Low-frequency extension of the Backus averaging method

2011 ◽  
Author(s):  
A. Stovas ◽  
Y. Roganov
Keyword(s):  
Averaging Method ◽  
Low Frequency ◽  
Backus Averaging

Comparison of seismic anisotropy of sedimentary strata at low- and high-frequency limits

Geophysics ◽  
2019 ◽  
Vol 85 (1) ◽  
pp. MR1-MR10 ◽  
Author(s):  
Fuyong Yan ◽  
De-Hua Han ◽  
Tongcheng Han ◽  
Xue-Lian Chen
Keyword(s):  
High Frequency ◽  
Seismic Anisotropy ◽  
Sedimentary Rocks ◽  
Low Frequency ◽  
Layered Media ◽  
Ray Theory ◽  
Frequency Limit ◽  
Anisotropic Properties ◽  
Velocity Anisotropy ◽  
Backus Averaging

The layer-induced seismic anisotropy of sedimentary strata is frequency-dependent. At the low-frequency limit, the effective anisotropic properties of the layered media can be estimated by the Backus averaging model. At the high-frequency limit, the apparent anisotropic properties of the layered media can be estimated by ray theory. First, we build a database of laboratory ultrasonic measurement on sedimentary rocks from the literature. The database includes ultrasonic velocity measurements on sandstones and carbonate rocks, and velocity-anisotropy measurements on shales. Then, we simulate the sedimentary strata by randomly selecting a certain number of rock samples and using their laboratory measurement results to parameterize each layer. For each realization of the sedimentary strata, we estimate the effective and apparent seismic anisotropy parameters using the Backus average and ray theory, respectively. We find that, relative to Backus averaging, ray theory usually underestimates the Thomsen parameters [Formula: see text] and [Formula: see text], and overestimates [Formula: see text]. For an effective layered medium consisting of isotropic sedimentary rocks, the differences are significant. These differences decrease when shales with intrinsic seismic anisotropy are included. For the same sedimentary strata, the seismic wave should perceive stronger seismic anisotropy than the ultrasonic wave.

Vertical propagation of low-frequency waves in finely layered media

Geophysics ◽  
2006 ◽  
Vol 71 (3) ◽  
pp. T87-T94 ◽  
Author(s):  
Alexey Stovas ◽  
Børge Arntsen
Keyword(s):  
Reflection Coefficient ◽  
Low Frequency ◽  
Effective Medium ◽  
Frequency Limit ◽  
Data Matching ◽  
Anisotropy Effect ◽  
Vertical Propagation ◽  
The Matrix ◽  
Backus Averaging ◽  
Time Average

Multiple scattering in finely layered sediments is important for interpreting stratigraphic data, matching well-log data with seismic data, and seismic modeling. Two methods have been used to treat this problem in seismic applications: the O’Doherty-Anstey approximation and Backus averaging. The O’Doherty-Anstey approximation describes the stratigraphic-filtering effects, while Backus averaging defines the elastic properties for an effective medium from the stack of the layers. It is very important to know when the layered medium can be considered as an effective medium. In this paper, we only investigate vertical propagation. Therefore, no anisotropy effect is taken into consideration. Using the matrix-propagator method, we derive equations for transmission and reflection responses from the stack of horizontal layers. From the transmission response, we compute the phase velocity and compare the zero-frequency limit with the effective-medium velocity from Backus averaging. We also investigate how the transition from time-average medium to effective medium depends on contrast; i.e., strength of the reflection-coefficient series. Using numerical examples, we show that a transition zone exists between the effective medium (low-frequency limit) and the time-average medium (high-frequency limit), and that the width of this zone depends on the strength of the reflection-coefficient series.

Theoretical and numerical study of nonlinear acoustic absorbers for low frequency noise control

2021 ◽  
Vol 263 (1) ◽  
pp. 5600-5604
Author(s):  
Min Yang ◽  
Xianhui Li ◽  
Zenong Cai ◽  
Junjuan Zhao ◽  
Peng Zhang ◽  
...  
Keyword(s):  
Sound Absorption ◽  
Averaging Method ◽  
Numerical Study ◽  
Structural Parameters ◽  
Low Frequency ◽  
Frequency Noise ◽  
Flow Equations ◽  
State Response ◽  
Runge Kutta Method ◽  
Absorption Performance

In this paper, the sound absorption characteristics of cubic nonlinear sound-absorbing structures are analyzed by theoretical and numerical methods. The slow flow equations of the system are derived by using complexification averaging method, and the nonlinear equations which describe the steady- state response are obtained. The resulting equations are verified by comparing the results which respectively obtained from complexification-averaging method and Runge-Kutta method. It is helpful to optimize the structural parameters and further improve the sound absorption performance to study the variation of the sound absorption performance of cubic nonlinear structure with its structural parameters.

