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Korte, M, Constable C.  2011.  Improving geomagnetic field reconstructions for 0-3 ka. Physics of the Earth and Planetary Interiors. 188:247-259.   10.1016/j.pepi.2011.06.017   AbstractWebsite

Global geomagnetic field reconstructions on millennial time scales can be based on comprehensive paleomagnetic data compilations but, especially for older data, these still suffer from limitations in data quality and age controls as well as poor temporal and spatial coverage. Here we present updated global models for the time interval 0-3 ka where additions to the data basis mainly impact the South-East Asian, Alaskan, and Siberian regions. We summarize recent progress in millennial scale modelling, documenting the cumulative results from incremental modifications to the standard algorithms used to produce regularized time-varying spherical harmonic models spanning 1000 BC to 1990 AD: from 1590 to 1990 AD gauss coefficients from the historical gufm1 model supplement the paleomagnetic information; in addition to absolute paleointensities, calibrated relative paleointensity data from sediments are now routinely included; iterative data rejection and recalibration of relative intensity records from sediments ensure stable results; bootstrap experiments to generate uncertainty estimates for the model take account of uncertainties in both age and magnetic elements and additionally assess the impact of sampling in both time and space. Based on averaged results from bootstrap experiments, taking account of data and age uncertainties, we distinguish more conservative model estimates CALS3k.nb representing robust field structure at the core-mantle boundary from relatively high resolution models CALS3k.n for model versions n = 3 and 4. We assess the impact of newly available data and modifications to the modelling method by comparing the previous CALS3k.3, the new CALS3k.4, and the conservative new model, CALS3k.4b. We conclude that with presently available data it is not feasible to produce a model that is equally suitable for relatively high-resolution field predictions at Earth's surface and robust reconstruction of field evolution, avoiding spurious structure, at the core-mantle boundary (CMB). We presently consider CALS3k.4 the best high resolution model and recommend the more conservative lower resolution version for studies of field evolution at the CMB. (C) 2011 Elsevier B.V. All rights reserved.

Cromwell, G, Tauxe L, Staudigel H, Constable CG, Koppers AAP, Pedersen RB.  2013.  In search of long-term hemispheric asymmetry in the geomagnetic field : Results from high northern latitudes. Geochemistry Geophysics Geosystems. 14:3234-3249.   10.1002/ggge.20174   AbstractWebsite

Investigations of the behavior of the geomagnetic field on geological timescales rely on globally distributed data sets from dated lava flows. We present the first suitable data from the Arctic region, comprising 37 paleomagnetic directions from Jan Mayen (71 degrees N, 0.2-461 ka) and Spitsbergen (79 degrees N, 1-9.2 Ma) and five paleointensity results. Dispersion of the Arctic virtual geomagnetic poles over the last 2 Ma (27.34.0 degrees) is significantly lower than that from published Antarctic data sets (32.15.0 degrees). Arctic average virtual axial dipole moment (76.824.3 ZAm(2)) is high in comparison to Antarctica over the same time interval (34.88.2 ZAm(2)), although the data are still too sparse in the Arctic to be definitive. These data support a long-lived hemispheric asymmetry of the magnetic field, contrasting higher, more stable fields in the north with lower average strength and more variable field directions in the south. Such features require significant non-axial-dipole contributions over 10(5)-10(6) years.

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.

Davies, CJ, Constable CG.  2014.  Insights from geodynamo simulations into long-term geomagnetic field behaviour. Earth and Planetary Science Letters. 404:238-249.   10.1016/j.epsl.2014.07.042   AbstractWebsite

