Non-hyperbolic common reflection surface

The method of common reflection surface (CRS) extend conventional stacking of seismic traces over offset to multidimensional stacking over offset-midpoint surfaces. We propose a new form of the stacking surface, derived from the analytical solution for reflection traveltime from a hyperbolic reflector. Both analytical comparisons and numerical tests show that the new approximation can be significantly more accurate than the conventional CRS approximation at large offsets or at large midpoint separations while using essentially the same parameters. INTRODUCTION Seismic data stacking is (together with deconvolution and migration) one of the fundamental operations in seismic data analysis (Yilmaz, 2000). Conventional stacking operates on common-midpoint (CMP) gathers and stacks traces after a hyperbolic moveout. The method of multifocusing (MF), originally developed by Gelchinsky et al. (1999a,b) and modified to the common-reflection-surface (CRS) method by Jäger et al. (2001), stacks data from multiple CMP locations. As a result, the signalto-noise ratio is improved considerably. Both MF and CRS require estimation of multiple parameters in addition to the conventional stacking velocity. These parameters correspond to the slope and curvature of seismic events in the midpoint direction and have physical interpretation in terms of wavefront slopes and curvatures. Many successful applications of MF and CRS have been reported in the literature (Landa et al., 1999; Gurevich et al., 2002; Menyoli et al., 2004; Heilmann et al., 2006; Gierse et al., 2006; Hoecht et al., 2009). The CRS method employs a multiparameter hyperbolic approximation of the reflection traveltime surface (Tygel and Santos, 2007). The hyperbolic approximation can be justified from a truncated Taylor series expansion of the squared traveltime around a reference ray. As such, it is always accurate at small deviations from the central ray. However, it loses its accuracy at large offsets or large midpoint separations. In this paper, we propose a new nonhyperbolic approximation. The form of this approximation follows from an analytical equation for reflection traveltime from a hyperbolic reflector. The idea of approximation reflection traveltimes by approximating reflector surfaces was first proposed by Moser and Landa (2009) and Landa et al. Fomel & Kazinnik 2 Nonhyperbolic CRS (2010). However, these publications did not provide a closed-form representation of the stacking surface. By analyzing the accuracy of the proposed nonhyperbolic approximation on a number of examples, we show that the proposed approximation can significantly extend the accuracy range of CRS. HYPERBOLIC AND NONHYPERBOLIC CRS If P (t,m, h) represents the prestack seismic data as a function of time t, midpoint m and half-offset h, then conventional stacking can be described as S(t0,m0) = ∫ P (θ(h; t0),m0, h) dh , (1) where S(t,m) is the stack section, and θ(h; t0) is the moveout approximation, which may take a form of a hyperbola θ(h; t0) = √ t0 + 4h2 v2 (2) with v as an effective velocity parameter or, alternatively, a more complicated nonhyperbolic functional form, which involves other parameters (Fomel and Stovas, 2010). The MF or CRS stacking takes a different form, Ŝ(t0,m0) = ∫∫ P ( θ̂(m−m0, h; t0),m, h ) dmdh , (3) where the integral over midpoint m is typically carried out only over a limited neighborhood of m0. The multifocusing approximation of Gelchinsky et al. (1999a) takes the form θ̂MF (d, h; t0) = t0 + T(+)(d, h) + T(−)(d, h) , (4) where, in the notation of Tygel et al. (1999), T(±) = √ 1 + 2K(±) (d± h) sin β +K (±) (d± h)2 − 1 V0K(±) , (5) K(±) = KN ± σKNIP 1± σ , (6) and σ(d, h) = h d+KNIP sin β(d2 − h2) . (7) The four parameters {KN , KNIP , β, V0} have clear physical interpretations in terms of the wavefront and ray geometries (Gelchinsky et al., 1999a). V0 represents the velocity at the surface and is typically assumed known