The instantaneous apparent resistivity tensor: a visualization scheme for LOTEM electric field measurements

Long-offset transient electromagnetic (LOTEM) data have traditionally been represented as early- and late-time apparent resistivities. Time-varying electric field data recorded in a LOTEM survey made with multiple sources can be represented by an 'instantaneous apparent resistivity tensor'. Three independent, coordinate-invariant, time-varying apparent resistivities can be derived from this tensor. For dipolar sources, the invariants are also independent of source orientation. In a uniform-resistivity half-space, the invariant given by the square root of the tensor determinant remains almost constant with time, deviating from the half-space resistivity by a maximum of 6 per cent. For a layered half-space, a distance–time pseudo-section of the determinant apparent resistivity produces an image of the layering beneath the measurement profile. As time increases, the instantaneous apparent resistivity tensor approaches the direct current apparent resistivity tensor. An approximate time-to-depth conversion can be achieved by integrating the diffusion depth formula with time, using the determinant apparent resistivity at each instant to represent the resistivity of the conductive medium. Localized near-surface inhomogeneities produce shifts in the time-domain apparent resistivity sounding curves that preserve the gradient, analogous to static shifts seen in magnetotelluric soundings. Instantaneous apparent resistivity tensors calculated for 3-D resistivity models suggest that profiles of LOTEM measurements across a simple 3-D structure can be used to create an image that reproduces the main features of the subsurface resistivity. Where measurements are distributed over an area, maps of the tensor invariants can be made into a sequence of images, which provides a way of 'time slicing' down through the target structure.     

Recovering magnetic susceptibility from electromagnetic data over a one-dimensional earth

While the inversion of electromagnetic data to recover electrical conductivity has received much attention, the inversion of those data to recover magnetic susceptibility has not been fully studied. In this paper we invert frequency-domain electromagnetic (EM) data from a horizontal coplanar system to recover a 1-D distribution of magnetic susceptibility under the assumption that the electrical conductivity is known. The inversion is carried out by dividing the earth into layers of constant susceptibility and minimizing an objective function of the susceptibility subject to fitting the data. An adjoint Green's function solution is used in the calculation of sensitivities, and it is apparent that the sensitivity problem is driven by three sources. One of the sources is the scaled electric field in the layer of interest, and the other two, related to effective magnetic charges, are located at the upper and lower boundaries of the layer. These charges give rise to a frequency-independent term in the sensitivities. Because different frequencies penetrate to different depths in the earth, the EM data contain inherent information about the depth distribution of susceptibility. This contrasts with static field measurements, which can be reproduced by a surface layer of magnetization. We illustrate the effectiveness of the inversion algorithm on synthetic and field data and show also the importance of knowing the background conductivity. In practical circumstances, where there is no a priori information about conductivity distribution, a simultaneous inversion of EM data to recover both electrical conductivity and susceptibility will be required.     

Three-dimensional massively parallel electromagnetic inversion—I. Theory

An iterative solution to the non-linear 3-D electromagnetic inverse problem is obtained by successive linearized model updates using the method of conjugate gradients. Full wave equation modelling for controlled sources is employed to compute model sensitivities and predicted data in the frequency domain with an efficient 3-D finite-difference algorithm. Necessity dictates that the inverse be underdetermined, since realistic reconstructions require the solution for tens of thousands of parameters. In addition, large-scale 3-D forward modelling is required and this can easily involve the solution of over several million electric field unknowns per solve. A massively parallel computing platform has therefore been utilized to obtain reasonable execution times, and results are given for the 1840-node Intel Paragon. The solution is demonstrated with a synthetic example with added Gaussian noise, where the data were produced from an integral equation forward-modelling code, and is different from the finite difference code embedded in the inversion algorithm     

Interpretation of transient electromagnetic soundings over three-dimensional structures for the central-loop configuration

Summary. An assessment is made of the bias of fitting constrained layered-earth models to transient electromagnetic data obtained over 3-D structures. In this assessment we use the central-loop configuration and show that accurate estimates of the depth of burial of 3-D structures can be obtained with layered-earth model fitting. However, layered-earth interpretations are not reliable for estimating depth extents and resistivities of 3-D structures. When layered earths are used for interpretation, it is advantageous in some cases to use data based on the magnetic field instead of the voltage. A magnetic-field definition of apparent resistivity, in contrast to a definition based on the voltage, eliminates apparent-resistivity overshoots and undershoots in the data. A resistivity undershoot in the data can produce an extraneous and misleading layer in an interpretation of a 3-D resistive structure. Due to 3-D effects, apparent-resistivity soundings (magnetic field and voltage) may rise so steeply at late times that it may not be possible to fit a sounding to a reasonable layered-earth model. Truncating such a sounding, over a buried conductor, allows for a reasonable layered-earth fit and an accurate estimate of the depth to the conductor. However, the resistivity of the conductor is overestimated.
Measurements of the horizontal field in the central-loop configuration can map 3-D structures, provided the sensor is located accurately at the centre of the transmitting loop. Horizontal-field calculations show that the transients peak on the flanks of a 3-D structure, but are depressed over the structure's centre. Weak transient responses flanked by two large transient responses, which are opposite in sign, locate the structure. The sign reversal is caused by a corresponding reversal in the currents that are channelled through or deflected away from conductive or resistive structures, respectively.     

