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.     

Two years of magnetotelluric measurements in Abitibi, western Québec, using a telephone line

Continuous magnetotelluric measurements were made over a period of 600 days, with 100-m-, 30-km- and 100-km-long dipoles and a period range of 40–4000 s. Data analysis for different dipole lengths indicates the presence of static shift at various scales. It is shown that the longer the telluric dipole, the less statically shifted the resistivity curves; nevertheless, static shifts can still be present due to geological structures causing anomalies exhibiting wavelengths comparable to the dipole length. Also, a relationship is observed between the coherence and the main impedance components. This relation is explained in terms of signal-to-noise ratio. A way to reduce the bias on the impedance estimates is suggested. The apparent resistivities and phases computed from three different impedance estimates using 100-km-long dipoles are then compared to those observed in similar studies made near our observation region.     

Magnetotelluric soundings over a Precambrian contact in Australia

Summary. The Willyama Complex near Broken Hill represents a major exposure of the Precambrian terrain in Australia. However, the eastern limits are obscured by recent cover in the Murray/Darling basin. The nature and extent of the contact with the younger basement to the east is unknown.
Magnetotelluric data have now been obtained as an aid to regional mapping. Profiles are based on observations at 14 sites in a traverse east of Broken Hill. Apparent resistivities presented as pseudosections reflect the major structural units identified from surface geology. Continuity between sites is indicated from similarity in apparent resistivities and lateral changes east of the Block are generally gradational.
Characteristic divergences in orthogonal resistivity components are consistent with 2-D structure. The foliated Precambrian complex appears to form a semi-vertical contact at the margin of the Murray/Darling basin, thus generating the required anisotropy at depths from 15–35 km. Seismic evidence indicates an equivalent structural contact possibly extending into the mantle.     

Effects of galvanic distortion on magnetotelluric data over a three-dimensional regional structure

Although the galvanic distortion due to local, near-surface inhomogeneities is frequency-independent, its effect on the magnetotelluric data becomes, in a 3-D structure, frequency-dependent. Therefore, both the apparent resistivity and the phase responses are disturbed, and a correction should be carried out prior to the 3-D interpretation in order to retrieve the 3-D regional impedance tensor. In many cases, the structure is 2-D for depths corresponding to a first range of periods and 3-D for longer periods (called 2-D/3-D). For these cases, a simple method which allows us to retrieve the 3-D regional impedance tensor (except the static shift) is presented. The method proposed uses the Groom & Bailey decomposition of the distortion matrix for the short periods. Three examples are presented: two using synthetic data and one employing real data. These examples show the effect of the galvanic distortion over a regional 2-D/3-D model and the retrieval of the regional transfer functions from the distorted ones.     

Magnetotelluric Soundings In the Canadian Rocky Mountains

Summary. Magnetotelluric soundings have been made at 25 stations in the Rocky Mountain Trench (RMT) and Main Ranges near 53° N, close to the centre of a major conductivity anomaly which had been mapped in a magnetovariation array study. Most stations covered the frequency range 0.01–500 Hz and three stations 0.0002–500 Hz. the resistivity tensor shows low to moderate anisotropy in the RMT, but is strongly 2-D or 3-D in the Rocky Mountains. Apparent resistivities as a function of frequency are displayed in pseudosections along the Trench and along a transverse profile across the RMT and into the Main Ranges. In preparation for 2-D modelling, 1-D inversions have been used to construct resistivity-depth sections satisfying both magnitudes and phases of the MT responses. These show very low resistivities, in the range 1—10Ωm, in the upper crust under the RMT and even lower values under the Main Ranges. the latter values give strong confirmation of the Northern Rockies conductor reported by Bingham, Cough & Ingham and are in agreement with models of the conductors fitted to long-period magnetovariation fields by Ingham, Gough & Parkinson. the MT results here reported add some essential depth and resistivity information. It is suggested that the conductors beneath the Rocky Mountains Main Ranges and Trench constitute a thickening at the edge of the Canadian Cordilleran Regional (CCR) conductor. Gough has argued that a wide variety of geophysical and geological parameters indicate high temperatures and partial melting in the mantle under the CCR conductor. At the upper crustal depths penetrated in this magnetotelluric study, it is considered more probable that the high conductivity is caused by hot, saline water of mantle origin rather than silicate melt. the CCR in general may have two layers of fluid producing its high conductivity, silicate melt below and saline hot water above.     

