Temporal windowing and inverse transform of the wavefield in the Laplace-Fourier domain

Geophysics ◽  
2013 ◽  
Vol 78 (5) ◽  
pp. R207-R222 ◽  
Author(s):  
Sangmin Kwak ◽  
Hyunggu Jun ◽  
Wansoo Ha ◽  
Changsoo Shin
Keyword(s):  
Fourier Transform ◽  
Time Window ◽  
Waveform Inversion ◽  
Velocity Model ◽  
Fourier Domain ◽  
Gain Function ◽  
The Fourier Transform ◽  
The Time Domain ◽  
Derivatives Of ◽  
Complex Damping

Temporal windowing is a valuable process, which can help us to focus on a specific event in a seismogram. However, applying the time window is difficult outside the time domain. We suggest a windowing method which is applicable in the Laplace-Fourier domain. The window function we adopt is defined as a product of a gain function and an exponential damping function. The Fourier transform of a seismogram windowed by this function is equivalent to the partial derivative of the Laplace-Fourier domain wavefield with respect to the complex damping constant. Therefore, we can obtain a windowed seismogram using the partial derivatives of the Laplace-Fourier domain wavefield. We exploit the time-windowed wavefield, which is modeled directly in the Laplace-Fourier domain, to reconstruct subsurface velocity model by waveform inversion in the Laplace-Fourier domain. We present the windowed seismograms by introducing an inverse Laplace-Fourier transform technique and demonstrate the effect of temporal windowing in a synthetic Laplace-Fourier domain waveform inversion example.

A Deep Neural Network Model for Learning Runtime Frequency Response Function Using Sensor Measurements

2021 ◽  
Author(s):  
Yongzhi Qu ◽  
Gregory W. Vogl ◽  
Zechao Wang
Keyword(s):  
Neural Network ◽  
Neural Networks ◽  
Fourier Transform ◽  
Time Domain ◽  
Frequency Response Function ◽  
Frequency Component ◽  
Operating Conditions ◽  
Input Output ◽  
The Fourier Transform ◽  
The Time Domain

Abstract The frequency response function (FRF), defined as the ratio between the Fourier transform of the time-domain output and the Fourier transform of the time-domain input, is a common tool to analyze the relationships between inputs and outputs of a mechanical system. Learning the FRF for mechanical systems can facilitate system identification, condition-based health monitoring, and improve performance metrics, by providing an input-output model that describes the system dynamics. Existing FRF identification assumes there is a one-to-one mapping between each input frequency component and output frequency component. However, during dynamic operations, the FRF can present complex dependencies with frequency cross-correlations due to modulation effects, nonlinearities, and mechanical noise. Furthermore, existing FRFs assume linearity between input-output spectrums with varying mechanical loads, while in practice FRFs can depend on the operating conditions and show high nonlinearities. Outputs of existing neural networks are typically low-dimensional labels rather than real-time high-dimensional measurements. This paper proposes a vector regression method based on deep neural networks for the learning of runtime FRFs from measurement data under different operating conditions. More specifically, a neural network based on an encoder-decoder with a symmetric compression structure is proposed. The deep encoder-decoder network features simultaneous learning of the regression relationship between input and output embeddings, as well as a discriminative model for output spectrum classification under different operating conditions. The learning model is validated using experimental data from a high-pressure hydraulic test rig. The results show that the proposed model can learn the FRF between sensor measurements under different operating conditions with high accuracy and denoising capability. The learned FRF model provides an estimation for sensor measurements when a physical sensor is not feasible and can be used for operating condition recognition.

