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.     

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.     

Transient electromagnetic field above a permeable and conducting half-space

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.     

Three-dimensional induction in a non-uniform thin sheet at the surface of a uniformly conducting earth

Summary. A new method for solving problems in three-dimensional electromagnetic induction in which the Earth is represented by a uniformly conducting half-space overlain by a surface layer of variable conductance is presented. Unlike previous treatments of this type of problem the method does not require the fields to be separated into their normal and anomalous parts, nor is it necessary to assume that the anomalous region is surrounded by a uniform structure; the model may approach either an E- or a B -polarization configuration at infinity. The solution is expressed as a vector integral equation in the horizontal electric field at the surface. The kernel of the integral is a Green's tensor which is expressed in terms of elementary functions that are independent of the conductance. The method is applied to an illustrative model representing an island near a bent coastline which extends to infinity in perpendicular directions.     

Interpretation of long-offset transient electromagnetic data from the Odenwald area, Germany, using two-dimensional modelling

The standard 1-D inversion approach for the interpretation of transient electromagnetic (TEM) data usually fails in the presence of near-surface conductivity anomalies. Since multidimensional inversion codes are not routinely available, the only alternative to discarding the data may be trial-and-error forward modelling. We interpret data from a long-offset transient electromagnetic (LOTEM) survey which was carried out in 1995 in the Odenwald area, using 2-D finite-difference modelling. We focus on a subsegment of the LOTEM profile, which was shot with two different electric dipole transmitters. A model is found which consistently explains the electric and magnetic field data at eight locations for both transmitters. First, we introduce a conductive dyke under the receiver spread to explain sign reversals in the magnetic field transients. A conductive slab under one of the transmitters is required to obtain a reasonable quantitative fit for that transmitter. Consideration of the electric field data then requires a modification of the layered earth background. Finally, we study the response of a crustal conductor, which was the original target of the survey. The data are sensitive to the conductor, and for the investigated subset of the data the fits are slightly better without the conductive layer.     

Electromagnetic induction fields in the deep ocean north-east of Hawaii: implications for mantle conductivity and source fields

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.     

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.     

Electric field at the seafloor due to a two-dimensional ionospheric current

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.     

A comparison of analytic and numerical results for a two-dimensional control model in electromagnetic induction – I. B-polarization calculations

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.     

Electromagnetic fields in a halfspace scattered by a buried sphere by the method of transformation of local elements

Summary. A problem in modelling electromagnetic fields used in exploration geophysics is treated mathematically. Analytical expressions are obtained for the electric field due to a harmonic current in a horizontal loop on or above a conducting ground in which is buried a conductive and permeable sphere (ore body). The loop is coaxial with the sphere. For a general time-varying current in the loop, the analysis is carried to the stage where a Fourier inversion can be used to obtain readily the electric field in the time-domain. A new relationship between spherical and cylindrical wave functions is obtained as a transformation of local elements.
Solution of this problem has not been presented before in this form. Lee's solution of 1975 which uses an integral-equation formulation treats a similar problem without taking account of differences in magnetic permeability. The effects of magnetic permeability may have important and useful implications for geophysical explorations.     

Two-dimensional induction in a thin sheet of variable integrated conductivity at the surface of a uniformly conducting earth

Summary. The forward solution of the general two-dimensional problem of induction in a model earth comprising a uniformly conducting half-space covered by a thin sheet of variable integrated conductivity is obtained. Unlike some previous treatments of similar problems, the method presented here does not require the field to be separated into its normal and anomalous parts. Both the E - and B -polarization modes of induction are considered and in each case the solution is expressed in terms of the horizontal component of the electric field satisfying, on the surface of the conductor, a singular integral equation whose kernel is a well-known analytic function. A recently published solution of the coast effect is included as a special case. The numerical procedure for solving the integral equations is described and some illustrative calculations are presented.     

The complex-image method for calculating the magnetic and electric fields produced at the surface of the Earth by the auroral electrojet

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.     

