Rapid convergence of some seismic processing algorithms

Geophysics ◽  
1981 ◽  
Vol 46 (12) ◽  
pp. 1667-1672 ◽  
Author(s):  
J. H. Rosenbaum ◽  
G. F. Boudreaux

The rate of convergence of many numerical algorithms can be greatly improved by repeated application of the method of summation by parts. Cases of interest in seismology arise when we need to resample the spectrum of a function at unevenly spaced frequency values. Related examples include the time‐domain evaluation of the Hilbert transform and the extrapolation to the real axis of spectra evaluated in the complex domain. A formula of this type was first presented by Lanczos (1956). The validity of the algorithms depends upon the fact that summation by parts of the cardinal‐series representation is justified as long as a function is somewhat oversampled.

2012 ◽  
Vol 79 (5) ◽  
Author(s):  
P. H. Wen

In this paper the fundamental solutions for an infinite poroelastic moderately thick plate and analytical solutions for a circular plate saturated by a incompressible fluid are derived in the Laplace transform domain. In order to obtain the solutions in the time domain, the Durbin’s Laplace transform inverse method has been used with high accuracy. The formulations using the boundary integral equation method can be derived directly with these fundamental solutions. In addition, the analytical solutions for a circular plate can be used to validate the accuracy of numerical algorithms such as the boundary element method and the method of fundamental solution. The deflection, moment, and equivalent moment in the time domain for a circular plate, subjected to uniform load and a concentrated force are presented, respectively. The analytical solutions demonstrate that interaction between the solid and flow is significant.


Author(s):  
Michael Feldman ◽  
Simon Braun

Abstract A method for dynamic analysis of sophisticated nonlinear single-degree-of-freedom systems, based on the Hilbert transform in the time domain is described. Using the Hilbert transform together with the proposed method for system identification, we obtain both instantaneous modal parameters together with non-linear force characteristics during free vibration analysis under impulse excitation without long resonance testing. Using the Hilbert transform in the time domain is a new method of studying linear and non-linear vibrating systems exposed to impulse or shock inputs.


A technique is described for obtaining asymptotic formulae for the time-domain waveforms associated with double integrals of rapidly-varying isolated pulses. The method is an extension of the theory of the time-domain asymptotics of single integrals given by C. J. Chapman ( Proc. R. Soc. Lond. A 437, 25–40 (1992)), and represents a generalization of the method of stationary phase to integrals with non-sinusoidal integrands. Stationary points of the phase function in the domain of integration each give rise to a particular waveform shape in the time-history of the solution; the waveforms associated with interior extrema and saddle points are proportional to the original source function and the Hilbert transform of the source function respectively. Boundary stationary points also give rise to distinctive waveforms, proportional to fractional integral transforms of the source function. The transitional case of the coalescence of two interior stationary points is considered in some detail; an asymptotic formula describing the coalescence is found, and the limiting behaviour of this formula after coalescence is calculated, i. e. the residual waveform after the annihilation of the stationary points. Asymptotic and numerical results are compared for an example integral, and good agreement is found even for moderate values of the asymptotic parameter.


Author(s):  
Michael Feldman

This paper describes a new technique, called the Hilbert Vibration Decomposition method, dedicated to decomposition of non-stationary wideband dynamic signals. Using the Hilbert transform in the time domain, we extract a number of elementary oscillating components of the initial signal, who’s both the instantaneous frequency and envelope can vary in time. Modeling examples of decomposition of non-stationary signals are included.


1992 ◽  
Vol 2 (4) ◽  
pp. 615-620
Author(s):  
G. W. Series
Keyword(s):  

2018 ◽  
Vol 12 (7-8) ◽  
pp. 76-83
Author(s):  
E. V. KARSHAKOV ◽  
J. MOILANEN

Тhe advantage of combine processing of frequency domain and time domain data provided by the EQUATOR system is discussed. The heliborne complex has a towed transmitter, and, raised above it on the same cable a towed receiver. The excitation signal contains both pulsed and harmonic components. In fact, there are two independent transmitters operate in the system: one of them is a normal pulsed domain transmitter, with a half-sinusoidal pulse and a small "cut" on the falling edge, and the other one is a classical frequency domain transmitter at several specially selected frequencies. The received signal is first processed to a direct Fourier transform with high Q-factor detection at all significant frequencies. After that, in the spectral region, operations of converting the spectra of two sounding signals to a single spectrum of an ideal transmitter are performed. Than we do an inverse Fourier transform and return to the time domain. The detection of spectral components is done at a frequency band of several Hz, the receiver has the ability to perfectly suppress all sorts of extra-band noise. The detection bandwidth is several dozen times less the frequency interval between the harmonics, it turns out thatto achieve the same measurement quality of ground response without using out-of-band suppression you need several dozen times higher moment of airborne transmitting system. The data obtained from the model of a homogeneous half-space, a two-layered model, and a model of a horizontally layered medium is considered. A time-domain data makes it easier to detect a conductor in a relative insulator at greater depths. The data in the frequency domain gives more detailed information about subsurface. These conclusions are illustrated by the example of processing the survey data of the Republic of Rwanda in 2017. The simultaneous inversion of data in frequency domain and time domain can significantly improve the quality of interpretation.


2019 ◽  
Vol 629 ◽  
pp. A112 ◽  
Author(s):  
B. M. Giuliano ◽  
A. A. Gavdush ◽  
B. Müller ◽  
K. I. Zaytsev ◽  
T. Grassi ◽  
...  

Context. Reliable, directly measured optical properties of astrophysical ice analogues in the infrared and terahertz (THz) range are missing from the literature. These parameters are of great importance to model the dust continuum radiative transfer in dense and cold regions, where thick ice mantles are present, and are necessary for the interpretation of future observations planned in the far-infrared region. Aims. Coherent THz radiation allows for direct measurement of the complex dielectric function (refractive index) of astrophysically relevant ice species in the THz range. Methods. We recorded the time-domain waveforms and the frequency-domain spectra of reference samples of CO ice, deposited at a temperature of 28.5 K and annealed to 33 K at different thicknesses. We developed a new algorithm to reconstruct the real and imaginary parts of the refractive index from the time-domain THz data. Results. The complex refractive index in the wavelength range 1 mm–150 μm (0.3–2.0 THz) was determined for the studied ice samples, and this index was compared with available data found in the literature. Conclusions. The developed algorithm of reconstructing the real and imaginary parts of the refractive index from the time-domain THz data enables us, for the first time, to determine the optical properties of astrophysical ice analogues without using the Kramers–Kronig relations. The obtained data provide a benchmark to interpret the observational data from current ground-based facilities as well as future space telescope missions, and we used these data to estimate the opacities of the dust grains in presence of CO ice mantles.


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