scholarly journals Integral equation modeling of three‐dimensional magnetotelluric response

Geophysics ◽  
1981 ◽  
Vol 46 (2) ◽  
pp. 182-197 ◽  
Author(s):  
Sam C. Ting ◽  
Gerald W. Hohmann
Keyword(s):  
Integral Equation ◽  
Cross Section ◽  
Apparent Resistivity ◽  
Transfer Functions ◽  
Three Dimensional ◽  
Equation Modeling ◽  
Impedance Tensor ◽  
Surface Charges ◽  
Text Mode ◽  
Magnetic Transfer

We have adapted a three‐dimensional (3-D) volume integral equation algorithm to magnetotelluric (MT) modeling. Incorporating an integro‐difference scheme increases accuracy somewhat. Utilizing the two symmetry planes of a buried prismatic body and a normally incident plane wave source greatly reduces required computation time and storage. Convergence checks and comparisons with one‐dimensional (1-D) and two‐dimensional (2-D) models indicate that our results are valid. We show theoretical surface anomalies due to a 3-D prismatic conductive body buried in a half‐space earth. Instead of studying the electric and magnetic fields, we have obtained impedance tensor and magnetic transfer functions by imposing two different source polarizations. Manipulation of the impedance tensor and magnetic transfer functions yields the following MT quantities: apparent resistivity and phase, impedance polar diagrams, tipper direction and magnitude, principal directions, skew, and ellipticity. With our preliminary analyses of these MT quantities, we have found that three‐dimensionality is usually revealed by all of them. Furthermore, we have recognized two pairs of complementary parameters: apparent resistivity and phase, and skew and ellipticity. Because of surface charges at conductivity boundaries, low‐frequency 3-D responses are much different from 1-D and 2-D responses. Thus, in many cases 3-D models are required for interpreting MT data. Although an overall 3-D MT interpretation is still not practical due to high computer costs and the complicated structure of the true earth, combined 2-D and 3-D modeling can be applied to yield a gross 3-D structure, which is composed of a cross‐section and its strike extent. In doing so, we suggest that the cross‐section be obtained from higher frequency 2-D E‐perpendicular [Formula: see text] mode modeling, and that the strike extent be derived by matching with lower frequency E‐parallel [Formula: see text] mode results due to corresponding 3-D models. In addition, we have indicated that some simple 3-D features, e.g., location above conductive zone, corners, and symmetry lines, can easily be recognized from the surface MT response.

Groundwater exploration using combined controlled-source and radiomagnetotelluric techniques

Geophysics ◽  
2005 ◽  
Vol 70 (1) ◽  
pp. G8-G15 ◽  
Author(s):  
Laust B. Pedersen ◽  
M. Bastani ◽  
L. Dynesius
Keyword(s):  
Transfer Functions ◽  
Three Dimensional ◽  
Groundwater Exploration ◽  
Impedance Tensor ◽  
Two Dimensional ◽  
Controlled Source ◽  
Reflection Seismic ◽  
Seismic Sections ◽  
Clay Layers ◽  
Clay Lenses

Radiomagnetotelluric (RMT) (14–250 kHz) combined with controlled-source magnetotelluric (CSMT) (1–12 kHz) measurements were applied to the exploration of groundwater located in sandy formations at depths as great as 20 m below thick clay lenses. A combination of approximately 30 radio frequencies and controlled-source frequencies is essential for penetrating the thick clay layers. The electromagnetic transfer functions of impedance tensor and tipper vectors point toward a structure that is largely two-dimensional, although clear three-dimensional effects can be observed where the sandy formation is close to the surface. The determinant of the impedance tensor was chosen for inversion using two-dimensional models. The final two-dimensional model fits the data to within twice the estimated standard errors, which is considered quite satisfactory, given that typical errors are on the level of 1% on the impedance elements. Comparison with bore-hole results and shallow-reflection seismic sections show that the information delivered by the electromagnetic data largely agrees with the former and provides useful information for interpreting the latter by identifying lithological boundaries between the clay and sand and between the sand and crystalline basement.

