Integral equation methods at very low frequency

2012 ◽  
Author(s):  
Felipe Vico-Bondia ◽  
Miguel Ferrando-Bataller ◽  
Alejandro Valero-Nogueira ◽  
Daniel Sanchez-Escuderos
Keyword(s):  
Integral Equation ◽  
Low Frequency ◽  
Very Low Frequency ◽  
Integral Equation Methods

A goal-oriented framework for rapid integral-equation-based simulation of borehole resistivity measurements of 3D hydraulic fractures

Geophysics ◽  
2017 ◽  
Vol 82 (2) ◽  
pp. D123-D133 ◽  
Author(s):  
Kai Yang ◽  
Ali E. Yılmaz ◽  
Carlos Torres-Verdín
Keyword(s):  
Integral Equation ◽  
Method Of Moments ◽  
Low Frequency ◽  
Hydraulic Fractures ◽  
Approximate Methods ◽  
Electrically Conductive ◽  
Resistivity Measurements ◽  
Integral Equation Methods ◽  
Approximate Integral ◽  
Quantities Of Interest

We have developed a goal-oriented framework for fast integral-equation-based simulation of low-frequency borehole resistivity measurements of 3D arbitrarily shaped hydraulic fractures. The framework explores the possibility of applying various approximate integral-equation methods to simulate borehole electromagnetic (EM) measurements acquired in the vicinity of 3D hydraulic fractures generated with electrically conductive proppant. It includes four approximate methods that are progressively more accurate, costly, and rigorous. Each method is used to approximate the method-of-moments solution of the integral equation and to evaluate/extract quantities of interest, e.g., bucked signals detected at receivers. When compared with rigorous fast Fourier transform (FFT)-accelerated method-of-moments solutions, the numerical results obtained with the four methods indicate the following (1) All of the approximate methods capture the main features of the quantities of interest, e.g., the shape of detected signals. (2) Different approximate methods exhibit different accuracies and efficiencies in the simulation of EM scattering from various 3D fractures. (3) The identified approximate method achieves accurate results (error [Formula: see text]) while reducing the simulation time by a factor of 2–1000 compared with the FFT-accelerated rigorous method. Thus, our approximate simulation framework is a promising candidate for evaluating the Jacobian matrix in the fast inversion of borehole EM measurements to detect and assess the geometry of 3D hydraulic fractures generated with electrically conductive proppant.

Very low‐frequency electromagnetic response of a perfectly conducting half‐plane in a layered half‐space

Geophysics ◽  
1982 ◽  
Vol 47 (7) ◽  
pp. 1059-1067 ◽  
Author(s):  
M. Poddar
Keyword(s):  
Integral Equation ◽  
Half Space ◽  
Half Plane ◽  
Tilt Angle ◽  
Low Frequency ◽  
Method Of Successive Approximations ◽  
Very Low Frequency ◽  
Depth Of Burial ◽  
Layered Half Space ◽  
Conductivity Contrast

Following Dmitriev (1961), a rigorous theoretical solution for the problem of scattering by a perfectly conducting inclined half‐plane in a layered half‐space in a plane‐wave field has been obtained. The solution is in the form of a Fredholm integral equation of the second kind, where the unknown is the Laplace transform of scattering current in the half‐plane. The integral equation is solved numerically by the method of successive approximations. The scattered fields at the surface of the half‐space are found by integrating the half‐space Green’s function over the transform of the scattering current. The effects of depth of burial and inclination of the half‐plane, conductivity contrast between the overburden and the substratum, and thickness of the overburden are studied in some detail. As expected, the tangent of the tilt angle and the ellipticity of the ellipse of magnetic polarization decrease rapidly with increasing depth of burial, conductivity contrast, and thickness. Inclination introduces asymmetry in the anomaly profile besides affecting its magnitude. Depth of exploration is greater for the ellipticity than for the tilt angle. A target depth equal to half of the skin depth in the substratum is the limiting depth of detection in the very low‐frequency, electromagnetic (VLF-EM) method. An interpretation scheme for VLF-EM field data is presented, based on peak‐to‐peak separation and difference between peaks of the two polarization parameters.

The Attenuation of Very Low Frequency Brain Oscillations in Transitions from a Rest State to Active Attention

2009 ◽  
Vol 23 (4) ◽  
pp. 191-198 ◽  
Author(s):  
Suzannah K. Helps ◽  
Samantha J. Broyd ◽  
Christopher J. James ◽  
Anke Karl ◽  
Edmund J. S. Sonuga-Barke
Keyword(s):  
Brain Activity ◽  
Sampling Rate ◽  
Low Frequency ◽  
Future Research ◽  
Adhd Symptoms ◽  
Default Mode ◽  
Very Low Frequency ◽  
Wide Range ◽  
Interference Hypothesis ◽  
Mode Interference

Background: The default mode interference hypothesis ( Sonuga-Barke & Castellanos, 2007 ) predicts (1) the attenuation of very low frequency oscillations (VLFO; e.g., .05 Hz) in brain activity within the default mode network during the transition from rest to task, and (2) that failures to attenuate in this way will lead to an increased likelihood of periodic attention lapses that are synchronized to the VLFO pattern. Here, we tested these predictions using DC-EEG recordings within and outside of a previously identified network of electrode locations hypothesized to reflect DMN activity (i.e., S3 network; Helps et al., 2008 ). Method: 24 young adults (mean age 22.3 years; 8 male), sampled to include a wide range of ADHD symptoms, took part in a study of rest to task transitions. Two conditions were compared: 5 min of rest (eyes open) and a 10-min simple 2-choice RT task with a relatively high sampling rate (ISI 1 s). DC-EEG was recorded during both conditions, and the low-frequency spectrum was decomposed and measures of the power within specific bands extracted. Results: Shift from rest to task led to an attenuation of VLFO activity within the S3 network which was inversely associated with ADHD symptoms. RT during task also showed a VLFO signature. During task there was a small but significant degree of synchronization between EEG and RT in the VLFO band. Attenuators showed a lower degree of synchrony than nonattenuators. Discussion: The results provide some initial EEG-based support for the default mode interference hypothesis and suggest that failure to attenuate VLFO in the S3 network is associated with higher synchrony between low-frequency brain activity and RT fluctuations during a simple RT task. Although significant, the effects were small and future research should employ tasks with a higher sampling rate to increase the possibility of extracting robust and stable signals.

VLF/LF (Very Low Frequency/Low Frequency) Reflection Properties of the Low Latitude Ionosphere

1988 ◽  
Author(s):  
Wayne I. Klemetti ◽  
Paul A. Kossey ◽  
John E. Rasmussen ◽  
Maria Sueli Da Silveira Macedo Moura
Keyword(s):  
Low Frequency ◽  
Low Latitude ◽  
Very Low Frequency ◽  
Low Latitude Ionosphere
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