Practical aspects of reflectivity modeling

Geophysics ◽  
1987 ◽  
Vol 52 (10) ◽  
pp. 1355-1364 ◽  
Author(s):  
Subhashis Mallick ◽  
L. Neil Frazer
Keyword(s):  
Velocity Gradient ◽  
Complex Frequency ◽  
Seismic Velocities ◽  
Frequency Dependent ◽  
Filon Method ◽  
Frequency Independent ◽  
Reflectivity Function ◽  
Sum Formula ◽  
Common Problems ◽  
Vector Computers

This paper is intended to help those not familiar with the “lore” of layered earth modeling to avoid some common problems. In the computation of the reflectivity function, an easily incorporated phase‐integral approximation is used away from turning points when the velocity gradient is smaller than the frequency. Hanning windows, or segments thereof, work well for both the slowness integral and the frequency integral. For the quadrature of the slowness integral the Filon method of Frazer is easily coded and vectorizes well; Levin’s Filon method and the Clenshaw‐Curtis‐Filon method of Xu and Mal are more difficult to vectorize, but more powerful because they require fewer evaluations of the reflectivity function. A modification of Strick’s power law is a convenient way to calculate complex frequency‐dependent seismic velocities. The complex frequency technique for avoiding time aliasing is explained by use of the Poisson sum formula. In writing code for vector computers, such as the CRAY, if frequency‐independent velocities are used, the frequency loop should be deepest, whereas if frequency‐dependent velocities are used, then the p‐loops should be deepest.

Impact of Frequency Dependence of Gas Labyrinth Seal Rotordynamic Coefficients on Centrifugal Compressor Stability

2010 ◽  
Author(s):  
Giuseppe Vannini ◽  
Manish R. Thorat ◽  
Dara W. Childs ◽  
Mirko Libraschi
Keyword(s):  
Molecular Weight ◽  
Numerical Model ◽  
Control Volume ◽  
Labyrinth Seal ◽  
Labyrinth Seals ◽  
Frequency Dependent ◽  
Rotordynamic Coefficients ◽  
Frequency Independent ◽  
Rotor Surface ◽  
Independent Model

A numerical model developed by Thorat & Childs [1] has indicated that the conventional frequency independent model for labyrinth seals is invalid for rotor surface velocities reaching a significant fraction of Mach 1. A theoretical one-control-volume (1CV) model based on a leakage equation that yields a reasonably good comparison with experimental results is considered in the present analysis. The numerical model yields frequency-dependent rotordynamic coefficients for the seal. Three real centrifugal compressors are analyzed to compare stability predictions with and without frequency-dependent labyrinth seal model. Three different compressor services are selected to have a comprehensive scenario in terms of pressure and molecular weight (MW). The molecular weight is very important for Mach number calculation and consequently for the frequency dependent nature of the coefficients. A hydrogen recycle application with MW around 8, a natural gas application with MW around 18, and finally a propane application with molecular weight around 44 are selected for this comparison. Useful indications on the applicability range of frequency dependent coefficients are given.

A Generalized Dual-Frequency Transformer for Two Arbitrary Complex Frequency-Dependent Impedances

2009 ◽  
Vol 19 (12) ◽  
pp. 792-794 ◽  
Author(s):  
Yongle Wu ◽  
Yuanan Liu ◽  
Shulan Li ◽  
Cuiping Yu ◽  
Xin Liu
Keyword(s):  
Complex Frequency ◽  
Dual Frequency ◽  
Frequency Dependent ◽  
Arbitrary Complex

scholarly journals A Blind Calibration Model for I/Q Imbalances of Wideband Zero-IF Receivers

Electronics ◽  
2020 ◽  
Vol 9 (11) ◽  
pp. 1868
Author(s):  
Xiaoye Peng ◽  
Zhiyu Wang ◽  
Jiongjiong Mo ◽  
Chenge Wang ◽  
Jiarui Liu ◽  
...  
Keyword(s):  
Frequency Domain ◽  
Finite Impulse Response ◽  
Low Cost ◽  
Classification Rule ◽  
Statistical Characteristics ◽  
High Signal ◽  
Calibration Model ◽  
Frequency Dependent ◽  
Frequency Independent ◽  
Blind Calibration

Frequency-dependent I/Q imbalance and frequency-independent I/Q imbalance are the major impairments in wideband zero-IF receivers, and they both cannot be ignored. In this paper, a blind calibration model is designed for compensating these I/Q imbalances. In order to accurately estimate the imbalance parameters with low cost, a classification rule is proposed according to the frequency-domain statistical characteristics of the received signal. The calibration points in the frequency-domain are divided into two groups. Then, the amplitude imbalance and the frequency-dependent phase imbalance are derived from the group of signal points and, separately, the frequency-independent phase imbalance is calculated from the group of noise points. In the derivation of the frequency-dependent phase imbalance, a general fitting model suitable for all signal points is proposed, which does not require special calculations for either DC point or fs/2 point. Then, a finite impulse response (FIR) real-valued filter is designed to correct the impairments of received signal. The performances of the proposed calibration model are evaluated through both simulations and experiments. The simulation results show the image rejection ratio (IRR) improvement to around 35–45 dBc at high signal-to-noise ratio (SNR). Based on the mismatched data of the ADRV9009 evaluation board, the experimental results exhibit the IRR improvement of both multi-tone and wideband signals to about 30 dBc.

