Anisotropy has significant effect on traveltime cross‐borehole tomography. Even relatively weak anisotropy cannot be ignored if accurate velocity estimates are desired, since isotropic traveltime tomography treats anisotropy as inhomogeneity. Traveltime data in our examples were synthetically generated by a ray‐tracing code for anisotropic media, and the computed quasi‐P‐wave traveltimes were subsequently inverted using the “dual tomography” technique (Carrion, 1991). The results of the tomographic inversion show typical artifacts due to the anisotropy, and that accurate imaging is impossible without taking the anisotropy into account.
The geophysical community admits that geotomography is a workable way to obtain accurate estimates of seismic velocity in complex structures. So‐called dual tomography improves the resolution of computed tomograms and thus can be applied to different geophysical and nongeophysical problems. Dual tomography emerges as a generalized approach to linearized constrained inversion. Dual inversion transforms a generalized constrained optimization problem being formulated in the physical space of seismic velocities to a dual unconstrained problem posed in the vector space of Lagrangian multipliers. It is a parametric optimization problem, with unknown solutions that are always perpendicular to the null‐space of a tomographic matrix. Imposed constraints improve the resolution and act as if the angular aperture coverage was extended. Thus dual tomography is able to quash image blurring associated with incomplete angular recording. Dual tomography does not require accurate knowledge of initial model. One starts with an arbitrary homogeneous medium and updates the medium in the course of iterations. Dual tomography does not require direct inversion of matrices and therefore it is relatively fast. Preliminary results indicate that dual tomography typically yields better images compared with algebraic reconstruction technique (ART), simultaneous reconstruction technique (SIRT), and other conventional techniques especially for limited angular aperture experiments typically used in seismic exploration.
This paper describes a new method for recovering velocity profiles utilizing both phase and amplitude information including wide‐angle arrivals, post‐ and precritical reflections. This method is based on a double spatial transformation with a minimization procedure. The first transformation is slant stacking of the observed wave field (seismogram). The second is projecting the slant stacked wave field into the domain of horizontal slowness p and depth z. In this domain the inverse problem is reduced to finding the critical path [Formula: see text] where V(z) is the true velocity of the compressional waves. A numerical algorithm based on a minimization technique is used to find the critical path, which is equivalent to the set of turning points of the critically reflected rays. When this path is found, then the following criteria are satisfied: (1) most of the energy is concentrated away from the precritical region; (2) the computed reflection coefficients reach their maximum on this path; and (3) for horizontally stratified media or CMP data, the reflectors are aligned in the p-z domain. In tests, this method has been shown to recover the velocity profile from both synthetic and real data. It is shown that the method is able to recover accurately velocity profiles even if only part of the data are given. For example, only part of the data are available when low‐ and high‐frequency components are missing or when the data are truncated in lateral extent due to the finite length of the recording system. Moreover, the method is able to handle virtually any vertical velocity gradients in a medium; therefore, it can be applied to complicated geologic structures. The method does not require elimination of multiples, but it is not applicable to the case of a medium with a large lateral velocity gradient. It can be used even for an elastic medium when the mode‐converted energy is not small.
Seismic data are non-linearly related to model parameters such as seismic velocities. However, seismic inversion is usually considered in a linear approximation. Such techniques as the Born inversion were recently applied to seismic data.Non-linear inversion is more complicated and involves extensive calculations. Non-linear inversion was developed in the frame work of an unconstrained optimization procedure. It uses as a priori information an initial model and probability distribution functions in the data and model spaces (This a priori information is called 'soft' bounds).In this paper, we propose a new technique for solving a constrained non-linear inversion. This technique will allow us to use a priori information not only in terms of 'soft' bounds, but 'hard' bounds as well (usually giving more stable and accurate solutions).Non-linear inversion is considered as an iterative procedure which involves a dual transform at each iteration. A dual transform allows for considering the problem in terms of the Lagrangian multipliers. The number of Lagrangian multipliers is equal to the number of available data and thus, significantly reduces the dimension of the problem (this is true for underdetermined problems only). However, the most important property of the dual transform is that it allows us to consider a constrained problem as an unconstrained problem.Another important property is that proper constraints incorporate small-wave numbers in the generalized inversion. It is shown that conventional (unconstrained non-linear inversion) is a special case of the constrained non-linear inversion developed in this paper if the truncation operator is represented by the identity matrix.
An inversion technique which can be used to estimate Q-profiles by inversion of reflection seismic data in one dimension is presented. It is shown that, in general, Q-profiles cannot be uniquely estimated from normal-incidence reflection seismograms if Q is an arbitrary function of frequency. The authors discuss a technique to reconstruct the quality parameter Q as a function of depth under the assumption that Q does not depend on frequency and the velocity profile is known a priori.
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