A study of diurnal variation of the electromagnetic field in northern France using ancient recordings

Summary. From 1883 to 1901 magnetic elements were continuously recorded at the French Saint-Maur observatory. From 1893 to 1895, Earth potentials along two 15 km long orthogonal lines were also recorded. Moreover, from 1884 to 1885, Blavier,'Ingénieur des Télégraphes', used several some hundred kilométre long telegraphic lines to measure and record Earth potentials. Using this set of data we will study the daily variations of the telluric and magnetic fields and the way according to which these two fields are correlated.
The observed magnetotelluric tensor is antisymmétric when the long telluric lines are considered. It is not the case for the short lines. But, making use of a correction derived from the formalism developed by Le Mouel & Menvielle in the static distortion approximation, one can derive an impedance whose phase is equal to the phase of the impedance derived from the long line data.     

New equipment and processing for magnetotelluric remote reference observations

Robust estimates of magnetotelluric and geomagnetic response functions are determined using the coherency and expected uniformity of the magnetic source field as quality criteria. The method is applied on data sets of three simultaneously recording sites. For the data acquisition we used a new generation of geophysical equipment (S.P.A.M. MkIII), which comprises novel concepts of parallel computing and networked, digital data transmission. The data-processing results show that the amount of noise on the horizontal components of the magnetic field varies considerably in time, between sites and over the frequency range. The removal of such contaminated data beforehand is essential for most data-processing schemes, as the magnetic channels are usually assumed to be free of noise. The standard remote reference method is aimed at reducing bias in response function estimates. However, this does not necessarily improve their precision as our results clearly show. With our method, on the other hand, we can filter out source field irregularities, thereby providing suitable working conditions for the robust algorithm, and eventually obtain considerably improved results. Contrary to previous concepts, we suggest rejecting as much data as feasible in order to concentrate on the remaining parts of high-quality observations.     

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.     

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.     

Finite-difference modelling of magnetotelluric fields in two-dimensional anisotropic media

An algorithm for the numerical modelling of magnetotelluric fields in 2-D generally anisotropic block structures is presented. Electrical properties of the individual homogeneous blocks are described by an arbitrary symmetric and positive-definite conductivity tensor. The problem leads to a coupled system of partial differential equations for the strike-parallel components of the electromagnetic field. E _x, and H _x These equations are numerically approximated by the finite-difference (FD) method, making use of the integro-interpolation approach. As the magnetic component H _x, is constant in the non-conductive air, only equations for the electric mode are approximated within the air layer. The system of linear difference equations, resulting from the FD approximation, can be arranged in such a way that its matrix is symmetric and band-limited, and can be solved, for not too large models, by Gaussian elimination. The algorithm is applied to model situations which demonstrate some non-trivial phenomena caused by electrical anisotropy. In particular, the effect of 2-D anisotropy on the relation between magnetotelluric impedances and induction arrows is studied in detail.     

A quantitative methodology to extract regional magnetotelluric impedances and determine the dimension of the conductivity structure

A methodology to determine quantitatively the dimensionality of the dominant conducting structures and the resolution of the structural parameters in magnetotelluric data is presented. In addition, the method recovers the regional impedance responses when the regional structure can be characterized, at least approximately, as 1- or 2-D. The methodology is based upon three general models of the MT tensor, each of which has a distinct parameterization and physical interpretation. A weighted statistical residual describes quantitatively the fit of the model response to the data within the scatter of the measured data and hence permits: (1) tests of dimensionality, (2) determination of the appropriate strike angle, and (3) recovery of the regional responses.
The method has been tested extensively with synthetic data and proven to be successful. These synthetic studies give insight into the different physical parameterizations and the stability of the parameters determined. We describe and illustrate some of these synthetic studies. With field data, the methodology is not always as straightforward, but its application to a great many sites has proven valuable. Data from two closely spaced sites, which are both affected by strong but very different 3-D effects, are analysed to illustrate the geological significance of the results. The analyses reveal and recover regional responses within the data which indicate the presence of electrical anisotropy located deep in the crust and upper mantle. Analyses of the entire data set, of which these two form a part, confirm this finding.     

Deep crustal geoelectric structure beneath the Northumberland Basin

Summary. In terms of lateral variations in conductivity structure, the southern Southern Uplands and Northumberland Basin are characterized by a region of attenuated vertical magnetic fields with small spatial gradients reflecting the presence of a substantial conducting zone. Five magnetotelluric data sets from the region have been analysed to provide accurate and unbiased estimates of the impedance tensor. The response data are used to investigate the deep geoelectric crustal structure of the region. Three appropriate sets of response data have been subjected to two construction algorithms for 1-D inversion. The geoelectric profiles recovered identify a deep crustal conducting zone underlying the Northumberland Basin. The zone, modelled as a layered structure, dips steeply from mid-crustal depths underneath the Northumberland Basin to lower crustal depths to the NW. The structure thus correlates, in location and geometry, with a deep crustal reflecting wedge detected offshore by a deep seismic reflection profile.     

An algebraic formulation of geophysical inverse problems

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.     

Joint Inversion of Geophysical Data

Summary. By jointly inverting several different kinds of geophysical measurements at a site we avoid some of the ambiguity inherent in the individual methods. We show how this can be done for the combination of DC resistivity and magnetotelluric measurements on a layered medium by considering a simple 3-layer model. The combination resolves the resistivity of the thin resistive second layer, even though neither of the two methods can do so alone.
The method is then applied to field data from a shallow sedimentary basin. A blind zone occurs beneath a thick near-surface conductive shale. By a study of the eigenvalue structure of the model it can be seen that resolution in this zone would be slightly enhanced by higher frequency magnetotelluric data, but additional DC data at larger spacing would yield no improvement.     