and constant around the central ray. One important property of the MF approximation is that, in a constant Fomel & Kazinnik 3 Nonhyperbolic CRS velocity medium with velocity V0, it can accurately describe both reflections from a plane dipping interfaces and diffractions from point diffractors. The CRS approximation (Jäger et al., 2001) is θ̂CRS(d, h; t0) = √ F (d) + b2 h2 , (8) where F (d) = (t0 + a1 d) 2 + a2 d , and the three parameters {a1, a2, b2} are related to the multifocusing parameters as follows: a1 = 2 sin β V0 , (9) a2 = 2 cos β KN t0 V0 , (10) b2 = 2 cos β KNIP t0 V0 . (11) Equation (8) is equivalent to a truncated Taylor expansion of the squared traveltime in equation (4) around d = 0 and h = 0. In comparison with MF, CRS possesses a fundamental simplicity, which makes it easy to extend the method to 3-D. However, it looses the property of accurately describing diffractions in a constant-velocity medium. We propose the following modification of approximation (8): θ̂(d, h; t0) = √ F (d) + c h2 + √ F (d− h)F (d+ h) 2 , (12) where c = 2 b2 + a 2 1 − a2. Equation (12), which we call non-hyperbolic common reflection surface, is derived in Appendix A. A truncated Taylor expansion of the squared traveltime from equation (12) around d = 0 and h = 0 is equivalent to equation (8). There are two important special cases: 1. If a2 = 0 or KN = 0, equation (12) becomes equivalent to equation (8), with F (d) = (t0 + a1 d) . In a constant-velocity medium, this case corresponds to reflection from a planar reflector. 2. If a2 = b2 or KNIP = 0, equation (12) becomes equivalent to θ̂(d, h; t0) = √ F (d− h) + √ F (d+ h) 2 . (13) In a constant-velocity medium, this case corresponds to a point diffractor. Fomel & Kazinnik 4 Nonhyperbolic CRS 3-D extension In the case of 3-D multi-azimuth acquisition, both d and h become two-dimensional vectors. A natural way to extend approximation (8) is to replace it with θ̂CRS(d,h; t0) = √ F (d) + hT B2 h , (14) where F (d) = (t0 + d T a1) 2 + d A2 d, a1 is a two-dimensional vector, and A2 and B2 are two-by-two symmetric matrices (Tygel and Santos, 2007). A similar approach works for extending approximation (12) to θ̂(d,h; t0) = √ F (d) + hT C h + √ F (d− h)F (d + h) 2 , (15) where C = 2 B2 + a1 a T 1 −A2. In the 3-D case, we have not found a simple connection between approximation (15) and the analytical reflection traveltime for a 3-D hyperbolic reflector. ACCURACY COMPARISONS Analytical Example The first example we use to compare the accuracy of different approximations is that of a circular reflector under a homogeneous overburden. As shown in Appendix B, the exact traveltime in this case can be derived analytically in a parametric form. Obtaining a non-parametric closed-form expression in this case would require a solution of a high-order algebraic equation (Landa et al., 2010). Figure 1 compares the accuracy of CRS and nonhyperbolic CRS approximations for a range of offsets and midpoints. We display the relative absolute error as a function of offset to depth ratio and midpoint separation to depth ratio for a range of offsets and midpoints. The central midpoint is taken at the same horizontal distance from the center of the circle as the depth. The CRS approximation (8) develops an error both at large offsets and at large midpoint separations. The proposed non-hyperbolic CRS approximation (12) shows a significantly smaller error in the full range of offsets and midpoints. In our experiments, the multifocusing approximation (4) was even more accurate in this example. However, because of its different functional form, we focus our analysis on comparing CRS and non-hyperbolic CRS. Numerical Example In our next test, we generate a reflection traveltime surface by modeling reflection seismic data from a Gaussian-shape reflector, shown in Figure 2(a) by Kirchhoff modeling. The velocity changes linearly with depth. We extract the traveltime surface, Fomel & Kazinnik 5 Nonhyperbolic CRS