Archaeological investigation by electrical resistivity tomography: a preliminary study

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This paper describes a method in which vertical resistivity sections are generated tomographically from measurements on a linear array of equally spaced electrodes inserted at the ground surface. The array is multiplexed to a resistivity meter which gathers one set of all possible independent apparent resistivity measurements and the geophysical section is then reconstructed by backprojecting these weighted data, along equipotentials, into the subsurface. The technique has been evaluated numerically and in field trials over shallow archaeological structures at Fountains Abbey.     

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.     

Analysing seismic-source mechanisms by linear-programming methods

Summary. Linear-programming methods are powerful and efficient tools for objectively analysing seismic focal mechanisms and are applicable to a wide range of problems, including tsunami warning and nuclear explosion identification. The source mechanism is represented as a point in the six-dimensional space of moment-tensor components. Each observed polarity provides an inequality constraint, linear with respect to the moment tensor components, that restricts the solution to a half-space bounded by a hyperplane passing through the origin. The intersection of these half-spaces is the convex set of all acceptable solutions. Using linear programming, a solution consistent with the polarity constraints can be obtained that maximizes or minimizes any desired linear function of the moment tensor components; the dilatation, the thrust-like nature, and the strike-slip-like nature of an event are examples of such functions. The present method can easily be extended to fit observed seismic-wave amplitudes (either signed or absolute) subject to polarity constraints, and to assess the range of mechanisms consistent with a set of measured amplitudes.     

Deep Schlumberger sounding and the crustal resistivity structure of central Australia

Summary. Three 200 km Schlumberger resistivity soundings have been conducted over the central Australian shield, using telephone lines to obtain the large electrode spacings. These represent the first crustal scale controlled source electrical study to be carried out in this continent. A computer controlled data acquisition system was used which allowed precise measurements to be made with only modest emission currents (0.1–0.5 A).
The three soundings, centred on the towns of Renner Springs, Wauchope and Aileron, showed the southern part of the study area (the Arunta Block) to be an order of magnitude more resistive than the more northerly section (the Tennant Creek Block). This difference correlates with the higher heat flow of the Tennant Creek Block. A lowering of apparent resistivity at large electrode spacings for one sounding (Wauchope) is taken to indicate the presence of a low resistivity layer in the middle crust, at a depth less than 20 km. However, the effect of the highly conductive overburden characteristic of inland Australia, combined with the large transverse resistance of the crust, prevented the other two soundings from detecting such a layer. Without support from these two soundings, it is impossible to be sure that the lowered resistivity at Wauchope is not caused merely by lateral variations in near-surface resistivity.
The data also show that crustal resistivities are much lower than the expected values for dry rock, whether or not a low resistivity layer is included in the model. This implies a widespread occurrence of free water in the crust, with greater amounts occurring at depth if the low resistivity zone exists.     

Preferential flow: first results of a full-scale flow model

The main goal of a joint project undertaken by the geophysical and hydrological research units of Kiel University is to study preferential flow in a large open-air, full-scale model, looking in particular at near-surface penetration and flow of water through the unsaturated vadose zone. An artificial irrigation device is installed in place of natural rain, and a homogeneous sand body is used instead of natural soil. This provides a reference model for future field experiments. Inside the sand body there are a large number of geophysical and hydrological sensors to measure DC resistivity (using various electrode configurations), water content and water potential (using TDR and tensiometer instruments, respectively). A ground-penetrating radar (GPR) system is installed at the surface, whereas at the bottom several containers and a thin gravel layer are embedded to measure the flow arrival and the discharge of water. Irrigation is varied in intensity, time, area, and salt content (tracer).
  Results of the first six experiments show that the percolation of intruding water can be followed by all techniques and percolation is finally controlled by the discharge measurements. These display some undulations and variations of the water 'front' and agree with the measurements of all other sensors. The redundancy achieved by the use of multiple methods was intended to enable an assessment of the reliability of the techniques used. The true values of electrical resistivity before and after irrigation reflect the distribution of water saturation within the sand body. A numerical 3-D inversion of the apparent resistivity provides information regarding future field experiments, in which it will be possible to install only some of the sensors in order to preserve the natural structure of the soil.     

