2-D or 3-D interpretation of conductivity anomalies: example of the Rhine-Graben conductivity anomaly

Summary. Evidence of a conductivity anomaly in the Rhine-Graben was first given about 15 years ago and consequently led to the definition of various models of induction in the region for periods ranging from a few minutes to a few hours. These models reflect two antagonistic ways of explaining the observed anomalous variations of the magnetic field: direct induction in a two-dimensional (2-D) structure or static distortion of telluric currents by the resistive crystalline Vosges (France) and Schwarzwalde (Germany) massifs. We discuss the two approaches using a simple formalism. In particular, we show that the self-induction related to the anomalous currents flowing in the Rhine-Graben is negligible for periods larger than 1000 s, and that, even though the static distortion of telluric currents does account for the observed anomaly, 2-D models can explain some of its features. We also show how the channelled currents are induced in the large sedimentary basins surrounding the area under study.
An experimental verification of this result is given.     

H-polarization induction over an ocean edge coupled to the mantle by a conducting crust

Summary. The Wiener—Hopf technique is used to obtain an exact analytical solution for the problem of H -polarization induction over the edge of a perfectly conducting thin sheet, representing an ocean, electrically connected to a perfectly conducting mantle through a slab of finite conductivity and thickness, which represents the Earths crust. It is shown that the induced currents resulting from this type of induction process are drawn up into the sea from the cust and mantle with the greatest concentration of current near the ocean edge. The surface impedance over the land surface, is calculated for various mantle depths and is shown to increase sharply as the coastline is approached. The magnetic field along the ocean floor is also plotted as a function of distance from the coastline, and the results are found to agree very well with those calculated previously by approximate and numerical methods.     

Modelling of solar quiet magnetic field variations near a conductivity anomaly

Summary. A simplified model of the solar quiet-time ionospheric current system is used to calculate the induced currents in a model earth. The conductivity is assumed to be constant below a depth of about 400 km and zero above that depth. The current induced in the north—south conductivity anomaly under the Rocky Mountains is then estimated from the time-varying potential difference between points at 30 and 45° latitude at the surface of the conducting sphere. The purpose of these calculations is to investigate whether variations in the latitude of the northern hemisphere current system vortex will substantially alter the relationship between the observed magnetic field components at the Earth's surface and the local magnetic field gradient caused by the conductivity anomaly. We find that a 10° shift in the latitude of the ionospheric current focus causes a change of 6 per cent or less in the transfer function from the field components to the gradient in the total field. Thus such latitude shifts cannot explain much of the magnetic field gradient variation at periods near 24 hr that has been observed near Boulder, Colorado.     

Electromagnetic response of an ocean-coast model to E-polarization induction

summary . An ocean-coast model which consists of a uniformly conducting half-space screened by a perfectly conducting half-plane (the model ocean) is studied. On the land the electric field decreases continuously to zero as the coast is approached. The horizontal magnetic field component is found to vary rapidly, but remains finite; the vertical component on the other hand, increases to infinity at the coast. On the surface of the model ocean as well as on the sea floor, electric field and vertical magnetic field are both nil, but the horizontal magnetic field becomes singular as the seashore is approached. This horizontal magnetic field however, is different on the sea floor and at the ocean surface, because the integrated ocean current is finite, even growing to infinity as the shore is approached. The very large ocean currents near the shore act as an extremely long line antenna, which radiates far afield. This antenna feature explains the very long range of the ocean-coast effects observed under E -polarization induction, compared to the corresponding H -polarization effects where no such antenna-like feature occurs. A similarly large difference of ranges can be expected for all shallow structures with large lateral conductivity contrasts. The present study may therefore be of some interest in relation to geomagnetic depth soundings by the inductive and magnetotelluric methods, as well as in understanding the ocean-coast effect known for some time from records of coastal observatories.     

Uplift by thermal expansion of the lithosphere

Summary. If one can measure the anomalous horizontal magnetic field associated with a (locally bounded) two-dimensional conductivity anomaly, the transfer function which results from correlating the vertical with the anomalous horizontal magnetic field directly indicates the depth to an equivalent line-current. A. numerical model can be used to illustrate this. If three-dimensional effects (including current channelling) produce the current anomaly, interpretation in terms of conductive structure would be less clear. It has been claimed (Babour & Mosnier etc.) that such three-dimensional effects are experimentally observed in the highly coherent transfer functions determined from differential geomagnetic sounding experiments. These effects are, specifically, the 'linear polarization' of the anomalous fields, and the invariance of the phase of the measured anomalous field across the anomaly. It is suggested in this present paper that both these effects can be explained in terms of simple local induction models.
If the embedded two-dimensional anomaly is sufficiently close to the Earth's surface, the transfer function (between the vertical and the total horizontal field) contains more information than is usually interpreted. With this in mind, the magnetovariational data collected by Rooney & Hutton in the Kenyan Rift is re-examined.     

