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A
Amit, H, Korte M, Aubert J, Constable C, Hulot G.  2011.  The time-dependence of intense archeomagnetic flux patches. Journal of Geophysical Research-Solid Earth. 116   10.1029/2011jb008538   AbstractWebsite

The long-term temporal behavior of intense geomagnetic flux patches at the core-mantle boundary and the relation with lower mantle lateral heterogeneity are under debate. We apply an algorithm to detect centers of intense flux patches and track their time-evolution in a recent archeomagnetic field model in order to study the kinematics of such intense magnetic flux patches on millennial timescale. We find that most intense flux patches appear near the edge of the tangent cylinder. Quasi-stationary periods with small oscillations of patches occur more than drifts. Detailed comparison of the archeomagnetic patches' behavior with that seen in numerical dynamos with tomographic heat flux boundary conditions suggests that core-mantle thermal coupling could be the cause of a statistical preference for some longitudes on the long term, which does not exclude significant time spent away from the preferred longitudes. This could explain the roughly coincident locations of high-latitude patches in the historical geomagnetic field with that of the time-average paleomagnetic field together with the much weaker patches intensity in the latter. Alternating eastward and westward drifts are also observed. The drifts are more westward than eastward, especially in the southern hemisphere, indicating that the time-average zonal core flow may also be driven by core-mantle thermal coupling. An average patch lifetime of similar to 300 years is found, which we hypothesize may indicate the vortex lifetime in the outer core.

Avery, MS, Constable CG, Davies CJ, Gubbins D.  2019.  Spectral methods for analyzing energy balances in geodynamo simulations. Physics of the Earth and Planetary Interiors. 286:127-137.   10.1016/j.pepi.2018.10.002   AbstractWebsite

The geomagnetic field displays complicated variations over a broad range of frequencies. These variations can be decomposed by frequency and linked to physical processes using frequency domain spectral methods. These spectral methods are well developed but have not previously been applied to study the energy balance of geodynamo simulations. We illustrate their potential by analyzing output from numerical dynamo simulations that have previously been studied for their apparently Earth-like properties. We show that high coherence between variations in axial dipole energy at the outer boundary of the simulation and total magnetic energy within the fluid shell occur at frequencies below similar to 0.1 kyr(-1). This suggests that paleomagnetically-observable signals with periods exceeding 10 kyrs contain information about magnetic energy changes in the bulk core. We then use spectral analysis to investigate differences in the rate of growth and decay of the axial dipole field. This behaviour, characterised by rapid growth and slow decay, is observed when signals with frequencies higher than 0.03 kyr(-1) have been filtered out. The origin of this asymmetric growth and decay is assessed using coherence spectra between rates of change in kinetic and magnetic energy, ohmic and viscous dissipation, and work done by the buoyancy and Lorentz forces. We show that asymmetry is associated with an imbalance between ohmic dissipation and work done by the Lorentz force; when changes in magnetic energy are more coherent with ohmic dissipation the field grows rapidly and decay slowly. Variations in Ohmic dissipation reflect changes in field strength in our models, while changes in viscous dissipation are associated with amplitude fluctuations of the large-scale flow that exists on millennial timescales. Our work shows that spectral analysis coupling observable and global products of the dynamo process can elucidate the physical origin of periodic processes occurring on timescales exceeding 10 kyrs.

Avery, MS, Gee JS, Constable CG.  2017.  Asymmetry in growth and decay of the geomagnetic dipole revealed in seafloor magnetization. Earth and Planetary Science Letters. 467:79-88.   10.1016/j.epsl.2017.03.020   AbstractWebsite

