Geomagnetic secular acceleration

Summary. A spherical harmonic model of the second time-derivative of the geomagnetic field is determined, for the first time, directly from measures of the secular acceleration based on observatory annual mean data. The data span the interval 1964.5–1975.5, and 165 observatories are included. The model comprises the 32 coefficients of degree and order up to 6 that are significant at the 5 per cent level. Its primary purpose is to aid in the reduction of data to epoch for the 1980 series of navigational charts. The model is compared with earlier estimates of secular acceleration, derived by less direct methods.     

Geomagnetic evidence for fluid upwelling at the core-mantle boundary

Summary. Previous studies, both geomagnetic and seismic, have been unable to show conclusively whether or not there is fluid upwelling at the core-mantle boundary. Here a new method is developed, in which an attempt is made to invert geomagnetic secular variation data measured at the Earth's surface for a frozen-flux purely toroidal core-mantle boundary (CMB) velocity field, under the assumption that the mantle is electrically insulating and flux is frozen in at the CMB. These data have previously been inverted for the core-mantle boundary radial secular variation, from which the appropriate fit between model and data is known. Two different main field models were used to assess the effect of uncertainty in its radial component at the CMB. The conclusions were the same in both cases: frozen-flux purely toroidal motions provide a poor fit. A statistical test allows very firm rejection of the hypothesis that the residuals are not significantly larger, whereas there is no statistical difference between the residuals of inversions for radial secular variation and frozen-flux velocity fields at the CMB if upwelling and down-welling is included. The inherent non-uniqueness in the velocity field obtained is not of concern, since only their statistical properties are utilized and no physical significance is attached to the flows obtained.     

Testing a palaeomagnetic study for the averaging of secular variation

Summary. It is shown that the interval estimate for K given by Cox (1969) is inappropriate as a test for determining whether secular variation has been averaged out in a particular study. The appropriate test is presented in two formulations, one for use with χ² tables and the other for use with F tables.
The well known secular variation models are investigated and it is shown that Model A due to Irving & Ward (1964) is, statistically, the most efficient predictor. Finally, for the 83 palaeomagnetic results selected by Brock (1971), it is shown that the most efficient predictor for the secular variation K is k '= 18.1 (1 + 3 sin²λ), where λ is the palaeolatitude, and the uncertainty in this predicted value is discussed.     

Geomagnetic secular variation and secular acceleration in the Red Sea area

Summary. Estimates of the secular variation in the Red Sea over the period 1959 to 1972 have been obtained from an analysis of marine magnetic data. A total of 318 crossings of ships' tracks were used to determine the mean secular variation for the intervals 1959–72, 1959–65 and 1965–72. The mean secular variation 1959–72 shows a marked northward increase from approximately -10 nT/yr at 13°N to +27 nT/yr at 24°N. North of this, the data suggest a small decrease to + 25 nT/yr at 27° N. These values are consistent with the secular variation recorded at the nearby geomagnetic observatory at Helwan, Egypt, but less than those predicted by the 1965 IGRF for the same period.
Comparison of the mean secular variations for 1959–65 and 1965–72 yields a rough estimate of the secular acceleration of - 1.5 nT yr⁻². Analysis of the cross-over information, corrected for the latitude dependence of the secular variation, shows a regular decrease in the secular change over the period 1959–72 at all latitudes of about -1 nTyr⁻². This secular acceleration makes a substantial contribution to the overall secular change in the Red Sea and as such must be included in the correction of magnetic data covering more than a few years.     

