Offset-continuation stacking

We introduce a data-driven stacking technique that transforms 2D/2.5D prestack multicoverage data into a common-offset (CO) section. We refer to this new process, which is based on the offsetcontinuation operation (OCO), as offset-continuation stacking or briefly OCO stack. Similarly to the CMP and CRS stacks, the OCO stack does not rely on an a-priori velocity model but provides velocity information itself. The original OCO method is a seismic configuration transform designed to simulate a seismic section as if obtained with a certain source-receiver offset using the data measured with another offset. Since OCO is dependent on the velocity model used in the process, it can be combined with stacking techniques for a set of models, thus allowing for the extraction of velocity information. The algorithm is based on so-called OCO trajectories, which are related to the concepts of image waves and velocity rays. We theoretically relate the OCO trajectories to the kinematic properties of OCO image waves that describe the continuous transformation of the common-offset reflection event from one offset to another. Based on OCO trajectories, we then formulate a horizon-based velocity analysis method, where root mean square (RMS) velocities and local event slopes are determined by stacking along event horizons. INTRODUCTION By definition, the Offset-Continuation Operation (OCO) is an operator that transforms common offset (CO) seismic gathers from one constant offset to another (Deregowski and Rocca, 1981). It is an important tool for imaging in a complex medium. Possible applications of OCO include velocity analysis, commonreflection point (CRP) stacking, dip moveout (DMO), migration to zero offset (MZO), interpolation of missing data, amplitude variation with offset (AVO) studies, and geometrical-spreading correction (see, e.g., Salvador and Savelli, 1982; Bolondi et al., 1982, 1984; Fomel, 1994, 2003; Santos et al., 1997). Since OCO is a configuration transform, its objective is to simulate a seismic section using as input the data measured with another configuration. As discussed by Hubral et al. (1996a) and mathematically demonstranted by Tygel et al. (1996), any configuration transform can be thought of as being composed of a migration and a subsequent demigration after changing a configuration parameter. Configurations transforms have already been used for several purposes in seismic processing such as MZO (Tygel et al., 1998; Bleistein et al., 1999), source continuation operation (SCO) (Bagaini and Spagnolini, 1993, 1996), azimuth moveout (AMO) (Biondi et al., 1998), DMO (Hale, 1984; Canning and Gardner, 1996; Collins, 1997; Black et al., 1993), common-source (CS)-DMO (Schleicher and Bagaini, 2004), data reconstruction (Bagaini et al., 1994; Stolt, 2002; Chemingui and Biondi, 2002), and velocity analysis (Silva, 2005; Coimbra et al., 2012). For data of very low signal-to-noise ratio (S/N) or acquisitions with very low fold, conventional common-midpoint (CMP) processing might not provide stacked sections of sufficient quality. In such situations, alternative processing sequences are necessary to improve the data quality. The OCO stack represents such an alternative path for the processing of reflection-seismic data. Its key element is the construction of common-offset stacked sections together with coherency sections and sections of kinematic 60 Annual WIT report 2012 and dynamic wavefield attributes. The OCO stacking surface is composed of so-called OCO trajectories (Coimbra et al., 2012). Such a trajectory requires only two parameters (local event slope and stacking velocity) to describe the seismic reflection event in the multi-coverage data. Neighbouring trajectories can be located by event tracking in the stacked section or described by a third curvature-related parameter. Using these parameters, the method stacks the data along a predicted traveltime curve that approximates the CRP event. Since the parameters, and thus the predicted traveltime curve, are updated from the data at each offset, the approximation is better than by conventional methods that adjust the approximate traveltime expression at some initial point. The purpose of this paper is to establish a consistent processing chain that is based entirely on the OCO stack, relying on identical assumptions at all steps. METHOD The OCO stack is a multiparameter stacking procedure similar to its relatives, the CMP and CRS stacks and multifocusing. It automatically determines stacking attributes based on a coherence measure applied at every common-offset sample of the data. Since these attributes vary with time for the same event, the OCO stacked section is free of normal moveout (NMO)-stretch (Perroud and Tygel, 2004). The main advantages of the OCO stack are twofold. Firstly, it is not limited to a zero-offset stacked section like the CMP stack. Secondly, for the 2D/2.5D case as discussed here, the OCO stack needs at most two parameter in addition to stacking velocity, even for the construction of stacked common-offset sections. There are other multiparameter stacking methods (Gelchinsky et al., 1999; Jäger et al., 2001; Zhang et al., 2001; Hertweck et al., 2007; Fomel and Kazinnik, 2012), which are based on stacks data from multiple CMP locations. As a result, they considerably improve the signal-to-noise ratio. However, these methods require the estimation of more data parameters than conventional CMP processing, in addition to the conventional stacking velocity. For instance, the zero-offset CRS method requires two additional parameters and common-offset CRS requires four of them. Moreover, due to multi-coverage some events such as diffractions, far-offset faults and strong dips can disappear. In this section, we derive the theoretical basis for the OCO stack. It is based on the kinematic behaviour of the OCO transformation as described by the OCO image-wave equation (Hubral et al., 1996b). Image-wave for OCO The OCO image-wave equation was derived through image-wave theory from the kinematic behaviour of the OCO transformation (Hubral et al., 1996b). It is a second order linear partial differential equation, which can be written as ht ( ∂U ∂h2 + 4 V 2 ∂U ∂t2 ) + ( t + 4h V 2 ) ∂U ∂h∂t − ht U ∂ξ2 = Θ ( ξ, t, h, U, ∂U ∂ξ , ∂U ∂t , ∂U ∂h ) . (1) Equation (1) describes the behavior of an artificial (non-physical) process of transforming reflection seismic data U(ξ, t, h) in the offset-midpoint-time domain as a certain kind of “wave propagation”. In this case it is the record of the seismic reflection that “propagates”as a function of half-offset h. In equation (1), ξ and t are the midpoint and time coordinate of the reflection event under consideration. The velocity V is assumed to be a constant average velocity that is known a priori. We will refer to V as the OCO velocity. Its relationship to the RMS velocity is discussed below. Equation (1) belongs to the class of linear hyperbolic equations, when t > 0 and h 6= 0, with the halfoffset h acting as propagation variable (i.e., equivalent to time in conventional wave propagation). Equation (1) describes a wave-like propagation in the offset direction that Hubral et al. (1996b) termed image-wave propagation. We use the OCO image-wave equation (1) to obtain the trajectory of single point under variation of the half-offset. Formally, we can think of the solution to equation (1) as being approximated by an expression that is analogous to the one used in ray theory, i.e., the leading term of a high-frequency asymptotic (WKBJtype) approximation for a reflected wave recorded on a seismogram of the form U(ξ, t, h) = A(ξ, t)F (h−H(ξ, t)), (2) Annual WIT report 2012 61