Interpretation of 3-D effects in long‐offset transient electromagnetic (LOTEM) soundings in the Münsterland area/Germany

Geophysics ◽  
1992 ◽  
Vol 57 (9) ◽  
pp. 1127-1137 ◽  
Author(s):  
Andreas Hördt ◽  
Vladimir L. Druskin ◽  
Leonid A. Knizhnerman ◽  
Kurt‐Martin Strack
Keyword(s):  
Integral Equation ◽  
Thin Sheet ◽  
Three Dimensional ◽  
Equation Modeling ◽  
Data Sets ◽  
Sign Reversal ◽  
Near Surface ◽  
Transient Electromagnetic ◽  
Long Offset ◽  
Geological Situation

The interpretation of long‐offset transient electromagnetic (LOTEM) data is usually based on layered earth models. Effects of lateral conductivity variations are commonly explained qualitatively, because three‐dimensional (3-D) numerical modeling is not readily available for complex geology. One of the first quantitative 3-D interpretations of LOTEM data is carried out using measurements from the Münsterland basin in northern Germany. In this survey area, four data sets show effects of lateral variations including a sign reversal in the measured voltage curve at one site. This sign reversal is a clear indicator of two‐dimensional (2-D) or 3-D conductivity structure, and can be caused by current channeling in a near‐surface conductive body. Our interpretation strategy involves three different 3-D forward modeling programs. A thin‐sheet integral equation modeling routine used with inversion gives a first guess about the location and strike of the anomaly. A volume integral equation program allows models that may be considered possible geological explanations for the conductivity anomaly. A new finite‐difference algorithm permits modeling of much more complex conductivity structures for simulating a realistic geological situation. The final model has the zone of anomalous conductivity aligned below a creek system at the surface. Since the creeks flow along weak zones in this area, the interpretation seems geologically reasonable. The interpreted model also yields a good fit to the data.

scholarly journals Exploring effects in tippers at island geomagnetic observatories due to realistic depth- and time-varying oceanic electrical conductivity

2021 ◽  
Vol 73 (1) ◽  
Author(s):  
Rafael Rigaud ◽  
Mikhail Kruglyakov ◽  
Alexey Kuvshinov ◽  
Katia J. Pinheiro ◽  
Johannes Petereit ◽  
...  
Keyword(s):  
Transfer Functions ◽  
Sea Water ◽  
Three Dimensional ◽  
Oceanic Lithosphere ◽  
Induction Effect ◽  
Electromagnetic Modeling ◽  
Period Range ◽  
Model Studies ◽  
Magnetic Transfer ◽  
The One

AbstractVertical magnetic transfer functions (tippers) estimated at island observatories can constrain the one-dimensional (1-D) conductivity distribution of the oceanic lithosphere and upper mantle. This is feasible due to the bathymetry-dependent ocean induction effect (OIE), which originates from lateral conductivity contrasts between ocean and land and leads to non-zero tippers even for 1-D conductivity distributions below the ocean. Proper analysis of island tippers requires accurate three-dimensional (3-D) modeling of the OIE, for which so far was performed assuming constant sea water electric conductivity with depth. In this study, we explore—using rigorous 3-D electromagnetic modeling—to what extent realistic, depth-dependent, oceanic conductivity affects island tippers. The modeling is performed for 11 island observatories around the world in the period range $$10^{-1}$$ 10 - 1 to $$10^{4}$$ 10 4 s. We also investigate the effect of seasonal variations of the oceanic conductivity and to which extent this could explain the observed systematic seasonal variation of tippers. Our model studies suggest that for most of the considered island observatories the effect from depth-varying oceanic conductivity is tangible and exceeds the error floor of 0.025, which usually is assigned to tippers during their inversion. The effect varies significantly with location, depending on regional bathymetry. Contrarily, the effects from seasonally varying oceanic conductivity were found to be too small to be worth consideration.

scholarly journals Exploring Effects in Tippers at Island Geomagnetic Observatories Due to Realistic Depth- and Time-varying Oceanic Electrical Conductivity

2020 ◽  
Author(s):  
Rafael Rigaud ◽  
Mikhail Kruglyakov ◽  
Alexey Kuvshinov ◽  
Katia J Pinheiro ◽  
Johannes Petereit ◽  
...  
Keyword(s):  
Transfer Functions ◽  
Sea Water ◽  
Three Dimensional ◽  
Oceanic Lithosphere ◽  
Induction Effect ◽  
Electromagnetic Modeling ◽  
Period Range ◽  
Model Studies ◽  
Magnetic Transfer ◽  
The One