Geomagnetic intensity fluctuations provide important constraints on time-scales associated with dynamical processes in the outer core. PADM2M is a reconstructed time series of the 0-2 Ma axial dipole moment (ADM). After smoothing to reject high frequency variations PADM2M's average growth rate is larger than its decay rate. The observed asymmetry in rates of change is compatible with longer term diffusive decay of the ADM balanced by advective growth on shorter time scales, and provides a potentially useful diagnostic for evaluating numerical geodynamo simulations. We re-analyze the PADM2M record using improved low-pass filtering to identify asymmetry and quantify its uncertainty via bootstrap methods before applying the new methodology to other kinds of records. Asymmetry in distribution of axial dipole moment derivatives is quantified using the geomagnetic skewness coefficient, sg. A positive value indicates the distribution has a longer positive tail and the average growth rate is greater than the average decay rate. The original asymmetry noted by Ziegler and Constable (2011) is significant and does not depend on the specifics of the analysis. A long-term record of geomagnetic intensity should also be preserved in the thermoremanent magnetization of oceanic crust recovered by inversion of stacked profiles of marine magnetic anomalies. These provide an independent means of verifying the asymmetry seen in PADM2M. We examine three near bottom surveys: a 0 to 780 ka record from the East Pacific Rise at 19 degrees S, a 0 to 5.2 Ma record from the Pacific Antarctic Ridge at 51 degrees S, and a chron C4Ar-C5r (9.3-11.2 Ma) record from the NE Pacific. All three records show an asymmetry similar in sense to PADM2M with geomagnetic skewness coefficients, s(g) > 0. Results from PADM2M and C4Ar-C5r are most robust, reflecting the higher quality of these geomagnetic records. Our results confirm that marine magnetic anomalies can carry a record of the asymmetric geomagnetic field behavior first found for 0-2 Ma in PADM2M, and show that it was also present during the earlier time interval from 9.3-11.2 Ma. (C) 2017 The Authors. Published by Elsevier B.V.

B
Backus, G, Parker RL, Constable C.  1996.  Foundations of geomagnetism. :xiv,369p.., Cambridge England ; New York: Cambridge University Press AbstractWebsite

The main magnetic field of the Earth is a complex phenomenon. To understand its origins in the fluid of the Earth's core, and how it changes in time requires a variety of mathematical and physical tools. This book presents the foundations of geomagnetism, in detail and developed from first principles. The book is based on George Backus' courses for graduate students at the University of California, San Diego. The material is mathematically rigorous, but is logically developed and has consistent notation, making it accessible to a broad range of readers. The book starts with an overview of the phenomena of interest in geomagnetism, and then goes on to deal with the phenomena in detail, building the necessary techniques in a thorough and consistent manner. Students and researchers will find this book to be an invaluable resource in the appreciation of the mathematical and physical foundations of geomagnetism.

Backus, G, Parker RL, Constable C.  2005.  Foundations of geomagnetism. :xiv,369p.., Cambridge ; New York: Cambridge University Press AbstractWebsite
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Brown, MC, Donadini F, Korte M, Nilsson A, Korhonen K, Lodge A, Lengyel SN, Constable CG.  2015.  GEOMAGIA50.v3: 1. general structure and modifications to the archeological and volcanic database. Earth Planets and Space. 67:1-31.   10.1186/s40623-015-0232-0   AbstractWebsite

Background: GEOMAGIA50.v3 is a comprehensive online database providing access to published paleomagnetic, rock magnetic, and chronological data from a variety of materials that record Earth's magnetic field over the past 50 ka. Findings: Since its original release in 2006, the structure and function of the database have been updated and a significant number of data have been added. Notable modifications are the following: (1) the inclusion of additional intensity, directional and metadata from archeological and volcanic materials and an improved documentation of radiocarbon dates; (2) a new data model to accommodate paleomagnetic, rock magnetic, and chronological data from lake and marine sediments; (3) a refinement of the geographic constraints in the archeomagnetic/volcanic query allowing selection of particular locations; (4) more flexible methodological and statistical constraints in the archeomagnetic/volcanic query; (5) the calculation of predictions of the Holocene geomagnetic field from a series of time varying global field models; (6) searchable reference lists; and (7) an updated web interface. This paper describes general modifications to the database and specific aspects of the archeomagnetic and volcanic database. The reader is referred to a companion publication for a description of the sediment database. Conclusions: The archeomagnetic and volcanic part of GEOMAGIA50.v3 currently contains 14,645 data (declination, inclination, and paleointensity) from 461 studies published between 1959 and 2014. We review the paleomagnetic methods used to obtain these data and discuss applications of the data within the database. The database continues to expand as legacy data are added and new studies published. The web-based interface can be found at http://geomagia.gfz-potsdam.de

