The behavior of the apparent resistivity phase spectrum in the case of a polarizable prism in an unpolarizable half‐space

Geophysics ◽  
1984 ◽  
Vol 49 (9) ◽  
pp. 1534-1540 ◽  
Author(s):  
Heikki Soininen
Keyword(s):  
Frequency Dependence ◽  
Half Space ◽  
Time Constant ◽  
Apparent Resistivity ◽  
Dispersion Model ◽  
Three Dimensional ◽  
Phase Spectrum ◽  
True Time ◽  
Polarizable Body ◽  
Lower Frequencies

In the application of the broadband induced polarization method, it is necessary to know how a petrophysical resistivity spectrum is transformed into an apparent spectrum measured in the field. Investigated in the present work was the forming of an apparent spectrum in the case of a polarizable three‐dimensional prism embedded in an unpolarizable half‐space for gradient and dipole‐dipole arrays. The computations were done numerically using the integral equation technique. The frequency dependence of the resistivity of the prism was depicted by means of the Cole‐Cole dispersion model. With this simple model geometry, the phase spectra of apparent resistivity resemble quite closely in functional form the original petrophysical phase spectrum of the Cole‐Cole dispersion model. The apparent spectra have shifted on the log‐log scale downward, owing to geometric attenuation, and toward lower frequencies. The apparent Cole‐Cole parameters have been inverted from the apparent spectra. The apparent chargeability is generally noticeably smaller, owing to the geometric attenuation, than the chargeability of the original petrophysical spectrum. The apparent frequency dependence, on the other hand, is very close to the value of the original frequency dependence. The shift of the apparent phase spectrum toward lower frequencies partly compensates for the decrease in the apparent time constant caused by attenuation of the spectrum. The apparent time constant is thus close to the true time constant of the petrophysical spectrum. It is therefore possible in principle to obtain by direct inversion from an apparent spectrum measured in the field a reasonable estimate of the frequency dependence and time constant of the true spectrum of a polarizable body.

The behavior of the apparent resistivity phase spectrum in the case of two polarizable media

Geophysics ◽  
1985 ◽  
Vol 50 (5) ◽  
pp. 810-819 ◽  
Author(s):  
Heikki Soininen
Keyword(s):  
Frequency Dependence ◽  
Half Space ◽  
Time Constant ◽  
Apparent Resistivity ◽  
Dispersion Model ◽  
Phase Spectrum ◽  
The Body ◽  
Spectral Parameters ◽  
The Difference ◽  
True Time

I employed numerical modeling to examine the formation of the apparent resistivity phase spectrum first of a polarizable prism situated in a polarizable half‐space, and second of two polarizable prisms joined in an unpolarizable half‐space. The calculations were done using the integral equation technique. The frequency dependence of the resistivity of the polarizable medium is depicted by means of the Cole‐Cole dispersion model. The effect of a weakly polarizable half‐space may be handled by simply adding the phase angle of the half‐space to the apparent phase due to the body. The apparent spectral parameters can be inverted by fitting the sum of two Cole‐Cole dispersion model phase spectra to the apparent phase spectrum. Of the parameters describing the prism, the apparent chargeability is smaller than the chargeability of the original petrophysical spectrum because of geometric attenuation. The apparent frequency dependence, on the other hand, is very close to the value of the original frequency dependence. The apparent time constant is commonly also near the true time constant of the petrophysical spectrum. The values of the apparent spectral parameters of the polarizable half‐space are all close to their petrophysical or true values. The apparent spectrum of two polarizable prisms builds up in a complex fashion. Nevertheless, by measuring the spectra at a number of points along a profile crossing over two formations differing in time constant, the various components can be discriminated from the apparent spectrum even if the difference in time constant is small. As the conductivity contrast decreases, the share of the spectrum of the formation in the apparent spectrum increases. Similarly, the formation with the smaller time constant is in a more advantageous position than the body with the greater time constant.

On: “Mineral discrimination and removal of inductive coupling with multifrequency IP” by W. H. Pelton, S. H. Ward, P. G. Hallof, W. R. Sill, and P. H. Nelson (GEOPHYSICS, 43, 588–609).

Geophysics ◽  
1984 ◽  
Vol 49 (9) ◽  
pp. 1556-1557
Author(s):  
Heikki Soininen
Keyword(s):  
Apparent Resistivity ◽  
Dispersion Model ◽  
Constant Factor ◽  
Constant Independent ◽  
Real Constant ◽  
Frequency Range ◽  
Dilution Factor ◽  
Amplitude Spectra ◽  
Polarizable Body ◽  
Phase Spectra

The authors discussed the behavior of the resistivity spectra by means of the Cole‐Cole dispersion model. They also discussed the corrections with which the petrophysical resistivity spectrum can be reduced into an apparent resistivity spectrum caused by a polarizable body embedded in an unpolarizable environment. The application of the Cole‐Cole dispersion model is a marked step forward in spectral IP analysis. However, closer attention must be paid to the assumptions and approaches on which the authors base the relations between the petrophysical and apparent spectra. The authors based their relations between the true and apparent spectra on the use of the dilution factor [Formula: see text]. In accordance with the definition by Seigel (1959), they assumed that [Formula: see text] is a real constant (independent of frequency) over the whole frequency range under consideration. First consider the justification for the assumption of the existence of a constant factor [Formula: see text] in the light of an example calculated for phase spectra. Similar considerations could also be made with the aid of amplitude spectra.