Abstract Vertical magnetic transfer functions (tippers) estimated at island observatories can constrain the one-dimensional (1-D) conductivity distribution of the oceanic lithosphere and upper mantle. This is feasible due to the bathymetry-dependent ocean induction effect (OIE), which originates from lateral conductivity contrasts between ocean and land and leads to non-zero tippers even for 1-D conductivity distributions below the ocean. Proper analysis of island tippers requires accurate three-dimensional (3-D) modeling of the OIE, for which so far was performed assuming constant sea water electric conductivity with depth. In this study we explore using rigorous 3-D electromagnetic modeling - to what extent realistic, depth-dependent, oceanic conductivity affects island tippers. The modeling is performed for eleven island observatories around the world in the period range 10-1 to 104 sec. We also investigate the effect of seasonal variations of the oceanic conductivity and to which extent this could explain the observed systematic seasonal variation of tippers. Our model studies suggest that for most of the considered island observatories the effect from depth-varying oceanic conductivity is tangible and exceeds the error floor of 0:025, which usually is assigned to tippers during their inversion. The effect varies significantly with location, depending on regional bathymetry. Contrarily, the effects from seasonally varying oceanic conductivity were found to be too small to be worth consideration.

Transformation of VLF anomaly maps into apparent resistivity and phase

Geophysics ◽  
2003 ◽  
Vol 68 (2) ◽  
pp. 497-505 ◽  
Author(s):  
Michael Becken ◽  
Laust B. Pedersen
Keyword(s):  
Magnetic Field ◽  
Plane Wave ◽  
Electric Field ◽  
Transfer Functions ◽  
Magnetic Data ◽  
Impedance Tensor ◽  
Wave Source ◽  
Layered Earth ◽  
Magnetic Transfer ◽  
The Magnetic Field

We investigate a transformation of magnetic transfer functions into the tangential‐electric mode part of the impedance tensor in the scope of the plane‐wave electromagnetic tensor–VLF method. The transformation, which is applicable to any 2D data representing the response of arbitrary 3D geoelectric structures, overcomes the difficulties of quantitative interpretation of magnetic transfer functions, which predominantly provide a measure of the lateral changes of the electrical conductivity in the earth. We require densely sampled magnetic transfer functions of one frequency as input data. These may be decomposed into their normal and anomalous parts (deviation from the response of a layered earth) for a unit external plane‐wave source field using the Hilbert transform relationship between the magnetic field components. Faraday's law then directly provides the anomalous toroidal electric field. Unfortunately, there is no chance to estimate the normal electric field from magnetic data, since the magnetic field is not sensitive to a layered earth. This constant must be provided as a boundary condition, e.g., from one ground measurement, to derive an impedance tensor and related apparent resistivities and phases.

scholarly journals Three‐dimensional mise‐à‐la‐masse modeling applied to mapping fracture zones

Geophysics ◽  
1986 ◽  
Vol 51 (1) ◽  
pp. 98-113 ◽  
Author(s):  
Craig W. Beasley ◽  
Stanley H. Ward
Keyword(s):  
Cross Section ◽  
Apparent Resistivity ◽  
Three Dimensional ◽  
The Body ◽  
Near Miss ◽  
Geothermal System ◽  
Volume Integral ◽  
Fracture Zones ◽  
Current Case ◽  
Similar Shape

A numerical scheme applying the method of volume integral equations has been developed for borehole‐to‐borehole and borehole‐to‐surface modeling of the apparent resistivity response of a thin conductive body in a half‐space; the inhomogeneity simulates a fracture zone in a geothermal system. The algorithm is applicable for the direct‐current case when the buried electrode is either inside (mise‐à‐la‐masse) or outside (near‐miss) the body. In implementing the scheme, the integral equation is transformed into a matrix equation as a result of discretizing the inhomogeneity into rectangular cells. All physical properties are assumed constant within each cell. The rectangular cells are used through‐out execution of the algorithm. The computed surface and subsurface apparent resistivity responses are examined for bodies of similar shape and size but with different orientations: (1) vertical, (2) horizontal, (3) dipping at 60 degrees, and (4) dipping at 30 degrees. The four bodies produce apparent resistivity cross‐section plots which differ little from each other except in orientation. Varying the depth to the top of a body does not significantly alter the subsurface apparent resistivity response in the vicinity of the body. In both section and plan views, estimates of orientation, areal extent, and dip can often be made. The maximum depth at which a body can be located and still produce a detectable surface anomaly is dependent upon the position of the buried electrode and upon the contrast in conductivity. Locating the buried electrode just outside the body does not significantly alter the results from those when the electrode is embedded in the inhomogeneity. However, the similarity between the results of these two cases decreases as the distance between the electrode and the body is increased.