Brown, MC, Donadini F, Nilsson A, Panovska S, Frank U, Korhonen K, Schuberth M, Korte M, Constable CG.  2015.  GEOMAGIA50.v3: 2. A new paleomagnetic database for lake and marine sediments. Earth Planets and Space. 67   10.1186/s40623-015-0233-z   AbstractWebsite

Background: GEOMAGIA50.v3 for sediments is a comprehensive online database providing access to published paleomagnetic, rock magnetic, and chronological data obtained from lake and marine sediments deposited over the past 50 ka. Its objective is to catalogue data that will improve our understanding of changes in the geomagnetic field, physical environments, and climate. Findings: GEOMAGIA50.v3 for sediments builds upon the structure of the pre-existing GEOMAGIA50 database for magnetic data from archeological and volcanic materials. A strong emphasis has been placed on the storage of geochronological data, and it is the first magnetic archive that includes comprehensive radiocarbon age data from sediments. The database will be updated as new sediment data become available. Conclusions: The web-based interface for the sediment database is located at http://geomagia.gfz-potsdam.de/geomagiav3/SDquery.php. This paper is a companion to Brown et al. (Earth Planets Space doi:10.1186/s40623-015-0232-0,2015) and describes the data types, structure, and functionality of the sediment database.

Buffett, BA, Ziegler L, Constable CG.  2013.  A stochastic model for palaeomagnetic field variations. Geophysical Journal International. 195:86-97.   10.1093/gji/ggt218   AbstractWebsite

Regeneration of the Earth's magnetic field by convection in the liquid core produces a broad spectrum of time variation. Relative palaeointensity measurements in marine sediments provide a detailed record over the past 2 Myr, but an explicit reconstruction of the underlying dynamics is not feasible. A more practical alternative is to construct a stochastic model from estimates of the virtual axial dipole moment. The deterministic part of the model (drift term) describes time-averaged behaviour, whereas the random part (diffusion term) characterizes complex interactions over convective timescales. We recover estimates of the drift and diffusion terms from the SINT2000 model of Valet et al. and the PADM2M model of Ziegler et al. The results are used in numerical solutions of the Fokker-Planck equation to predict statistical properties of the palaeomagnetic field, including the average rates of magnetic reversals and excursions. A physical interpretation of the stochastic model suggests that the timescale for adjustments in the axial dipole moment is set by the dipole decay time tau(d). We obtain tau(d) = 29 kyr from the stochastic models, which falls within the expected range for the Earth's core. We also predict the amplitude of convective fluctuations in the core, and establish a physical connection to the rates of magnetic reversals and excursions. Chrons lasting longer than 10 Myr are unlikely under present-day conditions. However, long chrons become more likely if the diffusion term is reduced by a factor of 2. Such a change is accomplished by reducing the velocity fluctuations in the core by a factor of root 2, which could be attributed to a shift in the spatial pattern of heat flux from the core or a reduction in the total core heat flow.

