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1.
Summary. Laplace and Bessel Transforms are used to solve for the transient behaviour of the electromagnetic fields after switching off a steady current in a grounded infinitesimal horizontal dipole on the surface of a uniformly conducting half-space. Simple analytic expressions, which are valid for times sufficiently long after the switch that displacement terms can be ignored, are obtained on the surface of the half-space for the electric field and the time derivative of the magnetic field. At the instant of switching an infinitesimally long image becomes established directly under the source dipole. It is the diffusion of this image which gives the vertical magnetic field and horizontal electric fields their transient behaviour. During the transient, there is also a decaying charge distribution on the surface.  相似文献   

2.
Summary. The transient fields resulting from an abrupt current switch-off in a vertically oriented finite loop and a magnetic dipole above a permeable and conducting half-space have been investigated by Fourier methods, utilizing an appropriate Green's function. Two partial fields are identified, one 'radiative' and evanescent and one 'diffusive'. Asymptotic formulae for the diffusive field above the interface, applicable for late times, are developed, and from these the effects of height of the transmitting source and the permeability contrast are calculated. It is shown that for 'late enough' times and at sufficient distances the dipole formulae provide adequate approximations to the field due to a finite loop.  相似文献   

3.
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.  相似文献   

4.
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.  相似文献   

5.
Summary. We have analysed a thirty-six day recording of the natural electric and magnetic field variations obtained on the deep ocean floor north-east of Hawaii. The electromagnetic fields are dominated by tides which have an appreciable oceanic component, especially in the east electric and north magnetic components. The techniques of data analysis included singular value decomposition (SVD) to remove uncorrelated noise. There are three degrees of freedom in the data set for periods longer than five hours, indicating a correlation of the vertical magnetic field and the horizontal components, suggesting source field inhomogeneity. Tensor response functions were calculated using spectral band averaging with both SVD and least squares techniques and rotated to the principal direction. One diagonal component, determined mainly by the north electric and east magnetic fields, is not interpretable as a one-dimensional induction phenomenon. The other diagonal term of the response function indicates a rapid rise in conductivity to 0.05 mho m−1 near 160 km. No decrease in conductivity below this depth is resolvable. Polarization analysis of the magnetic field indicates moving source fields with a wavelength near 5000 km. Model studies suggest that the two dimensionality in the response function may be caused by motion in the ionospheric current system.  相似文献   

6.
The relation between the seafloor electric field and the surface magnetic field is studied. It is assumed that the fields are created by a 2-D ionospheric current distribution resulting in the E-polarization. The layered earth below the sea water is characterized by a surface impedance. The electric field at the seafloor can be expressed either as an inverse Fourier transform integral over the wavenumber or as a spatial convolution integral. In both integrals the surface magnetic field is multiplied by a function that depends on the depth and conductivity of the sea water and on the properties of the basement. The fact that surface magnetic data are usually available on land, not at the sea surface, is also considered. Test computations demonstrate that the numerical inaccuracies involved in the convolution method are negligible. The theoretical equations are applied to calculate the seafloor electric fields due to an ionospheric line current or associated with real magnetic data collected by the IMAGE magnetometer array in northern Europe. Two different sea depths are considered: 100 m (the continental shelf) and 5 km (the deep ocean). It is seen that the dependence of the electric field on the oscillation period is weaker in the 5 km case than for 100 m.  相似文献   

7.
Summary. Various numerical procedures have been developed to study the electric currents which are produced by induction in the ocean. In nearly all these methods difficulties arise near the coastline where the electric field changes significantly over a small distance. In this paper a new one-sided correction for the electric field near the edge of the ocean is obtained. The results contain a multiplicative constant which may be determined by a suitable matching technique at an oceanic point. The anomalous magnetic field due to this edge correction is also given.  相似文献   

8.
A conservative staggered-grid finite difference method is presented for computing the electromagnetic induction response of an arbitrary heterogeneous conducting sphere by external current excitation. This method is appropriate as the forward solution for the problem of determining the electrical conductivity of the Earth's deep interior. This solution in spherical geometry is derived from that originally presented by Mackie et al. (1994 ) for Cartesian geometry. The difference equations that we solve are second order in the magnetic field H , and are derived from the integral form of Maxwell's equations on a staggered grid in spherical coordinates. The resulting matrix system of equations is sparse, symmetric, real everywhere except along the diagonal and ill-conditioned. The system is solved using the minimum residual conjugate gradient method with preconditioning by incomplete Cholesky decomposition of the diagonal sub-blocks of the coefficient matrix. In order to ensure there is zero H divergence in the solution, corrections are made to the H field every few iterations. In order to validate the code, we compare our results against an integral equation solution for an azimuthally symmetric, buried thin spherical shell model ( Kuvshinov & Pankratov 1994 ), and against a quasi-analytic solution for an azimuthally asymmetric configuration of eccentrically nested spheres ( Martinec 1998 ).  相似文献   

