Multifocusing homeomorphic imaging: Part 2. Multifold data set and multifocusing

Abstract

In this paper, we continue the construction of the multifocusing technique which is devoted to correlation and stacking of body wavefields, and determination of their kinematic attributes and amplitudes. In the first part of the work (Gelchinsky et al., 2000a), we obtain the local time correction formula for a pair of traces (the first is fixed and the second is any trace recorded in the vicinity of the first). The formula contains certain parameters, common for all traces and belonging to the so-called spherical vicinity and a pair of dual curvatures of two cross-sections of a ray tube surrounding the central ray and associated with the considered pair of traces. It was proved that an infinite family of the dual curvatures associated with the fixed central ray can be parameterized. The parameterization formulae contain, as parameters, the dual curvatures of the pair of fundamental ray tubes. The formula variable determining each ray tube is measured along the central ray. The parameterization formulae are only determined on the central ray. In order to use the formulae for the time correction, it is necessary to continue them in the vicinity of the central ray. A main idea behind this continuation is the establishment of a unique correspondence between each pair of traces consisting of the fixed central trace and another current trace recorded in its vicinity by a multifold acquisition system and a certain ray tube surrounding a central ray. More specifically, the establishment of this correlation means finding a formula for the parameterization variable for any trace recorded in the vicinity of a central trace. If the variable is known, then the values of the dual curvatures for a specific ray tube can be calculated using the parameterization formulae. In the first stage, the equation establishing the functional dependence between offsets of source and receiver is derived. This equation contains, as a parameter, the parameterization variable mentioned above. The equation derived is applied to the determination of special configurations of source–receiver pairs situated on two straight lines in the vertical plane. In the next stage, the solution of the equation with respect to the parameterization variable is found. The formula obtained facilitates calculation of the value of the variable for any trace, for which offsets of source and receiver are given and the parameters of the ray tube family are fixed. These parameters are the angles of departure and entry, the pair of two dual curvatures for two fixed fundamental ray tubes, if configuration with a nonzero offset central ray is considered. In the case of a zero offset normal central ray, the parameters are the angle of entry, and the Common Evolute Element (CEE) and Common Reflecting Element (CRE) curvatures. We also present a kinematic analysis of the obtained formulae. In particular, we show that the parameterization variable has a geometrical meaning as a focusing parameter. In order to make the consideration of the vicinity of a central ray more applicable, the Multifocusing Stacking Chart is proposed. It is shown that all traces recorded by an arbitrary acquisition system could be time-corrected. The number of traces corrected by multifocusing is the product of a multifolding degree and the number of traces in the CSP seismogram. Thus, in the case of modern acquisition systems, the number of stacked traces may vary from many hundreds to dozens of thousands. The flow chart of multifocusing correlation and the stacking algorithm is presented and discussed. Its output is a set of time sections presenting optimally stacked wavefields, angle of entry (anglegram), CEE and CRE radii (CEE and CRE radiusgrams) and maximum semblance (semblancegram) as 2D functions of coordinates of central traces and zero times. Thus, the procedure of optimal correlation and stack facilitates transformation of a set of hundreds or thousands of traces recorded near each central trace into an ensemble of time sections of different types presenting kinematics and averaged attributes of wavefields.