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Cronin, M, Tauxe L, Constable C, Selkin P, Pick T.  2001.  Noise in the quiet zone. Earth and Planetary Science Letters. 190:13-30.   10.1016/s0012-821x(01)00354-5   AbstractWebsite

We have carried out a detailed paleomagnetic investigation of two stratigraphically overlapping sections from the Scaglia Bianca Formation (similar to 85-89.5 Ma) in the Umbria-Marche area in central Italy. Sampling was conducted over 32 in and 7 in intervals at La Roccaccia and Furlo respectively. After AF cleaning the majority of specimens show the expected normal magnetic field orientation, however a number of specimens are directionally anomalous. Some of these deviant specimens are accompanied by apparent spikes or dips in normalized intensity. A detailed investigation of rock magnetics shows that most of these deviations are not a sign of excursionary geomagnetic field behavior, but rather correspond to specimens with distinct rock magnetic characteristics and are therefore rock magnetic 'noise'. Such specimens should not be interpreted as records of the geomagnetic field. Our experience suggests that detailed rock magnetic and magnetic fabric analysis should be done on all anomalous directions prior to interpreting them as geomagnetic field behavior. After elimination of rock magnetic noise in the Scaglia Bianca data sets, there is a high degree of agreement in direction and to a lesser extent relative intensity between correlative portions of the two sections. We therefore offer this data set as a robust record of geomagnetic field behavior during the 4.5 Myr interval represented by the La Roccaccia section. A statistical analysis of the relative intensity observations suggests that this period of the Cretaceous Normal Superchron is characterized by a normalized variability in paleointensity (standard deviation about 28% of the mean value) that is significantly lower than seen during the Oligocene over intervals in which reversals or tiny wiggles occur (typically about 50%). The directional stability results in virtual geomagnetic pole dispersion compatible with that found in volcanic rocks from around the same latitude and ranging in age from 80 to 110 Ma. (C) 2001 Elsevier Science B.V. All rights reserved.

Cromwell, G, Constable CG, Staudigel H, Tauxe L, Gans P.  2013.  Revised and updated paleomagnetic results from Costa Rica. Geochemistry Geophysics Geosystems. 14:3379-3388.   10.1002/ggge.20199   AbstractWebsite

Paleomagnetic results from globally distributed lava flows have been collected and analyzed under the time-averaged field initiative (TAFI), a multi-institutional collaboration started in 1996 and designed to improve the geographic and temporal coverage of the 0-5 Ma paleomagnetic database for studying both the time-averaged field and its very long-term secular variations. Paleomagnetic samples were collected from 35 volcanic units, either lava flows or ignimbrites, in Costa Rica in December 1998 and February 2000 from the Cordilleras Central and Guanacaste, the underlying Canas, Liberia and Bagaces formations and from Volcano Arenal. Age estimates range from approximately 40 ka to slightly over 6 Ma. Although initial results from these sites were used in a global synthesis of TAFI data by Johnson et al. (2008), a full description of methodology was not presented. This paper documents the definitive collection of results comprising 28 paleomagnetic directions (24 normal, 4 reversed), with enhanced precision and new geological interpretations, adding two paleointensity estimates and 19 correlated Ar-40/Ar-39 radiometric ages. The average field direction is consistent with that of a geocentric axial dipole and dispersion of virtual geomagnetic poles (17.34.6 degrees) is in general agreement with predictions from several statistical paleosecular variation models. Paleointensity estimates from two sites give an average field strength of 26.3 T and a virtual axial dipole moment of 65 ZAm(2). The definitive results provide a useful augmentation of the global database for the longer term goal of developing new statistical descriptions of paleomagnetic field behavior.

Cromwell, G, Johnson CL, Tauxe L, Constable CG, Jarboe NA.  2018.  PSV10: A global data set for 0-10 Ma time-averaged field and paleosecular variation studies. Geochemistry Geophysics Geosystems. 19:1533-1558.   10.1002/2017gc007318   AbstractWebsite