SAF file: It's really three dimensional file

2018 ◽  
Vol 14 (1) ◽  
pp. 1
Author(s):  
Prof. Dr. Jamal Aziz Mehdi
Keyword(s):  
Cross Section ◽  
Root Canal ◽  
Three Dimensional ◽  
Circular Cross Section ◽  
Circular Motion ◽  
Root Canal Treatment ◽  
Metal Core ◽  
Rotating Blade

The biological objectives of root canal treatment have not changed over the recentdecades, but the methods to attain these goals have been greatly modified. Theintroduction of NiTi rotary files represents a major leap in the development ofendodontic instruments, with a wide variety of sophisticated instruments presentlyavailable (1, 2).Whatever their modification or improvement, all of these instruments have onething in common: they consist of a metal core with some type of rotating blade thatmachines the canal with a circular motion using flutes to carry the dentin chips anddebris coronally. Consequently, all rotary NiTi files will machine the root canal to acylindrical bore with a circular cross-section if the clinician applies them in a strictboring manner

ANALYSIS OF THE GENERAL EQUATIONS OF THE TRANSVERSE VIBRATION OF A PIECEWISE UNIFORM VISCOELASTIC PLATE

2020 ◽  
Vol 7 (3) ◽  
pp. 52-56
Author(s):  
MMATMATISA JALILOV ◽  
◽  
RUSTAM RAKHIMOV ◽  
Keyword(s):  
Cross Section ◽  
Transverse Vibration ◽  
Three Dimensional ◽  
Viscoelastic Plate ◽  
Constant Thickness ◽  
Regional Problem ◽  
Rigid Contact ◽  
Under Construction ◽  
Derivatives Of ◽  
Theoretical Results

This article discusses the analysis of the general equations of the transverse vibration of a piecewise homogeneous viscoelastic plate obtained in the “Oscillation of inlayer plates of constant thickness” [1]. In the present work on the basis of a mathematical method, the approached theory of fluctuation of the two-layer plates, based on plate consideration as three dimensional body, on exact statement of a three dimensional mathematical regional problem of fluctuation is stood at the external efforts causing cross-section fluctuations. The general equations of fluctuations of piecewise homogeneous viscoelastic plates of the constant thickness, described in work [1], are difficult on structure and contain derivatives of any order on coordinates x, y and time t and consequently are not suitable for the decision of applied problems and carrying out of engineering calculations. For the decision of applied problems instead of the general equations it is expedient to use confidants who include this or that final order on derivatives. The classical equations of cross-section fluctuation of a plate contain derivatives not above 4th order, and for piecewise homogeneous or two-layer plates the elementary approached equation of fluctuation is the equation of the sixth order. On the basis of the analytical decision of a problem the general and approached decisions of a problem are under construction, are deduced the equation of fluctuation of piecewise homogeneous two-layer plates taking into account rigid contact on border between layers, and also taking into account mechanical and rheological properties of a material of a plate. The received theoretical results for the decision of dynamic problems of cross-section fluctuation of piecewise homogeneous two-layer plates of a constant thickness taking into account viscous properties of their material allow to count more precisely the is intense-deformed status of plates at non-stationary external loadings.

Boundary element method in numerical study of three-dimensional Stokes flows in channels of arbitrary shape

2012 ◽  
Vol 9 (1) ◽  
pp. 94-97
Author(s):  
Yu.A. Itkulova
Keyword(s):  
Boundary Element Method ◽  
Boundary Element ◽  
Cross Section ◽  
Numerical Study ◽  
Three Dimensional ◽  
Cylindrical Tube ◽  
Variable Cross Section ◽  
Variable Cross ◽  
Periodic Flows ◽  
Element Method

In the present work creeping three-dimensional flows of a viscous liquid in a cylindrical tube and a channel of variable cross-section are studied. A qualitative triangulation of the surface of a cylindrical tube, a smoothed and experimental channel of a variable cross section is constructed. The problem is solved numerically using boundary element method in several modifications for a periodic and non-periodic flows. The obtained numerical results are compared with the analytical solution for the Poiseuille flow.

Sign in / Sign up

Export Citation Format

Share Document