Analysis of the characteristics‐integration method for one‐dimensional wave inversion

Geophysics ◽  
1988 ◽  
Vol 53 (8) ◽  
pp. 1034-1044 ◽  
Author(s):  
Nei‐Mao Chen ◽  
Yu‐Hua Chu ◽  
John T. Kuo
Keyword(s):  
Wave Equation ◽  
Inverse Method ◽  
Integration Method ◽  
Low Frequency ◽  
Real Data ◽  
Frequency Component ◽  
One Dimensional ◽  
Reflection Data ◽  
Wave Inversion ◽  
Impedance Profile

Basing our work on the one‐dimensional (1-D) wave equation, we present an inverse method which we call the characteristics‐integration method. The method is derived from integration along characteristic families of straight lines of the wave equation in the time domain. With the source function known and reflection data recorded on the surface, the characteristics‐integration method can efficiently and economically recover the subsurface impedance profile, provided that the structure is inhomogeneous only in the depth direction. In general, when seismic data are contaminated by noise, the characteristics‐integration method, like any other 1-D inverse method, suffers from instability. We find that, for a smoothly varying impedance profile, the instability of inversions using characteristic methods depends heavily on the bandwidth of the source wavelet. We devised a resampling technique to stabilize the inverse scheme and to suppress the growth of errors. Numerical examples, including data contaminated by noise, data missing the low‐frequency component, and real data cases, show the feasibility of recovering impedance profiles using the characteristics‐integration method.

Propagation of Low-Frequency Elastic Disturbances in a Composite Material

1974 ◽  
Vol 41 (1) ◽  
pp. 97-100 ◽  
Author(s):  
W. Kohn
Keyword(s):  
Composite Material ◽  
Wave Equation ◽  
Dispersion Relation ◽  
Displacement Field ◽  
Low Frequency ◽  
Local Strain ◽  
Airy Functions ◽  
Low Frequencies ◽  
One Dimensional ◽  
Long Wavelength

In the limit of low frequencies the displacement u(x, t) in a one-dimensional composite can be written in the form of an operator acting on a slowly varying envelope function, U(x, t): u(x, t) = [1 + v1(x)∂/∂x + …] U(x, t). U(x, t) itself describes the overall long wavelength displacement field. It satisfies a wave equation with constant, i.e., x-independent, coefficients, obtainable from the dispersion relation ω = ω(k) of the lowest band of eigenmodes: (∂2/∂t2 − c¯2∂2/∂x2 − β∂4/∂x4 + …) U(x, t) = 0. Information about the local strain, on the microscale of the composite laminae, is contained in the function v1(x), explicitly expressible in terms of the periodic stiffness function, η(x), of the composite. Appropriate Green’s functions are constructed in terms of Airy functions. Among applications of this method is the structure of the so-called head of a propagating pulse.

scholarly journals Estimating broad trend of acoustic impedance profile from observed seismic reflection data using first principles only

2020 ◽  
Vol 17 (3) ◽  
pp. 475-483
Author(s):  
Animesh Mandal ◽  
Santi Kumar Ghosh
Keyword(s):  
First Principles ◽  
Acoustic Impedance ◽  
Seismic Reflection ◽  
Low Frequency ◽  
Singular Value ◽  
Earth Model ◽  
Depth Range ◽  
Seismic Reflection Data ◽  
Reflection Data ◽  
Impedance Profile

Abstract Estimation of broad features or the low-frequency part of acoustic impedance from conventional reflection data is an essential yet challenging step for quantitative interpretation of seismic data due to its band-limited nature. A missing low-frequency part leads to non-uniqueness in the solution as well as placing restrictions in recovering the absolute impedance values. The current industry practice fills this gap by assuming either an initial impedance model or statistical restrictions on such a model. Doing away with such assumptions but using only first principles (Zoeppritz's equations) and homogeneous layered earth model, we have formulated a set of linear equations that are then solved for an unknown reflection co-efficient using singular value decomposition (SVD) approach with time sampled seismic trace as the input data. The present work demonstrates the effectiveness of reconstructing a broad and smooth impedance profile from first principles and even from acquired seismic reflection data. It also illustrates the method's success with real data, while determining in one go the unknown scale factor linking the true and the relative seismic amplitudes, and the smallest singular value to be retained in the solution from only the knowledge of the average value of the acoustic impedance over the depth range in question. Thus, the salient feature of this work is the ability to reconstruct an approximate impedance profile from field data without the aid of an initial model or statistical assumption on the reflectivity series. This approximate impedance profile can serve as a reliable initial input for more refined inversion or geologic interpretation.

scholarly journals ABOUT CLASSIFICATION OF ECG SIGNALS BASED ON HIGH-FREQUENCY WAVELET COMPONENTS

2016 ◽  
pp. 63-71
Author(s):  
Paulina Podkur ◽  
Paulina Podkur ◽  
Nikolay Smolentsev ◽  
Nikolay Smolentsev
Keyword(s):  
Myocardial Infarction ◽  
High Frequency ◽  
Wavelet Decomposition ◽  
Low Frequency ◽  
Feature Space ◽  
Frequency Component ◽  
High Frequency Component ◽  
Ecg Signal ◽  
One Dimensional ◽  
Frequency Components

