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Hulot, G, Finlay CC, Constable CG, Olsen N, Mandea M.  2010.  The Magnetic Field of Planet Earth. Space Science Reviews. 152:159-222.   10.1007/s11214-010-9644-0   AbstractWebsite

The magnetic field of the Earth is by far the best documented magnetic field of all known planets. Considerable progress has been made in our understanding of its characteristics and properties, thanks to the convergence of many different approaches and to the remarkable fact that surface rocks have quietly recorded much of its history. The usefulness of magnetic field charts for navigation and the dedication of a few individuals have also led to the patient construction of some of the longest series of quantitative observations in the history of science. More recently even more systematic observations have been made possible from space, leading to the possibility of observing the Earth's magnetic field in much more details than was previously possible. The progressive increase in computer power was also crucial, leading to advanced ways of handling and analyzing this considerable corpus of data. This possibility, together with the recent development of numerical simulations, has led to the development of a very active field in Earth science. In this paper, we make an attempt to provide an overview of where the scientific community currently stands in terms of observing, interpreting and understanding the past and present behavior of the so-called main magnetic field produced within the Earth's core. The various types of data are introduced and their specific properties explained. The way those data can be used to derive the time evolution of the core field, when this is possible, or statistical information, when no other option is available, is next described. Special care is taken to explain how information derived from each type of data can be patched together into a consistent description of how the core field has been behaving in the past. Interpretations of this behavior, from the shortest (1 yr) to the longest (virtually the age of the Earth) time scales are finally reviewed, underlining the respective roles of the magnetohydodynamics at work in the core, and of the slow dynamic evolution of the planet as a whole.

Korte, M, Constable C.  2003.  Continuous global geomagnetic field models for the past 3000 years. Physics of the Earth and Planetary Interiors. 140:73-89.   10.1016/j.pepi.2003.07.013   AbstractWebsite

Several global geomagnetic field models exist for recent decades, but due to limited data availability models for several centuries to millennia are rare. We present a continuous spherical harmonic model for almost 3 millennia from 1000 B.C. to 1800 A.D., based on a dataset of directional archaeo- and paleomagnetic data and axial dipole constraints. The model, named Continuous Archaeomagnetic and Lake Sediment Geomagnetic Model for the last 3k years (CALS3K.1), can be used to predict both the field and secular variation. Comparisons and tests with synthetic data lead to the conclusion that CALS3K.1 gives a good general, large-scale representation of the geomagnetic field, but lacks small-scale structure due to the limited resolution of the sparse dataset. In future applications the model can be used for comparisons with additional, new data for that time span. For better resolved regions, the agreement of data with CALS3K.1 will provide an idea about the general compatibility of the data with the field and secular variation in that region of the world. For poorly covered regions and time intervals we hope to iteratively improve the model by comparisons with and inclusion of new data. Animations and additional snapshot plots of model predictions as well as the model coefficients and a FORTRAN code to evaluate them for any time can be accessed under The whole package is also stored in the Earthref digital archive at (C) 2003 Elsevier B.V. All rights reserved.

Love, JJ, Constable CG.  2003.  Gaussian statistics for palaeomagnetic vectors. Geophysical Journal International. 152:515-565.   10.1046/j.1365-246X.2003.01858.x   AbstractWebsite

With the aim of treating the statistics of palaeomagnetic directions and intensities jointly and consistently, we represent the mean and the variance of palaeomagnetic vectors, at a particular site and of a particular polarity, by a probability density function in a Cartesian three-space of orthogonal magnetic-field components consisting of a single (unimodal) non-zero mean, spherically-symmetrical (isotropic) Gaussian function. For palaeomagnetic data of mixed polarities, we consider a bimodal distribution consisting of a pair of such symmetrical Gaussian functions, with equal, but opposite, means and equal variances. For both the Gaussian and bi-Gaussian distributions, and in the spherical three-space of intensity, inclination, and declination, we obtain analytical expressions for the marginal density functions, the cumulative distributions, and the expected values and variances for each spherical coordinate (including the angle with respect to the axis of symmetry of the distributions). The mathematical expressions for the intensity and off-axis angle are closed-form and especially manageable, with the intensity distribution being Rayleigh-Rician. In the limit of small relative vectorial dispersion, the Gaussian (bi-Gaussian) directional distribution approaches a Fisher (Bingham) distribution and the intensity distribution approaches a normal distribution. In the opposite limit of large relative vectorial dispersion, the directional distributions approach a spherically-uniform distribution and the intensity distribution approaches a Maxwell distribution. We quantify biases in estimating the properties of the vector field resulting from the use of simple arithmetic averages, such as estimates of the intensity or the inclination of the mean vector, or the variances of these quantities. With the statistical framework developed here and using the maximum-likelihood method, which gives unbiased estimates in the limit of large data numbers, we demonstrate how to formulate the inverse problem, and how to estimate the mean and variance of the magnetic vector field, even when the data consist of mixed combinations of directions and intensities. We examine palaeomagnetic secular-variation data from Hawaii and Reunion, and although these two sites are on almost opposite latitudes, we find significant differences in the mean vector and differences in the local vectorial variances, with the Hawaiian data being particularly anisotropic. These observations are inconsistent with a description of the mean field as being a simple geocentric axial dipole and with secular variation being statistically symmetrical with respect to reflection through the equatorial plane. Finally, our analysis of palaeomagnetic acquisition data from the 1960 Kilauea flow in Hawaii and the Holocene Xitle flow in Mexico, is consistent with the widely held suspicion that directional data are more accurate than intensity data.

