Tomographic velocity model estimation using finite-offset kinematic wavefield attributes

I present a tomographic inversion scheme that makes use of finite-offset (FO) kinematic wavefront attributes to determine smooth, laterally inhomogeneous, isotropic 2D subsurface velocity models. The kinematic wavefield attributes are associated with rays starting at a common reflection point (CRP) in the subsurface. An optimum model is found iteratively by minimizing the misfit between measured and modeled data in the least-squares sense. The required forward modeled data are obtained during each iteration by dynamic ray tracing. Fréchet derivatives contained in the tomographic matrix are calculated by ray perturbation theory. The tomographic inversion is successfully tested on a synthetic example. INTRODUCTION A tomographic inversion scheme for the construction of smooth 2D isotropic velocity models for depth imaging based on kinematic wavefield attributes was recently established by Duveneck (2002). The method makes use of the curvature of normal incidence point (NIP) wavefronts, emergence angles of selected zerooffset (ZO) central rays and the corresponding one-way traveltimes. All kinematic wavefield attributes used in this inversion can be extracted in a robust way from seismic prestack data with the Common-ReflectionSurface (CRS) Stack (Mann et al., 1999). An alternative formulation (Duveneck et al., 2003) uses first and second order spatial derivatives of traveltime in place of angles and curvatures. In this paper the method is extended to the finite-offset (FO) case. Both alternative formulations are explained and their relationship is described. KINEMATIC WAVEFIELD ATTRIBUTES Kinematic wavefield attributes are all kinematic characteristics of a selected wave observable at the surface. The traveltime of a selected wave can be approximated around an observation point on the surface by parabolic and hyperbolic traveltime expansions (Schleicher et al., 1993). They provide the first and second order traveltime derivatives at the observation point. Beside the traveltime t itself this are the wavefield attributes used further in this paper. They can be related to hypothetical experiments yielding the same attributes. In the following only P-waves propagating in isotropic inhomogeneous media are considered. The associated rays are perpendicular to the wavefront. Only primary reflection events are taken into consideration. In order to introduce the wavefront attributes used in the inversion process I will start with the ZO-case. Then I will extend the ZO case to finite offsets. ZO-case In a ZO-experiment, the central ray associated with a selected wavefront hits the reflecting interface normally. Therefore the point of incidence on the reflector is called normal-incidence-point (NIP). The downAnnual WIT report 2003 129 and up-going ray segments are identical. The kinematic reflection response of a reflector segment around NIP is approximated by the ZO-CRS operator. This operator is expressed by the emergence angle β of the central ray and the curvatures of two eigenwaves measured at the surface. The near surface velocity v is assumed to be constant around the observation point. The two eigenwaves are associated with two hypothetical experiments. A point source at NIP provides the so-called NIP wave (Hubral, 1983) emerging on the ground surface with the curvature KNIP . An exploding reflector segment around NIP provides the so-called normal wave (Hubral, 1983) emerging at the observation point with the curvature KN . The emergence angle β is related to the first-order spatial derivative of the one-way traveltime with respect to the midpoint coordinate xm as follows: ∂T ∂xm = sinβ v , (1) where T = t/2 denotes the one-way traveltime and xm = 1 2 (xs + xG) is the location of the midpoint at the measurement surface. The source and receiver locations are denoted by xS and xG respectively. The x-axis of the coordinate system is assumed to be tangential at the measurement surface in the considered measurement point. The curvatureKNIP of the emerging wave due to the point source at NIP is related to the second-order spatial derivative of the one-way traveltime with respect to the half-offset h as follows: ∂T ∂h2 = KNIP v cosβ , (2) where h is defined by h = 12 (xG − xS). Please notice that the NIP wave experiment does not include any reflector characteristics. Thus it is possible to place a point source in a subsurface velocity model and to calculate the approximate kinematic traveltime response at the surface. Adjusting the velocity model and the location of the point source provides the same traveltime response actually extracted from the data by means of the CRS-stack. This is the basic idea of the inversion presented by Duveneck (2002). Extension to finite-offsets I consider a point source at the same depth point as in the ZO-case. Let me denote this point by CRP (common-reflection point). Separating the source and receiver location under the condition that the connecting central ray meets Snell’s law at the CRP provides all the source/receiver pairs pertaining to the CRP-trajectory. This trajectory connects all points in the three-dimensional prestack data volume pertaining to the same depth point. The two ray segments connecting the CRP with the source location xS and the receiver location xG are no longer identical now. One has to distinguish also between the emerging wavefronts at the source and receiver location due to the point source at the CRP. This two wavefronts are the finite-offset equivalents of the NIP wave discussed earlier. The relationship between emergence angles and wavefront curvatures to first and second order spatial derivatives is analogous to the ZO-case. I make the necessary distinction using subscripts S and G (source and geophone). In the following I make use of five kinematic wavefront attributes in the FO-case (see Figure 1): I have to use the measured traveltime T because it is not known how it is distributed on the two ray segments. Its first derivatives with respect to the source and receiver locations are given by pS = ∂T ∂xS and pG = ∂T ∂xG Further I assume the spatial second-order derivatives MS = ∂ T ∂x2 and MG = ∂T ∂x2 associated with the two emerging wavefronts due to a point source at CRP to be known. THE TOMOGRAPHIC APPROACH The wavefront attributes described above are independent of characteristics of the corresponding reflection point because they are related to theoretical experiments using only a point source placed in depth. Thus they allow to neglect all reflector effects in the tomographic approach. Billette and Lambaré (1998) discuss various approaches to tomography including finite-offset ray-pairs. In contrast to me they are not including second-order spatial derivatives. One possible approach discussed by them is to begin at the surface. The ray starting directions are known from the data. Ray-tracing can be performed until the traveltimes along the two ray segments equal the measured traveltime T . Using the correct velocity model the two rays should end in the same depth point. It is rather difficult to define a measure of misfit for this crossing 130 Annual WIT report 2003 condition. It is more suitable to place a point source in depth and to perform ray tracing until the two rays reach the surface. The forward modeled traveltime T and angles of emergence or alternatively the corresponding first-order traveltime derivatives can be compared to the measured quantities. A measure of misfit can easily be formulated in the least-squares sense. It is ensured that the rays start at the same point, a condition which is also met in reality. Therefore this approach is physically more suitable. One has to define for each ray-pair a starting position and two initial directions at the CRP in the subsurface. The directions can be given as angles to the vertical or as horizontal slowness components px and p G x . I make use of slowness components because they are directly involved in the ray-tracing system (see the Appendix). Additionally to Billette and Lambaré (1998) I make also use of second order traveltime derivatives associated with the waves due to the point source at the CRP, i.e I require the emerging wavefronts to focus at the CRP if propagated back from the surface to depth . Thus I do not use a measure of misfit between first order but between second order traveltime expansions at the measurement surface. For an overview of data and model components view Figure 1. PSfrag replacements z x (xS , pS , MS) (xG, pG, MG)