Detailed knowledge of the long-term spatial configuration and temporal variability of the geomagnetic field is lacking because of insufficient data for times prior to 10 ka. We use realisations from suitable numerical simulations to investigate three important questions about stability of the geodynamo process: is the present field representative of the past field; does a time-averaged field actually exist; and, supposing it exists, how long is needed to define such a field. Numerical geodynamo simulations are initially selected to meet existing criteria for morphological similarity to the observed magnetic field. A further criterion is introduced to evaluate similarity of long-term temporal variations. Allowing for reasonable uncertainties in the observations, observed and synthetic axial dipole moment frequency spectra for time series of order a million years in length should be fit by the same power law model. This leads us to identify diffusion time as the appropriate time scaling for such comparisons. In almost all simulations, intervals considered to have good morphological agreement between synthetic and observed field are shorter than those of poor agreement. The time needed to obtain a converged estimate of the time-averaged field was found to be comparable to the length of the simulation, even in non-reversing models, suggesting that periods of stable polarity spanning many magnetic diffusion times are needed to obtain robust estimates of the mean dipole field. Long term field variations are almost entirely attributable to the axial dipole; nonzonal components converge to long-term average values on relatively short timescales (15-20 kyr). In all simulations, the time-averaged spatial power spectrum is characterised by a zigzag pattern as a function of spherical harmonic degree, with relatively higher power in odd degrees than in even degrees. We suggest that long-term spatial characteristics of the observed field may emerge on averaging times that are within reach for the next generation of global time-varying paleomagnetic field models. (C) 2014 Elsevier B.V. All rights reserved.

Jackson, A, Parker RL, Sambridge M, Constable C, Wolf AS.  2018.  The inverse problem of unpolarized infrared spectroscopy of geological materials: Estimation from noisy random sampling of a quadratic form. American Mineralogist. 103:1176-1184.   10.2138/am-2018-6152   AbstractWebsite

We address the problem of unpolarized light spectroscopy of geological materials. Using infrared radiation, the aim of this technique is to learn about the absorbing species, such as hydroxyl. The use of unoriented samples leads to the need to perform a rigorous statistical analysis, so that the three principal absorbances of the crystal can be retrieved. We present here such an analysis based on a derivation of the probability density function for a single random measurement. Previous methods for retrieval of the absorbances are shown to be suboptimal, producing biased results that are sometimes even unphysical (e.g., negative estimates for an inherently positive quantity). The mathematical structure of the problem is developed to use the maximum likelihood estimation method, and we show how to optimize for the three absorbance parameters. This leads to good parameter retrieval on both synthetic and real data sets.

Constable, C, Korte M.  2006.  Is Earth's magnetic field reversing? Earth and Planetary Science Letters. 246:1-16.   10.1016/j.epsl.2006.03.038   AbstractWebsite

Earth's dipole field has been diminishing in strength since the first systematic observations of field intensity were made in the mid nineteenth century. This has led to speculation that the geomagnetic field might now be in the early stages of a reversal. In the longer term context of paleomagnetic observations it is found that for the current reversal rate and expected statistical variability in polarity interval length an interval as long as the ongoing 0.78 Myr Brunhes polarity interval is to be expected with a probability of less than 0.15, and the preferred probability estimates range from 0.06 to 0.08. These rather low odds might be used to infer that the next reversal is overdue, but the assessment is limited by the statistical treatment of reversals as point processes. Recent paleofield observations combined with insights derived from field modeling and numerical geodynamo simulations suggest that a reversal is not imminent. The current value of the dipole moment remains high compared with the average throughout the ongoing 0.78 Myr Brunhes polarity interval; the present rate of change in Earth's dipole strength is not anomalous compared with rates of change for the past 7 kyr; furthermore there is evidence that the field has been stronger on average during the Brunhes than for the past 160 Ma, and that high average field values are associated with longer polarity chrons. There is no evidence from recent millennial scale time-varying paleofield models to indicate that the field is entering a polarity transition. Nevertheless, it remains a reasonable supposition that the magnetic field will eventually reverse even though the time scale is unpredictable. A more immediate concern is that ongoing secular variation in the magnetic field may be expected to moderate the current high dipole strength on centennial to millennial time scales: it would not be surprising if it dropped substantially, returning closer to the average without necessarily reversing. This could have important consequences for space weather, and also highlights the need for improved understanding of the impact of geomagnetic field strength on the production rates of cosmogenic isotopes that are used to estimate past solar variability. (c) 2006 Elsevier B.V. All rights reserved.