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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.

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, CG.  2011.  Modelling the geomagnetic field from syntheses of paleomagnetic data. Physics of the Earth and Planetary Interiors. 187:109-117.   10.1016/j.pepi.2011.05.004   AbstractWebsite

This review examines results from time-varying geomagnetic field models that span several thousand years, and from variations in dipole moment strength up to million year time scales. For the past 400 years, twin magnetic flux lobes bordering the inner core tangent cylinder in both northern and southern hemispheres dominate the geomagnetic field and appear more or less fixed in location. In contrast, the millennial scale view shows that such features are quite mobile and subject to morphological changes on time scales of a few centuries to a thousand years, possibly reflecting large scale reorganization of core flow. The lobes rarely venture into the Pacific hemisphere, and average fields over various time scales generally reveal two or three sets of lobes, of diminished amplitude. Thus millennial scale models are suggestive of thermal core-mantle coupling generating a weak bias in the average field rather than a strong inhibition of large scale field changes. The recovery of variations in dipole moment on million year time scales allows frequency domain analyses to search for characteristic time scales for core dynamics that might be associated with excursion and reversal rate, time taken for reversals, or any signs of control by Earth's orbital parameters. The spectrum is characteristically red for the time interval 0-160 Ma, suggesting non-stationarity associated with average reversal rate changes, probably reflecting the impact of superchrons and a continually evolving core. Distinct regimes of power law decay with frequency may reflect different physical processes contributing to the secular variation. Evidence for non-stationarity at shorter time-scales is also present in dipole moment variations over 0-2 Ma with average growth rate faster than the decay process. Rates of change of dipole moment and rapid local field variations found in the paleomagnetic record are evaluated in the context of the 400 year historical record and the spectrum of geomagnetic variations for 0-160 Ma. (C) 2011 Elsevier 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.

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.