Subspace inversion of electromagnetic data: application to mid-ocean-ridge exploration

Controlled-source electromagnetic (CSEM) surveys have the ability to provide tomo-graphic images of electrical conductivity within the Earth. the interpretation of such data sets has long been hampered by inadequate modelling and inversion techniques. In this paper, a subspace inversion technique is described that allows electric dipole-dipole data to be inverted for a 2-D electrical conductivity model more efficiently than with existing techniques. the subspace technique is validated by comparison with conventional inversion methods and by inverting data collected over the East Pacific Rise in 1989. A model study indicates that, with adequate data, a variety of possible mid-ocean-ridge conductivity models could be distinguished on the basis of a CSEM survey.     

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.     

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.     

Inversion of time domain three-dimensional electromagnetic data

We present a general formulation for inverting time domain electromagnetic data to recover a 3-D distribution of electrical conductivity. The forward problem is solved using finite volume methods in the spatial domain and an implicit method (Backward Euler) in the time domain. A modified Gauss–Newton strategy is employed to solve the inverse problem. The modifications include the use of a quasi-Newton method to generate a pre-conditioner for the perturbed system, and implementing an iterative Tikhonov approach in the solution to the inverse problem. In addition, we show how the size of the inverse problem can be reduced through a corrective source procedure. The same procedure can correct for discretization errors that inevidably arise. We also show how the inverse problem can be efficiently carried out even when the decay time for the conductor is significantly larger than the repetition time of the transmitter wave form. This requires a second processor to carry an additional forward modelling. Our inversion algorithm is general and is applicable for any electromagnetic field  ( E , H , d B / dt )  measured in the air, on the ground, or in boreholes, and from an arbitrary grounded or ungrounded source. Three synthetic examples illustrate the basic functionality of the algorithm, and a result from a field example shows applicability in a larger-scale field example.     

New advances in three-dimensional controlled-source electromagnetic inversion

New techniques for improving both the computational and imaging performance of the three-dimensional (3-D) electromagnetic inverse problem are presented. A non-linear conjugate gradient algorithm is the framework of the inversion scheme. Full wave equation modelling for controlled sources is utilized for data simulation along with an efficient gradient computation approach for the model update. Improving the modelling efficiency of the 3-D finite difference (FD) method involves the separation of the potentially large modelling mesh, defining the set of model parameters, from the computational FD meshes used for field simulation. Grid spacings and thus overall grid sizes can be reduced and optimized according to source frequencies and source–receiver offsets of a given input data set. Further computational efficiency is obtained by combining different levels of parallelization. While the parallel scheme allows for an arbitrarily large number of parallel tasks, the relative amount of message passing is kept constant. Image enhancement is achieved by model parameter transformation functions, which enforce bounded conductivity parameters and thus prevent parameter overshoots. Further, a remedy for treating distorted data within the inversion process is presented. Data distortions simulated here include positioning errors and a highly conductive overburden, hiding the desired target signal. The methods are demonstrated using both synthetic and field data.     

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.     

On crosswell diffusive time-domain electromagnetic tomography

We investigate the reconstruction of a conductive target using crosswell time-domain electromagnetic tomography in the diffusive limit. The work is a natural extension of our ongoing research in the modification of time-domain methods for the rugged marine mid-ocean-ridge environment, an environment characterized by extreme topography and pronounced variations in crustal conductivity on all scales. We have proved both in theory and in practice that 'traveltime', the time taken for an electromagnetic signal to be identified at a receiver following a change of current in the transmitter, is an excellent, robust estimator of average conductivity on a path between transmitter and receiver. A simple estimate of the traveltime for a parallel electric dipole-dipole system is the time at which the derivative of the electric field with respect to logarithmic time at the receiver reaches its maximum. We have derived the fundamental relationship between the traveltime and the conductivity of the medium for a uniform whole-space. We have applied the concept of the traveltime inversion to the related crosswell problem and demonstrated reconstructions of finite targets based on tomographic analyses. Results show that the crosswell time-domain electromagnetic tomography can supply useful information, such as the location and shape of a conductive target.     

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.     

Calculation of electromagnetic sensitivities in the time domain

The speed of calculating sensitivities for 3-D conductivity structures for time- domain electromagnetic methods is significantly improved by applying the reciprocity theorem directly in the time domain. The sensitivities are obtained by convolving the electric field in the subsurface due to a transmitter at the surface with the electric field impulse response due to another transmitter, which replaces the original receiver. The acceleration compared to the classical perturbation method is approximately P/R , where P is the number of model parameters and R is the number of receiver positions. If the sensitivity has to be calculated very close to the receiver, approximate sensitivities can be obtained using an integral condition. Comparisons with the classical perturbation approach show that the method gives accurate results. Examples using transmitter–receiver configurations from a long-offset transient electromagnetics survey demonstrate the usefulness of sensitivities for the evaluation of resolution properties.     

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.     