Integral equation solution for the transient electromagnetic response of a three‐dimensional body in a conductive half‐space

Geophysics ◽  
1985 ◽  
Vol 50 (5) ◽  
pp. 798-809 ◽  
Author(s):  
William A. SanFilipo ◽  
Gerald W. Hohmann
Keyword(s):  
Integral Equation ◽  
Frequency Domain ◽  
Half Space ◽  
Time Constant ◽  
Free Space ◽  
Numerical Scheme ◽  
Three Dimensional ◽  
Basis Functions ◽  
The Body ◽  
Spatial Discretization

The time‐domain integral equation for the three‐dimensional vector electric field is formulated as a convolution of the scattering current with the tensor Green’s function. The convolution integral is divided into a sum of integrals over successive time steps, so that a numerical scheme can be formulated with a time stepping approximation of the convolution of past values of the solution with the system impulse response. This, together with spatial discretization, leads to a matrix equation in which previous solution vectors are multiplied by a series of matrices and fed back into the system by adding to the primary field source vector. The spatial discretization, based on a modification of the usual pulse basis formulation in the frequency domain, includes an additional subset of divergence‐free basis functions generated by integrating the Green’s function around concentric closed rectangular paths. The inductive response of the body is more accurately modeled with these additional basis functions, and a meaningful solution can be obtained for a body in free space. The resulting algorithm produces good results even for large conductivity contrasts. Internal checks, including convergence with respect to spatial and temporal discretization, and reciprocity, demonstrate self‐consistency of the numerical scheme. Independent checks include (a) comparison with results computed for a prism in free space, (b) comparison with results computed for a thin plate, (c) comparison of our conductive half‐space algorithm with an asymptotic solution for a sphere, and (d) comparison with results from inverse Fourier transformation of values computed using a frequency‐domain integral equation algorithm. Qualitative features of the results show that the relative importance of current channeling and confined eddy currents induced in the body depends upon both conductivity contrast and geometry. If the free‐space time constant is less than the time window during which currents in the host have not yet propagated well beyond the body, current channeling dominates the response. In such cases, simple superposition of free‐space results and the background is a poor approximation. In cases where the host currents diffuse beyond the body in a time less than the free‐space time constant of the body, the total response is approximately the sum of the free‐space and background (half‐space) responses.

scholarly journals Simulation and measurement of air quality in the traffic congestion area

2021 ◽  
Vol 31 (1) ◽  
Author(s):  
Shin Yu ◽  
Chang Tang Chang ◽  
Chih Ming Ma
Keyword(s):  
Air Quality ◽  
Traffic Congestion ◽  
Traffic Management ◽  
Reference Model ◽  
Dispersion Model ◽  
Fluid Model ◽  
Three Dimensional ◽  
Vehicle Speed ◽  
Management Plans ◽  
Dimensional Distribution

AbstractThe traffic congestion in the Hsuehshan tunnel and at the Toucheng interchange has led to traffic-related air pollution with increasing concern. To ensure the authenticity of our simulation, the concentration of the last 150 m in Hsuehshan tunnel was simulated using the computational fluid dynamics fluid model. The air quality at the Toucheng interchange along a 2 km length highway was simulated using the California Line Source Dispersion Model. The differences in air quality between rush hours and normal traffic conditions were also investigated. An unmanned aerial vehicle (UAV) with installed PM2.5 sensors was developed to obtain the three-dimensional distribution of pollutants. On different roads, during the weekend, the concentrations of pollutants such as SOx, CO, NO, and PM2.5 were observed to be in the range of 0.003–0.008, 7.5–15, 1.5–2.5 ppm, and 40–80 μg m− 3, respectively. On weekdays, the vehicle speed and the natural wind were 60 km h− 1 and 2.0 m s− 1, respectively. On weekdays, the SOx, CO, NO, and PM2.5 concentrations were found to be in the range of 0.002–0.003, 3–9, 0.7–1.8 ppm, and 35–50 μg m− 3, respectively. The UAV was used to verify that the PM2.5 concentrations of vertical changes at heights of 9.0, 7.0, 5.0, and 3.0 m were 45–48, 30–35, 25–30, and 50–52 μg m− 3, respectively. In addition, the predicted PM2.5 concentrations were 40–45, 25–30, 45–48, and 45–50 μg m− 3 on weekdays. These results provide a reference model for environmental impact assessments of long tunnels and traffic jam-prone areas. These models and data are useful for transportation planners in the context of creating traffic management plans.