Directional Image Filtering Based on the Fourier Transform

2014 ◽  
Vol 19 (2-3) ◽  
pp. 7-13
Author(s):  
Przemysław Korohoda ◽  
Joanna Grabska-Chrząstowska ◽  
Jaromir Przybyło
Keyword(s):  
Fourier Transform ◽  
Rotation Angle ◽  
Image Filtering ◽  
Suggested Method ◽  
Fourier Domain ◽  
Fourier Space ◽  
Direct Rotation ◽  
The Fourier Transform ◽  
Directional Image

Abstract An algorithm to design the small size 2-D filter masks with arbitrarily selected rotation angle has been proposed. The classical filter mask of size 3 × 3 is obtained from the reference Fourier space characteristics, rotated in the Fourier domain. The efficiency of the suggested method was illustrated with examples based on the Sobel gradient mask and two test images. Comparative computations indicated that the accuracy of the filtering result with use of the small size filters is noticeably better when the filter has been designed with use of the Fourier characteristics rotation than after direct rotation of the mask in the pixel domain.

Discrete Time, Discrete Frequency

2021 ◽  
pp. 106-155
Author(s):  
Victor Lazzarini
Keyword(s):  
Signal Processing ◽  
Fourier Transform ◽  
Discrete Time ◽  
Time Domain ◽  
Continuous Time ◽  
Time Varying ◽  
Discrete Frequency ◽  
The Fourier Transform ◽  
Definition Of ◽  
The Time Domain

This chapter is dedicated to exploring a form of the Fourier transform that can be applied to digital waveforms, the discrete Fourier transform (DFT). The theory is introduced and discussed as a modification to the continuous-time transform, alongside the concept of windowing in the time domain. The fast Fourier transform is explored as an efficient algorithm for the computation of the DFT. The operation of discrete-time convolution is presented as a straight application of the DFT in musical signal processing. The chapter closes with a detailed look at time-varying convolution, which extends the principles developed earlier. The conclusion expands the definition of spectrum once more.

Fast rigid-body refinement for molecular-replacement techniques

1992 ◽  
Vol 25 (2) ◽  
pp. 281-284 ◽  
Author(s):  
E. E. Castellano ◽  
G. Oliva ◽  
J. Navaza
Keyword(s):  
Fourier Transform ◽  
Rigid Body ◽  
Least Squares ◽  
Fourier Coefficients ◽  
Molecular Replacement ◽  
Present Formulation ◽  
Judicious Choice ◽  
The Fourier Transform ◽  
Derivatives Of ◽  
Patterson Search

A method for the least-squares rigid-body refinement of a general electron density model is described. The present formulation is different from a previously reported one in the computation of the derivatives of the calculated Fourier coefficients, which are derived analytically here. This, together with a judicious choice of the Fourier transform search arrays, makes the procedure extremely fast and sufficiently accurate. Although originally designed simply to optimize the values of the positional parameters obtained by Patterson search techniques, the method proved to be extremely efficient as an aid for evaluation of the correctness of potential molecular-replacement solutions.

Parameters Estimation and Waveform Reconstruction of Chirp Signal Based on FRFT

2014 ◽  
Vol 989-994 ◽  
pp. 3993-3996 ◽  
Author(s):  
Yan Jun Wu ◽  
Gang Fu ◽  
Fei Liu
Keyword(s):  
Fourier Transform ◽  
Fractional Fourier Transform ◽  
Signal Reconstruction ◽  
Parameters Estimation ◽  
The Other ◽  
Fourier Domain ◽  
Chirp Signal ◽  
The Fourier Transform ◽  
Definition Of ◽  
Noisy Background

The fractional Fourier transform (FRFT) is a generalization of the Fourier transform. The article first introduces the definition of FRFT transformation; then analyzed FRFT Chirp signal based on this humble proposed restoration Chirp signal in a noisy background in two ways: one is based on parameter estimation, and the other is based on the scores Fourier domain filtering to achieve signal reconstruction; Finally, simulation verify the feasibility of the above analysis.