Rapid relaxation inversion of CSAMT data

In this paper an inversion algorithm for controlled-source audio frequency magnetotelluric data is presented. This algorithm combines 2.5-D finite element forward modelling with the concepts of rapid relaxation inversion of magnetotelluric data. The inversion uses the same technique to compute sensitivities as the rapid relaxation inversion, and these approximate sensitivities are validated by comparison with exact 2.5-D sensitivities. The comparison shows that the approximate sensitivities are similar in shape to the exact sensitivities when transmitter–receiver offsets are greater than one skin depth in the Earth. The magnitudes of the two sensitivities differ but the variations with depth are similar. Tests of the algorithm on synthetic data and field data provide promising results.     

Spectral-finite-element approach to two-dimensional electromagnetic induction in a spherical earth

We present a spectral-finite-element approach to the 2-D forward problem for electromagnetic induction in a spherical earth. It represents an alternative to a variety of numerical methods for 2-D global electromagnetic modelling introduced recently (e.g. the perturbation expansion approach, the finite difference scheme). It may be used to estimate the effect of a possible axisymmetric structure of electrical conductivity of the mantle on surface observations, or it may serve as a tool for testing methods and codes for 3-D global electromagnetic modelling. The ultimate goal of these electromagnetic studies is to learn about the Earth's 3-D electrical structure.
Since the spectral-finite-element approach comes from the variational formulation, we formulate the 2-D electromagnetic induction problem in a variational sense. The boundary data used in this formulation consist of the horizontal components of the total magnetic intensity measured on the Earth's surface. In this the variational approach differs from other methods, which usually use spherical harmonic coefficients of external magnetic sources as input data. We verify the assumptions of the Lax-Milgram theorem and show that the variational solution exists and is unique. The spectral-finite-element approach then means that the problem is parametrized by spherical harmonics in the angular direction, whereas finite elements span the radial direction. The solution is searched for by the Galerkin method, which leads to the solving of a system of linear algebraic equations. The method and code have been tested for Everett & Schultz's (1995) model of two eccentrically nested spheres, and good agreement has been obtained.     

Transient electromagnetic fields about an infinitesimally long grounded horizontal electric dipole on the surface of a uniform half-space

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.     

A magnetotelluric study of the Alpine Fault, New Zealand

Magnetotelluric soundings have been made at seven locations on a 4  km profile crossing the Alpine Fault in the South Island of New Zealand. The 'distortion' techniques of Groom & Bailey (1989 ) and Lilley (1998a , b ) have been used to derive regional apparent resistivity and phase curves that correspond to electromagnetic induction in orientations parallel and perpendicular to the fault. 2-D inversion of the regional responses reveals that a narrow (<1  km wide) conductive zone is associated with the Alpine Fault. This conductor is most probably related to the heating of deep circulating meteoric water in a region in which enhanced temperatures occur at shallow depth due to the tectonic uplift of the Southern Alps.     

Three-dimensional topography corrections of magnetotelluric data

Topographic effects due to irregular surface terrain may prevent accurate interpretation of magnetotelluric (MT) data. Three-dimensional (3-D) topographic effects have been investigated for a trapezoidal hill model using an edge finite-element method. The 3-D topography generates significant MT anomalies, and has both galvanic and inductive effects in any polarization. This paper presents two different correction algorithms, which are applied to the impedance tensor and to both electric and magnetic fields, respectively, to reduce topographic effects on MT data. The correction procedures using a homogeneous background resistivity derived from a simple averaging method effectively decrease distortions caused by surface topography, and improve the quality of subsurface interpretation. Nonlinear least-squares inversion of topography-corrected data successfully recovers most of structures including a conductive or resistive dyke.     

Separation of local and regional information in distorted GDS response functions by hypothetical event analysis

Electromagnetic investigations are usually intended to examine regional structures where induction takes place at a given period range. However, the regional information is often distorted by galvanic effects at local conductivity boundaries. Bahr (1985) and Groom & Bailey (1989) developed a physical distortion model for decomposing the MT impedance tensor, based upon local galvanic distortion of a regional 2-D electromagnetic field. We have extended their method to predict the magnetic variation fields created at an array of sites. The magnetic response functions at periods around 1000 s may be distorted by large-scale inhomogeneities in the upper or middle crust. In this period range, the data measured by a magnetometer array contain common information that can be extracted if the data set is treated as a unit, for example by using hypothetical event analysis. With this technique it is always possible to recover the regional strike direction from distorted data, even if a strong, spatially varying regional vertical field component is present in the data set. The determination of the regional impedance phases, on the other hand, is far more sensitive to deviations from the physical distortion model.
The approach has been used to investigate the Iapetus data set. For the array, which covers an area of 200  km × 300  km in northern England/southern Scotland, the technique revealed a common regional strike azimuth of ca . N125° E in the period range 500–2000  s. This direction differs from the strike indicated by the induction arrows, which seem influenced mainly by local current concentrations along the east–west-striking Northumberland Trough and a NE–SW-striking mid-crustal conductor. Both impedance phases are positive and differ by ca . 10°, which supports the assumptions of distortion fields in the data set and that the regional structure is 2-D.