Johnson, CL, Constable CG.  1998.  Persistently anomalous Pacific geomagnetic fields. Geophysical Research Letters. 25:1011-1014.   10.1029/98gl50666   AbstractWebsite

A new average geomagnetic field model for the past 3kyr (ALS3K) helps bridge a large temporal sampling gap between historical models and more traditional paleomagnetic studies spanning the last 5 Myr. A quasi-static feature seen historically in the central Pacific has the opposite sign in ALS3K; its structure is similar to, but of larger amplitude than, that in the time-averaged geomagnetic field for the last 5 Myr. Anomalous geomagnetic fields exist beneath the Pacific over timescales ranging from 10(2)-10(6) years. It is unlikely that bias over such long time scales arises from electromagnetic screening, but conceivable that the Lorentz force is influenced by long wavelength thermal variations and/or localized regions of increased electrical conductivity (associated with compositional anomalies and possibly partial melt). This is consistent with recent seismic observations of the lower mantle.

RygaardHjalsted, C, Constable CG, Parker RL.  1997.  The influence of correlated crustal signals in modelling the main geomagnetic field. Geophysical Journal International. 130:717-726.   10.1111/j.1365-246X.1997.tb01866.x   AbstractWebsite

Algorithms used in geomagnetic main-field modelling have for the most part treated the noise in the field measurements as if it were white. A major component of the noise consists of the field due to magnetization in the crust and it has been realized for some time that such signals are highly correlated at satellite altitude. Hence approximation by white noise, while of undoubted utility, is of unknown validity. Langel, Estes & Sabaka (1989) were the first to evaluate the influence of correlations in the crustal magnetic field on main-field models. In this paper we study two plausible statistical models for the crustal magnetization described by Jackson (1994), in which the magnetization is a realization of a stationary, isotropic, random process. At a typical satellite altitude the associated fields exhibit significant correlation over ranges as great as 15 degrees or more, which introduces off-diagonal elements into the covariance matrix, elements that have usually been neglected in modelling procedures. Dealing with a full covariance matrix for a large data set would present a formidable computational challenge, brit fortunately most of the entries in the covariance matrix are so small that they can be replaced by zeros. The resultant matrix comprises only about 3 per cent non-zero entries and thus we can take advantage of efficient sparse matrix techniques to solve the numerical system. We construct several main-field models based on vertical-component data from a selected 5 degrees by 5 degrees data set derived from the Magsat mission. Models with and without off-diagonal terms are compared. For one of the two Jackson crustal models, k(3), we find significant changes in the main-field coefficients, with maximum discrepancies near degree 11 of about 27 per cent. The second crustal spectrum gives rise to much smaller effects for the data set used here, because the correlation lengths are typically shorter than the data spacing. k(4) also significantly underpredicts the observed magnetic spectrum around degree 15. We conclude that there is no difficulty in computing main-field models that include off-diagonal terms in the covariance matrix when sparse matrix techniques are employed; we find that there may be important effects in the computed models, particularly if we wish to make full use of dense data sets. Until a definitive crustal field spectrum has been determined, the precise size of the effect remains uncertain. Obtaining such a statistical model should be a high priority in preparation for the analysis of future low-noise satellite data.

Obrien, MS, Constable CG, Parker RL.  1997.  Frozen-flux modelling for epochs 1915 and 1980. Geophysical Journal International. 128:434-450.   10.1111/j.1365-246X.1997.tb01566.x   AbstractWebsite