C
Clement, BM, Constable CG.  1991.  Polarity Transitions, Excursions and Paleosecular Variation of the Earths Magnetic-Field. Reviews of Geophysics. 29:433-442. AbstractWebsite
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Constable, CG, Tauxe L, Parker RL.  1998.  Analysis of 11 Myr of geomagnetic intensity variation. Journal of Geophysical Research-Solid Earth. 103:17735-17748.   10.1029/98jb01519   AbstractWebsite

We have conducted a detailed exploratory analysis of an II million year long almost continuous record of relative geomagnetic paleointensity from a sediment core acquired on Deep Sea Drilling Project Leg 73, at Site 522 in the South Atlantic. We assess the quality of the paleointensity record using spectral methods and conclude that the relative intensity record is minimally influenced by climate variations. Isothermal remanence is shown to be the most effective normalizer for these data, although both susceptibility and anhysteretic remanence are also adequate. Statistical analysis shows that the paleointensity variations follow a gamma distribution, and are compatible with predictions from modified paleosecular variation models and global absolute paleointensity data. When subdivided by polarity interval, the variability in paleointensity is proportional to the average, and further, the average is weakly correlated with interval length. Spectral estimates for times from 28.77 until 22.74 Ma, when the reversal rate is about 4 Myr(-1), are compatible with a Poisson model in which the spectrum of intensity variations is dominated by the reversal process in the frequency range 1-50 Mgr(-1) In contrast, between 34.7 and 29.4 Ma, when the reversal rate is about 1.6 Myr(-1), the spectra indicate a different secular variation regime. The magnetic field is stronger, and more variable, and a strong peak in the spectrum occurs at about 8 Myr(-1). This peak magi be a reflection of the same signal as recorded by the small variations known as tiny wiggles seen in marine magnetic anomaly profiles.

Constable, C.  1992.  Link Between Geomagnetic Reversal Paths And Secular Variation Of The Field Over The Past 5 MY. Nature. 358:230-233.   10.1038/358230a0   AbstractWebsite

PALAEOMAGNETIC records provide information about the behaviour of the geomagnetic field during reversals1,2. Existing records are incompatible with transitional field configurations that are either entirely dipolar or entirely zonal (dependent only on latitude)3,4. Recent compilations5-8 have indicated that the transitional paths of virtual geomagnetic poles (VGPs) for the past few reversals are located preferentially within two antipodal longitudinal bands, suggesting that simple but non-zonal field configurations dominate during reversals. Here I point out that one of the longitudinal bands coincides with that expected from the reversal of a non-axial-dipole field exactly like that present today; the other requires only a sign change in the non-axial-dipole terms of today's field. Evidence for persistent non-zonal contributions to the field has generally9-13 (but not always14,15) been regarded as not statistically significant in the light of poor data distributions. I show here that a non-zonal bias, similar to that observed in reversal data, is evident in data on secular variation of the field over the past 5 Myr, even after normalization according to site locations. These results suggest that the time-averaged field does indeed contain persistent (but not constant) non-zonal contributions.

Constable, CG.  1988.  Parameter-Estimation In Non-Gaussian Noise. Geophysical Journal-Oxford. 94:131-142.   10.1111/j.1365-246X.1988.tb03433.x   AbstractWebsite

Least squares (LS) estimation of model parameters is widely used in geophysics. If the data errors are Gaussian and independent the LS estimators will be maximum likelihood (ML) estimators and will be unbiased and of minimum variance. However, if the noise is not Gaussian, e.g. if the data are contaminated by extreme outliers, LS fitting will result in parameter estimates which may be biased or grossly inaccurate. When the probability distribution of the errors is known it is possible, using the maximum likelihood method, to obtain consistent and efficient (minimum variance) estimates of parameters. In some cases the distribution of the noise may be determined empirically, and the resulting distribution used in the ML estimation. A procedure for doing this is described here. Hourly values of geomagnetic observatory data are used to illustrate the technique. These data sets contain a number of periodic components, whose amplitudes and phases are geophysically interesting. Geomagnetic storms and other phenomena in the record make the noise distribution long-tailed, asymmetric and variable with location. Using an iterative procedure, one can model the form of these distributions using smoothing splines. For these data ML estimation yields quite different results from standard robust and LS procedures. The technique has the potential for widespread application to other problems involving the recovery of a known form of signal from non-Gaussian noise.

