Penetration of Nonstationary Ionospheric Electric Fields into Lower Atmospheric Layers in the Global Electric Circuit Model

The problem of the penetration of nonstationary ionospheric electric fields into the lower atmospheric layers is considered based on the model of the global electric circuit in the Earth’s atmosphere. For the equation of the electric field potential, a solution that takes into account exponential variation in the electrical conductivity with height has been obtained. Analysis of the solution made it possible to reveal three cases of the dependence of the solution on height. The first case (the case of high frequencies) corresponds to the Coulomb approximation, when the electrical conductivity of the atmosphere can be neglected. In the case of low frequencies (when the frequency of changes in the ionosphere potential is less than the quantity reciprocal to the time of electric relaxation of the atmosphere), a quasi-stationary regime, in which the variation in the electric potential of the atmosphere is determined by the electric conduction currents, occurs. In the third case, due to the increase in the electrical conductivity of the atmosphere, two spherical regions appear: with the Coulomb approximation in the lower region and conduction currents in the upper one. For these three cases, formulas for estimating the electric field strength near the Earth’s surface have been obtained.     

ASPECTS OF CHARGE ACCUMULATION IN d. c. RESISTIVITY EXPERIMENTS1

When an electric current is introduced to the earth, it sets up a distribution of charges both on and beneath the earth's surface. These charges give rise to the anomalous potential measured in the d. c. resistivity experiment. We investigate different aspects of charge accumulation and its fundamental role in d. c. experiments. The basic equations and boundary conditions for the d. c. problem are first presented with emphasis on the terms involving accumulated charges which occur wherever there is a non-zero component of electric field parallel to the gradient of conductivity. In the case of a polarizable medium, the polarization charges are also present due to the applied electric field, yet they do not change the final field distribution. We investigate the precise role of the permittivity of the medium. The charge buildup alters the electric fields and causes the refraction of current lines; this results in the well-known phenomenon of current channelling. We demonstrate this by using charge density to derive the refraction formula at a boundary. An integral equation for charge density is presented for whole-space models where the upper half-space is treated as an in-homogeneity with zero conductivity. The integral equation provides a tool with which the charge accumulation can be examined quantitatively and employed in the practical forward modelling. With the aid of this equation, we investigate the effect of accumulated charges on the earth's surface and show the equivalence between the half-space and whole-space formulations of the problem. Two analytic examples are presented to illustrate the charge accumulation and its role in the d. c. problem. We investigate the relationship between the solution for the potential via the image method and via the charge density. We show that the essence of the image method solution to the potential problem is to derive a set of fictitious sources which produce the same potential as does the true charge distribution. It follows that the image method is viable only when the conductivity structure is such that the effect of the accumulated charge can be represented by a set of point images. As numerical examples, we evaluate quantitatively the charge density on the earth's surface that arises because of topography and the charge density on a buried conductive prism. By these means, we demonstrate the use of the boundary element technique and charge density in d. c. forward modelling problems.     

Penetration of electric field from the near ground atmospheric layer to the ionosphere

A mathematical model has been proposed for describing quasi-stationary atmospheric electric fields with approximate, but fairly accurate allowance for ionospheric conductivity. It is shown that some well-known models of electric field penetration from the Earth into the ionosphere have been deemed inadequate, though they work well in the atmosphere below 50 km. In these models, the arbitrarily specified boundary condition in the upper boundary of the atmosphere omits the existing good conductor or adds not existent conductor. The maximum possible field in our model is far less than in models where ionospheric conductivity is not taken into account, but vastly larger than in models based on the approximation with infinite Pedersen conductivity in the upper ionosphere.     

Electromagnetic evidence for lateral inhomogeneities within the Earth's upper mantle

The composition of the upper mantle is of great significance to our understanding of plate tectonics and global evolution. Information about the physical properties of the Earth at upper mantle depths, including lateral variations in electrical conductivity, can be deduced from measurements of the electric and magnetic fields at the Earth's surface. Electromagnetic methods appear to give poorer resolution than do some other methods, for example seismics, but as they are sensitive to quite different properties of a medium they provide a different and complementary class of information.The basic theory of electromagnetic sounding methods is briefly reviewed below, and evidence regarding lateral conductivity inhomogeneities in the Earth's upper mantle is examined. While lateral electrical conductivity inhomogeneities appear to be the rule rather than the exception, the interpretation of electromagnetic data still presents difficulties and the results from many regions are not as yet unambiguous. Where the data are of sufficient resolution, a rapid increase in electrical conductivity can usually be identified within the upper mantle. The depth to this highly conductive zone is different in different tectonic environments, but is broadly consistent between analogous but widely separated tectonic environments. A comparatively shallow conducting region is found beneath the ocean lithosphere. The depth of this region is dependent on lithospheric age. Many of the more shallow conducting regions in both continental and oceanic environments are associated with high heat flow values and seismic low velocity zones. These highly conducting regions may be zones of partial melt.     