Electromagnetic induction studies in the Eyre Peninsula, South Australia

Magnetic field fluctuations have been recorded by an array of portable three-component magnetometers at 60 sites across the Eyre Peninsula in South Australia between December 1993 and March 1995. An additional 54 magnetometer data records, collected prior to 1989 and described by Milligan (1989) and Milligan, White & Chamalaun (1989), were included in the analysis. A major conductive feature in the crust, first noted by White & Milligan (1984) as the Eyre Peninsula Anomaly (EPA), is re-examined to assess its continuity to the north of the original arrays and to investigate its relationship with major tectonic features.
Magnetic-field time-series were converted to induction arrows in the frequency domain. These induction arrows were initially inverted using the minimum-structure 2-D Occam approach to estimate the electrical conductance of the crust. Following this, thin-sheet forward modelling was used to examine the relationship between the conductance and the dominant tectonic features. The principal results of the modelling are that a narrow conductive feature extends inland from the coast about 160 km before terminating, and the conductance is in the range 3000 to 10 000 S, which decreases inland.
A strong correlation exists between the electrical conductance of the Eyre Peninsula and Bouguer gravity anomalies, and in particular the EPA is coincident with a significant Bouguer gravity gradient. There is also good agreement between the locations of the foci of earthquakes of magnitude greater than 4.0 and the EPA. We believe that the anomaly is associated with a geological fracture in the Precambrian upper crust as a result of crustal extension prior to the rifting of Australia from Antarctica in the Jurassic (160 Ma).     

Geomagnetic induction in a heterogeneous sphere: fully three-dimensional test computations and the response of a realistic distribution of oceans and continents

Long-period geomagnetic data can resolve large-scale 3-D mantle electrical conductivity heterogeneities which are indicators of physiochemical variations found in the Earth's dynamic mantle. A prerequisite for mapping such heterogeneity is the ability to model accurately electromagnetic induction in a heterogeneous sphere. A previously developed finite element method solution to the geomagnetic induction problem is validated against an analytic solution for a fully 3-D geometry: an off-axis spherical inclusion embedded in a uniform sphere. Geomagnetic induction is then modelled in a uniform spherical mantle overlain by a realistic distribution of oceanic and continental conductances. Our results indicate that the contrast in electrical conductivity between oceans and continents is not primarily responsible for the observed geographic variability of long-period geomagnetic data. In the absence of persistent high-wavenumber magnetospheric disturbances, this argues strongly for the existence of large-scale, high-contrast electrical conductivity heterogeneities in the mid-mantle. Lastly, for several periods the geomagnetic anomaly associated with a mid-mantle spherical inclusion is calculated. A high-contrast inclusion can be readily detected beneath the outer shell of oceans and continents. A comparison between observed and computed c responses suggests that the mid-mantle contains more than one order of magnitude of lateral variability in electrical conductivity, while the upper mantle contains at least two orders of magnitude of lateral variability in electrical conductivity.     

E-polarization induction in two thin half-sheets

Summary. An analytical solution is obtained for the E-polarization problem of electromagnetic induction in two adjacent half-sheets underlain by a uniform conducting half-space. In this mode the inducing magnetic field is assumed horizontal, uniform and perpendicular to the discontinuity. The same model was previously solved under B-polarization by Dawson & Weaver. The present solution then completes the study of two-dimensional induction in the described model. Further, it extends both the analytic E-polarization solution of Weidelt by the inclusion of an underlying conductor and that of Raval, Weaver & Dawson by the inclusion of arbitrary conductance values for the two surface sheets. The solution may be used as an idealized model of the coast effect and allows detailed study of the field behaviour near the discontinuity. The horizontal magnetic field on each side of the surface layer has a finite jump discontinuity at the interface and the vertical magnetic field exhibits a logarithmic singularity there. If the right-hand conductance (say) becomes infinite, the horizontal magnetic field exhibits an algebraic singularity as the coastline is approached from the right, while the vertical magnetic field does likewise from the left. Calculations are presented for the same two models as discussed in B-polarization by Dawson & Weaver and the results are compared to values obtained from a more general numerical scheme. The electric current distribution inside the conducting half-space is depicted for the second model.     

Electromagnetic induction in the Earth in electrical contact with the oceans

Summary. Integral equations are formulated to solve the problem of electromagnetic induction in the Earth in the presence of thin finitely conducting oceans. Unlike earlier models, the oceans are assumed to be in electrical contact with the mantle and leakage of oceanic electric current is therefore possible. A powerful iterative method is developed to solve the very large system of equations and is applied successfully to a model problem. A feature of the integral equations is that they lead to a good first approximation. When fed into the iterative scheme this enhances the rate of convergence. Variable mesh size also appears possible.     