Constable, C.  2007.  Non-dipole field. Encyclopedia of geomagnetism and paleomagnetism. ( Gubbins D, Herrero-Bervera E, Eds.).:701-704., Dordrecht: Springer Abstract
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Constable, C, Johnson C.  2005.  A paleomagnetic power spectrum. Physics of the Earth and Planetary Interiors. 153:61-73.   10.1016/j.pepi.2005.03.015   AbstractWebsite

We construct a power spectrum of geomagnetic dipole moment variations or their proxies that spans the period range from some tens of million down to about 100 years. Empirical estimates of the spectrum are derived from the magnetostratigraphic time scale, from marine sediment relative paleointensity records, and from a time varying paleomagnetic field model for the past 7 kyr. The spectrum has the most power at long periods, reflecting the influence of geomagnetic reversals and in general decreases with increasing frequency (decreasing period). The empirical spectrum is compared with predictions from simple models. Discrepancies between the observed and predicted spectra are discussed in the context of: (i) changes in reversal rate, (ii) overall average reversal rate, (iii) cryptochrons, (iv) the time taken for a reversal to occur, and (v) long term paleosecular variations and average estimates of the field strength and variance from other sources. (c) 2005 Elsevier B.V. All rights reserved.

Constable, C, Parker R.  1991.  Deconvolution of Long-Core Paleomagnetic Measurements - Spline Therapy for the Linear Problem. Geophysical Journal International. 104:453-468.   10.1111/j.1365-246X.1991.tb05693.x   AbstractWebsite

The magnetization of long cores of sedimentary material is often measured in a pass-through magnetometer, whose output is the convolution of the desired function with the broad impulse response of the system. Because of inevitable measurement noise and the inherent poor conditioning of the inverse problem, any attempt to estimate the true magnetization function from the observations must avoid unnecessary amplification of small-scale features which would otherwise dominate the model with deceptively large undulations. We propose the construction of the smoothest possible magnetization model satisfying the measured data to within the observational error. By means of a cubic spline basis in the representations of both the unknown magnetization and the empirically measured response, we facilitate the imposition of maximum smoothness on the unknown magnetization. For our purposes, the smoothest model is the one with the smallest 2-norm of the second derivative, the same criterion used in the construction of cubic spline interpolators. The approach is tested on a marine core that was subsequently sectioned and measured in centimetre-sized individual specimens, with highly satisfactory results. An empirical estimate of the resolution of the method indicates a three-fold improvement in the processed record over the original signal. We illuminate the behaviour of the numerical scheme by showing the relation between our smoothness-maximizing procedure and a more conventional filtering approach. Our solution can indeed be approximated by convolution with a special set of weights, although the approximation may be poor near the ends of the core. In an idealized system we study the question of convergence of the deconvolution process, by whether the model magnetization approaches the true one when the experimental error and other system parameters are held constant, while the spacing between observations is allowed to become arbitrarily small. We find our procedure does in fact converge (under certain conditions) but only at a logarithmic rate. This suggests that further significant improvement in resolution cannot be achieved by increased measurement density or enhanced observational accuracy.

Constable, CG, McElhinny MW.  1985.  Holocene Geomagnetic Secuar Variation Records From Northeastern Australian Lake-Sediments. Geophysical Journal of the Royal Astronomical Society. 81:103-120.   10.1111/j.1365-246X.1985.tb01353.x   AbstractWebsite

Secular variation records have been obtained from cores from Lakes Barrine and Eacham, two north-eastern Australian volcanic crater lakes. The results from several cores have been stratigraphically correlated and then stacked and smoothed. The chronology provided by radiocarbon dating indicates that the Lake Eacham sequence spans the last 5700 calendar years. The time-scale for the Lake Barrine record is less well constrained but it appears to cover about 1600 to 16200 yr BP.VGP paths for the sites show two periods of anticlockwise motion between about 5710 and 3980 BP and 10500 and 8800 BP. These times correspond to periods of anticlockwise motion in south-eastern Australian records (Barton & McElhinny) and Argentine records (Creer et al.), to within the uncertainties of the assigned time-scales.

