142 5 D INTERPOLATION AND REGULARIZATION WITH WAVEFRONT ATTRIBUTES

RTM and FWI usually require regularly spaced data. However, as some natural and anthropogenic reasons, real 3D seismic data, sometimes, are irregularly and sparsely sampled, which will restrict the usage of the RTM and FWI in actual applications. As an interpolation technique to regularize and enhance the quality of data with low signal-to-noise (S/N), the partial 3D common-reflectionsurface (CRS) method was proposed. However, two potential problems still need to be addressed while extending this technique into a 5D interpolation and regularization. The fundamental one is that the CRS wavefront attributes would be better determined simultaneously by a global search strategy. The second one is that an azimuth-based regularization would be considered in different azimuth direction, which will benefit for a further anisotropic traveltime operator. With the 3D SEG/EAGE C3WA data as an example, our results indicated that the partial 3D CRS can perform well in the 5D interpolation and regularization if the above two problems are addressed. INTRODUCTION 3D prestack data are generally recorded in 5D data space. As some natural and anthropogenic factors, the 3D prestack data are often irregularly and sparsely sampled during data acquisition, which would affect the quality of many wave-equation migration and inversion methods, such as reverse time migration (RTM) (e.g., Baysal et al., 1983), full wavefield inversion (FWI) (see e.g., Virieux and Operto, 2009). In order to resolve this problem, one simply method is to introduce an interpolation approach. As a traveltime-based interpolation technique to regularize and enhance the quality of data with low signal-to-noise (S/N), the partial common-reflection-surface (CRS) method was proposed by Baykulov and Gajewski (2009), which can use geological information from neighboring CMP gathers, thus significantly increase the S/N after the interpolation. In the 3D case, the partial 3D CRS (Baykulov and Gajewski, 2010) was proposed. It is actually a 5D interpolation technique in the time domain. However, two potential problems are still needed to be resolved in this technique. The fundamental one is that the CRS wavefront attributes would be better determined by a global search strategy. The second one is that, in each 3D CMP gather, an azimuthbased regularization will be considered. The azimuth-based regularization will benefit for an anisotropic traveltime operator. In order to perform a global CRS search, the differential evolution (DE) algorithm (Storn and Price, 1997) is applied. As examples, the 3D SEG/EAGE C3WA data are applied. THEORY The partial 3D CRS traveltime operator can be derived from the classical hyperbolic 3D CRS traveltime approximation (e.g., Müller, 2003): thyp = (t0 + w T z md) 2 + md Nmd + h Mh, (1) Annual WIT report 2016 143 where wz includes the near-surface velocity (v0), the dip (α) and azimuth (λ) of the reference ray. The midpoint parameter satisfies: md = m−m0, where m0 is a considered CMP position, and m is a neighboring midpoint of the considered CMP. Parameter h is half offset. The 2 × 2 symmetrical matrices M and N are related to the curvatures of the normal-incidence-point (NIP) wavefront and the normal (N) wavefront, respectively. The superscript T denotes the transpose of a matrix. To constructe the partial 3D CRS operator, the ZO traveltime t0,p of the partial 3D CRS operator is required. For one sample A on an arbitrary trace with half offset h = (offx, offy), the t0,p can be found or fitted by t22(offx, offy) = t 2 11 + h Mh. (2) We assume the ZO traveltime t0,p = t11 if the calculated t22 satisfies: (t − t22) ≤ fmin, where t11 is the trial ZO traveltime along the ZO trace, and t22 is the hyperbolic traveltime calculated by Equation 2. The coefficient fmin can be automatically computed by the wavefront attributes of two neighboring ZO samples around the fitted t0,p. A refinement of the t0,p with sample A is unnecessary if the ZO attributes obtained from t0,p is worse or if the wavefront attributes at t0,p have a strong local variation. The wavefront attributes at t0,p can be read from the attribute files after the t0,p is determined. Hence we construct the partial 3D CRS opeator by thyp,p = (t0,p + w T z md) 2 + md Nmd + h Mh. (3) Equation 3 is a 5D interpolation operator in the 5D data space: one temporal dimension and four spatial dimensions. With the 5D interpolation operator, one can interpolate a simple at A (m0,x, m0,y , offx, offy, t), where md = (m0,x, m0,y). METHOD Differential Evolution We perform a global CRS wavefront attribute search with the DE algorithm, which is a variant of evolutionary algorithms solved an optimization problem by the evolutionary ideas of natural selection. The DE algorithm has been used in many fields and has already shown good convergence properties (Price et al., 2006). The general DE mutation operator is given by Ni,j = Pr1,j + F (Pr2,j − Pr3,j), (4) where r1, r2 and r3 are three random agents picked from the generation. They must be distinct from each other as well as distinct from the considered agent. Generally, the differential weight F is set between [0 2]. For each agent i, the mutant operator is [Ui,j = { Ni,j pj < CR or j = L, Pi,j Otherwise. ] (5) whereCR is the crossover probability set between [0 1], pj ∈ [0 1], and L is an integer randomly generated between [0 7]. In this work, the parameters F and CR are estimated from the data. Alternatively, a good choice of the F and CR for various optimization problems was suggested by Pedersen (2010). The accuracy of the wavefront attributes generated by the DE algorithm is checked by the semblance (e.g., Neidell and Taner, 1971). In each generation, the agents are updated with a higher semblance and the set of wavefront attributes with the best semblance is kept for the next generation. Azimuth-based regularization Azimuth-based regularization can be considered in a 3D CMP gather. The trace location to be regularized in each considered CMP gather is given by h = (offx, offy). For each azimuth direction iaz ∗ δλ in the CMP gather, seeing Figure 1, we have offy = tan(0 + iaz ∗ δλ) ∗ offx, where δλ is azimuth interval, and iaz indicates different azimuth direction. The offx is given by offx = k ∗ δhx, where δhx is the half-offset increment in the x direction, and integer k = 0, 1, 2, . , intmax(hx)/δhx. The intmax() retures 144 Annual WIT report 2016 the integer part of the maximum hx. With the relationship of offx and offy, a full, wide or narrow azimuth regularization within [0 2π] can be chosen freely in the considered 3D CMP gather (see Figure 1). The δλ and δhx should be predetermined by users in different data. With the 3D SEG/EAGE C3WA data as an example, the azimuth range regularized here only set between [π/4, π/2] in each 3D CMP gather (see Figure 1c), where we use δλ = 15 ∗ π/180 and δhx = 40 m. The streamer-line direction is referenced as azimuth = π/2 since the 3D SEG/EAGE C3WA data has long offset in hy direction.