Global and Regional Electricity Components in Undisturbed Midlatitude Lower Atmosphere

The physical mechanisms determining the variability of the vertical profiles of electrical conductivity, space charge density, and electric field in the undisturbed midlatitude lower atmosphere are discussed. The influence of the global and local mesoscale processes on the variability of electrical conductivity and the main component of the atmospheric electric field is estimated. The sunrise effect is studied, estimates are obtained for the charge accumulation rate in the column of the lower atmosphere and the corresponding growth rate of the field strength close to the ground. It is shown that the increase in the average charge density is mainly due to the breakdown of the stable stratification of the atmospheric boundary layer and transformation of the vertical profile of electrical conductivity following the convective mixing of a radon and its daughter products.     

Electromagnetic fields in a steel-cased borehole

The development of an electromagnetic numerical modelling scheme for a magnetic dipole in an arbitrary casing segment in an inhomogeneous conductivity background has been difficult, due to the very high electrical conductivity and magnetic permeability contrasts between the steel casing and the background medium. To investigate the effect of steel casing efficiently, we have developed an accurate but simple finite‐element modelling scheme to simulate electromagnetic fields in a medium of cylindrically symmetric conductivity structures. In order to preserve the cylindrical symmetry in the resulting electromagnetic fields, a horizontal loop current source is used throughout. One of the main advantages of the approach is that the problem is scalar when formulated using the azimuthal electric field, even if the casing is both electrically conductive and magnetically permeable. Field calculations have been made inside the cased borehole as well as in another borehole which is not cased. Careful analyses of the numerical modelling results indicate that the anomaly observed in a cross‐borehole configuration is sensitive enough to be used for tomographic imaging.     

Cauchy integral analogues for the separation and continuation of electromagnetic fields within conducting matter

The main results in the theory of the interpretation of geopotential fields are generalized to the case of arbitrary variable electromagnetic fields by means of elaborating electrodynamic analogues for the integral of the Cauchy type.The generalized Kertz method for separating a variable electromagnetic field into parts related to the sources located in different regions of space is elaborated on the basis of this technique. The generalized Kertz method allows the selection of external and internal, normal and anomalous parts of the geomagnetic field, as well as the separation of geomagnetic anomalies into the surface and deep components caused by conductivity inhomogeneities in the Earth's crust and upper mantle.The theory of analytical continuation of variable electromagnetic fields in a conducting medium is also developed in the present work using the technique of analogues for the integral of the Cauchy type. It is shown that analytical continuation of a field downwards permits the determination of the location and form of deep geoelectric inhomogeneities according to the configuration of the isolines of flux functions for magnetic and electric fields.     

POTENTIAL FIELD OF A STATIONARY ELECTRIC CURRENT USING FREDHOLM'S INTEGRAL EQUATIONS OF THE SECOND KIND*

An integral equation method is described for solving the potential problem of a stationary electric current in a medium that is linear, isotropic and piecewise homogeneous in terms of electrical conductivity. The integral equations are Fredholm's equations of the ‘second kind’ developed for the potential of the electric field. In this method the discontinuity-surfaces of electrical conductivity are divided into ‘sub-areas’ that are so small that the value of their potential can be regarded as constant. The equations are applied to 3-D galvanic modeling. In the numerical examples the convergence is examined. The results are also compared with solutions derived with other integral equations. Examples are given of anomalies of apparent resistivity and mise-a-la-masse methods, assuming finite conductivity contrast. We show that the numerical solutions converge more rapidly than compared to solutions published earlier for the electric field. This results from the fact that the potential (as a function of the location coordinate) behaves more regularly than the electric field. The equations are applicable to all cases where conductivity contrast is finite.     

Propagation effects on the electric field time derivatives generated by return strokes in lightning flashes

The effects of propagation over finitely conducting ground on the features of radiation component of the electric field time derivatives are investigated. The results show that the peak, the half-width and the risetime of the electric field time derivative change significantly in propagating over finitely conducting ground. Furthermore, any correlation that may exist between various parameters could also change significantly due to propagation effects. Consequently, in return stroke model validations using experimentally measured fields, remote sensing of return stroke current time derivatives using measured electric field time derivatives and in the calculation of induced voltages generated by lightning flashes in electrical installations the distortions caused by propagation effects on the electric field time derivatives cannot be neglected.     