Constable, C, Tauxe L.  1990.  The Bootstrap for Magnetic Susceptibility Tensors. Journal of Geophysical Research-Solid Earth and Planets. 95:8383-8395.   10.1029/JB095iB06p08383   AbstractWebsite

In studies of the anisotropy of susceptibility or remanence of paleomagnetic samples it is conventional to specify the anisotropy in terms of the parameters of the anisotropy ellipsoids, namely the directions of the principal axes of the ellipsoid and their associated eigenvalues. Confidence intervals for these parameters have in the past often been estimated by using a linearization scheme to propagate the effect of small changes through the eigenvalue decomposition. The validity of these approximations is explored using a Monte-Carlo simulation from measurements that are presumed normally distributed, showing that there are circumstances in which the linearization scheme gives confidence intervals that are much too small. Q-Q plots indicate that the common assumption that the noise in the measurements is Gaussian does not always hold. Because of these shortcomings in the conventional technique we propose using a bootstrap resampling scheme to find empirically the distribution of uncertainties in the results. Confidence intervals for the eigenvalues are found directly from their empirical distributions. For the principal axes, approximate elliptical regions of confidence on the unit sphere are parameterized in terms of the Kent or FB5 distribution. The number of modes observed in the distribution of eigenvalues obtained by bootstrapping is used to classify the shape of the susceptibility ellipsoid as spherical, oblate, prolate or triaxial. The empirical nature of the bootstrap technique allows the extension of the analysis of uncertainties to parameters derived from the principal susceptibilities, such as percentage anisotropy or shape factor.

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.

Constable, CG, Parker RL, Stark PB.  1993.  Geomagnetic-Field Models Incorporating Frozen-Flux Constraints. Geophysical Journal International. 113:419-433.   10.1111/j.1365-246X.1993.tb00897.x   AbstractWebsite

Techniques for modelling the geomagnetic field at the surface of Earth's core often penalize contributions at high spherical harmonic degrees to reduce the effect of mapping crustal fields into the resulting field model at the core-mantle boundary (CMB). Ambiguity in separating the observed field into crustal and core contributions makes it difficult to assign error bounds to core field models, and this makes it hard to test hypotheses that involve pointwise values of the core field. The frozen-flux hypothesis, namely that convective terms dominate diffusive terms in the magnetic-induction equation, requires that the magnetic flux through every patch on the core surrounded by a zero contour of the radial magnetic field remains constant, although the shapes, areas and locations (but not the topology) of these patches may change with time. Field models exactly satisfying the conditions necessary for the hypothesis have not yet been constructed for the early part of this century. We show that such models must exist, so testing the frozen-flux hypothesis becomes the question of whether the models satisfying it are geophysically unsatisfactory on other grounds, for example because they are implausibly rough or complicated. We introduce an algorithm to construct plausible fleld models satisfying the hypothesis, and present such models for epochs 1945.5 and 1980. Our algorithm is based on a new parametrization of the field in terms of its radial component B(r) at the CMB. The model consists of values of B(r) at a finite set of points on the CMB, together with a rule for interpolating the values to other points. The interpolation rule takes the specified points to be the vertices of a spherical triangle tessellation of the CMB, with B(r) varying linearly in the gnomonic projections of the spherical triangles onto planar triangles in the planes tangent to the centroids of the spherical triangles. This parametrization of B(r) provides a direct means of constraining the integral invariants required by the frozen-flux hypothesis. Using this parametrization, we have constructed field models satisfying the frozen-flux hypothesis for epochs 1945.5 and 1980, while fitting observatory and survey data for 1945.5 and Magsat data for 1980. We use the better constrained 1980 CMB field model as a reference for 1945.5: we minimize the departure of the 1945.5 CMB field model from a regularized 1980 CMB field model, while constraining the 1945.5 model to have the same null-flux curves and flux through those curves as the 1980 model. The locations, areas and shapes of the curves are allowed to change. The resulting 1945.5 CMB field model is nearly as smooth as that for 1980, fits the data adequately, and satisfies the conditions necessary for the frozen-flux hypothesis.