Magnetotelluric image of the crust and upper mantle in the backarc of the northwestern Argentinean Andes

Magnetotelluric data from the backarc of the Central Andes in NW Argentinawere re-examined by employing impedance tensor decomposition and 2-D inversion and modelling techniques. The data in the period range of 50–15 000 s were collected on a profile of 220 km length reaching from the Eastern Cordillera across the Santa Barbara System to the Andean foreland of the Argentinean Chaco.
After a dimensionality analysis, data from most sites were treated as regional 2-D. The exception was the eastern section of the profile, where the magnetotelluric transfer functions for periods ≤ 1000 s reflect a 3-D earth. Application of two tensor decomposition schemes yielded a regional strike direction of N–S, which is the azimuth of the Central Andean mountain chains. Several 2-D models were obtained by pseudo- and full 2-D Occam inversion schemes. Special emphasis was placed on the inversion of phase data to reduce the influence of static shifts in the apparent resistivity data. The smooth inversion models all show a good conductor at depth. A final model was then calculated using a finite element forward algorithm.
The most prominent feature of the resulting model is a conductor which rises from depths of 180 km below the Chaco region to 80 km beneath the Santa Barbara System and the Eastern Cordillera. Its interpretation as a rise of the electrical asthenosphere is supported by seismic attenuation studies. Magnetotelluric results, surface heat-flow distribution in the area, and the electrical properties of crustal and mantle rocks suggest that the upper mantle is predominantly ductile beneath the Eastern Cordillera and the western Santa Barbara System. This generally agrees with anelastic seismic attenuation models of the area and is useful in discriminating between models of Q quality factor distribution.     

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.     

Induction in a thin sheet of variable conductance at the surface of a stratified earth - I. Two-dimensional theory

Summary. An existing 2-D integral equation method for modelling electromagnetic induction in a thin sheet at the surface of a uniform half-space can be generalized to deal with a layered half-space by the inclusion of an extra term in the integral equation. The results obtained are shown to be in excellent agreement with finite difference solutions to the same modelling problem.     

Diffraction of P, S and Rayleigh waves by three-dimensional topographies

The diffraction of P, S and Rayleigh waves by 3-D topographies in an elastic half-space is studied using a simplified indirect boundary element method (IBEM). This technique is based on the integral representation of the diffracted elastic fields in terms of single-layer boundary sources. It can be seen as a numerical realization of Huygens principle because diffracted waves are constructed at the boundaries from where they are radiated by means of boundary sources. A Fredholm integral equation of the second kind for such sources is obtained from the stress-free boundary conditions. A simplified discretization scheme for the numerical and analytical integration of the exact Green's functions, which employs circles of various sizes to cover most of the boundary surface, is used.
The incidence of elastic waves on 3-D topographical profiles is studied. We analyse the displacement amplitudes in the frequency, space and time domains. The results show that the vertical walls of a cylindrical cavity are strong diffractors producing emission of energy in all directions. In the case of a mountain and incident P, SV and SH waves the results show a great variability of the surface ground motion. These spatial variations are due to the interference between locally generated diffracted waves. A polarization analysis of the surface displacement at different locations shows that the diffracted waves are mostly surface and creeping waves.     

Magnetotelluric studies in the Market Weighton area of eastern England

Summary Magnetotelluric measurements at periods from 30 to 1000s were made at eight locations in the Market Weighton (MW) area, along an east—west profile across gravity and magnetic anomalies. Dimensional parameters were developed for assessing the structural dimensionality of the electrical conductivity of the Earth from the data. One-dimensional inversion modelling techniques were employed to interpret the data at each site, and four-layer models were obtained to explain the main structure of the crust in the area studied. If it is assumed that all strata are unmagnetized then the results show that there is a highly resistive layer in the crust, the thickness of the highly resistive layer ranges from 12 km in the east to 44 km in the west with a large change in the middle near the MW site. A structural boundary lying north—south near MW was also indicated by the principal directions of rotated apparent resistivities and transfer functions. Both electrical conductivity and magnetic permeability contrast in the ground were considered in an attempt to interpret the observed variations in apparent resistivity at different periods.     