Palaeosecular variation in Central Mexico over the last 30 000 years: the record from lavas

13 lava flows of known age (ages from ¹⁴C dating), which have been erupted in the last 30 000 years, have been studied to determine the palaeosecular variation of the geomagnetic field in Central Mexico. Samples were taken from two different monogenetic volcanic fields: the Michoacan-Guanajuato volcanic field (six sites) and the Chichinautzin Formation (seven sites), both part of the Transmexican Volcanic Belt. The lavas were studied in detail using rock magnetic methods (magnetic susceptibility at room temperature, low-temperature susceptibility behaviour, hysteresis loops, Curie temperatures), combined with reflected light microscopy, in order to deduce their magnetic mineralogy and the domain states of the magnetic minerals. The magnetic carriers are titanomagnetites, which show differing degrees of high-temperature deuteric oxidation, and seem to be predominantly pseudo-single domain (PSD), though in many cases are probably a mixture of domain states. Mean palaeomagnetic directions and palaeointensity values using Shaw and Thellier techniques were obtained using several specimens from each flow. Our data seem to indicate a sharp easterly swing in declination about 5000 years ago, which is also observed in lake sediments from Central Mexico. The calculated values of the virtual dipole moment (VDM) range from 3.1 to 14.9 × 10²² A m². Our data indicate that the virtual dipole moment seems to have increased gradually in magnitude over the last 30 kyr, with a peak at about 9000 years BP. These are features that have been observed in other parts of the globe and are probably caused by variations in the dipole part of the geomagnetic field.     

Laboratory studies of depositional DRM

Summary. Acquisition of magnetic remanence in slurries of fme grained organic muds settling in long tubes is investigated using a cryogenic magnetometer. The average settling behaviour of remanence carrying grains relative to the whole sediment gives information about the relative magnetic grain size spectrum, whereas the response of settled deposits to vibration gives an indication of the degree of alignment of particles and their average shape.
Two classes of behaviour are apparent in both the time and field dependence of detrital remanent magnetization (DRM) acquisition. Dilute slurries settling in the Earth's field (analogous to detrital sedimentation) acquire a remanence which reaches a maximum after about 2 day, whereas for concentrated slurries (analogous to slumped or bioturbated sediments) this takes only a matter of minutes. The field dependence of DRM in dilute slurries is in plausible quantitative agreement with Stacey's extension of the classical Langevin expression for the susceptibility of a paramagnetic gas, whereas concentrated slurries show a quasi-linear dependence of DRM on the applied field. Inclination errors are generally absent, but do appear in fields less than about 0.5 Oe, and when the magnetic fraction settles out preferentially. A weak negative dependence of DRM on temperature is found, but the results are too crude to provide a further test of Stacey's theory.
Remanence acquisition in slurries settled in zero field indicates that short term post-depositional magnetization processes are relatively unimportant in slurries that have ceased to compact rapidly. A large increase in remanence from naturally occurring sediments to slowly redeposited long cores, to rapidly settled slurries is best explained as a feature of compaction.     

Geomagnetic field analysis - I. Stochastic inversion

Summary. Most of the Earth's magnetic field and its secular change originate in the core. Provided the mantle can be treated as an electrical insulator, stochastic inversion enables surface observations to be analysed for the core field. A priori information about the variation of the field at the core boundary leads to very stringent conditions at the Earth's surface. The field models are identical with those derived from the method of harmonic splines (Shure, Parker & Backus) provided the a priori information is specified appropriately.
The method is applied to secular variation data from 106 magnetic observatories. Model predictions for fields at the Earth's surface have error estimates associated with them that appear realistic. For plausible choices of a priori information the error of the field at the core is unbounded, but integrals over patches of the core surface can have finite errors. The hypothesis that magnetic fields are frozen to the core fluid implies that certain integrals of the secular variation vanish. This idea is tested by computing the integrals and their standard and maximum errors. Most of the integrals are within one standard deviation of zero, but those over the large patches to the north and south of the magnetic equator are many times their standard error, because of the dominating influence of the decaying dipole. All integrals are well within their maximum error, indicating that it will be possible to construct core fields, consistent with frozen flux, that satisfy the observations.     