The geomagnetic field over the past 5 million years

The modern geomagnetic field is usually expressed as a spherical harmonic expansion. Although the palaeomagnetic record is very incomplete in both space and time, sufficient data are available from a span of ages to generate time-averaged spherical harmonic field models with many degrees of freedom. Here three data sets are considered: directional measurements from lavas, inclination measurements from ocean sediments, and intensity measurements from lavas. Individual data are analysed, as well as site-averages, using the same methods that have been developed for the modern field, to give models for the past 5 Myr. The normal-polarity field model has an axial-dipole intensity similar to that of the modern-day field, whilst the equatorial-dipole component is very much smaller. The field is not axisymmetric, but shows flux concentrations at the core's surface under Canada and Siberia similar to those observed in the field over historical timescales. Tests on synthetic data show that it is unlikely that these similarities result from the overprinting of the palaeomagnetic field due to inadequate cleaning of the samples. The reverse-polarity field model does not show such obvious features, but this may be due to the sparsity of the data.
The patterns observed in the normal-polarity field, with persistent features in the northern hemisphere and a smooth southern hemisphere, could be explained if the present pattern of secular variation is typical of the past several million years. This would reveal itself as large variations over time in the direction of the magnetic vector in regions of high secular variation, with relatively little change over quieter regions. However, we have been unable to find any evidence for a geographical pattern of secular variation in the data.     

Estimability analysis of geodetic, astrometric and geodynamical quantities in very long baseline interferometry

Summary. Within the framework of Newtonian kinematics VLBI observations are analysed with respect to estimability of geodetic and astrometric quantities. An Earth model of either rigid or deformable type is designed; instrumental clock offsets and clock drifts are included. Observational patterns are studied in all detail reviewed in seven tables. Appendix A is an introduction to the set-up of the observational model for a deformable Earth both in terms of coordinate-free and coordinate-related geometry. Appendix B is a study of the invariance characteristics of VLBI observations. Interrelations of three fundamental quantities, length unit, time unit and velocity of light are discussed. An overall result of an Earth model of deformable type is the need of simultaneous observations to more than one source; VLBI time delay observations cannot distinguish between secular changes of network size (expansion or shrinking) and a common secular drift (deceleration or acceleration) of clocks used.     

A definitive model of the geomagnetic field and its secular variation for 1965 — I. Derivation of model and comparison with the IGRF

Summary. Using a very large body of post-1955 data, a spherical harmonic model of the geomagnetic field and its secular variation is derived for 1965.0. This model is compared with the original International Geomagnetic Reference Field (IGRF) and with individual models used, or proposed for use, in producing the IGRF. Positions of the dip-poles, the geomagnetic poles and the eccentric dipole are derived from the model, together with their rates of change, and comparisons are made with other estimates of these positions.     

Some results of the investigation of geomagnetic field variation in the Urals and the Carpathians

Summary. A precision magnetic survey for the investigation of current activity in the Earth's lithosphere has been carried out in the Urals and in the Carpathians. As a result of this research three types of time variation of the total field were discovered. These are:
(1) The normal field variation reflecting the general pattern of secular variation. The difference of initial and repeat observation where only this type of variation operates, is rather small and usually does not exceed 0.2–0.3 nT. The field changes in such regions can be used only to evaluate the observation errors and to provide the regional pattern of secular variation.
(2) The slow but localized'anomalous field'change from year to year corresponding, presumably, to anomalies of a tectonomagnetic nature. The normal pattern of the secular variation field here is disturbed by sources located in the upper part of the lithosphere.
(3) Irregular time changes of the field with rather large amplitudes (up to 10–20 nT). Repeated observations of such anomalies show that the field changes significantly here even during one day. Both in the Urals and Carpathians these anomalies form extended elongated structures with widths up to 10–30 km. These anomalies usually coincide with those deep faults where the strongest recent crustal movements have been determined by means of geodetic observations. The analysis of the results of precision geomagnetic surveys in the Urals and in the Carpathians shows that geomagnetic investigations can be used for the exploration of tectonically active zones.     