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.     

Error propagation in non-linear delay-time tomography

Delay-time tomography can be either linearized or non-linear. In the case of linearized tomography, an error due to the linearization is introduced. If the tomography is performed in a non-linear fashion, the theory used is more accurate from the physical point of view, but if the data have a statistical error, a noise bias in the model is introduced due to the non-linear propagation of errors. We investigate the error propagation of a weakly non-linear delay-time tomography example using second-order perturbation theory. This enables us to compare the linearization error with the noise bias. We show explicitly that the question of whether a non-linear inversion methods leads to a better estimation of the model parameters than a linearized method is dependent on the signal-to-noisc ratio. We also show that, in cases of poor data quality, a linearized inversion method leads to a better estimation of the model parameters than a non-linear method.     

Calculating palaeomagnetic poles for oceanic plates

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The information available for determining palaeopoles for oceanic plates includes data of four types: (1) palaeomagnetic measurements of inclinations but not declinations from core samples; (2) palaeopoles determined from magnetic anomalies over seamounts; (3) phase shifts of linear magnetic anomalies; and (4) palaeoequators determined from geological analysis of sediment cores. A method is presented which combines these different kinds of data and their respective confidence limits to yield a best fit pole, a confidence ellipse, estimates of the data importances and goodness of fit parameters. Using this method we calculate the Campanian pole position for the Pacific plate which is shown by a chi-squared test to be a reasonable fit to the data. We conclude that the method should be a useful tool for calculating pole paths for oceanic plates such as the Pacific and Indian plates.     

Multiple-taper spectral analysis of terrestrial free oscillations: part II

In Part I of this paper, we derived a set of data tapers designed to minimize the spectral leakage of decaying sinusoids immersed in white noise. Multiplying a long-period seismic record by K of these tapers creates K time series. A decaying sinusoid is fit to these K time series in the frequency domain at a number of chosen frequencies by a least-squares procedure. The fit is tested at each frequency using a statistical F -test. In Part I, we demonstrated that the multiple-taper method is a more sensitive detector of decaying sinusoids than the conventional direct spectral-estimate.
In this paper, we present a number of extensions to the multiple-taper method. We explain how the technique can be modified to estimate the harmonic components of records containing gaps. We discuss how sinusoids at frequencies between FFT bin frequencies can be detected, and how this method can be combined with conventional multi-station stacking procedures.     

Long-period seismic source inversions using global tomographic models

We investigate the effect of laterally varying earth structure on centroid moment tensor inversions using fundamental mode mantle waves. Theoretical seismograms are calculated using a full formulation of surface wave ray theory. Calculations are made using a variety of global tomographic earth models. Results are compared with those obtained using the so-called great-circle approximation, which assumes that phase corrections are given in terms of mean phase slowness along the great circle, and which neglects amplitude effects of heterogeneity. Synthetic tests suggest that even source parameters which fit the data very well may have large errors due to incomplete knowledge of lateral heterogeneity. The method is applied to 31 shallow, large earthquakes. For a given earthquake, the focal mechanisms calculated using different earth models and different forward modelling techniques can significantly vary. We provide a range of selected solutions based on the fit to the data, rather than one single solution. Difficulties in constraining the dip-slip components of the seismic moment tensor often produce overestimates of seismic moment, leading to near vertical dip-slip mechanisms. This happens more commonly for earth models not fitting the data well, confirming that more accurate modelling of lateral heterogeneity can help to constrain the dip-slip components of the seismic moment tensor.     

Multiple-taper spectral analysis of terrestrial free oscillations: part I

We present a new method for estimating the frequencies of the Earth's free oscillations. This method is an extension of the techniques of Thomson (1982) for finding the harmonic components of a time series. Optimal tapers for reducing the spectral leakage of decaying sinusoids immersed in white noise are derived. Multiplying the data by the best K tapers creates K time series. A decaying sinusoid model is fit to the K time series by a least squares procedure. A statistical F -test is performed to test the fit of the decaying sinusoid model, and thus determine the probability that there are coherent oscillations in the data. The F -test is performed at a number of chosen frequencies, producing a measure of the certainty that there is a decaying sinusoid at each frequency. We compare this method with the conventional technique employing a discrete Fourier transform of a cosine-tapered time-series. The multiple-taper method is found to be a more sensitive detector of decaying sinusoids in a time series contaminated by white noise.     

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.     

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.     

Natural electromagnetism in the Rhine Graben

Summary. Together with experiments by differential geomagnetic soundings, magnetotelluric soundings have been performed in the Rhine Graben. The results of the former have been presented and discussed before. We try to show that it is not necessary to call upon the notion of induction in surrounding areas to justify the characteristics of the observed anomalous magnetic fields. Theoretical calculations of these fields on structures defined by the interpretation of magnetotelluric sounds, or by geological and other geophysical data, agree very well with experimental data. We conclude that magnetotelluric experiments and interpretations are valid as the Rhine Graben is a well-known region allowing us to verify our interpretation.     

Automatic model for finding the one-dimensional magnetotelluric problem

Summary. A method is described for finding a resistivity model that fits given magnetotelluric data in the one-dimensional case. The procedure is automatic and objective in that no a priori model structure is imposed. Starting with a uniform half space derived directly from the data, the procedure gradually transforms the half space to one with a continuous and smooth resistivity distribution whose response fits the measured data. The method is illustrated by application to two magnetotelluric data sets.     

Geoelectromagnetic induction in a heterogeneous sphere:a new three-dimensional forward solver using a conservative staggered-grid finite difference method

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 ).