A general correspondence principle for time-domain electromagnetic wave and diffusion fields

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.     

Head waves in laterally heterogeneous media

Summary. A layer of constant thickness over a half-space is assumed, and the propagation of head waves is considered for the following two cases: (1) the P -wave velocity varies in the layer in the horizontal direction, and is constant in the half-space: (2) the P -wave velocity varies in the half-space in the horizontal direction, and is constant in the layer. In each case the horizontal velocity gradient is assumed to remain constant. The wave propagation is investigated in the direction of the gradient (direct profile), and opposite to it (reverse profile). Formulae for the travel times and the amplitudes are obtained on the basis of ray-theoretical considerations. Conditions are discussed for the discrimination in a field experiment between the case of a sloping boundary separating the homogeneous media, and the case of an intrinsic horizontal velocity gradient.     

The change in Sq(H) amplitude on Abnormal Quiet Days

Summary. Analysis of geomagnetic data has shown that the superposed northward magnetic field, which reduces the S _q( H ) amplitude at northern mid-latitude stations on Abnormal Quiet Days, and increases the amplitude at stations on the equatorward side of the S _q focus, builds up in amplitude over four to five days before the AQD occurs, and subsides over a similar period after the AQD. It is inferred indirectly that the azimuthal component B_y of the interplanetary magnetic field varies similarly. Data for the opposite meridian show that the imposed field reverses to a southward direction at lower latitudes. The inferred currents to account for these fields are believed to flow in the ionosphere, but to arise from magnetospheric electric fields induced by the solar wind-transported IMF.     

Seismic scattering in the upper crystalline crust based on evidence from sonic logs

Telluric distortion occurs when electric charges accumulate along near-surface inhomogeneities. At low frequencies, the electric currents associated with these charges can be neglected compared to currents induced deeper in the Earth. At higher frequencies, the magnetic fields associated with these currents may be significant. Some parameters describing the distortion magnetic fields can be estimated from measured magneto-telluric impedance matrices. For regional magnetic fields aligned with regional strike directions, parameters associated with the distortion magnetic field component parallel to the regional magnetic field are undeterminable, whereas parameters associated with the distortion magnetic field component perpendicular to the regional magnetic field can be estimated. Optimal estimates are straightforward even for the realistic case of measurement errors that are correlated between elements of a measured impedance matrix. In a simple example of a 1-D anisotropic model with anisotropy direction varying with depth, the modelling of distortion magnetic fields results in regional impedance estimates corresponding more closely to the responses of uncoupled isotropic models, allowing sensible interpretation of an additional one and a half decades of data.     

The use of equivalent current systems in the interpretation of Geomagnetic Deep Sounding data

Summary. Many geomagnetic variation anomalies are probably caused by the channelling, through small-scale bodies, of electric currents induced in much larger conductors elsewhere. Consequently, the direct interpretation of anomalous magnetic fields by modelling the electromagnetic response of conductive structures may give misleading results. It is suggested that, rather than attempting to proceed directly from the electromagnetic fields to conductivity models, we should instead take the intermediate step of determining the distribution of anomalous current flow.
Maps of the anomalous fields over a conductive structure can be generated from inter-station transfer functions. If it is assumed that the internal currents are concentrated in a thin sheet at a specified depth, the equivalent current system in the sheet can be computed directly from the vertical magnetic field. The most straightforward method of performing this calculation is to compute the Fast Fourier Transform of the magnetic field data, and then to apply a wavenumber filter.
The presence of any vertical currents invalidates the thin sheet model. However, if the spatial distribution of a horizontal component of the anomalous magnetic field is also known, the presence of any vertical currents can be detected directly, and their position determined. The value of the methods is illustrated by applying them to the interpretation of a Geomagnetic Deep Sounding survey of the Kenya rift valley.