The frozen-flux hypothesis for the Earth's liquid core assumes that convective terms dominate diffusive terms in the induction equation governing the behaviour of the magnetic field at the surface of the core. While highly plausible on the basis of estimates of physical parameters, the hypothesis has been questioned in recent work by Bloxham, Gubbins & Jackson (1989) who find it to be inconsistent with their field models for most of the century. To study this question we improve the method of Constable, Parker & Stark (1993), which tests the consistency of magnetic observations with the hypothesis by constructing simple, flux-conserving core-field models fitting the data at pairs of epochs. We introduce a new approach that fixes the patch configurations at each of the two epochs before inversion, so that each configuration is consistent with its respective data set but possesses the same patch topology. We expand upon the inversion algorithm, using quadratic programming to maintain the proper flux sign within patches; the modelling calculations are also extended to include data types that depend non-linearly on the model. Every test of a hypothesis depends on the characterization of the observational uncertainties; we undertake a thorough review of this question. For main-field models, the primary source of uncertainty comes from the crustal field. We base our analysis on one of Jackson's (1994) statistical models of the crustal magnetization, adjusted to bring it into better conformity with our data set. The noise model permits us to take into account the correlations between the measurements and requires that a different weighting be given to horizontal and vertical components. It also indicates that the observations should be fit more closely than has been the practice heretofore. We apply the revised method to Magsat data from 1980 and survey and observatory data from 1915.5, two data sets believed to be particularly difficult to reconcile with the frozen-flux hypothesis. We compute a pair of simple, flux-conserving models that fit the averaged data from each epoch. We therefore conclude that present knowledge of the geomagnetic fields of 1980 and 1915.5 is consistent with the frozen-flux hypothesis.

Johnson, CL, Constable CG.  1997.  The time-averaged geomagnetic field: global and regional biases for 0-5 Ma. Geophysical Journal International. 131:643-+.   10.1111/j.1365-246X.1997.tb06604.x   AbstractWebsite

Palaeodirectional data from lava flows and marine sediments provide information about the long-term structure and variability in the geomagnetic held. We present a detailed analysis of the internal consistency and reliability of global compilations of sediment and lava-flow data. Time-averaged field models are constructed for normal and reverse polarity periods for the past 5 Ma, using the combined data sets. Non-zonal models are required to satisfy the lava-flow data, but not those from sediments alone. This is in part because the sediment data are much noisier than those from lavas, but is also a consequence of the site distributions and the way that inclination data sample the geomagnetic field generated in the Earth's core. Different average held configurations for normal and reverse polarity periods are consistent with the palaeomagnetic directions; however, the differences are insignificant relative to the uncertainty in the average field models. Thus previous inferences of non-antipodal normal and reverse polarity field geometries will need to be re-examined using recently collected high-quality palaeomagnetic data. Our new models indicate that current global sediment and lava-flow data sets combined do not permit the unambiguous detection of northern hemisphere flux lobes in the 0-5 Ma time-averaged field, highlighting the need for the collection of additional high-latitude palaeomagnetic data. Anomalous time-averaged held structure is seen in the Pacific hemisphere centred just south of Hawaii. The location of the anomaly coincides with heterogeneities in the lower mantle inferred from seismological data. The seismic observations can be partly explained by lateral temperature variations; however, they also suggest the presence of lateral compositional variations and/or the presence of partial melt. The role of such heterogeneities in influencing the geomagnetic held observed at the Earth's surface remains an unresolved issue, requiring higher-resolution time-averaged geomagnetic field models, along with the integration of future results from seismology, mineral physics and numerical simulations.

Johnson, CL, Constable CG.  1995.  The Time-Averaged Geomagnetic-Field As Recorded By Lava Flows Over The Past 5 Million-Years. Geophysical Journal International. 122:489-519.   10.1111/j.1365-246X.1995.tb07010.x   AbstractWebsite

A recently compiled lava flow data base spanning the last 5 million years is used to investigate properties of the time-averaged geomagnetic field. More than 90 per cent of the power in the palaeofield can be accounted for by a geocentric axial dipole; however, there are significant second-order structures in the held. Declination and inclination anomalies for the new data base indicate that the main second-order signal is the 'far-sided' effect, and there is also evidence for non-zonal structure. VGP (virtual geomagnetic pole) latitude distributions indicate that, over the last 5 million years, normal and reverse polarity morphologies are different, and that any changes in the normal polarity field morphology are undetectable, given the present data distribution. Regularized non-linear inversions of the palaeomagnetic directions support all these observations. We test the hypothesis that zonal models for the time-averaged field are adequate to describe the data and find that they are not. Non-zonal models are needed to fit the data to within the required tolerance level. Normal and reverse polarity held models obtained are significantly different. Field models obtained for the Brunhes epoch data alone are much smoother than those obtained from combining an the normal polarity data; simulations indicate that these differences can be explained by the less extensive data distribution for the Brunhes epoch. The field model for all of the normal polarity data (LN1) contains features observed in the historical field maps, although the details differ. LN1 suggests that, although the two northern hemisphere flux lobes observed in the historical field are stationary to a first-order approximation, they do show changes in position and amplitude. A. third, less pronounced flux lobe is observed in LN1 over central Europe. The lack of structure ih the southern hemisphere is due in part to the paucity of data. Jackknife estimates of the field models for different subsets of the data suggest that a few sites contribute significant structure to the final field models. More conservative estimates of the time-averaged field morphology are obtained by removing these sites.