Daily Variation and Secular Variation of the Geomagnetic Field from Shipboard Observations in the Gulf of Aden

Summary The problems of reducing geomagnetic observations from ships at sea in areas influenced by the effect of the equatorial electrojet are discussed. In particular, observations within the Gulf of Aden have been corrected for daily variation and secular variation for the purposes of constructing a contoured magnetic anomaly chart.
An empirical formula is given with which the range of daily variation at different latitudes within the Gulf was estimated for the purpose of correcting the data for daily variation. The observed secular variation, which was used to correct the data, is—11 γ/yr. which differs from the secular variation of +19 γ/yr. in the Gulf of Aden given by the recently adopted International Geomagnetic Reference Field (Zmuda 1969).     

Modelling of sedimentary basins from gravity anomalies with variable density contrast

Summary. The decrease of density contrast with depth in sedimentary basins could be approximated by a quadratic function. The sedimentary basin is viewed as a number of outcropping vertical prisms put in juxtaposition at each observation point. Equations are derived for the gravity anomaly of a vertical prism for a quadratic density function. The initial model is obtained by calculating the thicknesses of the prisms at each station by an infinite slab formula. The differences between the observed anomalies and the anomalies of the initial model at each station are used to modify the initial model. A computer program is developed for initial estimates and refinement of the model. The method is applied to interpret the gravity anomaly of San Jacinto Graben, California, where sediments filling the graben have a density contrast that can be approximated to a quadratic function. Finally the convergence of the method is discussed.     

Centennial to millennial geomagnetic secular variation

A time-varying spherical harmonic model of the palaeomagnetic field for 0–7 ka is used to investigate large-scale global geomagnetic secular variation on centennial to millennial scales. We study dipole moment evolution over the past 7 kyr, and estimate its rate of change using the Gauss coefficients of degree 1 (dipole coefficients) from the CALS7K.2 field model and by two alternative methods that confirm the robustness of the predicted variations. All methods show substantial dipole moment variation on timescales ranging from centennial to millennial. The dipole moment from CALS7K.2 has the best resolution and is able to resolve the general decrease in dipole moment seen in historical observations since about 1830. The currently observed rate of dipole decay is underestimated by CALS7K.2, but is still not extraordinarily strong in comparison to the rates of change shown by the model over the whole 7 kyr interval. Truly continuous phases of dipole decrease or increase are decadal to centennial in length rather than longer-term features. The general large-scale secular variation shows substantial changes in power in higher spherical harmonic degrees on similar timescales to the dipole. Comparisons are made between statistical variations calculated directly from CALS7K.2 and longer-term palaeosecular variation models: CALS7K.2 has lower overall variance in the dipole and quadrupole terms, but exhibits an imbalance between dispersion in   g ¹₂  and   h ¹₂  , suggestive of long-term non-zonal structure in the secular variations.     

An improved geomagnetic data selection algorithm for global geomagnetic field modelling

A spherical harmonic degrees 60, global internal field model is described (called BGS/G/L/0706). This model includes a degree 15 core and piecewise-linear secular variation model and is derived from quiet-time Ørsted and Champ satellite data sampled between 2001.0 and 2005.0. For the satellite data selection, a wide range of geomagnetic index and other data selection filters have been used to best isolate suitably quiet magnetospheric and ionospheric conditions. Only a relatively simple, degree one spherical harmonic, external field model is then required. It is found that a new 'Vector Magnetic Disturbance' index ( VMD ), the existing longitude sector A indices, the auroral zone index IE , and the polar cap index PC are better than Kp and Dst at rejecting rapidly varying external field signals at low, middle, auroral and polar latitudes. The model quality is further enhanced by filling spatial and temporal gaps in the quiet data selection with a second selection containing slightly more disturbed data. It is shown that VMD provides a better parametrization than Dst of the large-scale, rapidly changing, external field. The lithospheric field model between degrees 16 and 50 is robust and displays good coherence with other recently published models for this epoch. BGS/G/L/0706 also shows crustal anomalies consistent with other studies, although agreement is poorer in the southern polar cap. Intermodel coherency reduces above about degree 40, most likely due to incompletely filtered signals from polar ionospheric currents and auroral field aligned currents. The absence of the PC index for the southern hemisphere for 2003 onwards is a particular concern.     

