We have extended the method of stationary spiking deconvolution of seismic data to the context of nonstationary signals in which the nonstationarity is due to attenuation processes. As in the stationary case, we have assumed a statistically white reflectivity and a minimum-phase source and attenuation process. This extension is based on a nonstationary convolutional model, which we have developed and related to the stationary convolutional model. To facilitate our method, we have devised a simple numerical approach to calculate the discrete Gabor transform, or complex-valued time-frequency decomposition, of any signal. Although the Fourier transform renders stationary convolution into exact, multiplicative factors, the Gabor transform, or windowed Fourier transform, induces only an approximate factorization of the nonstationary convolutional model. This factorization serves as a guide to develop a smoothing process that, when applied to the Gabor transform of the nonstationary seismic trace, estimates the magnitude of the time-frequency attenuation function and the source wavelet. By assuming that both are minimum-phase processes, their phases can be determined. Gabor deconvolution is accomplished by spectral division in the time-frequency domain. The complex-valued Gabor transform of the seismic trace is divided by the complex-valued estimates of attenuation and source wavelet to estimate the Gabor transform of the reflectivity. An inverse Gabor transform recovers the time-domain reflectivity. The technique has applications to synthetic data and real data.

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After silicone resilient denture liner treatment with certain perborate-containing denture cleansers, a greater amount of components could leach from the liner leading to a loss of color if the liner surface is rough.

We study the meet irreducible ideals (ideals I so that I = J ∩ K implies I = J or I = K) in certain direct limit algebras. The direct limit algebras will generally be strongly maximal triangular subalgebras of AF C * -algebras, or briefly, strongly maximal TAF algebras. Of course, all ideals are closed and two-sided.These ideals have a description in terms of the coordinates, or spectrum, that is a natural extension of one description of meet irreducible ideals in the upper triangular matrices. Additional information is available if the limit algebra is an analytic subalgebra of its C * -envelope or if the analytic algebra is trivially analytic with an injective 0-cocycle. In the latter case, we obtain a complete description of the meet irreducible ideals, modeled on the description in the algebra of upper triangular matrices. This applies, in particular, to all full nest algebras.One reason for interest in the meet irreducible ideals of a strongly maximal TAF algebra is that each meet irreducible ideal is the kernel of a nest representation of the algebra (Theorem 2.4). A nest representation of an operator algebra A is a norm continuous representation of A acting on a Hilbert space with the property that the lattice of closed invariant subspaces for the representation is totally ordered. These representations were introduced in [L1] as analogues for a general operator algebra of the irreducible representations of a C * -algebra. The meet irreducible ideals seem analogous to the primitive ideals in a C * -algebra. Indeed, in a C * -algebra, the meet irreducible ideals are precisely the primitive ideals [L3, Theorem 2.1]. This analogy can be extended by noting that the meet irreducible ideals form a topological space under the hull-kernel topology and every ideal is the intersection of the meet * Partially supported by an NSF grant. † Partially supported by an NSERC grant.

Spatial transformation of an irregularly sampled data series to a regularly sampled data series is a challenging problem in many areas such as seismology. The discrete Fourier analysis is limited to regularly sampled data series. On the other hand, the least-squares spectral analysis (LSSA) can analyze an irregularly sampled data series. Although the LSSA method takes into account the correlation among the sinusoidal basis functions of irregularly spaced series, it still suffers from the problem of spectral leakage: Energy leaks from one spectral peak into another. We have developed an iterative method called antileakage LSSA to attenuate the spectral leakage and consequently regularize irregular data series. In this method, we first search for a spectral peak with the highest energy, and then we remove (suppress) it from the original data series. In the next step, we search for a new peak with the highest energy in the residual data series and remove the new and the old components simultaneously from the original data series using a least-squares method. We repeat this procedure until all significant spectral peaks are estimated and removed simultaneously from the original data series. In addition, we address another problem, which is random noise attenuation in the data series, by applying a certain confidence level for significant peaks in the spectrum. We determine the robustness of our method on irregularly sampled synthetic and real data sets, and we compare the results with the antileakage Fourier transform and arbitrary sampled Fourier transform.