Quadratic normal moveouts in isotropic media : a quick tutorial

The design of traveltime moveouts for optimal stacking and inversion of kinematic parameters has always been a subject of much interest in seismic processing. A most prominent role is played by second-order Taylor expressions of reflection traveltimes, commonly known as parabolic and hyperbolic, around a zero-offset (ZO) ray. Here we refer to these expressions as (generalized) quadratic normal moveouts. For a common midpoint (CMP) gather and assuming a planar measurement surface, one has the well-known normal moveout (NMO) that is given in terms of a single parameter, the NMO-velocity. For a supergather of non-symmetrical source and receiver pairs around a fixed ZO point, the simple NMO traveltime is replaced by the generalized quadratic (parabolic or hyperbolic) normal moveout, that depends on three (or eight) parameters in the 2D (or 3D) situation. The simplest expression for the generalized moveout, as used in the CRS method, assumes a planar measurement surface and locally constant near-surface velocities. Corresponding expressions that do not consider these simplifying assumptions, although more complicated, may be required in more complex situations. In this work we present an organized and didactic tutorial on the formulation and derivation of the generalized quadratic normal moveouts in isotropic media. INTRODUCTION Traveltime expressions that are able to well approximate reflection events and also convey useful information of such events have always been of key interest in seismic data processing. A common feature to all traveltime formulas, simply referred as moveouts in the seismic literature, is the dependence of a certain number of parameters, that are estimated by means of coherence analysis directly applied on the seismic data. More specifically, the parameters are the ones that maximize the stack energy along the moveout. Besides providing a best-possible stack, the traveltime (or kinematic) parameters play an important role in deriving further imaging information, such as a background velocity model for migration, geometricalspreading compensation and others. Of particular importance are the moveouts of rays around the ZO reflection ray. The most familiar of such moveouts, the Normal Moveout (NMO), considers, in its two-dimensional version, a common midpoint (CMP) gather of sources and receivers along a horizontal seismic line. The reflection traveltimes along offset rays not far from the zero-offset (ZO) ray at the CMP are approximated by the one-parameter formula (Dix, 1955) T (h) = √ T 2 0 + C h 2 . (1) In the above equation, T is the traveltime from the source to the reflector and back to the receiver, T0 is the ZO traveltime at the CMP, h is the half-offset between shot and receiver. Finally, C = 4/V 2 NMO , (2) where VNMO is the NMO-velocity, is the single parameter that is to be inverted from the CMP data. Note that the square of the NMO equation (1) and readily be seen as a second-order Taylor expansion with respect to half-offset. 244 Annual WIT report 2004 Under the same conditions as above, a more accurate equation than the NMO equation (1) is the twoparameter Shifted Hyperbola formula (de Bazelaire, 1988; Castle, 1994) T (h) = T0(1−A) + √ (A T0) +B h2 . (3) The two parameters A and B,that are to be inverted from the CMP data, bear a relationship the previous single NMO-parameter C, namely C = B/A. (4) Also observe that the Shifted Hyperbola equation (3) reduces to the NMO equation (1) if we take A = 1. Still considering the 2D CMP situation, nonhyperbolic moveouts have been introduced to account for larger offsets in isotropic or weakly transversely isotropic media. The moveout expressions are given in the form of a three-parameter continued fractions expression (see, e.g., Tsvankin and Thomsen (2002), Fomel and Gretchka (2001)) T (h) = T 2 0 + Ch 2 + Dh 1 +Eh2 , (5) where C is the NMO parameter. More details on the meaning and properties of the above nonhyperbolic moveout can be found in Fomel and Gretchka (2001) and references therein. In the last years, more general moveout formulas have been developed, which are not restricted to the CMP configuration and, moreover, take into account a possibly irregular topography at the measurement surface. The point of departure for some of such formulas is to apply a second-order Taylor approximation to the traveltime with respect to the distances of source and receiver from the ZO point. The procedure leads to the so-called parabolic or hyperbolic traveltime moveout, as used, for example in the CRS method. In the following, the general second-order Taylor approximations of the traveltime around the ZO ray will be simply called quadratic normal moveouts. It is to be mentioned, however, that different approximations, not based on Taylor expansions are also well established. This is the case, for example, of the Multifocus Moveout (Gelchinsky et al., 1999; Tygel et al., 1999). In particular, the Shifted-Hyperbola equation (3) is a particular case of such expressions. An important advantage of Taylor moveouts is their general formulation, also valid for both 2D and 3D situations. Taylor parabolic and hyperbolic moveouts in 3D for arbitrary source and receiver location are well described in Ursin (1982). That paper uses the simple and direct formalism of Taylor expansions to quickly derive and elegant expressions. Under the assumption that sources and receivers lie on given smooth surfaces, the corresponding moveout expressions are obtained in Schleicher et al. (1993) using the surface-to-surface formalism of Bortfeld (1989). The connection to Ursin’s expressions are also described in Schleicher et al. (1993). Both papers assume locally constant velocities (that is, negligible velocity gradients) at the source and receiver points. Moreover, the effects of a possibly rugged topography, although implicit in the expressions in Ursin’s paper, are not explicitly considered in neither in Ursin (1982) nor in Schleicher et al. (1993). Using the formalism of paraxial ray theory, which includes ray and surface-to-surface propagator matrices, Cervený (2001) provides a complete treatment of the quadratic (parabolic and hyperbolic) moveouts, without a particular attention to the case of normal moveouts. Although the arguments and results described in Cervený (2001) are, of course, very correct and complete, the presentation demands, perhaps, too much of an ordinary reader, especially those more involved with practical applications. After Hubral (1983), the appealing and useful concepts of the normal (N) and normal-incident-point (NIP) waves were incorporated in the Taylor formulation of the reflection moveouts in the vicinity of the ZO ray. For instance, for sources and receivers along a horizontal line in the vicinity of a central (ZO) point, the 2D ZO CRS method uses the hyperbolic normal moveout (see, e.g., Tygel et al. (1997),Müller (1999)) T (m,h) = ( T0 + 2 sinβ v0 mx )2 + 2 T0 cos 2 β v0 [KN m 2 +KNIP h ] . (6) Here, m and h denote the midpoint (relative to the central point) and half-offset coordinates of the source and receiver pair, T0 is the ZO traveltime at the central point. Moreover, β, KN and KNIP , referred to as the ZO CRS parameters, denote the emergence angle of the ZO ray with respect to the surface normal and the curvatures of the Nand NIP-waves, respectively, all these quantities evaluated at the central point. Annual WIT report 2004 245 Finally, v0 is the medium velocity at the central point. An implicit assumption of the above formula is that the velocity gradient at the central point is negligible. Müller (1999) derives the above formula by comparing the substituting the propagator matrix of a ZO ray in terms of the curvatures KN and KNIP given in Hubral (1983) into the hyperbolic traveltime expression in Schleicher et al. (1993). It is to be remarked that also the (nonhyperbolic) multifocus moveout with respect to the ZO ray can be also expressed in terms of the ZO CRS parameters, namely (Tygel et al., 1997) T (xs, xg) = Rs v0 [√ 1 + 2 sinβ Rs xs + xs R2 s − 1 ]