Globally distributed paleomagnetic data from discrete volcanic sites have previously been used for statistical studies of paleosecular variation and the structure of the time-averaged field. We present a new data compilation, PSV10, selected from high-quality paleodirections recorded over the past 10 Ma and comprising 2,401 sites from 81 studies. We require the use of modern laboratory and processing methods, a minimum of four samples per site, and within-site Fisher precision parameter, k(w), 50. Studies that identify significant tectonic effects or explicitly target transitional field states are excluded, thereby reducing oversampling of transitional time intervals. Additionally, we apply two approaches using geological evidence to minimize effects of short-term serial correlation. PSV10 is suitable for use in new global geomagnetic and paleomagnetic studies as it has greatly improved spatial coverage of sites, especially at equatorial and high latitudes. VGP dispersion is latitudinally dependent, with substantially higher values in the Southern Hemisphere than at corresponding northern latitudes when no VGP cutoff is imposed. Average inclination anomalies for 10 degrees latitude bins range from about +32 degrees to -7.52 degrees for the entire data set, with the largest negative values occurring at equatorial and mid-northern latitudes. New 0-5 Ma TAF models (LN3 and LN3-SC) based on selections of normal polarity data from PSV10 indicate a Non-zonal variations in field structure are observed near the magnetic equator and in regions of increased radial flux at high latitudes over the Americas, the Indian Ocean, and Asia.

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.

Constable, CG.  2003.  Geomagnetic Reversals: Rates, Timescales, Preferred Paths, Statistical Models, and Simulations. Earth's core and lower mantle: Fluid mechanics of astrophysics and geophysics. ( Jones CA, Soward AM, Zhang K, Eds.).:77-99., London ; New York: Taylor & Francis Abstract
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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.  1985.  Eastern Australian Geomagnetic-Field Intensity Over the Past 14000 yr. Geophysical Journal of the Royal Astronomical Society. 81:121-130.   10.1111/j.1365-246X.1985.tb01354.x   AbstractWebsite

Two north-eastern Australian volcanic crater lake cores have been used to obtain relative intensity estimates for the geomagnetic field. ARM imparted in a low DC bias field has been used as a normalizing parameter. The intensity fluctuations in the two lakes are in excellent agreement with each other and with south-eastern Australian archaeointensity data over their coeval time spans. This strongly suggests that the same sources are influencing the geomagnetic secular variation throughout eastern Australia at this time. The relative intensity records go back to about 14000yr BP thereby extending currently available recent Australian intensity records by some 7000 yr.

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, 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.  1990.  A Simple Statistical-Model For Geomagnetic Reversals. Journal of Geophysical Research-Solid Earth and Planets. 95:4587-4596.   10.1029/JB095iB04p04587   AbstractWebsite

The diversity of paleomagnetic records of geomagnetic reversals now available indicate that the field configuration during transitions cannot be adequtely described by simple zonal or standing field models. A new model described here is based on statistical properties inferred from the present field and is capable of simulating field transitions like those observed. Some insight is obtained into what one can hope to learn from paleomagnetic records. In particular, it is crucial that the effects of smoothing in the remanence acquisition process be separated from true geomagnetic field behavior. This might enable us to determine the time constants associated with the dominant field configuration during a reversal.

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, 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, 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, CG, Constable SC.  2004.  Satellite magnetic field measurements: applications in studying the deep earth. The state of the planet : frontiers and challenges in geophysics. ( Sparks RSJ, Hawkesworth CJ, Eds.).:147-160., Washington, DCS.l.: American Geophysical Union ;International Union of Geodesy and Geophysics   10.1029/150GM13   Abstract
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Constable, C.  1992.  The Bootstrap for Magnetic-Susceptibility Tensors - Reply. Journal of Geophysical Research-Solid Earth. 97:13997-13998.   10.1029/92jb00745   AbstractWebsite
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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, CG.  2007.  Centennial to millennial-scale geomagnetic field variations. Treatise on geophysics. 5( Kono M, Schubert G, Eds.).:337-372., Amsterdam ; Boston: Elsevier Abstract
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Constable, CG, Johnson CL, Lund SP.  2000.  Global geomagnetic field models for the past 3000 years: transient or permanent flux lobes? Philosophical Transactions of the Royal Society of London Series a-Mathematical Physical and Engineering Sciences. 358:991-1008. AbstractWebsite

