Shalom Raz

Orthonormal shift-invariant wavelet packet decomposition and representation

Abstract In this work, a shifted wavelet packet (SWP) library, containing all the time shifted wavelet packet bases, is defined. A corresponding shift-invariant wavelet packet decomposition (SIWPD) search algorithm for a ‘best basis’ is introduced. The search algorithm is representable by a binary tree, in which a node symbolizes an appropriate subspace of the original signal. We prove that the resultant ‘best basis’ is orthonormal and the associated expansion, characterized by the lowest information cost, is shift-invariant. The shift invariance stems from an additional degree of freedom, generated at the decomposition stage and incorporated into the search algorithm. The added dimension is a relative shift between a given parent node and its respective children nodes. We prove that for any subspace it suffices to consider one of two alternative decompositions, made feasible by the SWP library. These decompositions correspond to a zero shift and a 2−l relative shift where l denotes the resolution level. The optimal relative shifts, which minimize the information cost, are estimated using finite depth subtrees. By adjusting their depth, the quadratic computational complexity associated with SIWPD may be controlled at the expense of the attained information cost down to O (Nlog2N).

Three-dimensional velocity profile inversion from finite-offset scattering data

The reconstruction of three-dimensional (3-D) velocity variations from finite-offset scattering data is formulated. Reduction to the limiting cases of zero and small offset distances as well as the case of one-dimensional (1-D) stratification is given. An inherent increase in complexity is cited and interpreted. The relationship of the proposed inversion to the F-K migration is discussed.

Direct reconstruction of velocity and density profiles from scattered field data

A direct Born inversion algorithm is presented which separates the effects of stratified medium velocity and density gradients, thus facilitating the simultaneous reconstruction of the velocity and density profiles from scattered field data. A significant but not unexpected conclusion is that velocity and density variations cannot be separated and resolved from data acquired by a single source‐receiver configuration. Measurements at a minimum of two offset angles are required.

Orthonormal shift-invariant adaptive local trigonometric decomposition

Abstract In this paper, an extended library of smooth local trigonometric bases is defined, and an appropriate fast “best-basis” search algorithm is introduced. When compared with the standard local cosine decomposition (LCD), the proposed algorithm is advantageous in three respects. First, it leads to a best-basis expansion that is shift-invariant. Second, the resulting representation is characterized by a lower information cost. Third, the polarity of the folding operator is adapted to the parity properties of the segmented signal at the end-points. The shift invariance stems from an adaptive relative shift of expansions in distinct resolution levels. We show that at any resolution level l it suffices to examine and select one of two relative shift options — a zero shift or a 2 −l−1 shift. A variable folding operator, whose polarity is locally adapted to the parity properties of the signal, further enhances the representation. The computational complexity is manageable and comparable to that of the LCD.

Adaptive suppression of Wigner interference-terms using shift-invariant wavelet packet decompositions

The Wigner distribution (WD) possesses a number of desirable mathematical properties relevant to time—frequency analysis. However, the presence of interference terms renders the WD of multicomponent signals extremely diƒcult to interpret. In this work, we propose adaptive suppression of interference terms using the shift-invariant wavelet packet decomposition. A prescribed signal is expanded on its best basis and transformed into the Wigner domain. Subsequently, the interference terms are eliminated by adaptively thresholding the cross-WD of interactive basis functions, according to their amplitudes and distance in an idealized time—frequency plane. We define a distance measure that weighs the Euclidean distance with the local distribution of the signal. The amplitude and distance thresholds control the cross-term interference, the useful properties of the distribution, and the computational complexity. The properties of the resultant modified …igner distribution (MWD) are investigated, and its performance in eliminating interference terms, while still retaining high-energy resolution, is compared with that of other existing approaches. It is shown that the proposed MWD is directly applicable to resolving multicomponent signals. Each component is determined as a partial sum of basis functions over a certain equivalence class in the time—frequency plane. ( 1999 Elsevier Science B.V. All rights reserved. Zusammenfassung Die Wigner-Verteilung (WD) besitzt eine Reihe wu‹ nschenswerter mathematischer Eigenschaften, die fu‹ r eine Zeit— Frequenzanalyse von Bedeutung sind. Allerdings erschwert das Auftreten von Kreuztermen die Interpretation von WD mehrkomponentiger Signale extrem. In dieser Arbeit stellen wir eine adaptive Unterdru‹ ckung von Kreuztermen unter Verwendung der »erschiebungsinvarianten …avelet-Paket-Zerlegung vor. Ein vorgeschriebenes Signal wird auf seine beste Basis erweitert und in den Wigner-Bereich transformiert. Anschlie{end werden die Kreuzterme durch einen adaptiven Schwellwertvergleich mit der Kreuz-WD wechselwirkender Basisfunktionen eliminiert, entsprechend ihrer Amplituden und Abstand in einer idealisierten Zeit—Frequenzebene. Wir definieren ein Abstandsma{, das den euklidischen Abstand mit der lokalen Verteilung des Signals gewichtet. Der Amplituden - und Abstandsschwellwert kontrolliert die Kreuztermsto‹ rung, die nu‹ tzlichen Eigenschaften der Verteilung und den Rechenaufwand. Die Eigenschaften der resultierenden modifizierten …igner-»erteilung (MWD) werden untersucht und ihre Leistungsfa‹ higkeit zur Eliminierung der Kreuzterme bei noch immer hoher Energieauflo‹ sung wird mit der anderer Ansa‹ tze verglichen. Es wird gezeigt,