Effect of anisotropy of the earth on the impedance measurements

Summary Electromagnetic field response of a homogeneous uniaxially anisotropic conducting medium has been investigated in the presence of a vertical electric dipole placed at a heighth above the conducting surface. Expressions for the horizontal components of electric and magnetic fields and hence that of vertical impedance of the field, as a function of the distance of the place of observation from the source, have been derived. Numerical computations have been done and curves are given to illustrate the influence of anisotropy on the amplitude values of the impedance on the surface of the conducting medium.N.G.R.I. Contribution number 71–279.     

Calculation of atmospheric electric fields penetrating from the ionosphere

The spatial distributions of electric fields and currents in the Earth’s atmosphere are calculated. Electric potential distributions typical of substorms and quiet geomagnetic conditions are specified in the ionosphere. The Earth is treated as a perfect conductor. The atmosphere is considered as a spherical layer with a given height dependence of electrical conductivity. With the chosen conductivity model and an ionospheric potential of 300 kV with respect to the Earth, the electric field near the ground is vertical and reaches 110 Vm⁻¹. With the 60-kV potential difference in the polar cap of the ionosphere, the electric field disturbances with a vertical component of up to 13 V m⁻¹ can occur in the atmosphere. These disturbances are maximal near the ground. If the horizontal scales of field nonuniformity are over 100 km, the vertical component of the electric field near the ground can be calculated with the one-dimensional model. The field and current distributions in the upper atmosphere can be obtained only from the three-dimensional model. The numerical method for solving electrical conductivity problems makes it possible to take into account conductivity inhomogeneities and the ground relief.     

Direct current electric potential computation in an inhomogeneous and arbitrarily anisotropic layered earthl

We consider the calculation of the electrical field quantities, electric potential and the vertical component of the total volume density of electric current, in a horizontally layered, piecewise homogeneous and arbitrarily anisotropic earth due to a system of direct current point sources. By applying Fourier transformation with respect to the horizontal space coordinates to the static field equations, the field quantities are obtained as the solutions of the system of transform-domain differential equations in the vertical (depth) coordinates. A recurrence scheme has been given to compute the tranform-domain field quantities at any depth. The corresponding space-domain quantities are then obtained by inverse Fast Fourier Transformation (FFT). A complete computer program has been developed for computing the electric potentials at any depth of the layered earth, which is composed of an arbitrary number of anisotropic layers with arbitrary conductivity tensors. By considering the point sources at different depths from the surface, equipotential contours on the surface of arbitrarily anisotropic layered earth models are given.     

基于二次场电导率分块连续变化的三维可控源电磁有限元数值模拟

从电偶源三维地电断面可控源电磁法的二次电场边值问题及其变分问题出发,采用任意六面体单元对研究区域进行剖分,并且在单元分析中同时对电导率及二次电场进行三线性插值,实现电导率分块连续变化情况下,基于二次场的可控源电磁三维有限元数值模拟.这个新的可控源电磁三维正演方法可以模拟实际勘探中地下任意形状及电性参数连续变化的复杂模型.理论模型的计算结果表明,均匀大地计算的视电阻率误差和相位误差分别为0.002%和0.0005°.分层连续变化模型的有限元计算结果表明,其与对应的分层均匀模型解析结果有明显差异.三维异常体组合模型以及倾斜异常体等复杂模型的有限元计算结果也有效地反映了异常形态.     

ELECTROMAGNETIC FIELD OF SOURCES AT THE SURFACE OF A HOMOGENEOUS CONDUCTING HALFSPACE WITH HORIZONTAL ANISOTROPY: APPLICATION TO FISSURED MEDIA*

The electric and magnetic fields generated by horizontal electric and vertical magnetic dipoles lying on the surface of a conducting medium with horizontal anisotropy are investigated. Full expressions of their Fourier transforms are given, and the fields for a vertical magnetic dipole are calculated numerically. The radial and vertical magnetic components are found to be independent of the receiver-transmitter direction, whereas the other magnetic and electric components strongly vary with this direction. These results give useful criteria for defining the direction and amplitude of anisotropy from ground data; a ground experiment on fissured limestone was found to confirm the expected variations of the various field components. It is believed that this electromagnetic method can be used in order to provide information about the direction and amplitude of electric anisotropy.     

电性任意各向异性且分块连续变化CSAMT三维有限元数值模拟

考虑地球介质电导率任意各向异性且随空间位置连续变化的情况,本文实现了直接求解电磁场的可控源音频大地电磁测深（CSAMT）三维有限元数值模拟.首先给出了电导率任意各向异性介质中CSAMT二次电场满足的控制方程及其相应变分问题,然后采用任意六面体单元对研究区域进行剖分,在网格单元中对任意各向异性电导率进行线性插值,解决了实际工作中岩矿石电导率各向异性且连续变化的情况,将变分问题转化为线性代数方程组的求解.电导率各向异性且连续变化一维模型三维有限元数值模拟结果与电导率各向异性且分层均匀渐进模型解析解结果对比验证了方法的有效性;三维地电模型电导率随位置线性变化且各向同性、主轴各向异性、方位各向异性和倾斜各向异性的数值模拟结果表明,电导率各向异性且连续变化对CSAMT视电阻率和相位数据均有明显的影响.     