A generalized O’Doherty‐Anstey formula for waves in finely layered media

Geophysics ◽  
1994 ◽  
Vol 59 (11) ◽  
pp. 1750-1762 ◽  
Author(s):  
Serge A. Shapiro ◽  
Holger Zien ◽  
Peter Hubral
Keyword(s):  
Layered Medium ◽  
Perturbation Technique ◽  
Finite Thickness ◽  
Low Frequency ◽  
Variable Density ◽  
Analytic Description ◽  
Mean Values ◽  
Time Harmonic ◽  
Backus Averaging ◽  
Theory Result

We investigate the angle‐dependent plane wave transmissivity of a pressure wave in a random, multilayered, acoustic, variable velocity and variable density medium. The main result of our consideration is a simple, explicit analytic description of the influence of such a medium on the transmissivity kinematics and dynamics for the whole frequency range. We assume that the velocity and density dependencies on depth are typical realizations of random stationary processes. Moreover, the fluctuations in both values must be relatively small compared to their constant mean values (of the order of 30 percent or smaller). In our derivation, we combine the small perturbation technique with the localization and self‐averaging theory. We obtain the attenuation and the phase of the time‐harmonic transmissivity, as well as the pulse form of the transient transmissivity from an angle‐dependent combination of the auto‐ and crosscorrelation functions of both the sonic and density logs. Our results for the kinematics of the transmissivity yield the wellknown “Backus averaging” in the low‐frequency limit. Likewise, they provide the ray theory result as the high‐frequency asymptotic value. The analytic expression for the transmissivity can be viewed as a generalization of the O’Doherty‐Anstey formula. Numerical computations of the actual transmissivity show fluctuations around the theoretical prediction given by our formula, which is strictly valid only in the case of infinitely thick media. The larger the layered medium, the smaller are these fluctuations. They can be well estimated with a formula which we derive to describe the deviations between the analytic and the exact transmissivity obtained for a layered medium of finite thickness.

Elastic waves in finely layered sediments: The equivalent medium and generalized O’Doherty‐Anstey formulas

Geophysics ◽  
1996 ◽  
Vol 61 (5) ◽  
pp. 1282-1300 ◽  
Author(s):  
Sergei A. Shapiro ◽  
Peter Hubral
Keyword(s):  
Phase Velocity ◽  
Multiple Scattering ◽  
Random Media ◽  
Plane Waves ◽  
Low Frequency ◽  
Analytic Solutions ◽  
Seismic Velocities ◽  
Statistical Parameters ◽  
Practical Applications ◽  
Backus Averaging

We study the influence of elastic 1-D inhomogeneous random media (e.g., finely layered media with variable density and shear and compressional velocities) on the kinematics and dynamics of the transmitted obliquely incident P‐ and SV‐plane waves. Multiple scattering (resulting in localization and spatial dispersion of the elastic wavefield) is the main physical effect controlling the properties of the wavefield in such media. We analyze the wave propagation assuming the fluctuations of velocities and density to be small (of the order of 20% or smaller). We obtain explicit analytic solutions for the attenuation coefficient and phase velocity of the transmitted waves. These solutions are valid for all frequencies. They agree very well with results of numerical modeling. Our theory shows that fine elastic multilayering is characterized by a frequency‐dependent anisotropy. At typical acquisition frequencies this anisotropy differs significantly from the low‐frequency anisotropy described by the well‐known Backus averaging. The increase of the phase velocity with frequency is quantified. It can partly explain the difference between well‐log‐derived velocities and lower frequency seismic velocities [e.g., vertical seismic profiling (VSP) velocities] in terms of localization. The low‐ and high‐frequency asymptotical results for the phase velocity agree with those of Backus averaging and ray approximation, respectively. The theory describes the angle‐dependent attenuation caused by multiple scattering. The proposed formulas are simple enough to be used in many practical applications as, e.g., in an amplitude variation with offset (AVO) analysis. They can be implemented for taking into account the angle dependence of transmission effects, or they can be used in an inversion for statistical parameters of sediments.

Oscillations in a Prey-Predator-Superpredator System

1998 ◽  
Vol 06 (01) ◽  
pp. 17-33 ◽  
Author(s):  
H. Boudjellaba ◽  
T. Sari
Keyword(s):  
Food Chain ◽  
Growth Rates ◽  
High Frequency ◽  
Averaging Method ◽  
Low Frequency ◽  
High Frequency Oscillations

The behaviour of a food chain with three populations of the type prey-predator-superpredator is studied in the case where the growth rates of the three populations are highly diversified. The analysis shows that the system can have high-frequency oscillations, due to the interaction between prey and predator, which arise during a low-frequency cycle, due to the interaction between predator and superpredator. Using an averaging method, the dynamics of the superpredator is determined, during the high-frequency oscillations of the prey and predator.