Guided Surface Waves on an Elastic Half Space

1971 ◽  
Vol 38 (4) ◽  
pp. 899-905 ◽  
Author(s):  
L. B. Freund
Keyword(s):  
Wave Propagation ◽  
Surface Wave ◽  
Surface Waves ◽  
Half Space ◽  
Body Wave ◽  
Three Dimensional ◽  
Elastic Half Space ◽  
Normal Displacement ◽  
Rigid Barrier ◽  
Wave Disturbances

Three-dimensional wave propagation in an elastic half space is considered. The half space is traction free on half its boundary, while the remaining part of the boundary is free of shear traction and is constrained against normal displacement by a smooth, rigid barrier. A time-harmonic surface wave, traveling on the traction free part of the surface, is obliquely incident on the edge of the barrier. The amplitude and the phase of the resulting reflected surface wave are determined by means of Laplace transform methods and the Wiener-Hopf technique. Wave propagation in an elastic half space in contact with two rigid, smooth barriers is then considered. The barriers are arranged so that a strip on the surface of uniform width is traction free, which forms a wave guide for surface waves. Results of the surface wave reflection problem are then used to geometrically construct dispersion relations for the propagation of unattenuated guided surface waves in the guiding structure. The rate of decay of body wave disturbances, localized near the edges of the guide, is discussed.

Airborne resistivity and susceptibility mapping in magnetically polarizable areas

Geophysics ◽  
2000 ◽  
Vol 65 (2) ◽  
pp. 502-511 ◽  
Author(s):  
Haoping Huang ◽  
Douglas C. Fraser
Keyword(s):  
Half Space ◽  
Magnetic Permeability ◽  
Apparent Resistivity ◽  
Dynamic Range ◽  
Single Frequency ◽  
Measured Signal ◽  
Apparent Permeability ◽  
Quadrature Component ◽  
New Methods ◽  
Quadrature Response

The apparent resistivity technique using half‐space models has been employed in helicopter‐borne resistivity mapping for twenty years. These resistivity algorithms yield the apparent resistivity from the measured in‐phase and quadrature response arising from the flow of electrical conduction currents for a given frequency. However, these algorithms, which assume free‐space magnetic permeability, do not yield a reliable value for the apparent resistivity in highly magnetic areas. This is because magnetic polarization also occurs, which modifies the electromagnetic (EM) response, causing the computed resistivity to be erroneously high. Conversely, the susceptibility of a magnetic half‐space can be computed from the measured EM response, assuming an absence of conduction currents. However, the presence of conduction currents will cause the computed susceptibility to be erroneously low. New methods for computing the apparent resistivity and apparent magnetic permeability have been developed for the magnetic conductive half‐space. The in‐phase and quadrature responses at the lowest frequency are first used to estimate the apparent magnetic permeability. The lowest frequency should be used to calculate the permeability because this minimizes the contribution to the measured signal from conduction currents. Knowing the apparent magnetic permeability then allows the apparent resistivity to be computed for all frequencies. The resistivity can be computed using different methods. Because the EM response of magnetic permeability is much greater for the in‐phase component than for the quadrature component, it may be better in highly magnetic environments to derive the resistivity using the quadrature component at two frequencies (the quad‐quad algorithm) rather than using the in‐phase and quadrature response at a single frequency (the in‐phase‐quad algorithm). However, the in‐phase‐quad algorithm has the advantage of dynamic range, and it gives credible resistivity results when the apparent permeability has been obtained correctly.

Indentation of a Transversely Isotropic Thermoporoelastic Half-Space by a Rigid Circular Cylindrical Punch

2017 ◽  
Vol 84 (11) ◽  
Author(s):  
Yilan Huang ◽  
Guozhan Xia ◽  
Weiqiu Chen ◽  
Xiangyu Li
Keyword(s):  
Half Space ◽  
Original Problem ◽  
Three Dimensional ◽  
Transversely Isotropic ◽  
The Finite Element Method ◽  
Thermal Fields ◽  
Cylindrical Punch ◽  
The One ◽  
Physical Quantities ◽  
Potential Theory Method

Exact solutions to the three-dimensional (3D) contact problem of a rigid flat-ended circular cylindrical indenter punching onto a transversely isotropic thermoporoelastic half-space are presented. The couplings among the elastic, hydrostatic, and thermal fields are considered, and two different sets of boundary conditions are formulated for two different cases. We use a concise general solution to represent all the field variables in terms of potential functions and transform the original problem to the one that is mathematically expressed by integral (or integro-differential) equations. The potential theory method is extended and applied to exactly solve these integral equations. As a consequence, all the physical quantities of the coupling fields are derived analytically. To validate the analytical solutions, we also simulate the contact behavior by using the finite element method (FEM). An excellent agreement between the analytical predictions and the numerical simulations is obtained. Further attention is also paid to the discussion on the obtained results. The present solutions can be used as a theoretical reference when practically applying microscale image formation techniques such as thermal scanning probe microscopy (SPM) and electrochemical strain microscopy (ESM).

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