The Vertical Seismic Profile: Imaging Heterogeneous Media

The paraxial ray method and its use for rapidly computing Green's functions at a dense grid of points in a heterogeneous medium is described in Chapter 2. This application is important for the inverse scattering problems such as migration or Born inversion. The paraxial ray method is illustrated and tested for different velocity structures. In each case errors in determining travel times and amplitudes are quantified. The primary benefit of the Paraxial ray method over standard dynamic ray tracing methods is speed of computation. A trade-off exists between speed and accuracy.

In Chapter 3 the acoustic Kirchhoff migration formula for variable background parameters is derived for such media where the Green's functions can be accurately computed using the paraxial ray method. An efficient algorithm is presented which describes the implementation of Kirchhoff migration with the paraxial ray method. The Paraxial ray Kirchhoff migration is illustrated with several examples. The acoustic synthetic data for the examples are generated using both the Paraxial ray method and a finite difference method.

In Chapter 4 the elastic Kirchhoff migration formula is expressed in a form for implementation using the paraxial ray method. The elastic Kirchhoff migration formula requires three component displacement and stress data. Since stress data are not collected, and three component data are rarely collected, approximate migration formulas are derived for these cases. Examples are given which illustrate the use of the exact and approximate elastic migration formulas. The elastic synthetic data for the examples are generated using both the Paraxial ray method and the discrete wavenumber method. An application is made to a multi-offset Michigan VSP.

In Chapter 5 an approximate direct multi-parameter inversion is derived for the cross-hole and surface-to-borehole VSP geometries. The method requires an array of sources. The medium parameters are determined on both sides of the receiver array. A relationship between the medium parameters and the scattered field in the wavenumber-frequency domain is derived for both the two-parameter (bulk modulus and density) acoustic case and the three-parameter (bulk modulus, shear modulus, and density) elastic case. This relationship is obtained using the Born approximation. The relationship assumes small parameter perturbations to a constant background. The problem of identifying and reducing artifacts which are due to the linear nature of Born inversion is addressed in an appendix.