Beam stacking: A generalized preprocessing technique

Gaussian beams are well understood frequency‐ domain entities combining the directional properties of plane waves with an effectively finite region of support. These outstanding properties are retained not only on a prescribed observation plane, but throughout the propagation path. A preprocessing sequence aimed at transforming raw seismic data into beam stacks is proposed. That is, time‐harmonic Gaussian beams are synthesized, replacing the plane waves generated by conventional slant‐stacking procedures. The suggested scheme is characterized by an open parameter, essentially the beam width, whose selection is critical to ultimate success. Specific criteria for choosing this parameter can be given. In the limits of zero and infinite beam widths, beam stacks degenerate to the original raw data and to the conventional slant stacks, respectively. Although beam stacking is basically a frequency‐domain procedure, a transformation into the time domain, using frequency constituents within selected bands, may be ...

Multichannel Deconvolution of Seismic Signals Using Statistical MCMC Methods

In this paper, we propose two multichannel blind deconvolution algorithms for the restoration of two-dimensional (2D) seismic data. Both algorithms are based on a 2D reflectivity prior model, and use iterative multichannel deconvolution procedures which deconvolve the seismic data, while taking into account the spatial dependency between neighboring traces. The first algorithm employs in each step a modified maximum posterior mode (MPM) algorithm which estimates a reflectivity column from the corresponding observed trace using the estimate of the preceding reflectivity column. The second algorithm takes into account estimates of both the preceding and subsequent columns in the estimation process. Both algorithms are applied to synthetic and real data and demonstrate better results compared to those obtained by a single-channel deconvolution method. Expectedly, the second algorithm which utilizes more information in the estimation process of each reflectivity column is shown to produce better results than the first algorithm.

Adaptive time-frequency distributions via the shift-invariant wavelet packet decomposition

Utilizing the shift-invariant wavelet packet decomposition (SIWPD), various useful properties relevant to time-frequency analysis, including high energy concentration and suppressed interference terms, can be achieved simultaneously in the Wigner domain. A prescribed signal is expanded on its best basis and transformed into the Wigner domain. Subsequently, the interference terms are eliminated by adaptively thresholding the cross Wigner distribution of interactive basis functions, according to their amplitudes and distance in an idealized time-frequency plane. The properties of the resultant modified Wigner distribution (MWD) are investigated, and its performance in eliminating interference terms, while still retaining high energy resolution, is compared with that of other existing approaches. We demonstrate the effectiveness of the proposed MWD to resolving multicomponent signals. Each component is determined as a partial sum of basis-functions over a certain equivalence class in the time-frequency plane.

Eliminating interference terms in the Wigner distribution using extended libraries of bases

The Wigner distribution (WD) possesses a number of desirable mathematical properties relevant to time-frequency analysis. However, the presence of interference terms renders the WD of multicomponent signals extremely difficult to interpret. We propose an adaptive decomposition of the WD using extended libraries of orthonormal bases. A prescribed signal is expanded on a basis of adapted waveforms, that best match the signal components, and subsequently transformed into the Wigner domain. The interference terms are controlled by thresholding the cross WD of interactive basis functions according to their degree of adjacency in an idealized time-frequency plane. This measure is implicitly adapted to the local distribution of the signal, thus compensating for a global nonadaptive threshold. In particular we focus on a shift-invariant decomposition in an extended library of wavelet packets. The resulting modified distribution achieves high time-frequency resolution, and is superior in eliminating interference terms associated with bilinear distributions.

Time-frequency analysis and noise suppression with shift-invariant wavelet packets

Cross terms associated with bilinear distributions are not necessarily interpretable as interference terms. Any signal can be broken up in an infinite number of ways, each of which generates different cross terms. Therefore, it is important to choose an appropriate decomposition that separates the parts which are well delineated in the time-frequency plane. We have presented a modified Wigner distribution, where undesirable interference-terms can be eliminated while still retaining high energy concentration. A prescribed signal is expanded into a redundant library of orthonormal wavelet packet bases, from which the best decomposition is selected, and subsequently transformed into the Wigner domain. The discrimination between beneficial cross terms, which primarily enhance the useful properties of the time-frequency representation, and undesirable interference terms is determined according to the degree of adjacency and relative amplitudes of the interacting basis functions; only adjacent pairs whose coefficients are large enough are related to the same component of the signal. The balance between interference terms, concentration and computational complexity is achieved by adjusting the distance and amplitude thresholds.