Electromagnetic analogue model measurements and finite-difference numerical calculations of the response of a conducting slab to three different source fields

This work compares experimental analogue model measurements and finite-difference numerical calculations of the electric and magnetic fields for a highly conducting slab embedded in a poorly conducting host earth for three different source field configurations. Measurements and calculations were carried out for a uniform source, a sheet current source with a y exp(?ay) current intensity distribution, and a horizontal magnetic dipole source. The results indicate reasonable agreement with some exceptions between the analogue and numerical methods. The source field is found to have an important effect on the field anomalies at the interface of the highly conducting slab and the poorly conducting host medium.     

三维非均匀地幔中深源电场和磁场的谱理论(Ⅱ)——横向缓变摄动和极型场快速扩散

It is one of the classical problems in Earth′s electromagnetism that continuation of the observatory data of magnetic field into the conducting region. This paper built the coupling vector equations governing the poloidal, toroidal and potential fields in 3-D inhomogeneous conducting mantle. Considering the limitation on variable scale of inhomogeneities in global mantle from study of the Earth′s deep interior, a perturbation　theory of gradually lateral variation was presented. It is unnecessary that the 1-D spherical symmetric distribution of electric conductivity as zero-degree approximation. Serving as a example of solvability of the zero-degree approximation, it was demonstrated how the gradually lateral variation of electric conductivity effect on anti-diffusive problem of poloidal field in Earth′s mantel.     

三维非均匀地幔中深源电场和磁场的谱理论（Ⅱ）──-横向缓变摄动和极型场快速扩散

把磁场观测数据向导电区域延拓,是地球电磁学的经典问题之一．从准静态近似和非零矢势规范的电磁场方程出发,依据矢量的球面分解唯一性定理,本文建立支配三维非均匀电寻率分布全球地慢中的环型场、极型场和电位势场的耦合方程组烤虑地球深部研究对认识全球地幔非均匀性横向变化尺度的限制,提出横向缓变意义下三维非均匀地幔中电场和磁场的摄动理论,其零级近似不要求电导率分布一维球对称．作为零级近似可解的例证,研究了地幔深源极型场的反扩散问题,在利用地面磁场观测反演到核幔边界时,可以考虑地幄横向非均匀性影响．     

Elongated horizontal and vertical receivers in three-dimensional electromagnetic modelling and inversion

Electromagnetic geophysical methods often rely on measurements of naturally occurring or artificially impressed electric fields. It is technically impossible, however, to measure the electric field directly. Instead, the electric field is approximated by recording the voltage difference between two electrodes and dividing the obtained voltage by the distance between the electrodes. Typically, modelling and inversion algorithms assume that the electric fields are obtained over infinitely short point-dipoles and thus measured fields are assigned to a single point between the electrodes. Such procedures imply several assumptions: (1) The electric field between the two electrodes is regarded as constant or being a potential field and (2) the receiver dimensions are negligible compared to the dimensions of the underlying modelling grid. While these conditions are often fulfilled for horizontal electric fields, borehole sensors for recordings of the vertical electric field have dimensions in the order of ≈100 m and span several modelling grid cells. Observations from such elongated borehole sensors can therefore only be interpreted properly if true receiver dimensions and variations of electrical conductivity along the receiver are considered. Here, we introduce a numerical solution to include the true receiver geometry into electromagnetic modelling schemes, which does not rely on such simplifying assumptions. The algorithm is flexible, independent of the chosen numerical method to solve Maxwell's equations and can easily be implemented in other electromagnetic modelling and inversion codes. We present conceptual modelling results for land-based controlled source electromagnetic scenarios and discuss consideration of true receiver geometries for a series of examples of horizontal and vertical electric field measurements. Comparison with Ez data measured in an observation borehole in a producing oil field shows the importance of both considering the true length of the receiver and also its orientation. We show that misalignment from the vertical axis as small as 0.1° may seriously distort the measured signal, as horizontal electric field components are mapped into the desired vertical component. Adequate inclusion of elongated receivers in modelling and inversion can also help reducing effects of static shift when interpreting (natural source) magnetotelluric data.     

Hydroacoustic wave in the external alternating magnetic field

The electromagnetic fields induced by hydroacoustic waves, propagating in a liquid conducting medium in an alternating magnetic field, have been considered. The equation, relating the induced magnetic field to the undisturbed antenna field and acoustic wave parameters, has been obtained. The spatial—temporal pattern of the induced field has been constructed in the case when acoustic wave propagates along a direct line with an alternating current.