The new design method of the automated classifying system for electrocardiograms recognition (ECG) of healthy and ill patients, based only on high-frequency components of ECG signal with the use of statistical images recognition is offered. Cardiograms of two groups of patients were studied: healthy and those who came through myocardial infarction. The first step of classification method is ECG wavelet decomposition to the 4th level and allocation of four high-frequency ECG components. The choice of the 4th decomposing level is explained by the fact that the first four high-frequency components represent high ECG frequencies from 30 to 350 Hz, and low-frequency component represents the undistorted smoothed ECG signal cleared of high-frequency oscillations. In case of more deep signal expansion the following high-frequency component has frequency spectrum to 30 Hz, and low frequency component is significantly distorted. For each of the first four components of wavelet decomposition there is a number of ECG numerical signs, including energy, entropy and frequency characteristics, 21 signs in total. During the second step reduction of the dimensionality of the feature space by using scatter matrix is made for two chosen ECG groups. It has turned out that the reduced feature space is one-dimensional. Histograms of values of this one-dimensional feature for groups of healthy and ill patients are constructed. The third step is finding of the dividing constant which is able to distinguish both groups of ECG records. For testing 96 ECG records of patients with normal cardiograms and 120 ECG records of the patients who came through myocardial infarction are used. Only three features (3%) of 96 given features values of the first group are referred by the classifier to patients group and only 20 features (< 17%) of 120 given features values of the second patients group are referred by the classifier to ECG group of healthy patients. Considering that for each patient the system determines 12 features by 12 standard assignments, testing results show well classification accuracy.

COMPUTATION OF THE FLOW AND NEAR SOUND FIELDS OF A FREE SURFACE PIERCING CYLINDER

2009 ◽  
Vol 17 (04) ◽  
pp. 365-382 ◽  
Author(s):  
ELDAD J. AVITAL ◽  
GUANGXU YU ◽  
JOHN WILLIAMS
Keyword(s):  
Wave Equation ◽  
Free Surface ◽  
Sound Wave ◽  
Near Field ◽  
Low Frequency ◽  
Sound Field ◽  
Frequency Component ◽  
Sound Fields ◽  
Pressure Surface ◽  
Interface Surface

The near sound field generated due to a vertically mounted circular cylinder piercing a free surface in shallow water, is studied computationally using the Large Eddy Simulation (LES) approach and the sound wave equation. The flow is simulated in both the air and water phases. The interface surface is allowed to move and is simulated using the Volume of Fluid technique. The pressure distribution over the cylinder is fed back into the sound wave equation to calculate the near field. The interface surface is modeled as a zero pressure surface in the acoustic calculation and the bottom is taken as having infinite impedence modeling the case of a rigid floor. Two of acoustic calculations methods are used. In the first method, the interface surface is assumed to be fixed and the wave equation is solved in the frequency domain in a post-processing stage. In the second method, the evolution of the interface surface is taken into account and the wave equation is simulated simultaneously with the LES. Both solutions are analyzed and compared to show that the interface surface acts as a strong damper to the low frequency sound by damping the vortex Von Karman rollers as well as causing the low frequency component to be nonradiative. The variation of the near sound field with the water depth and Froude number is investigated and the propagation and damping characteristics are analyzed.

Thermally corrected solutions of the one-dimensional wave equation for the laser-induced ultrasound

2021 ◽  
Vol 130 (2) ◽  
pp. 025104
Author(s):  
Misael Ruiz-Veloz ◽  
Geminiano Martínez-Ponce ◽  
Rafael I. Fernández-Ayala ◽  
Rigoberto Castro-Beltrán ◽  
Luis Polo-Parada ◽  
...  
Keyword(s):  
Wave Equation ◽  
One Dimensional ◽  
The One ◽  
Dimensional Wave

scholarly journals An Improved Time Integration Method Based on Galerkin Weak Form with Controllable Numerical Dissipation

2021 ◽  
Vol 11 (4) ◽  
pp. 1932
Author(s):  
Weixuan Wang ◽  
Qinyan Xing ◽  
Qinghao Yang
Keyword(s):  
High Frequency ◽  
Integration Method ◽  
Time Integration ◽  
Low Frequency ◽  
Weak Form ◽  
High Accuracy ◽  
Numerical Dissipation ◽  
Frequency Range ◽  
Time Integration Method ◽  
Numerical Damping

Based on the newly proposed generalized Galerkin weak form (GGW) method, a two-step time integration method with controllable numerical dissipation is presented. In the first sub-step, the GGW method is used, and in the second sub-step, a new parameter is introduced by using the idea of a trapezoidal integral. According to the numerical analysis, it can be concluded that this method is unconditionally stable and its numerical damping is controllable with the change in introduced parameters. Compared with the GGW method, this two-step scheme avoids the fast numerical dissipation in a low-frequency range. To highlight the performance of the proposed method, some numerical problems are presented and illustrated which show that this method possesses superior accuracy, stability and efficiency compared with conventional trapezoidal rule, the Wilson method, and the Bathe method. High accuracy in a low-frequency range and controllable numerical dissipation in a high-frequency range are both the merits of the method.

