Chap-Intro The effect of velocity anisotropy on wave propagation in homogeneous and heterogeneous media has been the subject of numerous publications. Careful forward modeling has helped interpreters understand how velocity anisotropy manifests itself in field data. Fewer attempts have been made, however, to solve the inverse problem, namely, the estimation of the parameters that describe the complexity of velocity anisotropy. These parameters are the elastic constants. The estimation of elastic constants is important because it can aid lithologic discrimination and fracture orientation, reveal anisotropic properties of the medium not obvious in the data, and provide further imaging or full waveform inversion algorithms with background models that can be refined iteratively. This dissertation will focus on one solution to the inverse problem, namely, the estimation of elastic constants from seismic measurements. Anisotropy or heterogeneity? Rocks can be anisotropic for a variety of reasons. Some rocks, minerals for example, can be inherently anisotropic depending on how their molecules ...
Estimation of elastic constants
in homogeneous transversely isotropic media
(ps.gz 675K) , pdf, (src 844K)
Chap-Elastic The elastic constants that control - and SV -wave propagation in a transversely isotropic medium can be estimated by using - and SV -wave traveltimes from either cross-well or VSP geometries. This chapter explains the procedure, which consists of two steps. First, elliptical velocity models are used to fit the traveltimes near one axis. The result is four elliptical parameters that represent direct and normal moveout velocities near the chosen axis for - and SV -waves. Second, the elliptical parameters are used to solve a system of four equations and four unknown elastic constants. The system of equations is solved analytically, yielding simple expressions for the elastic constants as a function of direct and normal moveout velocities. For SH -waves, the estimation of the corresponding elastic constants is easier because the phase velocity is already elliptical. The procedure for homogeneous media introduced in this chapter is generalized to heterogeneous media as explained in chapter 5, by using ...
Kinematic ray tracing in anisotropic layered media
(ps.gz 77K) , pdf, (src 491K)
Chap-Raytr The first step toward estimating tomographically elliptical velocities in heterogeneous anisotropic media is to be able to do ray tracing. In this chapter, I review a procedure to trace rays in layered transversely isotropic models with dipping interfaces. Group velocities are used to propagate the ray across each homogeneous layer, and phase velocities are used to find out how a given ray changes its direction when impinging on an interface. The equation that relates the ray parameter of the incident ray with the angle of the emergent phase at each interface is studied in detail. Finally, examples of ray tracing in simple anisotropic models are shown. Introduction Estimating velocity anisotropy tomographically from cross-well traveltime data is a process that often has two kinds of nonlinearity: the anisotropy itself and the ray bending from one iteration to the next. Under certain conditions one of these nonlinearities can be neglected while the computations ...
Tomographic estimation of elliptical velocities
(ps.gz 200K) , pdf, (src 846K)
Chap-Tomo1 To estimate elastic constants from traveltime measurements it is first necessary to fit the traveltimes with elliptical velocity functions. As described in chapter 2, when the model is homogeneous and the axis of symmetry is vertical, only two parameters for each wave type are needed to describe the elliptical velocity functions. When the model is heterogeneous, however, the elliptical velocity functions need to be estimated at each point in space. Thus the problem of estimating them, which in homogeneous media is highly overdetermined, becomes in heterogeneous media a highly underdetermined problem. This chapter describes a tomographic approach for solving it. The underdeterminedness is resolved by using models for the heterogeneities that contain information about the expected variations in the medium. I show synthetic and field data examples that illustrate the application of the technique, which uses the anisotropic ray tracing algorithm developed in chapter 3. ...
Estimation of elastic constants in heterogeneous transversely
(ps.gz 309K) ,pdf, (src 653K)
Chap-Tomo2 Chapter 2 introduced the idea of fitting the traveltimes with elliptical velocity functions as a first step in the estimation of the elastic constants of a homogeneous TI medium. The techniques presented in chapters 3 and 4 ( anisotropic ray tracing and anisotropic tomography) generalize to heterogeneous media the method of fitting the data with elliptical velocity functions. After fitting the traveltimes, the next step is the mapping from elliptical velocities to elastic constants. In this chapter, I show how all these techniques work together in the estimation of elastic constants in heterogeneous TI media. Introduction As chapter 2 shows, obtaining the elastic constants of a homogeneous TI medium from P -, SV - and, SH -wave traveltimes is a two-step procedure. The first step is to obtain direct and normal moveout (NMO) velocities by separately fitting traveltimes from each wave type ...