Efficient 3D frequency response modeling with spectral accuracy by the rapid expansion method

Geophysics ◽  
2012 ◽  
Vol 77 (4) ◽  
pp. T117-T123 ◽  
Author(s):  
Chunlei Chu ◽  
Paul L. Stoffa
Keyword(s):  
Fourier Transform ◽  
Frequency Response ◽  
Time Domain ◽  
Wave Equations ◽  
Time Integration ◽  
Expansion Method ◽  
Rapid Expansion ◽  
The Fourier Transform ◽  
The Time Domain ◽  
Response Modeling

Frequency responses of seismic wave propagation can be obtained either by directly solving the frequency domain wave equations or by transforming the time domain wavefields using the Fourier transform. The former approach requires solving systems of linear equations, which becomes progressively difficult to tackle for larger scale models and for higher frequency components. On the contrary, the latter approach can be efficiently implemented using explicit time integration methods in conjunction with running summations as the computation progresses. Commonly used explicit time integration methods correspond to the truncated Taylor series approximations that can cause significant errors for large time steps. The rapid expansion method (REM) uses the Chebyshev expansion and offers an optimal solution to the second-order-in-time wave equations. When applying the Fourier transform to the time domain wavefield solution computed by the REM, we can derive a frequency response modeling formula that has the same form as the original time domain REM equation but with different summation coefficients. In particular, the summation coefficients for the frequency response modeling formula corresponds to the Fourier transform of those for the time domain modeling equation. As a result, we can directly compute frequency responses from the Chebyshev expansion polynomials rather than the time domain wavefield snapshots as do other time domain frequency response modeling methods. When combined with the pseudospectral method in space, this new frequency response modeling method can produce spectrally accurate results with high efficiency.

Prediction of Ground Vibration from High Speed Trains Using a Vehicle–Track–Ground Coupling Model

2016 ◽  
Vol 16 (08) ◽  
pp. 1550051 ◽  
Author(s):  
H. L. Yao ◽  
Z. Hu ◽  
Z. Lu ◽  
Y. X. Zhan ◽  
J. Liu
Keyword(s):  
Fourier Transform ◽  
High Speed ◽  
Dynamic Stiffness ◽  
Ground Vibration ◽  
Coupling Model ◽  
Vehicle Speed ◽  
High Speed Trains ◽  
Rail Irregularity ◽  
The Fourier Transform ◽  
The Time Domain

The dynamically induced ground vibration from high speed trains (HSTs) is investigated using a semi-analytical vehicle–track–ground coupling model. A multi-body vehicle is adopted along with rail irregularity considered in the model. The soil is simulated as a saturated poroelastic half-space with two elastic layers. The coupling system is solved in the transformed domain by applying the Fourier transform, and the dynamic stiffness matrix method is used to deal with the layered soil. The time-domain solutions are obtained by the inverse fast Fourier transform (FFT). The effects of the vehicle speed, observation location, rail irregularity, subgrade-bed stiffness, and vehicle type on the ground vibration are investigated thoroughly. The results show that all these factors can significantly affect the dynamically induced ground vibration.

scholarly journals Stress Waves in Composite Laminates Excited by Transverse Plane Shock Waves

1996 ◽  
Vol 3 (6) ◽  
pp. 419-433 ◽  
Author(s):  
G.R. Liu ◽  
K.Y. Lam ◽  
E.S. Chan
Keyword(s):  
Fourier Transform ◽  
Composite Laminates ◽  
Underwater Explosion ◽  
Stress Waves ◽  
Plane Shock ◽  
Wave Fields ◽  
Underwater Explosions ◽  
Reinforced Plastic ◽  
The Fourier Transform ◽  
The Time Domain

A simple 1-dimensional model is presented to investigate elastic stress waves in composite laminates excited by underwater explosion shocks. The focus is on the elastic dynamic stress fields in the composite laminate immediately after the action of the shock wave. In this model, the interaction between the laminate and the water is taken into account, and the effects of the laminate-water interaction on the stress wave fields in the laminate are investigated. In the formulation of the model, wave fields in the laminate and the water are the first obtained in the frequency domain and then transferred into the time domain using the Fourier transform techniques. A quadrature technique is used to deal with the Fourier transform integrals in which the integrands have very sharp peaks on the integral axis. Numerical examples for stress waves in a steel plate and a glass reinforced plastic sandwich laminate are presented. The technique and the results presented in this article may be used in the design of ship hull structures subjected to underwater explosions.

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