Frequency‐domain elastic wave modeling by finite differences: A tool for crosshole seismic imaging

Geophysics ◽  
1990 ◽  
Vol 55 (5) ◽  
pp. 626-632 ◽  
Author(s):  
R. Gerhard Pratt
Keyword(s):  
Frequency Domain ◽  
Time Domain ◽  
Equations Of Motion ◽  
Domain Modeling ◽  
Multiple Sources ◽  
State Equations ◽  
Migration Imaging ◽  
Time Domain Modeling ◽  
Frequency Components ◽  
The Time Domain

The migration, imaging, or inversion of wide‐aperture cross‐hole data depends on the ability to model wave propagation in complex media for multiple source positions. Computational costs can be considerably reduced in frequency‐domain imaging by modeling the frequency‐domain steady‐state equations, rather than the time‐domain equations of motion. I develop a frequency‐domain approach in this note that is competitive with time‐domain modeling when solutions for multiple sources are required or when only a limited number of frequency components of the solution are required.

Time domain waveform inversion—A frequency domain view: How well we need to match waveforms?

1991 ◽  
Vol 81 (6) ◽  
pp. 2351-2370
Author(s):  
Zoltan A. Der ◽  
Robert H. Shumway ◽  
Michael R. Hirano
Keyword(s):  
Time Domain ◽  
High Frequency ◽  
Waveform Inversion ◽  
Quantitative Measure ◽  
Domain Modeling ◽  
Low Frequencies ◽  
Waveform Modeling ◽  
Time Domain Modeling ◽  
Frequency Components ◽  
The Time Domain

Abstract Waveform modeling in the time domain matches the various frequency components of seismic signals unevenly; the agreement is better at low frequencies and becomes progressively worse towards higher frequencies. The net effect of this kind of time-domain modeling is that the resolution in the spatial details of the source is less than optimal since the high-frequency components of the signal with their short wavelengths to resolve finer details do not fit the data. These problems are demonstrated by numerical simulations and by the reanalysis of some aspects of the El Golfo earthquake in using a new seismic imaging technique based on a generalization of an f-k algorithm. This procedure computes a statistic that can be used to derive confidence limits of the parameters sought in the inversion, thus providing a quantitative measure of the uncertainties in the results.

A Proposal for the Time Domain Modeling of Split Air Conditioners for Consumer Reimbursement Studies

ENERGYO ◽  
2018 ◽  
Author(s):  
Paulo Henrique Oliveira Rezende ◽  
Afonso Bernardino Almeida Junior ◽  
Isaque Nogueira Gondim ◽  
José Carlos Oliveira
Keyword(s):  
Time Domain ◽  
Domain Modeling ◽  
Air Conditioners ◽  
Time Domain Modeling ◽  
The Time Domain

Time-domain modeling of electromagnetic diffusion with a frequency-domain code

Geophysics ◽  
2008 ◽  
Vol 73 (1) ◽  
pp. F1-F8 ◽  
Author(s):  
Wim A. Mulder ◽  
Marwan Wirianto ◽  
Evert C. Slob
Keyword(s):  
Fourier Transform ◽  
Frequency Domain ◽  
Time Domain ◽  
Hermite Interpolation ◽  
Computational Grid ◽  
Skin Depth ◽  
Domain Modeling ◽  
Time Domain Modeling ◽  
Robust Multigrid ◽  
Selection Of

We modeled time-domain EM measurements of induction currents for marine and land applications with a frequency-domain code. An analysis of the computational complexity of a number of numerical methods shows that frequency-domain modeling followed by a Fourier transform is an attractive choice if a sufficiently powerful solver is available. A recently developed, robust multigrid solver meets this requirement. An interpolation criterion determined the automatic selection of frequencies. The skin depth controlled the construction of the computational grid at each frequency. Tests of the method against exact solutions for some simple problems and a realistic marine example demonstrate that a limited number of frequencies suffice to provide time-domain solutions after piecewise-cubic Hermite interpolation and a fast Fourier transform.