Constable, CG, Parker RL.  1988.  Smoothing, Splines And Smoothing Splines - Their Application In Geomagnetism. Journal of Computational Physics. 78:493-508.   10.1016/0021-9991(88)90062-9   AbstractWebsite

We discuss the use of smoothing splines (SS) and least squares splines (LSS) in nonparametric regression on geomagnetic data. The distinction between smoothing splines and least squares splines is outlined, and it is suggested that in most cases the smoothing spline is, a preferable function estimate. However, when large data sets are involved, the smoothing spline may require a prohibitive amount of computation; the alternative often put forward when moderate or heavy smoothing is -desired is the least squares spline. This may not be capable of modeling the data adequately since the smoothness of the resulting function can be controlled only by the number and position of the knots. The computational efficiency of the least squares spline may be retained and its principal disadvantage overcome, by adding a penalty term in the square of the second derivative to the minimized functional. We call this modified form a penalized least squares spline, (denoted by PS throughout this work), and illustrate its use in the removal of secular trends in long observatory records of geomagnetic field components. We may compare the effects of smoothing splines, least squares splines, and penalized least squares splines by treating them as equivalent variable-kernel smoothers. As Silverman has shown, the kernel associated with the smoothing spline is symmetric and is highly localized with small negative sidelobes. The kernel for the least squares spline with the same fit to the data has large oscillatory sidelobes that extend far from the central region; it can be asymmetric even in the middle of the interval. For large numbers of data the penalized least squares spline can achieve essentially identical performance to that of a smoothing spline, but at a greatly reduced computational cost. The penalized spline estimation technique has potential widespread applicability in the analysis of geomagnetic and paleomagnetic data. It may be used for the removal of long term trends in data, when either the trend or the residual is of interest.

Constable, C.  2007.  Geomagnetic temporal spectrum. Encyclopedia of geomagnetism and paleomagnetism. ( Gubbins D, Herrero-Bervera E, Eds.).:353-355., Dordrecht: Springer Abstract
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Constable, CG, Tauxe L.  1987.  Paleointensity In The Pelagic Realm - Marine Sediment Data Compared With Archaeomagnetic And Lake Sediment Records. Geophysical Journal of the Royal Astronomical Society. 90:43-59.   10.1111/j.1365-246X.1987.tb00674.x   AbstractWebsite

Four box cores collected from the Ontong—Java plateau during the Eurydice expedition have been used to make relative geomagnetic palaeo-intensity measurements. Rock magnetic measurements on the sediments show that they are characterized by a uniform magnetic mineralogy, and that they are suitable for relative intensity estimates. These are obtained by normalizing the NRM by an ARM imparted in a low DC bias field. the palaeoceanographic event known as the preservation spike is used to establish a crude time-scale for the record so that it may be compared with other data from the same region, and also with global palaeointensity estimates. the marine sediment data are quite similar to Australian intensity data from lake sediments and archaeomagnetic sources, but as might be expected exhibit some obvious differences from the global record.

Constable, C.  2000.  On rates of occurrence of geomagnetic reversals. Physics of the Earth and Planetary Interiors. 118:181-193.   10.1016/s0031-9201(99)00139-9   AbstractWebsite

