, 2014; Tsvankin, 2012; Koren and Ravve, 2006; Buland et al., 2011; Ursin and Stovas, 2006).

Seeking to understand and quantify the first-order effects from lateral heterogeneity, Blias (1981) studied the second-order traveltime derivative in a 2-D medium with curved reflectors and variable velocities based on perturbations from 1-D isotropic medium. The result was used to analyze the effects of overburden velocity anomalies on stacking velocities (Blias, 2006; Blias and Gritsenko, 2003; Blias, 2009b) and led to a traveltime inversion approach, which honored the effects of lateral heterogeneity (Blias and Khatchatran, 2003). Several other developments along the same line exist in the Russian literature (e.g, Gritsenko and Chernjak, 2001; Blyas et al., 1984) and they were recently reviewed in Russian by Gritsenko (2013). These methods, however, remain applicable only to the case of multi-layered isotropic media.

Assuming that the slowness (1/velocity) varies slowly in the midpoint direction and can be approximated as a Taylor series, Lynn and Claerbout (1982) studied the second-order traveltime derivative in a single horizontal isotropic layer with laterally varying velocity. A similar idea was used by Grechka and Tsvankin (1999) on the group slowness to study the second-order traveltime derivative in one- and two-layered anisotropic models with laterally varying medium parameters. Takanashi and Tsvankin (2011) and Takanashi and Tsvankin (2012) extended the results of Grechka and Tsvankin (1999) to multi-layer anisotropic models with horizontal boundary interfaces (except at the target reflector) and proposed a correction algorithm for removing the effects of embedded velocity anomalies from reflection data. However, the effects from curved reflectors at intermediate interfaces are not considered in these methods.

Apart from moveout analysis, it is also important to note that there is another time-domain processing technique whose governing parameters can be related to one-way traveltime derivatives of some special ray—namely time migration. While moveout analysis relies on its connection to the one-way traveltime derivatives of the normal-incidence ray, time migration relies on its connection to the one-way traveltime derivatives of the image ray (Hubral, 1977). The former denotes the ray that has zero phase slownesses tangent to the reflector, whereas the latter denotes the ray the has zero phase slownesses tangent to the recording surface (Figure 1). For the image ray, the second-order derivative of the one-way (upward) traveltime is related to time-migration velocity, which serves as the basis for time-domain imaging (i.e, collecting contributions along the corresponding diffraction traveltime curve) and also for studying diffraction imaging (Fomel et al., 2007; Reshef and Landa, 2009). Both normal-incidence and image rays may also coincide in some special cases, for example, in a horizontal anisotropic layer with horizontal symmetry plane such as a transversely isotropic medium with vertical symmetry axis (VTI), an orthorhombic medium (ORT), and a monoclinic medium, where both rays become vertical. We note that several researchers have previously studied the process of time migration when underlying velocity models are ‘weakly’ laterally heterogeneous and some developments were reviewed by Cameron et al. (2007), Schleicher et al. (2007) and Iversen et al. (2012). However, in this study, we shall present an alternative approach to this problem based on an integrated use of Fermat’s principle and approximate lateral hetergogeneity effects.

vertical
Figure 1. The ray configuration as the basis for computing the traveltime derivatives. We use the vertical ray in a 1-D anisotropic medium at the $\mathbf {x}_0$ location as the reference for the normal-incidence ray and the image ray in the cases of reflection and diffraction traveltimes, respectively.

The fundamental dependency of time-domain processing techniques on the one-way traveltime derivatives of both the normal-incidence ray and the image ray encourages a thorough study on their behaviors under influences from lateral heterogeneity. In this paper, we focus on the forward problem and propose a general unified framework for computing one-way traveltime derivatives in the presence of 'weak' lateral heterogeneity from both curved interfaces and smoothly variable medium parameters in a multi-layered anisotropic medium. We rely on the fundamental ideas from Blias (1981), Lynn and Claerbout (1982), and Blyas et al. (1984) to extend the theory to work with perturbations from background 1-D anisotropic medium and demonstrate its connections to existing theories. Particularly, qe show that the effects from lateral heterogeneity in each anisotropic sublayer can be approximately quantified by the Taylor expansions of interface surfaces and layer group velocities with respect to the 1-D background. Their cumulative effect along the (normal-incidence or image) ray path can then be computed using an exact recursion derived from the Fermat's principle instead of an approximate summation previously used. We test our proposed theory by demonstrating improvements in reflection and diffraction traveltime predictions. In view of these results, we discuss potential applications of this theoretical study such as improving Dix inversion to honor the effects from lateral heterogeneity when inverting for interval parameters.