Large-scale Gauss-Newton inversion of transient controlled-source electromagnetic measurement data using the model reduction framework

Geophysics ◽  
2013 ◽  
Vol 78 (4) ◽  
pp. E161-E171 ◽  
Author(s):  
M. Zaslavsky ◽  
V. Druskin ◽  
A. Abubakar ◽  
T. Habashy ◽  
V. Simoncini
Keyword(s):  
Frequency Domain ◽  
Time Domain ◽  
Large Scale ◽  
Jacobian Matrix ◽  
Computational Cost ◽  
Measurement Data ◽  
Multiple Sources ◽  
Time Data ◽  
Controlled Source ◽  
The Time Domain

Transient data controlled-source electromagnetic measurements are usually interpreted via extracting few frequencies and solving the corresponding inverse frequency-domain problem. Coarse frequency sampling may result in loss of information and affect the quality of interpretation; however, refined sampling increases computational cost. Fitting data directly in the time domain has similar drawbacks, i.e., its large computational cost, in particular, when the Gauss-Newton (GN) algorithm is used for the misfit minimization. That cost is mainly comprised of the multiple solutions of the forward problem and linear algebraic operations using the Jacobian matrix for calculating the GN step. For large-scale 2.5D and 3D problems with multiple sources and receivers, the corresponding cost grows enormously for inversion algorithms using conventional finite-difference time-domain (FDTD) algorithms. A fast 3D forward solver based on the rational Krylov subspace (RKS) reduction algorithm using an optimal subspace selection was proposed earlier to partially mitigate this problem. We applied the same approach to reduce the size of the time-domain Jacobian matrix. The reduced-order model (ROM) is obtained by projecting a discretized large-scale Maxwell system onto an RKS with optimized poles. The RKS expansion replaces the time discretization for forward and inverse problems; however, for the same or better accuracy, its subspace dimension is much smaller than the number of time steps of the conventional FDTD. The crucial new development of this work is the space-time data compression of the ROM forward operator and decomposition of the ROM’s time-domain Jacobian matrix via chain rule, as a product of time- and space-dependent terms, thus effectively decoupling the discretizations in the time and parameter spaces. The developed technique can be equivalently applied to finely sampled frequency-domain data. We tested our approach using synthetic 2.5D examples of hydrocarbon reservoirs in the marine environment.

Quasi-analytical method for frequency-to-time conversion in CSEM applications

Geophysics ◽  
2012 ◽  
Vol 77 (5) ◽  
pp. E357-E363 ◽  
Author(s):  
Ali Moradi Tehrani ◽  
Evert Slob ◽  
Wim Mulder
Keyword(s):  
Frequency Domain ◽  
Time Domain ◽  
Diffusion Time ◽  
Basis Functions ◽  
Time Range ◽  
Late Time ◽  
Square Root ◽  
Domain Modeling ◽  
The Time Domain ◽  
Exponential Part

Frequency-to-time transformations are of interest to controlled-source electromagnetic methods when time-domain data are inverted for a subsurface resistivity model by numerical frequency-domain modeling at a selected, small number of frequencies whereas the data misfit is determined in the time domain. We propose an efficient, Prony-type method using frequency-domain diffusive-field basis functions for which the time-domain equivalents are known. Diffusive fields are characterized by an exponential part whose argument is proportional to the square root of frequency and a part that is polynomial in integer powers of the square root of frequency. Data at a limited number of frequencies suffice for the transformation back to the time. In the exponential part, several diffusion-time values must be chosen. Once a suitable range of diffusion-time values are found, the method is quite robust in the number of values used. The highest power in the polynomial part can be determined from the source and receiver type. When the frequency-domain data are accurately approximated by the basis functions, the time-domain result is also accurate. This method is accurate over a wider time range than other methods and has the correct late-time asymptotic behavior. The method works well for data computed for layered and 3D subsurface models.

Modeling and Transient Response of Cable Network

2012 ◽  
Vol 433-440 ◽  
pp. 2868-2873
Author(s):  
Xing Zhou ◽  
Zhen Yu Xiang ◽  
Er Wei Cheng ◽  
Li Si Fan
Keyword(s):  
Transmission Line ◽  
Time Domain ◽  
Modeling Method ◽  
Domain Model ◽  
Domain Modeling ◽  
Cable Network ◽  
Cable Networks ◽  
Time Domain Modeling ◽  
The Time Domain ◽  
Line Network

This paper introduces modeling method for calculating nodes responses of transmission-line network directly in the time domain. Arising from classical telegrapher equations, the time-domain model of transmission line is gained. The transmission-line model, together with transmission-line nodes model, form the network model. Using the time-domain modeling method, transient responses for two given networks are gained. The injection experiments to cable networks are done to validate the calculating results. The consistency of calculating results with measure results indicates the model is feasible.

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