Dual Inversion Research Articles

Iterative Algorithm For the Damped Minimum Norm, Least-squares And Constrained Problem In Seismic Tomography

The ray tomography problem can be reduced to a minimum norm, least-squares constrained problem. As is well known, the generalized inverse matrix methods, such as singular value decomposition (SVD) and damped least squares residual inversion (DLSQR, WLSQR), which work well for dense matrix problems, become expensive for a large, sparse problem in ray tomography. Some iterative algorithms (ART, SIRT) are directly applied to solve the system without considering a priori information and the non-uniqueness of the problem. The conjugate gradient (CG) method and the Paige-Saunders' algorithm are popular solutions only for the (undamped) LSQR problem. In this paper, the CG iterative algorithm is developed to solve the damped minimum norm, least squares constrained problem and the dual inversion problem in non-linear traveltime tomography. The suggested algorithms exhibit the following desirable features: (1) consideration is given to errors in both the primary estimation of the model and the traveltime data; (2) solutions are less sensitive to data error; (3) there is a reduction in the non-uniqueness of the solution using a priori information ('hard' and 'soft' bounds); and (4) the algorithms take advantage of the sparsity of the matrix and consume less computer time and memory space for a large problem. The quantitative comparisons between most of the popular ray tomography methods and our algorithms were based on three measures: the misfit with the true model; solution stability; and image quality. Studies were made under different levels of noise. The results indicate that our algorithm is an accurate, flexible and efficient method for non-linear traveltime tomography and has greater tolerance to noise than any standard tomography algorithm.

Dual tomography for imaging complex structures

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.

Noise reduction in single channel radio bearers employing privacy

Frequency inversion of the voice spectrum is a common method of providing simple privacy on single channel radio links. When combined with exponential modulation this method produces an increase in system noise of 8.00 dB. Alternative inversion methods are analyzed. The methods involve partitioning the voice spectrum into two halves, inverting one or both of these halves and, in some cases, incorporating a bandshift. The optimum method, entitled "dual inversion with down-shift," reduced the increase in system noise performance to 0.99 dB for a small increase in residual intelligibility. This was measured at 36 percent using a voice test incorporating digit identification. Frequency inversion had a residual intelligibility of 31 percent.