945 resultados para Simultaneous Diophantine approximations
Resumo:
This thesis presents two different forms of the Born approximations for acoustic and elastic wavefields and discusses their application to the inversion of seismic data. The Born approximation is valid for small amplitude heterogeneities superimposed over a slowly varying background. The first method is related to frequency-wavenumber migration methods. It is shown to properly recover two independent acoustic parameters within the bandpass of the source time function of the experiment for contrasts of about 5 percent from data generated using an exact theory for flat interfaces. The independent determination of two parameters is shown to depend on the angle coverage of the medium. For surface data, the impedance profile is well recovered.
The second method explored is mathematically similar to iterative tomographic methods recently introduced in the geophysical literature. Its basis is an integral relation between the scattered wavefield and the medium parameters obtained after applying a far-field approximation to the first-order Born approximation. The Davidon-Fletcher-Powell algorithm is used since it converges faster than the steepest descent method. It consists essentially of successive backprojections of the recorded wavefield, with angular and propagation weighing coefficients for density and bulk modulus. After each backprojection, the forward problem is computed and the residual evaluated. Each backprojection is similar to a before-stack Kirchhoff migration and is therefore readily applicable to seismic data. Several examples of reconstruction for simple point scatterer models are performed. Recovery of the amplitudes of the anomalies are improved with successive iterations. Iterations also improve the sharpness of the images.
The elastic Born approximation, with the addition of a far-field approximation is shown to correspond physically to a sum of WKBJ-asymptotic scattered rays. Four types of scattered rays enter in the sum, corresponding to P-P, P-S, S-P and S-S pairs of incident-scattered rays. Incident rays propagate in the background medium, interacting only once with the scatterers. Scattered rays propagate as if in the background medium, with no interaction with the scatterers. An example of P-wave impedance inversion is performed on a VSP data set consisting of three offsets recorded in two wells.
Resumo:
If E and F are real Banach spaces let Cp,q(E, F) O ≤ q ≤ p ≤ ∞, denote those maps from E to F which have p continuous Frechet derivatives of which the first q derivatives are bounded. A Banach space E is defined to be Cp,q smooth if Cp,q(E,R) contains a nonzero function with bounded support. This generalizes the standard Cp smoothness classification.
If an Lp space, p ≥ 1, is Cq smooth then it is also Cq,q smooth so that in particular Lp for p an even integer is C∞,∞ smooth and Lp for p an odd integer is Cp-1,p-1 smooth. In general, however, a Cp smooth B-space need not be Cp,p smooth. Co is shown to be a non-C2,2 smooth B-space although it is known to be C∞ smooth. It is proved that if E is Cp,1 smooth then Co(E) is Cp,1 smooth and if E has an equivalent Cp norm then co(E) has an equivalent Cp norm.
Various consequences of Cp,q smoothness are studied. If f ϵ Cp,q(E,F), if F is Cp,q smooth and if E is non-Cp,q smooth, then the image under f of the boundary of any bounded open subset U of E is dense in the image of U. If E is separable then E is Cp,q smooth if and only if E admits Cp,q partitions of unity; E is Cp,psmooth, p ˂∞, if and only if every closed subset of E is the zero set of some CP function.
f ϵ Cq(E,F), 0 ≤ q ≤ p ≤ ∞, is said to be Cp,q approximable on a subset U of E if for any ϵ ˃ 0 there exists a g ϵ Cp(E,F) satisfying
sup/xϵU, O≤k≤q ‖ Dk f(x) - Dk g(x) ‖ ≤ ϵ.
It is shown that if E is separable and Cp,q smooth and if f ϵ Cq(E,F) is Cp,q approximable on some neighborhood of every point of E, then F is Cp,q approximable on all of E.
In general it is unknown whether an arbitrary function in C1(l2, R) is C2,1 approximable and an example of a function in C1(l2, R) which may not be C2,1 approximable is given. A weak form of C∞,q, q≥1, to functions in Cq(l2, R) is proved: Let {Uα} be a locally finite cover of l2 and let {Tα} be a corresponding collection of Hilbert-Schmidt operators on l2. Then for any f ϵ Cq(l2,F) such that for all α
sup ‖ Dk(f(x)-g(x))[Tαh]‖ ≤ 1.
xϵUα,‖h‖≤1, 0≤k≤q
Resumo:
Simultaneous phase-shifting ellipsometry based on a grating beamsplitter is presented. In the corresponding setup, an orthogonal grating and analyzer array are used. The latter is composed of four separate analyzers arranged in a 2x2 grid, the polarization axes of which are set to 0, 45, 90, and 135 deg. A mask allows only four diffracted beams of the fist order, having the same optical intensities, to pass. Each beam is incident on one of the analyzers of the array. The intensities of the four beams are simultaneously detected by a quadrant detector. The ellipsometric parameters are obtained using the four intensity signals. The feasibility of simultaneous phase shifting ellipsometry is thus demonstrated. (C) 2008 Society of Photo-Optical Instrumentation Engineers.
Resumo:
We found that Ce3+:Lu2Si2O7 single crystals could be excited at 800 nm by using a femtosecond Ti:sapphire laser. The emission spectra of Ce3+:Lu2Si2O7 crystals were the same for one-photon excitation at 267 nm as for excitation at 800 nm. The emission intensity of Ce3+: Lu2Si2O7 crystals was found to depend on the cube of the laser power at 800 nm, consistent with simultaneous absorption of three 800 nm photons. The measured value of the three-photon absorption cross section is sigma'(3) = 2.44 x 10(-77) cm(6) s(2). (c) 2006 Optical Society of America.
Resumo:
We provide a comprehensive overview of many recent algorithms for approximate inference in Gaussian process models for probabilistic binary classification. The relationships between several approaches are elucidated theoretically, and the properties of the different algorithms are corroborated by experimental results. We examine both 1) the quality of the predictive distributions and 2) the suitability of the different marginal likelihood approximations for model selection (selecting hyperparameters) and compare to a gold standard based on MCMC. Interestingly, some methods produce good predictive distributions although their marginal likelihood approximations are poor. Strong conclusions are drawn about the methods: The Expectation Propagation algorithm is almost always the method of choice unless the computational budget is very tight. We also extend existing methods in various ways, and provide unifying code implementing all approaches.