Asymptotic derivation of a refined equation for an elastic beam resting on a Winkler foundation

2021 ◽  
pp. 108128652110238
Author(s):  
Barış Erbaş ◽  
Julius Kaplunov ◽  
Isaac Elishakoff
Keyword(s):  
Ad Hoc ◽  
Moving Load ◽  
Low Frequency ◽  
Elastic Beam ◽  
Winkler Foundation ◽  
One Dimensional ◽  
Beam Equations ◽  
Dimensional Equation ◽  
Phase And Group Velocities ◽  
Group Velocities

A two-dimensional mixed problem for a thin elastic strip resting on a Winkler foundation is considered within the framework of plane stress setup. The relative stiffness of the foundation is supposed to be small to ensure low-frequency vibrations. Asymptotic analysis at a higher order results in a one-dimensional equation of bending motion refining numerous ad hoc developments starting from Timoshenko-type beam equations. Two-term expansions through the foundation stiffness are presented for phase and group velocities, as well as for the critical velocity of a moving load. In addition, the formula for the longitudinal displacements of the beam due to its transverse compression is derived.

An improved method to estimate Q based on the logarithmic spectrum of moving peak points

2019 ◽  
Vol 7 (2) ◽  
pp. T255-T263 ◽  
Author(s):  
Yanli Liu ◽  
Zhenchun Li ◽  
Guoquan Yang ◽  
Qiang Liu
Keyword(s):  
Seismic Waves ◽  
Wave Attenuation ◽  
Real Data ◽  
Peak Frequency ◽  
Seismic Reflection Data ◽  
The Real ◽  
Time Frequency ◽  
Reflection Data ◽  
Text Filtering ◽  
The Impact

The quality factor ([Formula: see text]) is an important parameter for measuring the attenuation of seismic waves. Reliable [Formula: see text] estimation and stable inverse [Formula: see text] filtering are expected to improve the resolution of seismic data and deep-layer energy. Many methods of estimating [Formula: see text] are based on an individual wavelet. However, it is difficult to extract the individual wavelet precisely from seismic reflection data. To avoid this problem, we have developed a method of directly estimating [Formula: see text] from reflection data. The core of the methodology is selecting the peak-frequency points to linear fit their logarithmic spectrum and time-frequency product. Then, we calculated [Formula: see text] according to the relationship between [Formula: see text] and the optimized slope. First, to get the peak frequency points at different times, we use the generalized S transform to produce the 2D high-precision time-frequency spectrum. According to the seismic wave attenuation mechanism, the logarithmic spectrum attenuates linearly with the product of frequency and time. Thus, the second step of the method is transforming a 2D spectrum into 1D by variable substitution. In the process of transformation, we only selected the peak frequency points to participate in the fitting process, which can reduce the impact of the interference on the spectrum. Third, we obtain the optimized slope by least-squares fitting. To demonstrate the reliability of our method, we applied it to a constant [Formula: see text] model and the real data of a work area. For the real data, we calculated the [Formula: see text] curve of the seismic trace near a well and we get the high-resolution section by using stable inverse [Formula: see text] filtering. The model and real data indicate that our method is effective and reliable for estimating the [Formula: see text] value.

scholarly journals Infinitely many periodic solutions for a semilinear wave equation with x-dependent coefficients

2020 ◽  
Vol 26 ◽  
pp. 7
Author(s):  
Hui Wei ◽  
Shuguan Ji
Keyword(s):  
Wave Equation ◽  
Periodic Solutions ◽  
Spectral Properties ◽  
Seismic Waves ◽  
Forced Vibrations ◽  
One Dimensional ◽  
Semilinear Wave Equation ◽  
Spatial Interval ◽  
Rational Multiple ◽  
Infinitely Many Periodic Solutions

This paper is devoted to the study of periodic (in time) solutions to an one-dimensional semilinear wave equation with x-dependent coefficients under various homogeneous boundary conditions. Such a model arises from the forced vibrations of a nonhomogeneous string and propagation of seismic waves in nonisotropic media. By combining variational methods with an approximation argument, we prove that there exist infinitely many periodic solutions whenever the period is a rational multiple of the length of the spatial interval. The proof is essentially based on the spectral properties of the wave operator with x-dependent coefficients.

scholarly journals Shot- and angle-domain wave-equation traveltime inversion of reflection data: Theory

Geophysics ◽  
2015 ◽  
Vol 80 (4) ◽  
pp. U47-U59 ◽  
Author(s):  
Sanzong Zhang ◽  
Yi Luo ◽  
Gerard Schuster
Keyword(s):  
Wave Equation ◽  
Traveltime Inversion ◽  
Reflection Data
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