9.
Summary. Currents in the 1.28 × 103 km Alaska oil pipeline, induced from the ionospheric, auroral electrojet, were measured at three sites, near Fairbanks, Paxson and Valdez, Alaska, using a gradient configuration of two fluxgate magnetometers. The observed pipeline current magnitudes, which reached 50 A during times of mild geomagnetic activity, displayed a linear relationship with the electric earth potential. Using the induction relationship between the electric and magnetic fields and the typical spectral composition of the geomagnetic field at high latitudes, I obtained a spectral appearance of the current that shows a maximum in the range of 4.5- to 10-min period. Near Fairbanks the pipeline current amplitudes, I (Amperes), could be represented, approximately, by I = 0.65 B x T −0.5, where B x(nT) is the north—south geomagnetic field variation amplitude and T (min) is its apparent period. There is much less pipelines current at the sites south of Fairbanks. A previously established relationship between the local electric field and the planetary geomagnetic activity index, Ap , permitted a prediction of the pipeline current surge amplitudes in the Fairbanks region as approximately I = 5.0 Ap . Current surges larger than 500 A may be expected rather often in the Alaska pipeline during large geomagnetic storms.  相似文献   

10.
Summary. The electrical system of currents excited by a uniform electric field of arbitrary direction in an infinite plane sheet of uniform conductivity except for two non-overlapping circular areas is obtained analytically. Using the method introduced by Ashour, the magnetic field of the system is also obtained. The components of this additional field are expressed as line integrals which are suitable for computation. The results reduce, in the special case of one insertion, to those obtained earlier by Ashour & Chapman.
As an illustration, numerical results are obtained for the special case of two equal insertions of zero conductivity.
The analysis and results obtained are useful in estimating the modification of the currents flowing in an ocean and their magnetic field by two islands.  相似文献   

11.
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.  相似文献   

12.
Summary. A conducting slab of finite thickness divided into three segments of different conductivities and overlying a perfect conductor is proposed as a suitable two-dimensional 'control' model for testing the accuracy of the various numerical modelling programs that are available for calculating the fields induced in the Earth by an external, time-varying magnetic source. An analytic solution is obtained for this control model for the case of the magnetic field everywhere parallel to the conductivity boundaries ( B -polarization). Values of the field given by this solution for a particular set of model parameters are calculated at selected points on the surface and on a horizontal plane inside the conductor, and are tabulated to three figure accuracy for reference. They are used to check the accuracy of the results given by the finite difference program of Brewitt-Taylor & Weaver and the finite element program of Kisak & Silvester for the same model. Improved formulae for calculating the derived electric field components in B -polarization are first developed for incorporation in the finite difference program, and these give surface electric fields within 1 per cent of the analytic values, while all three field components inside the conductor are calculated to better than 96 per cent accuracy by the finite difference program. The results given by the finite element program are not quite so satisfactory. Errors somewhat greater than 10 per cent are present and although the program requires much less disk space it takes rather more CPU time to complete the calculations.  相似文献   

13.
Summary. Attention is drawn to a direct analytic method of calculating the quadrupole parameters of a planetary main magnetic field. Following a brief survey of the general theory of magnetic multipoles, an explicit algorithm is derived for calculating the quadrupole moment and the directions of the two quadrupole axes, given the five spherical harmonic coefficients of the second degree. It is shown that the direct analytic method of calculating the quadrupole parameters yield results for the geomagnetic quadrupole that are in exact agreement with those obtained by the more usual iterative procedure. Moreover, the direct analytic method has the distinct advantage that it does not require approximate values of the quadrupole parameters to be specified in advance. The direct analytic method of solution is used to calculate the quadrupole parameters for three different models of the main field of the Earth at epoch 1975; these computations provide some indication of the current uncertainties in the geomagnetic quadrupole parameters. It is also pointed out that a pseudo-quadrupole moment, which has been used to compare the quadrupole strengths of different planetary magnetic fields, is not strictly consistent with Maxwell's classical definition of a quadrupole moment. A precise physical definition of this pseudo-quadrupole moment is propounded.  相似文献   

