A six-parameter statistical model of the Earth's magnetic field

A six-parameter statistical model of the non-dipole geomagnetic field is fitted to 2597 harmonic coefficients determined by Cain, Holter & Sandee (1990) from MAGSAT data. The model includes sources in the core, sources in the crust, and instrument errors. External fields are included with instrument errors. The core and instrument statistics are invariant under rotation about the centre of the Earth, and one of the six parameters describes the deviation of the crustal statistics from rotational invariance. The model treats the harmonic coefficients as independent random samples drawn from a Gaussian distribution. The statistical model of the core field has a correlation length of about 500 km at the core-mantle boundary, too long to be attributed to a white noise source just below the boundary layers at the top of the core. The estimate of instrument errors obtained from the statistical model is in good agreement with an independent estimate based on tests of the instruments (Langel, Ousley & Berbert 1982).     

Comment on 'Estimation of resolution and covariance for large matrix inversions' by J. Zhang and G. A. McMechan

When inverting large matrices, iterative techniques are necessary because of their speed and low memory requirements, as opposed to singular value decomposition (SVD). Recently, there have been attempts to obtain information on the quality of the solutions calculated using conjugate gradient (CG) methods such as LSQR. The purpose of this note is to comment on the paper titled "Estimation of resolution and covariance for large matrix inversions' by Zhang & McMechan (1995), who extend Paige and Saunders' LSQR algorithm to obtain an orthonormal basis used to approximate resolution and covariance. We show that for larger problems, where the number of orthogonal vectors is several orders of magnitude smaller than the number of model parameters, the vectors obtained do not adequately span the range of the model space. We use a synthetic borehole experiment to illustrate the differences between the singular value spectrum obtained through the more complete method of SVD and the Ritz value spectrum that results from a simple extension of LSQR, We also present a trivial numerical example to illustrate the differences between Zhang & McMechan's approximate resolution matrix and the true resolution.     

A comparison of the stability of stationary and non-stationary magnetotelluric analysis methods

A Magnetotelluric (MT) sounding was carried out at a site in south-east Queensland, in the Clarence-Moreton Basin. The synoptic recordings were taken over a period of four months at sampling frequencies from 500 Hz to 5 × 10-5 Hz. The resulting data was analysed by the stationary cross-frequency and the Cone kernel time-frequency distribution (TFD) methods of MT analysis. The results were compared as apparent resistivities on a daily basis for frequencies above 1 Hz, as well as over all the available data. The TFD MT apparent-resistivity results were more stable and less noisy on an daily basis than the cross-frequency results. Similarly the TFD analysis gave less noisy results than the cross-frequency analysis when all available data was processed. Application of these new non-stationary analysis techniques to MT processing should decrease the bias error problem of the MT methods and so increase reliability and repeatability of MT soundings.     

A simple notation for differential vector expressions in orthogonal curvilinear coordinates

We present a simple notation for performing differential vector operations in orthogonal curvilinear coordinates, and for easily obtaining partial differential expressions in terms of the physical components. We express n th-order tensors as the summed products of the physical components and n th-order polyads of unit vectors (an extension of Gibbs dyadic notation convenient for a summation convention). By defining a gradient operator with partial derivatives balanced by the inverse scale factors, differential vector (or tensor) operations in orthogonal coordinates do not require the covariant/contravariant notation. Our primary focus is on spherical-polar coordinates, but we also derive formulae which may be applied to arbitrary orthogonal coordinate systems. The simpler case of cylindrical-polar coordinates is briefly discussed. We also offer a compact form for the gradient and divergence of general second-order tensors in orthogonal curvilinear coordinates, which are generally unavailable in standard handbooks. We show how our notation relates to that of tensor analysis/differential geometry. As the analysis is not restricted to Euclidean geometry, our notation may be extended to Riemannian surfaces, such as spherical surfaces, so long as an orthogonal coordinate system is utilized. We discuss the Navier-Stokes equation for the case of spatially variable viscosity coefficients.     

Glacial rebound of the British Isles—II. A high-resolution, high-precision model

Observations of ice movements across the British Isles and of sea-level changes around the shorelines during Late Devensian time (after about 25 000 yr BP) have been used to establish a high spatial and temporal resolution model for the rebound of Great Britain and associated sea-level change. The sea-level observations include sites within the margins of the former ice sheet as well as observations outside the glaciated regions such that it has been possible to separate unknown earth model parameters from some ice-sheet model parameters in the inversion of the glacio-hydro-isostatic equations. The mantle viscosity profile is approximated by a number of radially symmetric layers representing the lithosphere, the upper mantle as two layers from the base of the lithosphere to the phase transition boundary at 400 km, the transition zone down to 670 km depth, and the lower mantle. No evidence is found to support a strong layering in viscosity above 670 km other than the high-viscosity lithospheric layer. Models with a low-viscosity zone in the upper mantle or models with a marked higher viscosity in the transition zone are less satisfactory than models in which the viscosity is constant from the base of the lithosphere to the 670 km boundary. In contrast, a marked increase in viscosity is required across this latter boundary. The optimum effective parameters for the mantle beneath Great Britain are: a lithospheric thickness of about 65 km, a mantle viscosity above 670 km of about (4-5) 10²⁰ Pa s, and a viscosity below 670 km greater than 4 × 10²¹ Pa s.     

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.     

Numerical inversion of deformation caused by pressure sources: application to Mount Etna (Italy)

The interpretation of geodetic data in volcanic areas is usually based on analytical deformation models. Although numerical finite element (FE) modelling allows realistic features such as topography and crustal heterogeneities to be included, the technique is not computationally convenient for solving inverse problems using classical methods. In this paper, we develop a general tool to perform inversions of geodetic data by means of 3-D FE models. The forward model is a library of numerical displacement solutions, where each entry of the library is the surface displacement due to a single stress component applied to an element of the grid. The final solution is a weighted combination of the six stress components applied to a single element-source. The pre-computed forward models are implemented in a global search algorithm, followed by an appraisal of the sampled solutions. After providing extended testing, we apply the method to model the 1993–1997 inflation phase at Mt Etna, documented by GPS and EDM measurements. We consider four different forward libraries, computed in models characterized by homogeneous/heterogeneous medium and flat/topographic free surface. Our results suggest that the elastic heterogeneities of the medium can significantly alter the position of the inferred source, while the topography has minor effect.