A Study on the Leakage and Rotordynamic Performance of a Long Labyrinth Seal Under Mainly Air Conditions

2019 ◽  
Vol 141 (12) ◽  
Author(s):  
Min Zhang ◽  
Dara W. Childs
Keyword(s):  
Silicone Oil ◽  
Dynamic Stiffness ◽  
Excitation Frequency ◽  
Labyrinth Seal ◽  
System Stability ◽  
Test Fluid ◽  
Liquid Volume ◽  
Frequency Dependent ◽  
Frequency Independent ◽  
The Impact

Abstract This paper investigates the impact of liquid presence in air on the leakage and rotordynamic coefficients of a long (length-to-diameter ratio L/D = 0.747) teeth-on-stator labyrinth seal. The test fluid is a mixture of air and silicone oil (PSF-5cSt). Tests are carried out at inlet pressure Pi = 62.1 bars, three pressure ratios from 0.21 to 0.46, three speeds from 10 to 20 krpm, and six inlet liquid volume fractions (LVFs) from 0% to 15%. Complex dynamic-stiffness coefficients Hij are measured. The real parts of Hij are too frequency dependent to be fitted by frequency-independent stiffness and virtual-mass coefficients. Therefore, this paper presents frequency-dependent direct stiffness KΩ and cross-coupled stiffness kΩ. The imaginary parts of Hij produce frequency-independent direct damping C. Test results show that, under both pure- and mainly air conditions, the leakage mass flowrate m˙ of the test seal steadily increases as inlet LVF increases. KΩ is negative under all test conditions, and the magnitude of KΩ increases as inlet LVF increases, leading to a larger negative centering force on the associated compressor rotor. Under pure-air conditions, kΩ is a small negative value. Injecting oil into the air increases kΩ slightly and make the magnitude of kΩ closer to zero. Under mainly air conditions, increasing inlet LVF from 2% to 15% has little impact on kΩ. C normally increases as inlet LVF increases. The value of the effective damping Ceff = C − kΩ/Ω near 0.5ω is of significant interest to the system stability since an unstable centrifugal compressor may precess at approximately 0.5ω. Ω denotes the excitation frequency. The oil presence in the air has little impact on the value of Ceff near 0.5ω. Also, the liquid presence does not change the insensitiveness of m˙, KΩ, kΩ, C, and Ceff to change in ω; i.e., under both pure- and mainly air conditions, changes in ω has little impact on m˙, KΩ, kΩ, C, and Ceff.

Coupling Sound Absorbing Materials With an Air Cavity Using Frequency Dependent Bulk Material Properties

2016 ◽  
Author(s):  
Shung H. Sung ◽  
Donald J. Nefske
Keyword(s):  
Finite Element ◽  
Frequency Response ◽  
Material Properties ◽  
Complex Frequency ◽  
Air Cavity ◽  
Frequency Dependent ◽  
Absorbing Materials ◽  
Mode Method ◽  
Modal Solution ◽  
Solution Methodology

This paper presents the acoustic finite element method and the modal solution method for coupling sound absorbing materials with an air cavity to predict the sound pressure frequency response. The sound absorbing materials are represented with complex, frequency-dependent, effective mass-density and bulk-modulus properties obtained from the acoustic impedance of material samples. To couple the sound absorber cavity and air cavity, the boundary conditions at the interface between the cavities requires equality of pressure and equality of acoustic volume flow. Two modal solution methods are developed to compute the frequency response of the coupled system with frequency dependent material properties: the component mode method and the coupled mode method. The finite element and modal solution methodology is developed in a form readily adaptable for implementation in commercially available codes. The accuracy of the modal solution methodology is assessed for modeling a one-dimensional air tube terminated with absorbent material and the seats in an automobile passenger compartment.

Robust Empirical Time–Frequency Relations for Seismic Spectral Amplitudes, Part 2: Model Uncertainty and Optimal Parameterization

2020 ◽  
Author(s):  
Maryam Safarshahi ◽  
Igor B. Morozov
Keyword(s):  
Measurement Errors ◽  
Nonparametric Model ◽  
Model Parameters ◽  
Geometrical Spreading ◽  
Model Quality ◽  
Frequency Dependent ◽  
Time Frequency ◽  
Trade Offs ◽  
Frequency Independent ◽  
Seismic Amplitudes