Telluric measurements and differential geomagnetic soundings in the Rhinegraben (period range: 1–125 s)

Summary. In 1976, seven stations measuring the variations of the telluric and geomagnetic fields in the period range 1–125 s were operated in the southern part of the Rhinegraben. The study of the recordings shows that the telluric field is linearly polarized according to a direction perpendicular to that of the horizontal anomalous magnetic fields and that telluric and anomalous magnetic fields have the same time dependence. The conducted currents responsible for the anomaly flow probably into the superficial conductive layer.     

Sudden Local Conductivity Changes in the Ionosphere

Summary. An attempt is made to describe the transient magnetic change caused by a sudden local change of conductivity in the ionosphere. In a rough illustrative model, electric current is taken to be flowing uniformly in a thin plane rigid sheet of uniform isotropic integrated conductivity, I/ϱO e.m.u. A particular type of perturbing current system is defined, and the type of conductivity anomaly which could produce it is derived. It is shown that the problem is equivalent to that of the free decay of the perturbing current system. For a very small change of conductivity, the current system is found to spread radially at a uniform rate, while decaying. The case of a large conductivity anomaly is analysed by a numerical method. It is concluded that effects at a distance would be similar to those produced by a very small change, and that near the anomaly there would be as well, a local decaying current vortex system. It is found in particular, that at a large distance L cm, the magnetic change would commence immediately and reach an extremum after time of order L × 10⁻⁶ s.     

Forward modelling of direct current and low-frequency electromagnetic fields using integral equations

We present a semi-analytical, unifying approach for modelling the electromagnetic response of 3-D bodies excited by low-frequency electric and magnetic sources. We write the electric and magnetic fields in terms of power series of angular frequency, and show that to obey Maxwell's equations, the fields must be real when the exponent is even, and imaginary when it is odd. This leads to the result that the scattering equations for direct current fields and for fields proportional to frequency can both be explicitly formulated using a single, real dyadic Green's function. Although the underground current flow in each case is due to different physical phenomena, the interaction of the scattering currents is of the same type in both cases. This implies that direct current resistivity, magnetometric resistivity and electric and magnetic measurements at low induction numbers can all be modelled in parallel using basically the same algorithm. We make a systematic derivation of the quantities required and show that for these cases they can all be expressed analytically. The problem is finally formulated as the solution of a system of linear equations. The matrix of the system is real and does not depend on the type of source or receiver. We present modelling results for different arrays and apply the algorithm to the interpretation of field data. We assume the standard dipoledipole resistivity array for the direct current case, and vertical and horizontal magnetic dipoles for induction measurements. In the case of magnetometric resistivity we introduce a moving array composed of an electric dipole and a directional magnetometer. The array has multiple separations for depth discrimination and can operate in two modes. The mode where the predominant current flow runs along the profile is called MMR-TM. This mode is more sensitive to lateral variations in resistivity than its counterpart, MMR-TE, where the mode of conduction is predominantly perpendicular to the profile.     

B-polarization induction in two generalized thin sheets at the surface of a conducting half-space

Summary. A new closed-form solution is obtained analytically for a B- polarization induction problem of geophysical interest, in which a local region of the Earth is represented by a generalized thin sheet at the surface of and in electrical contact with a uniformly conducting half-space. The generalized sheet, first introduced by Ranganayaki & Madden, is a mathematical idealization of a double layer which consists, in this problem, of two adjacent half-planes with distinct conductances representing a surface conductivity discontinuity such as an ocean—coast boundary, underlain by a uniform sheet of finite integrated resistivity representing the lower crust. The resistive sheet exerts a considerable mathematical influence on the solution causing, under certain conditions, an additional pole to appear in one of the forms of contour integral by which the solution can be expressed; it also weakens or eliminates field singularities that would otherwise occur at the conductance discontinuity. A numerical calculation is made for model parameters typifying an ocean—coast boundary underlain by a highly resistive crust. It is found that the residue of the pole associated with the resistive sheet dominates the solution for this example, the main consequence of which is a huge increase in the horizontal range over which the induced currents adjust themselves between the different 'skin-effect' distributions at infinity on either side of the model. Moreover the solution shows that this 'adjustment distance' has a more complicated dependence on the conductance and integrated resistivity of the sheet than that given simply by the square root of their product which was the length parameter proposed by Ranganayaki & Madden.     