Calculating the two-dimensional magnetotelluric Jacobian in finite elements using reciprocity

To speed up the calculation of the field Jacobian for 2-D magnetoteliuric inversion using finite elements, the principle of electromagnetic reciprocity is applied. The governing relationship for the Jacobian of the field along strike is obtained by differentiating the Helmholtz equation with respect to the resistivity of each region in the finite-element mesh. The result is a similar Helmholtz equation for the Jacobian, with new sources distributed over all nodes within the parameter medium. However, according to the principle of electromagnetic reciprocity, the roles of sources and receivers are interchangeable. Utilizing reciprocity, the field values obtained from the original forward problem and for new unit sources imposed at the receivers are then utilized in the calculation of the Jacobian by simple multiplication and summation with finite-element terms at each rectangle in the mesh. For the auxiliary (across-strike) fields, the Jacobian terms are obtained by solving source vectors loaded with parabola coefficients used in the approximation to Maxwell's equations. Jacobian terms for the apparent resistivity ( p _a), the impedance phase (φ) and the vertical magnetic field ( K _zy) are then calculated utilizing the parallel- and auxiliary-field Jacobians. Comparison of Jacobian values obtained from reciprocity calculations and by differencing two forward solutions show that the reciprocity method is accurate and can be used to decrease the number of calculations required to obtain sensitivities by one to two orders of magnitude.     

Statistics and geometry of the eigenspectra of three-dimensional second-rank symmetric random tensors

Geoscientists have undertaken mapping of the Earth's crustal strain (or stress) fields using a great variety of field data. The output can be represented by a 3-D second-rank symmetric random strain tensor. The random principal strains-land rotations of the random tensor are frequently computed. The accuracy is calculated using a first-order approximation. The distribution aspects of the random principal strains and rotations have received almost no attention in Earth Sciences. A first-order approximation of accuracy may not be sufficient if the signal-to-noise ratio is small, as is often the case for geodetically derived random strain tensors. Therefore, the purpose of this paper is to investigate the distribution and estimation problems of the general 3-D second-rank tensor equation GΛG ^T= T , where T is a given 3-D second-rank symmetric random tensor, Λ a diagonal (3 × 3) random eigenvalue matrix, and G a (3 × 3) random orientation matrix, which is also orthogonal. Λ and G are to be estimated (or solved) from T . If some eigenvalues coincide, additional conditions are imposed on the eigenvectors so that they can be chosen uniquely. The joint probability density function (pdf) of the random eigenvalues and rotations will be worked out, given a joint pdf of the elements of random tensors T. Because the rotations are of special interest in Earth Sciences, we shall also derive the joint marginal pdf of random rotations. The geometry of eigenspectra will be studied. The biases of random eigenvalues and rotations will be derived, which have been neglected in the past. They can be very crucial in interpreting the pattern of a derived strain field, however, when applied to a real Earth Science problem. The variance-covariance matrices will be computed using a second-order approximation.     

Extreme sensitivity of crosshole electrical resistivity tomography measurements to geometric errors

The effects of geometric errors on crosshole resistivity data are investigated using analytical methods. Geometric errors are systematic and can occur due to uncertainties in the individual electrode positions, the vertical spacing between electrodes in the same borehole, or the vertical offset between electrodes in opposite boreholes. An estimate of the sensitivity to geometric error is calculated for each of two generic types of four-electrode crosshole configuration: current flow and potential difference crosshole (XH) and in-hole (IH). It is found that XH configurations are not particularly sensitive to geometric error unless the boreholes are closely spaced on the scale of the vertical separation of the current and potential electrodes. However, extremely sensitive IH configurations are shown to exist for any borehole separation. Therefore, it is recommended that XH configurations be used in preference to IH schemes. The effects of geometric error are demonstrated using real XH data from a closely spaced line of boreholes designed to monitor bioremediation of chlorinated solvents at an industrial site. A small fraction of the data had physically unrealistic apparent resistivities, which were either negative or unexpectedly large. However by filtering out configurations with high sensitivities to geometric error, all of the suspect data were removed. This filtering also significantly improved the convergence between the predicted and the measured resistivities when the data were inverted. In addition to systematic geometric errors, the measured data also exhibit a high level of random noise. Despite this, the resulting inverted images correspond reasonably closely with the known geology and nearby cone penetrometer resistivity profiles.     

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.     

The first-order statistical moment of the seismic moment tensor

Summary. If a complex earthquake is assumed to be a set of individual, randomly oriented elementary pure double couple sources, the solution for the seismic moment of the complex event projected on the mean trend of the fault will perforce be comprised of sources of both double couple and compensated linear vector dipole (CLVD) types. We investigate the statistical properties of these two components of seismic sources in terms of the invariants of the seismic moment tensor of a realistic set of synthetic earthquakes. It is very likely that the size of the CLVD component is two to three orders of magnitude smaller than that of the double couple component.