ABSTRACT In a companion article, Safarshahi and Morozov (2020) argued that construction of distance- and frequency-dependent models for seismic-wave amplitudes should include four general elements: (1) a sufficiently detailed (parametric or nonparametric) model of frequency-independent spreading, capturing all essential features of observations; (2) model parameters with well-defined and nonoverlapping physical meanings; (3) joint inversion for multiple parameters, including the geometrical spreading, Q, κ, and source and receiver couplings; and (4) the use of additional dataset-specific criteria of model quality, while fitting the logarithms of seismic amplitudes. Some of these elements are present in existing models, but, taken together, they are poorly understood and require an integrated approach. Such an approach was illustrated by detailed analysis of an S-wave amplitude dataset from southern Iran. The resulting model is based on a frequency-independent Q, and matches the data closer than conventional models and across the entire epicentral-distance range. Here, we complete the analysis of this model by evaluating the uncertainties and trade-offs of its parameters. Two types of trade-offs are differentiated: one caused by a (possibly) limited model parameterization and the second due to statistical data errors. Data bootstrapping shows that with adequate parameterization, attenuation properties Q, κ, and geometrical spreading parameters are resolved well and show moderate trade-offs due to measurement errors. Using the principal component analysis of these trade-offs, an optimal (trade-off free) parameterization of seismic amplitudes is obtained. By contrast, when assuming theoretical values for certain model parameters and using multistep inversion procedures (as commonly done), parameter trade-offs increase dramatically and become difficult to assess. In particular, the frequency-dependent Q correlates with the distribution of the source and receiver-site factors, and also with biases in the resulting median data residuals. In the new model, these trade-offs are removed using an improved parameterization of geometrical spreading, constant Q, and model quality constraints.

An investigation into the fundamental relationships between attenuation, phase dispersion, and frequency using seismic refraction profiles over sedimentary structures

Geophysics ◽  
1987 ◽  
Vol 52 (1) ◽  
pp. 72-87 ◽  
Author(s):  
R. S. Jacobson
Keyword(s):  
Frequency Dependence ◽  
Seismic Waves ◽  
Velocity Dispersion ◽  
Seismic Refraction ◽  
Systematic Investigation ◽  
Data Sets ◽  
Frequency Dependent ◽  
Frequency Independent ◽  
Apparent Attenuation ◽  
Seismic Refraction Data

Despite many attenuation measurements which indicate a linear functional frequency dependence of absorption or constant [Formula: see text] in sediments, several theories predict no such linear dependence. The primary justification for rejecting a first‐power frequency dependence of attenuation is that it implies that seismic waves cannot propagate causally. Seismic waves must also travel with some velocity dispersion to satisfy causality, yet there is a lack of velocity dispersion measurements in sediments. In‐situ attenuation is caused by two distinct mechanisms: anelastic heating, and scattering due to interbed multiples. Apparent, or scattering, attenuation can produce both frequency‐dependent and non‐frequency‐dependent effects. Accurate measurements of attenuation and velocity dispersion are difficult; it is not surprising that a systematic investigation into the frequency dependence of absorption and velocity has not been made. A reinvestigation into two seismic refraction data sets collected over thickly stratified deep‐sea fans indicates that [Formula: see text] should not be assumed to be independent of frequency. Further, significant frequency‐independent absorption is present, indicating a high degree of apparent attenuation. Phase, or velocity, dispersion was also measured, but the results are more ambiguous than those for attenuation, due to inherent limitations of digital signals. Nevertheless, the absorption and velocity dispersion results are largely compatible, suggesting that if apparent attenuation is observed, then the scattered waves propagate causally.

scholarly journals Frequency-dependent attenuation analysis of ground-penetrating radar data

Geophysics ◽  
2007 ◽  
Vol 72 (3) ◽  
pp. J7-J16 ◽  
Author(s):  
John H. Bradford
Keyword(s):  
Attenuation Coefficient ◽  
Ground Penetrating Radar ◽  
A Priori ◽  
Radar Data ◽  
Frequency Dependent ◽  
First Order ◽  
Reflection Data ◽  
Frequency Independent ◽  
Band Limited ◽  
Ground Penetrating

In the early 1990s, it was established empirically that, in many materials, ground-penetrating radar (GPR) attenuation is approximately linear with frequency over the bandwidth of a typical pulse. Further, a frequency-independent [Formula: see text] parameter characterizes the slope of the band-limited attenuation versus frequency curve. Here, I derive the band-limited [Formula: see text] function from a first-order Taylor expansion of the attenuation coefficient. This approach provides a basis for computing [Formula: see text] from any arbitrary dielectric permittivity model. For Cole-Cole relaxation, I find good correlation between the first-order [Formula: see text] approximation and [Formula: see text] computed from linear fits to the attenuation coefficient curve over two-octave bands. The correlation holds over the primary relaxation frequency. For some materials, this relaxation occurs between 10 and [Formula: see text], a typical frequency range for many GPR applications. Frequency-dependent losses caused by scattering and by the commonly overlooked problem of frequency-dependent reflection make it difficult or impossible to measure [Formula: see text] from reflection data without a priori understanding of the materials. Despite these complications, frequency-dependent attenuation analysis of reflection data can provide valuable subsurface information. At two field sites, I find well-defined frequency-dependent attenuation anomalies associated with nonaqueous-phase liquid contaminants.

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