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.

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.

2D frequency-domain elastic full-waveform inversion using time-domain modeling and a multistep-length gradient approach

Geophysics ◽  
2014 ◽  
Vol 79 (2) ◽  
pp. R41-R53 ◽  
Author(s):  
Kun Xu ◽  
George A. McMechan
Keyword(s):  
Fourier Transform ◽  
Frequency Domain ◽  
Time Domain ◽  
Grid Point ◽  
Waveform Inversion ◽  
Synthetic Data ◽  
Full Waveform Inversion ◽  
Domain Modeling ◽  
Full Waveform ◽  
Gradient Approach

To decouple the parameters in elastic full-waveform inversion (FWI), we evaluated a new multistep-length gradient approach to assign individual weights separately for each parameter gradient and search for an optimal step length along the composite gradient direction. To perform wavefield extrapolations for the inversion, we used parallelized high-precision finite-element (FE) modeling in the time domain. The inversion was implemented in the frequency domain; the data were obtained at every subsurface grid point using the discrete Fourier transform at each time-domain extrapolation step. We also used frequency selection to reduce cycle skipping, time windowing to remove the artifacts associated with different source spatial patterns between the test and predicted data, and source wavelet estimation at the receivers over the full frequency spectrum by using a fast Fourier transform. In the inversion, the velocity and density reconstructions behaved differently; as a low-wavenumber tomography (for velocities) and as a high-wavenumber migration (for density). Because velocities and density were coupled to some extent, variations were usually underestimated (smoothed) for [Formula: see text] and [Formula: see text] and correspondingly overestimated (sharpened) for [Formula: see text]. The impedances [Formula: see text] and [Formula: see text] from the products of the velocity and density results compensated for the under- or overestimations of their variations, so the recovered impedances were closer to the correct ones than [Formula: see text], [Formula: see text], and [Formula: see text] were separately. Simultaneous reconstruction of [Formula: see text], [Formula: see text], and [Formula: see text] was robust on the FE and finite-difference synthetic data (without surface waves) from the elastic Marmousi-2 model; satisfactory results are obtained for [Formula: see text], [Formula: see text], [Formula: see text], and the recovered [Formula: see text] and [Formula: see text] from their products. Convergence is fast, needing only a few tens of iterations, rather than a few hundreds of iterations that are typical in most other elastic FWI algorithms.

JOINT INTERPRETATION OF AIRBORNE ELECTROMAGNETIC DATA IN THE TIME DOMAIN AND FREQUENCY DOMAIN

2018 ◽  
Vol 12 (7-8) ◽  
pp. 76-83
Author(s):  
E. V. KARSHAKOV ◽  
J. MOILANEN
Keyword(s):  
Fourier Transform ◽  
Frequency Domain ◽  
Time Domain ◽  
Frequency Interval ◽  
Single Spectrum ◽  
Simultaneous Inversion ◽  
Homogeneous Half Space ◽  
Time Domain Data ◽  
The Time Domain

Тhe advantage of combine processing of frequency domain and time domain data provided by the EQUATOR system is discussed. The heliborne complex has a towed transmitter, and, raised above it on the same cable a towed receiver. The excitation signal contains both pulsed and harmonic components. In fact, there are two independent transmitters operate in the system: one of them is a normal pulsed domain transmitter, with a half-sinusoidal pulse and a small "cut" on the falling edge, and the other one is a classical frequency domain transmitter at several specially selected frequencies. The received signal is first processed to a direct Fourier transform with high Q-factor detection at all significant frequencies. After that, in the spectral region, operations of converting the spectra of two sounding signals to a single spectrum of an ideal transmitter are performed. Than we do an inverse Fourier transform and return to the time domain. The detection of spectral components is done at a frequency band of several Hz, the receiver has the ability to perfectly suppress all sorts of extra-band noise. The detection bandwidth is several dozen times less the frequency interval between the harmonics, it turns out thatto achieve the same measurement quality of ground response without using out-of-band suppression you need several dozen times higher moment of airborne transmitting system. The data obtained from the model of a homogeneous half-space, a two-layered model, and a model of a horizontally layered medium is considered. A time-domain data makes it easier to detect a conductor in a relative insulator at greater depths. The data in the frequency domain gives more detailed information about subsurface. These conclusions are illustrated by the example of processing the survey data of the Republic of Rwanda in 2017. The simultaneous inversion of data in frequency domain and time domain can significantly improve the quality of interpretation.

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