Topography Estimation of Visual Evoked Potentials Using a Combination of Mathematical Models

2013 ◽  
pp. 129-141
Author(s):  
Takenao Sugi ◽  
Kazuhiko Goto ◽  
Satoru Goto ◽  
Yoshinobu Goto ◽  
Takao Yamasaki ◽  
...  
Keyword(s):  
Power Spectrum ◽  
Evoked Potentials ◽  
Visual Evoked Potentials ◽  
Alpha Rhythm ◽  
Averaging Method ◽  
Second Harmonic ◽  
Low Frequency ◽  
Frequency Component ◽  
Topographical Distribution ◽  
Visual Evoked

This study proposes a method for estimating the topographical distribution of Visual Evoked Potentials (VEPs) from separated power spectrum components by a combination of models. VEPs with various temporal frequencies were recorded from nine healthy adults. The original power spectrum consisted of the VEP; background activities, artifacts, and other components were then obtained. To extract the VEP components from the original power spectrum, models corresponding to background activities, especially for posterior alpha rhythm, the low-frequency component and the high-frequency component, caused due to the EMG artifact, were constructed, and the relevant parameters were estimated. Finally, VEP components were calculated by subtracting them from the original power spectrum. The topographical distribution of the first harmonic (1F) and second harmonic (2F) components of the VEP were obtained by the proposed method. The estimation of the other components, aside from the VEPs, was also investigated. The merits and usefulness of the proposed method were analyzed with a comparison to the conventional stimulus-locked averaging method. The proposed method has several advantageous points compared to the conventional averaging method. Specifically, the posterior alpha rhythm and the EMG artifact were accounted for directly in the estimation of the VEP components. Therefore, an accurate estimation of the VEP components can be performed even the measurement of the components are prone to the error.

A Study on the Fine Structure of the Lateral Line Organ of the Sea Eel

1973 ◽  
Vol 31 ◽  
pp. 648-649
Author(s):  
K. Hama
Keyword(s):  
Fine Structure ◽  
Electron Microscope ◽  
Lateral Line ◽  
Low Frequency ◽  
Sensory Cells ◽  
Apical Surface ◽  
Voltage Electron ◽  
Sensory Epithelia ◽  
Low Frequency Vibration ◽  
Transmission Electron

The lateral line organs of the sea eel consist of canal and pit organs which are different in function. The former is a low frequency vibration detector whereas the latter functions as an ion receptor as well as a mechano receptor.The fine structure of the sensory epithelia of both organs were studied by means of ordinary transmission electron microscope, high voltage electron microscope and of surface scanning electron microscope.The sensory cells of the canal organ are polarized in front-caudal direction and those of the pit organ are polarized in dorso-ventral direction. The sensory epithelia of both organs have thinner surface coats compared to the surrounding ordinary epithelial cells, which have very thick fuzzy coatings on the apical surface.

Lectin Binding to Human Breast Tumor Cells

1976 ◽  
Vol 34 ◽  
pp. 34-35
Author(s):  
Robert E. Nordquist ◽  
J. Hill Anglin ◽  
Michael P. Lerner
Keyword(s):  
Carcinoma Cell ◽  
Ricinus Communis ◽  
Lectin Binding ◽  
Low Frequency ◽  
Breast Carcinoma Cell Line ◽  
Human Breast ◽  
Human Breast Tumor ◽  
Duct Carcinoma ◽  
Specific Antigens ◽  
Ricin Agglutinin

A human breast carcinoma cell line (BOT-2) was derived from an infiltrating duct carcinoma (1). These cells were shown to have antigens that selectively bound antibodies from breast cancer patient sera (2). Furthermore, these tumor specific antigens could be removed from the living cells by low frequency sonication and have been partially characterized (3). These proteins have been shown to be around 100,000 MW and contain approximately 6% hexose and hexosamines. However, only the hexosamines appear to be available for lectin binding. This study was designed to use Concanavalin A (Con A) and Ricinus Communis (Ricin) agglutinin for the topagraphical localization of D-mannopyranosyl or glucopyranosyl and D-galactopyranosyl or DN- acetyl glactopyranosyl configurations on BOT-2 cell surfaces.

Practical High Resolution Microscopy

1968 ◽  
Vol 26 ◽  
pp. 302-303
Author(s):  
P. A. Marsh ◽  
T. Mullens ◽  
D. Price
Keyword(s):  
High Resolution ◽  
Magnetic Fields ◽  
Low Frequency ◽  
Resolving Power ◽  
Careful Attention ◽  
Electron Microscopes ◽  
The Relationship ◽  
High Resolution Microscopy ◽  
New Facilities

It is possible to exceed the guaranteed resolution on most electron microscopes by careful attention to microscope parameters essential for high resolution work. While our experience is related to a Philips EM-200, we hope that some of these comments will apply to all electron microscopes.The first considerations are vibration and magnetic fields. These are usually measured at the pre-installation survey and must be within specifications. It has been our experience, however, that these factors can be greatly influenced by the new facilities and therefore must be rechecked after the installation is completed. The relationship between the resolving power of an EM-200 and the maximum tolerable low frequency interference fields in milli-Oerstedt is 10 Å - 1.9, 8 Å - 1.4, 6 Å - 0.8.

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