PSVMOD1.0 is a compilation of globally distributed palaeodirectional data from archaeomagnetic artefacts, lava flows, and lake sediments at 24 sites evaluated at 100 year intervals from 1000 BC to AD 1800. We estimate uncertainty in these measures of declination and inclination by comparison with predictions from standard historical models in time-intervals of overlap, and use the 100-year samples and their associated uncertainties to construct a sequence of minimum structure global geomagnetic field models. Global predictions of radial magnetic field at the core mantle boundary (CMB), as well as inclination and declination anomalies at the Earth's surface, provide an unprecedented view of geomagnetic secular variations over the past 3000 years, and demonstrate a consistent evolution of the field with time. Resolution of the models is poorest in the Southern Hemisphere, where only six of the 24 sites are located, several with incomplete temporal coverage. Low-flux regions seen in the historical field near the North Pole are poorly resolved, but the Northern Hemisphere flux lobes are clearly visible in the models. These lobes are not fixed in position and intensity, but they only rarely venture into the Pacific hemisphere. The Pacific region is seen to have experienced significant secular variation: a strong negative inclination anomaly in the region, like that seen in 0-5 Ma models, persists from 1000 BC until AD 1000 and then gradually evolves into the smaller positive anomaly seen today. On average bt tween 1000 BC and AD 1800, the non-axial-dipole contribution to the radial magnetic field at the core-mantle boundary is largest in the north-central Pacific, and beneath Central Asia, with clear non-zonal contributions. At the Earth's surface, average inclination anomalies are large and negative in the central Pacific, and most positive slightly to the east of Central Africa. Inclination anomalies decrease with increasing latitude. Average declinations are smallest in equatorial regions, again with strong longitudinal variations, largest negative departures are centred over Australia and Eastern Asia. Secular variation at the Earth's surface is quantified by standard deviation of inclination and declination about their average values, and at the CMB by standard deviation in radial magnetic field. All three show significant geographical variations, but appear incompatible with the idea that secular variation in the Pacific hemisphere is permanently attenuated by greatly enhanced conductivity in D " beneath the region.

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.  2007.  Dipole moment variation. Encyclopedia of geomagnetism and paleomagnetism. ( Gubbins D, Herrero-Bervera E, Eds.).:159-161., Dordrecht: Springer Abstract
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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, 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.  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.

Constable, S, Constable C.  2004.  Observing geomagnetic induction in magnetic satellite measurements and associated implications for mantle conductivity. Geochemistry Geophysics Geosystems. 5   10.1029/2003gc000634   AbstractWebsite

Currents induced in Earth by temporal variations in the external magnetic field have long been used to probe mantle electrical conductivity, but almost exclusively from sparsely distributed land observatories. Satellite-borne magnetometers, such as flown on Magsat, Orsted, and Champ, offer the prospect of improved spatial coverage. The approach we have taken is to isolate induction by harmonic Dst ("disturbance storm time'') excitation of the magnetospheric ring current in satellite magnetic measurements: this is done by removing the magnetic contributions of the main (core) magnetic field, the crustal magnetic field, and ionospheric fields (cause of the daily variation) using Sabaka et al.' s [2000, 2002] CMP3 comprehensive model. The Dst signal is then clearly evident in the midlatitude satellite passes lower than 50 degrees geomagnetic latitude. At higher latitudes, auroral and field aligned currents contaminate the data. We fit the internal and external components of the Dst signal for each equatorial pass, exploiting the fact that the geometry for the internal and external components is different for the azimuthal and radial vector components. The resulting timeseries of internal and external field variations shows that the Dst signals for the dawn passes are half those of the dusk passes. The sum of equatorial external and internal components of the field averaged over dawn and dusk passes provides an excellent estimate for the Dst index, and may in fact be superior when used as a proxy for the purposes of removing induced and magnetospheric fields from satellite magnetic data. We call this estimate satellite Dst. Cross spectral analysis of the internal and external timeseries shows both greater power and higher coherence in the dusk data. We processed the transfer function between internal and external dusk timeseries to provide globally-averaged, frequency dependent impedances that agree well with independently derived estimates. We estimate Earth's radial electrical conductivity structure from these impedances using standard regularized inversion techniques. A near-surface conductor is required, of thickness less than 10 km with a conductivity-thickness product almost exactly that of an average Earth ocean. Inversions suggest that an increase in conductivity at 440 km depth, predicted by recent laboratory measurements on high pressure phases of olivine, is not favored by the data, although, as in previous studies, the 670 km discontinuity between the upper and lower mantle is associated with a two orders of magnitude jump in conductivity. A new feature in our inversions is a further increase in lower mantle conductivity at a depth of 1300 km. A global map of the internal (induced) component of the magnetic field provides a qualitative estimate of three-dimensional (3-D) variations in Earth electrical conductivity, demonstrating graphically that the satellite data are responsive to lateral variations in electrical conductivity caused by the continents and oceans.

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.