Singular value decomposition for cross-well tomography
(ps.gz 32K) , pdf, (src 59K)
App-SVD Singular value decomposition is performed on the matrices that result in tomographic velocity estimation from cross-well traveltimes in isotropic and anisotropic media. For a simple recording geometry, this appendix shows the singular vectors in both data and model space along with their corresponding singular values. Introduction In ray theoretic traveltime tomography, the solution of a linear system of equations is the heart of the problem. Solving this linear system transforms variations in traveltimes into variations in model parameters. This transformation from data to model depends on the properties of the matrix that describes the linear system, and singular value decomposition (SVD) is the tool for studying such properties. SVD has been applied in the past to study the structure of the matrices involved in tomographic traveltime inversion problems. White (1989), Bregman et al. (1989), and Pratt and Chapman (1992) among others, present singular values and singular vectors in model space for cross-well ...
Coefficients of the equation that relates ray parameter
and scattered phase angles
(ps.gz 8K) , pdf, (src 0K)
App-coeff This appendix shows the explicit form of the coefficients in equation poly , the fourth order polynomial that relates the angles of the scattered phases and the incident ray parameter. The coefficients are a 0 = c 4 p 4 - T 0 2 , a 1 = -2T 0 T 1, a 2 = 2 b 2 c 2 p 4 - 4 d 2 p 4 - T 1 2 - 2 T 2 T 0, a 3 = -2 T 2 T 1, a 4 = p 4 b 4 - T 2 2 , eqnarray* where T 0 = 2 ( ) - p 2 (c 2 + 2 a 2), T 1 = 2 (2 ), T 2 = 2 ( ) - p 2 (b 2 + 2 a 2), a 2 = c 44 / b 2 = (c 11 - c 44 ) / c 2 = (c 33 - c 44 ) / ...
Traveltime in homogeneous elliptically
anisotropic media with a nonvertical
axis of symmetry
(ps.gz 11K) , pdf, (src 36K)
App-traveltime The expression for the ray velocity in a medium with elliptical velocity dependency is given by the expression where is the ray angle measured from the axis of symmetry (positive counterclockwise), and and are the velocities in the directions parallel and perpendicular to the axis of symmetry (Figure tilted-ellipses a). When the axis of symmetry is vertical, the angle that measures the direction of propagation of the ray with respect to the vertical is the same as the group velocity angle. When the axis of symmetry is rotated an angle (Figure tilted-ellipses b), the expression for the ray velocity (for the same ray direction) becomes where is the angle from the axis of symmetry to the ray (the group velocity angle). ...
Partial derivatives of the traveltime with respect
to the model parameters
(ps.gz 13K) , pdf, (src 1K)
App-derivatives In this appendix, I show the expressions for the partial derivatives of the traveltime [equation system ] with respect to the model parameters , where is a component of the vector as follows: First, the derivatives with respect to the interval parameters , , and ( ) are
chapter: not found.
(ps.gz 15K) , pdf, (src 4K)
Bibliography toc chapter Bibliography Aki, K., and Richards, P. G., 1980, Quantitative seismology, volume 1: W. H. Freeman Co. Arts, R. J., Rasolofosaon, P. N. J., and Zinszner, B. E., 1991, Complete inversion of the anisotropic elastic tensor in rocks: experiment versus theory: 61st Ann. Internat. Mtg., Soc. Expl. Geophys., Expanded Abstracts, 1538--1541. Auld, B. A., 1990, Acoustic fields and waves in solids, volumes 1 and 2: Robert E. Krieger Publishing Co. Avasthi, J. M., Nolen-Hoeksema, R. J., and El Rabaa, A. W. M., 1991, In-situ stress evaluation of the McElroy field, west Texas: SPE Formation Eval., September issue, 301--309. Babuska, V., and Cara, M., 1991, Seismic anisotropy in the earth: Kluwer Academic Publishers. Backus, G. E., 1962, Long-wave elastic anisotropy produced by horizontal layering: J. Geophys. Res., 67 , 4427--4440. Banik, N. C., 1984, Velocity anisotropy of shales and depth estimation in the North Sea basin: Geophysics, 49 , 1411--1419. ...