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Journal Article
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.

Tauxe, L, Constable C, Johnson CL, Koppers AAP, Miller WR, Staudigel H.  2003.  Paleomagnetism of the southwestern USA recorded by 0-5 Ma igneous rocks. Geochemistry Geophysics Geosystems. 4   10.1029/2002gc000343   AbstractWebsite

The issue of permanent nondipole contributions to the time-averaged field lies at the very heart of paleomagnetism and the study of the ancient geomagnetic field. In this paper we focus on paleomagnetic directional results from igneous rocks of the southwestern U. S. A. in the age range 0-5 Ma and investigate both the time-averaged field and its variability about the mean value. Several decades of work in the southwestern United States have resulted in the publication of paleomagnetic data from over 800 individual paleomagnetic sites. As part of a new investigation of the San Francisco Volcanics, we collected paleomagnetic samples from 47 lava flows, many of which have been previously dated. The new data combined with published data are highly scattered. Contributions to the scatter were considered, and we find that removal of data sets from tectonically active areas and judicious selection according to Fisher's [1953] precision parameter results in an axially symmetric data distribution with normal and reverse modes that are indistinguishable from antipodal. Monte Carlo simulations suggest that a minimum of 5 samples per site are needed to estimate the precision parameter sufficiently accurately to allow its use as a determinant of data quality. Numerical simulations from statistical paleosecular variation models indicate the need for several hundred paleomagnetic sites to get an accurate determination of the average field direction and are also used to investigate the directional bias that results from averaging unit vectors rather than using the full field vector. Average directions for the southwestern U. S. A. show small deviations from a geocentric axial dipole field, but these cannot be considered statistically significant. Virtual geomagnetic pole (VGP) dispersions are consistent with those from globally distributed observations analyzed by McElhinny and McFadden [1997]. However, a systematic investigation of the effect of imposing a cutoff on VGPs with large deviations from the geographic axis indicates that while it may reduce bias in calculating the average direction, such a procedure can result in severe underestimates of the variance in the geomagnetic field. A more satisfactory solution would be to use an unbiased technique for joint estimation of the mean direction and variance of the field distribution.

Lawrence, KP, Constable CG, Johnson CL.  2006.  Paleosecular variation and the average geomagnetic field at +/- 20 degrees latitude. Geochemistry Geophysics Geosystems. 7   10.1029/2005gc001181   AbstractWebsite

[1] We assembled a new paleomagnetic directional data set from lava flows and thin dikes for four regions centered on +/-20 degrees latitude: Hawaii, Mexico, the South Pacific, and Reunion. We investigate geomagnetic field behavior over the past 5 Myr and address whether geographical differences are recorded by our data set. We include inclination data from other globally distributed sites with the +/-20 degrees data to determine the best fitting time-averaged field (TAF) for a two-parameter longitudinally symmetric (zonal) model. Values for our model parameters, the axial quadrupole and octupole terms, are 4% and 6% of the axial dipole, respectively. Our estimate of the quadrupole term is compatible with most previous studies of deviations from a geocentric axial dipole (GAD) field. Our estimated octupole term is larger than that from normal polarity continental and igneous rocks, and oceanic sediments, but consistent with that from reversed polarity continental and igneous rocks. The variance reduction compared with a GAD field is similar to 12%, and the remaining signal is attributed to paleosecular variation (PSV). We examine PSV at +/-20 degrees using virtual geomagnetic pole (VGP) dispersion and comparisons of directional distributions with simulations from two statistical models. Regionally, the Hawaii and Reunion data sets lack transitional magnetic directions and have similar inclination anomalies and VGP dispersion. In the Pacific hemisphere, Hawaii has a large inclination anomaly, and the South Pacific exhibits high PSV. The deviation of the TAF from a GAD contradicts earlier ideas of a "Pacific dipole window,'' and the strong regional PSV in the South Pacific contrasts with the generally low secular variation found on short timescales. The TAF and PSV at Hawaii and Reunion are distinct from values for the South Pacific and Mexico, demonstrating the need for time-averaged and paleosecular variation models that can describe nonzonal field structures. Investigations of zonal statistical PSV models reveal that recent models are incompatible with the empirical +/-20 degrees directional distributions and cannot fit the data by simply adjusting relative variance contributions to the PSV. The +/-20 degrees latitude data set also suggests less PSV and smaller persistent deviations from a geocentric axial dipole field during the Brunhes.

Johnson, CL, Constable CG, Tauxe L, Barendregt R, Brown LL, Coe RS, Layer P, Mejia V, Opdyke ND, Singer BS, Staudigel H, Stone DB.  2008.  Recent investigations of the 0-5 Ma geomagnetic field recorded by lava flows. Geochemistry Geophysics Geosystems. 9   10.1029/2007gc001696   AbstractWebsite

We present a synthesis of 0 - 5 Ma paleomagnetic directional data collected from 17 different locations under the collaborative Time Averaged geomagnetic Field Initiative ( TAFI). When combined with regional compilations from the northwest United States, the southwest United States, Japan, New Zealand, Hawaii, Mexico, South Pacific, and the Indian Ocean, a data set of over 2000 sites with high quality, stable polarity, and declination and inclination measurements is obtained. This is a more than sevenfold increase over similar quality data in the existing Paleosecular Variation of Recent Lavas (PSVRL) data set, and has greatly improved spatial sampling. The new data set spans 78 degrees S to 53 degrees N, and has sufficient temporal and spatial sampling to allow characterization of latitudinal variations in the time-averaged field (TAF) and paleosecular variation (PSV) for the Brunhes and Matuyama chrons, and for the 0 - 5 Ma interval combined. The Brunhes and Matuyama chrons exhibit different TAF geometries, notably smaller departures from a geocentric axial dipole field during the Brunhes, consistent with higher dipole strength observed from paleointensity data. Geographical variations in PSV are also different for the Brunhes and Matuyama. Given the high quality of our data set, polarity asymmetries in PSV and the TAF cannot be attributed to viscous overprints, but suggest different underlying field behavior, perhaps related to the influence of long-lived core-mantle boundary conditions on core flow. PSV, as measured by dispersion of virtual geomagnetic poles, shows less latitudinal variation than predicted by current statistical PSV models, or by previous data sets. In particular, the Brunhes data reported here are compatible with a wide range of models, from those that predict constant dispersion as a function of latitude to those that predict an increase in dispersion with latitude. Discriminating among such models could be helped by increased numbers of low-latitude data and new high northern latitude sites. Tests with other data sets, and with simulations, indicate that some of the latitudinal signature previously observed in VGP dispersion can be attributed to the inclusion of low-quality, insufficiently cleaned data with too few samples per site. Our Matuyama data show a stronger dependence of dispersion on latitude than the Brunhes data. The TAF is examined using the variation of inclination anomaly with latitude. Best fit two- parameter models have axial quadrupole contributions of 2 - 4% of the axial dipole term, and axial octupole contributions of 1 - 5%. Approximately 2% of the octupole signature is likely the result of bias incurred by averaging unit vectors.