14.
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.  相似文献   

15.
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.  相似文献   

16.
Summary. The paper reports studies of the three-dimensional magnetospheric—ionospheric current systems which produced polar magnetic substorms on 1974 September 7 and September 18. The data were magnetic perturbation fields observed with a two-dimensional array of 23 three-component magnetometers located in western Canada beneath the auroral oval. In an earlier study of a substorm of September 11 (Bannister & Gough) the fields fitted calculated field for a Boström Type 1 current loop with field-aligned currents at east and west ends of the ionospheric segment, and with uniform current density across the width. The substorms here reported could not be modelled with uniform current density. An inverse method due to Oldenburg was therefore used to estimate current density distributions, and satisfactory fits of calculated to observed field resulted. Each substorm was modelled at six representative epochs. In general the principal ionospheric current seem by the array was westward. At four epochs of the September 7 substorm and throughout the September 18 substorm, significant eastward ionospheric current (or its equivalent in terms of the fields produced) was observed north of the westward electrojet. Northwestward bends in the ionospheric current segments were found at four epochs on September 7 and at three epochs on September 18. As in the September 11 substorm (Paper 1), these bends were either west of or close to magnetic midnight. In some cases the bends may follow the auroral oval, but in others they are sharper and may be associated with the Harang discontinuity. East of geomagnetic the ionospheric currents tend to run in a constant geomagnetic midnight latitude range. The developments of the three substorms, of September 7, 11 (Paper 1) and 18, are compared. They showed a variety of shifts in longitude, though all moved eastward relative to magnetic midnight.  相似文献   

17.
For studying the auroral electrojet and for examining the effects it can produce in power systems on the ground, it is useful to be able to calculate the magnetic and electric fields that the electrojet produces at the surface of the Earth. Including the effects of currents induced in the Earth leads to a set of integral expressions, the numerical computation of which is complicated and demanding of computer resources. An approximate solution can be achieved by representing the induced currents by an image current at a complex depth. We present a simple derivation of the complex-image expressions and use them to calculate the fields produced by the auroral electrojet at the surface of an earth represented by layered conductivity models. Comparison of these results with ones obtained using the exact integral solution show that the errors introduced are insignificant compared to the uncertainties in the parameters used. The complex-image method thus provides a simple, fast and accurate means of calculating the magnetic and electric fields.  相似文献   

18.
A general correspondence principle is presented that relates any time-domain electromagnetic diffusion field to an electromagnetic wavefield in a 'corresponding' configuration. The principle applies to arbitrarily inhomogeneous and anisotropic media and arbitrary transmitters and receivers. For the correspondence between the two types of electromagnetic fields to hold, the electric conductivity in the diffusive case and the permittivity in the wavefield case should have the same spatial variation, while the permeability distributions in space in the two cases are to be identical. Essential steps in the derivation of the correspondence principle are the use of the time Laplace transformation of causal signals, taken at real, positive values of the transform parameter, the Schouten-Van der Pol theorem in the theory of the Laplace transformation, and the reliance upon Lerch's theorem of the uniqueness of the interrelation between causal field quantities and their time-Laplace-transform representations at real, positive values of the transform parameter. Correspondence is then established between the tensorial Green's functions in the two cases, where the Green's functions are the point-receiver responses (either electric or magnetic field) to point-transmitter excitations (either electric- or magnetic-current source).
Through the correspondence principle, all transient electromagnetic wavefields (where losses are neglected) have as a counterpart a transient diffusive electromagnetic field (where the electric displacement current is neglected). The interrelation yields the tool to compare quantitatively the potentialities of the two types of fields in transient electromagnetic geophysical prospecting.
Finally, a general medium-parameter scaling law for time-domain electromagnetic wavefields is presented.  相似文献   

19.
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.  相似文献   

20.
Many geophysical inverse problems derive from governing partial differential equations with unknown coefficients. Alternatively, inverse problems often arise from integral equations associated with a Green's function solution to a governing differential equation. In their discrete form such equations reduce to systems of polynomial equations, known as algebraic equations. Using techniques from computational algebra one can address questions of the existence of solutions to such equations as well as the uniqueness of the solutions. The techniques are enumerative and exhaustive, requiring a finite number of computer operations. For example, calculating a bound to the total number of solutions reduces to computing the dimension of a linear vector space. The solution set itself may be constructed through the solution of an eigenvalue problem. The techniques are applied to a set of synthetic magnetotelluric values generated by conductivity variations within a layer. We find that the estimation of the conductivity and the electric field in the subsurface, based upon single-frequency magnetotelluric field values, is equivalent to a linear inverse problem. The techniques are also illustrated by an application to a magnetotelluric data set gathered at Battle Mountain, Nevada. Surface observations of the electric ( E y ) and magnetic ( H x ) fields are used to construct a model of subsurface electrical structure. Using techniques for algebraic equations it is shown that solutions exist, and that the set of solutions is finite. The total number of solutions is bounded above at 134 217 728. A numerical solution of the algebraic equations generates a conductivity structure in accordance with the current geological model for the area.  相似文献   

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