Multifocusing revisited ‐ inhomogeneous media and curved interfaces*

Abstract

A B S T R A C TWe review the multifocusing method for traveltime moveout approximation of multicoverage seismic data. Multifocusing constructs the moveout based on two notional spherical waves at each source and receiver point, respectively. These two waves are mutually related by a focusing quantity. We clarify the role of this focusing quantity and emphasize that it is a function of the source and receiver location, rather than a fixed parameter for a given multicoverage gather. The focusing function can be designed to make the traveltime moveout exact in certain generic cases that have practical importance in seismic processing and interpretation. The case of a plane dipping reflector (planar multifocusing) has been the subject of all publications so far. We show that the focusing function can be generalized to other surfaces, most importantly to the spherical reflector (spherical multifocusing). At the same time, the generalization implies a simplification of the multifocusing method. The exact traveltime moveout on spherical surfaces is a very versatile and robust formula, which is valid for a wide range of offsets and locations of source and receiver, even on rugged topography. In two-dimensional surveys, it depends on the same three parameters that are commonly used in planar multifocusing and the common-reflection surface (CRS) stack method: the radii of curvature of the normal and normal-incidence-point waves and the emergence angle. In three dimensions the exact traveltime moveout on spherical surfaces depends on only one additional parameter, the inclination of the plane containing the source, receiver and reflection point. Comparison of the planar and spherical multifocusing with the CRS moveout expression for a range of reflectors with increasing curvature shows that the planar multifocusing can be remarkably accurate but the CRS becomes increasingly inaccurate. This can be attributed to the fact that the CRS formula is based on a Taylor expansion, whereas the multifocusing formulae are double-square root formulae. As a result, planar and spherical multifocusing are better suited to model the moveout of diffracted waves. high-quality time imaging can provide a basis for interpretation in the processing sequence and often a useful one, even in case of poor data quality or strong structural complexity. Time imaging is basically model independent and does not require estimation or construction of a velocity model of the subsurface which is a crucial problem of seismic imaging. In the process of constructing a time image, useful additional products such as an root-mean-square (RMS) velocity and several important wavefield attributes can be obtained. Time-domain

Passive seismic source localization via common-reflection-surface attributes

“…Alongside the co-existing multifocusing approach and the estimation of local event slopes (Fomel, 2007), several higher-order CRS formulations have been introduced in recent years (Landa et al, 2010;Fomel and Kazinnik, 2013;Schwarz et al, 2014b). By extending the work of de Bazelaire (1988) and Höcht et al (1999), Schwarz et al (2014aSchwarz et al ( , 2015 demonstrated that all these higher-order expressions, including the commonly used hyperbolic CRS operator, can either be formulated for the optical projection or an effective analog of the true kinematics of the wave propagation.…”

Curvatures and inhomogeneities: An improved common-reflection-surface approach

“…Whereas Höcht et al (1999) present parametric expressions, Landa et al (2010), following Drexler and Gander (1998), provide the analytical solution to the circular reflector problem. Both lead to exact traveltimes for the constant velocity case.…”

“…The most intuitive way to extend the hyperbolic formulas, which are suited for planar reflectors, is to assume a locally circular reflector in a constant velocity medium. Whereas Fomel and Kazinnik (2012) present a closed-form solution for a hyperbolic reflector model, Höcht et al (1999) and Landa et al (2010) consider a circular or spherical reflecting interface.…”

“…Because these approaches are either computationally demanding or require trace interpolation, they are not well suited for efficient implementation. In conventional MF and the extended approach by Landa et al (2010), heterogeneity is handled by the application of a time shift, revealing the strong link of these methods to the shifted hyperbola by de Bazelaire (1988).…”

i-CRS: a new multiparameter stacking operator for an/isotropic media

“…Examples of such operators are the shifted hyperbola (de Bazelaire, 1988), multifocusing (MF, e.g., Landa et al, 2010), and the common reflection surface stack (CRS; Müller, 1999;Fomel and Kazinnik, 2009). These operators describe the traveltime surface for a reflected event in the short offset limit.…”