The magnetostratigraphic time scale provides a record of the occurrence of geomagnetic reversals. The temporal distribution of reversals may be modelled as the realization of an inhomogeneous renewal process; i.e., one in which the intensity, lambda(t), or reversal rate is a function of time. Variations in reversal rate occurring on time scales of tens of millions of years an believed to reflect changes in core-mantle boundary conditions influencing the structure of core flow and the field produced by the geodynamo. We present a new estimate for reversal rate variations as a function of time using nonparametric adaptive kernel density estimation and discuss the difficulties in making inferences on the basis of such estimates. Using a technique proposed by Hengartner and Stark (1992a; b; 1995), it is possible to compute confidence bounds on the temporal probability density function for geomagnetic reversals. The method allows the computation of a lower bound on the number of modes required by the observations, thus enabling a test of whether "bumps" are required features of the reversal rate function. Conservative 95% confidence intervals can then be calculated for the temporal location of a single mode or antimode of the probability density function. Using observations from the time interval 0-158 Ma, it is found that the derivative of the rate function must have changed sign at least once. The timing of this sign change is constrained to be between 152.56 and 22.46 Ma the 95% confidence level. Confidence bounds are computed for the reversal rate under the assumption that the observed reversals are a realization of an inhomogenous Poisson or other renewal process with an arbitrary monotonically increasing rate function from the end of the Cretaceous Normal Superchron (CNS) to the present, a zero rate during the CNS, and a monotonically decreasing rate function from M29R at 158 Ma to the onset of the CNS. It is unnecessary to invoke more than one sign change in the derivative of the rare function to fit the observations. There is no incompatibility between our results and a recent assertion that there is an asymmetry in average reversal rate prior to and after the CNS, when the CNS is assumed to be a period of zero reversal rate. Neither can we use our results to reject an alternative hypothesis that rates are essentially constant from 158 to 130 Ma, and from 25 Ma to the present. with an intermediate nonstationary segment. (C) 2000 Elsevier Science B.V. All rights reserved.

Constable, C, Tauxe L.  1996.  Towards absolute calibration of sedimentary paleointensity records. Earth and Planetary Science Letters. 143:269-274.   10.1016/0012-821x(96)00128-8   AbstractWebsite

Using relative paleointensity estimates derived from twelve globally distributed pelagic sediment cores, we assess whether they record a signal consistent with that expected from a dominant geocentric axial dipole, The cores span the Matuyama-Brunhes boundary and we normalize the observations by supposing that at the time the direction reverses the intensity low reflects only the non-axial-dipole contribution to the field. We further assume that this non-axial-dipole contribution to the field is invariant with geographic location. From absolute paleointensity compilations we estimate its size to be about 7.5 mu T; this supplies the calibration for the axial dipole signal away from the extreme low in intensity, The data predict the dipole field variation with latitude with similar accuracy to that observed in absolute paleointensity records, and show similar behavior when transformed to virtual axial dipole moments.

Constable, C.  2016.  Earth's electromagnetic environment. Surveys in Geophysics. 37:27-45.   10.1007/s10712-015-9351-1   AbstractWebsite

The natural spectrum of electromagnetic variations surrounding Earth extends across an enormous frequency range and is controlled by diverse physical processes. Electromagnetic (EM) induction studies make use of external field variations with frequencies ranging from the solar cycle which has been used for geomagnetic depth sounding through the 10-10 Hz frequency band widely used for magnetotelluric and audio-magnetotelluric studies. Above 10 Hz, the EM spectrum is dominated by man-made signals. This review emphasizes electromagnetic sources at 1 Hz and higher, describing major differences in physical origin and structure of short- and long-period signals. The essential role of Earth's internal magnetic field in defining the magnetosphere through its interactions with the solar wind and interplanetary magnetic field is briefly outlined. At its lower boundary, the magnetosphere is engaged in two-way interactions with the underlying ionosphere and neutral atmosphere. Extremely low-frequency (3 Hz-3 kHz) electromagnetic signals are generated in the form of sferics, lightning, and whistlers which can extend to frequencies as high as the VLF range (3-30 kHz).The roughly spherical dielectric cavity bounded by the ground and the ionosphere produces the Schumann resonance at around 8 Hz and its harmonics. A transverse resonance also occurs at 1.7-2.0 kHz arising from reflection off the variable height lower boundary of the ionosphere and exhibiting line splitting due to three-dimensional structure. Ground and satellite observations are discussed in the light of their contributions to understanding the global electric circuit and for EM induction studies.