Magnetometer array studies and deep Schlumberger soundings in the Damara orogenic belt, South West Africa

Summary. A zone of concentrated induced electric currents crossing parts of Zimbabwe, Botswana and South West Africa was discovered during a magnetovariational study conducted in 1972. In 1977, a second study was made with 27 recording magnetometers distributed across the width of South West Africa between latitudes 19 and 22°S. Several geomagnetic disturbances were recorded with high recording efficiencies. Three of these time sequences were digitized for analysis. Magnetograms and Fourier transform amplitude and phase maps in the period range 22–128min were used to delineate the westward continuation of the conductive structure revealed by the earlier investigation. The conductive zone runs approximately east-west from the Botswana border (21°E) to 17°E longitude. From here to the Atlantic coast it trends in a NE—SW direction. Anomalous fields, normalized to the horizontal field at a station recording the normal field, were used to obtain maximum depth estimates of around 45 km for the induced currents. Several deep Schlumberger soundings were done over the anomalous zone and the results showed that the conductive structure is, in places, only 3 km from the surface and that it has a resistivity of less than 20 Ωm. The resistivity of the upper crust outside the structure ranges from 5000 to more than 20000 Ωm. Some 14 post-Karoo alkaline igneous complexes occur along the course of the resistivity anomaly. These intrusive complexes represent the youngest igneous activity in the Damara Orogenic Belt and were most probably emplaced along a line of weakness in the lithosphere. The resistivity anomaly would seem to delineate this line of weakness.     

Significance of the sign changing of the imaginary arrows in geomagnetic induction investigation

Summary. In this investigation, we carry out a two-dimensional study of the dependence of the imaginary Parkinson arrows on the frequency of the inducing geomagnetic field. Our results demonstrate that the imaginary arrows reverse direction as the inducing period varies. Therefore, we consider that there is no way to fix a consistent sign convention for the imaginary arrows even when the time factor is taken into account. We find that in the twodimensional case there exists a characteristic period T _c at which the phase difference between the vertical and horizontal magnetic components is zero. It is anticipated that T _c is related to the parameters of the conductivity anomaly and the status of the half-space host.     

Induced magnetic field of the equatorial electrojet at the surface of the Earth

Summary. The northward component of the induced magnetic field due to the equatorial electrojet at the Earth's surface is calculated using a more realistic local time variaton of the external field due to the electrojet than is provided for by models of the electrojet currently used in induction calculations. It is seen that appreciable induction effects can be expected about an hour before local noon for the kind of local time variation considered. Our results are in qualitative agreement with direct observations of Earth currents in the equatorial region in Nigeria. At local times when observable induction effects are present, the magnetic field due to the electrojet is necessarily three-dimensional; hence in order to obtain the internal part directly from the observed total field due to the electrojet at the Earth's surface, a three-dimensional formulation is required.     

A study of the physics of telluric current flow at very low frequencies in the Earth's crust

Summary. Interpretation of the effects of natural electromagnetic induction is often in terms of models in which changes in resistivity at great distances from the point of measurement are not taken into account. In this paper we will try to show that this is not a valid approximation for very low-frequency telluric currents, flowing near the surface in the Earth's crust, when the material separating the crust from the mantle has a sufficiently high electrical resistance. In this case the shape and dimensions of the circuit, as well as the resistivity distribution along the whole length of the path followed by the currents, play a part in determining the induced electromagnetic field. A number of experimentally verifiable consequences follow from this.     

On the location of the ionospheric current systems responsible for the lunar and solar magnetic variations

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From a study of the sunspot cycle influence on the lunar ( L ) and solar ( S ) daily geomagnetic variations, Malin, Cecere & Palumbo have suggested that the electric currents responsible for these variations do not flow in the E -region as has hitherto been accepted. Indeed they suggested that the L current may even flow in the F 2 region. The consequences of such a suggestion are considered and it is shown that an L current in the F -region is highly unlikely from the dynamical point of view. The evidence for E -region currents is presented and it is suggested that the variation of E -region electron density used by Malin et al . to indicate conductivity changes in that region may not be a reliable indicator of such changes.     

基于地形动力学的海滩裂流安全性评价 ——以三亚大东海为例

文章基于海滩地形动力学模型建立了海滩安全性评价方法,认为低潮沙坝/裂流海滩类和沙坝类海滩出现裂流概率最大,沙坝消散类和有裂流的低潮台地类海滩裂流风险中等,没有裂流的低潮台地海滩、没有沙坝的消散海滩、超消散型、完全反射型海滩裂流风险很小。以三亚大东海为例,利用该评价方法分析了海滩溺水事故频发的原因,结果显示：1）该海滩状态以沙坝型和低潮沙坝/裂流海滩型为主,属于高风险海滩,此结论与当地救生实践吻合;2）该海滩溺水事故发生的根本原因为裂流危险性高,故容易发生溺水事故。