This paper presents an overview and a detailed description of the key logic steps and mathematical-physics framework behind the development of practical algorithms for seismic exploration derived from the inverse scattering series. There are both significant symmetries and critical subtle differences between the forward scattering series construction and the inverse scattering series processing of seismic events. These similarities and differences help explain the efficiency and effectiveness of different inversion objectives. The inverse series performs all of the tasks associated with inversion using the entire wavefield recorded on the measurement surface as input. However, certain terms in the series act as though only one specific task,and no other task, existed. When isolated, these terms constitute a task-specific subseries. We present both the rationale for seeking and methods of identifying uncoupled task-specific subseries that accomplish: (1) free-surface multiple removal; (2) internal multiple attenuation; (3) imaging primaries at depth; and (4) inverting for earth material properties. A combination of forward series analogues and physical intuition is employed to locate those subseries. We show that the sum of the four taskspecific subseries does not correspond to the original inverse series since terms with coupled tasks are never considered or computed. Isolated tasks are accomplished sequentially and, after each is achieved, the problem is restarted as though that isolated task had never existed. This strategy avoids choosing portions of the series, at any stage, that correspond to a combination of tasks,i.e.,

For the purposes of seismic propagation, a slip fault may be regarded as a surface across which the displacement caused by a seismic wave is discontinuous while the stress traction remains continuous. The simplest assumption is that this slip and the stress traction are linearly related. Such a linear slip interface condition is easily modeled when the fault is parallel to the finite-difference grid, but is more difficult to do for arbitrary nonplanar fault surfaces. To handle such situations we introduce equivalent medium theory to model material behavior in the cells of the finitedifference grid intersected by the fault. Virtually identical results were obtained from modeling the fault by (1) an explicit slip interface condition (fault parallel to the grid) and (2) using the equivalent medium theory when the finite-difference grid was rotated relative to the fault and receiver array. No additional computation time is needed except for the preprocessing required to find the relevant cells and their associated moduli. The formulation is sufficiently general to include faults in and between arbitrary anisotropic materials with slip properties that vary as a function of position.

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Zeroth-order ray theory is frequently used to calculate synthetic seismograms in media which are both anisotropic and inhomogeneous. One of the principal features of such media is that the polarization vectors of the two quasi-shear ( q S ) waves are determined by the nature of the anisotropy. Thus, a shear wave entering a region of anisotropy will generally be split into two separate polarizations. Ray theory predicts that these two waves will propagate independently, at different velocities, throughout the anisotropic region. Ray theory solutions also show that in inhomogeneous media, the polarization vectors will rotate along the ray. The rotations of these polarization vectors are strongly influenced by the symmetry and orientation of the anisotropy system, but only weakly depend upon the strength of the anisotropy. In contrast, in isotropic media the polarization of S-waves is determined from the initial conditions and only varies slowly due to the ray curvature. The polarization only changes in the ray direction and at any point does not rotate about the ray.In this paper we show that in the limit of infinitely weak anisotropy, solutions calculated using ray theory in anisotropic media conflict with the known results calculated for a similar isotropic medium. We show this fundamental breakdown in ray theory occurs because coupling between the @-waves is ignored in the zeroth approximation. Thus, the isotropic limit is not equivalent to the high-frequency limit of anisotropic ray theory. The coupling is particularly important in weakly anisotropic media, where the q S velocities are similar, but the same effect is still present in media exhibiting stronger anisotropy. This coupling must be taken into account when calculating waveforms.We show that this coupling may be modelled by treating the 'error' terms, produced by substituting a zeroth-order ray theory Green's function into the wave equation, as source terms distributed throughout the medium. For weakly anisotropic media where the qS ray paths are similar, this volume integral may be simplified using perturbation and asymptotic methods and evaluated as a simple integral along the ray path. In the isotropic limit this expression correctly describes the polarization of shear waves along the ray. This integral is easy to compute, requiring only quantities already used in ray tracing and traveltime calculations. A prior knowledge of the location, or even the existence of kiss, intersection, point or other singularities along the ray path, is not required for the method to give accurate results. We present some numerical examples for some simple cases previously investigated by less general or more expensive techniques.

Ray perturbation theory and the Born approximation have both been used extensively in seismological studies to describe the effects of a slowness perturbation on body and surface wavefields. The relationship between the expressions for the perturbed wavefield calculated using the two methods is investigated here. Using the symplectic symmetry of the ray equations we demonstrate the agreement, in the far field, of the two methods to first order in the slowness perturbation and to leading order in the asymptotic ray series. Thus it is shown that geometrical ray effects, like the traveltime perturbation, ray bending and focusing, are contained within the Born scattering formalism, provided these effects are small. The propagator formalism used to present the results is sufficiently general to include body and surface waves with a smoothly varying inhomogeneous elastic reference medium.