Rotational and elliptical splitting of the free oscillations of the Earth

Summary. We present a table of rotational and elliptical splitting parameters for earth model 1066A, including all terms through second order in rotation and first order in ellipticity. An algorithm for calculating the second-order Coriolis splitting by summing over all modes which are coupled to first order is given in detail. Coupling to secular (or zero frequency) modes, as well as the usual seismic modes, can provide significant contributions to these splitting parameters.     

Secular variation of the poloidal magnetic field at the core–mantle boundary

A region of enhanced conductivity at the base of the mantle is modelled by an infinitesimally thin sheet of uniform effective conductance adjacent to the core–mantle boundary. Currents induced in this sheet by the temporally varying magnetic field produced by the geodynamo give rise to a discontinuity in the horizontal components of the poloidal magnetic field on crossing the sheet, while the radial component is continuous across the sheet. Treating the rest of the mantle as an insulator, the horizontal components of the poloidal magnetic field and their secular variation at the top of the core are determined from geomagnetic field, secular variation and secular acceleration models. It is seen that for an assumed effective conductance of the sheet of 10⁸  S, which may be not unrealistic, the changes produced in the horizontal components of the poloidal field at the top of the core are usually ≤10 per cent, but corrections to the secular variation in these components at the top of the core are typically 40 per cent, which is greater than the differences that exist between different secular variation models for the same epoch. Given the assumption that all the conductivity of the mantle is concentrated into a thin shell, the present method is not restricted to a weakly conducting mantle. Results obtained are compared with perturbation solutions.     

A Late Holocene geomagnetic secular variation record from Erhai Lake, southwest China

A secular variation record of the geomagnetic field direction for the last 6.5  kyr has been obtained from the magnetization of sediment cores from Erhai Lake, southwest China. In order to make a comparison with this record, secular variation in east-central China was investigated by combining available magnetic field data from historical records and archaeomagnetic measurements since about 350 bc . The secular variation in Erhai Lake shows features consistent with the combined record, except for the oldest three observed declination swings in Sian from 720 to 900 ad . Many features of declination and inclination in China also occur in Japan. From 500 to 1000 ad , declination was westerly ranging from about −20° to −5° in Erhai Lake, east-central China, and Japan.     

Estimation of undiscovered deposits in quantitative mineral resource assessments—examples from Venezuela and Puerto Rico

Quantitative mineral resource assessments used by the United States Geological Survey are based on deposit models. These assessments consist of three parts: (1) selecting appropriate deposit models and delineating on maps areas permissive for each type of deposit; (2) constructing a grade-tonnage model for each deposit model; and (3) estimating the number of undiscovered deposits of each type. In this article, I focus on the estimation of undiscovered deposits using two methods: the deposit density method and the target counting method.In the deposit density method, estimates are made by analogy with well-explored areas that are geologically similar to the study area and that contain a known density of deposits per unit area. The deposit density method is useful for regions where there is little or no data. This method was used to estimate undiscovered low-sulfide gold-quartz vein deposits in Venezuela.Estimates can also be made by counting targets such as mineral occurrences, geophysical or geochemical anomalies, or exploration plays and by assigning to each target a probability that it represents an undiscovered deposit that is a member of the grade-tonnage distribution. This method is useful in areas where detailed geological, geophysical, geochemical, and mineral occurrence data exist. Using this method, porphyry copper-gold deposits were estimated in Puerto Rico.     

Geomagnetic field analysis - I. Stochastic inversion

Summary. Most of the Earth's magnetic field and its secular change originate in the core. Provided the mantle can be treated as an electrical insulator, stochastic inversion enables surface observations to be analysed for the core field. A priori information about the variation of the field at the core boundary leads to very stringent conditions at the Earth's surface. The field models are identical with those derived from the method of harmonic splines (Shure, Parker & Backus) provided the a priori information is specified appropriately.
The method is applied to secular variation data from 106 magnetic observatories. Model predictions for fields at the Earth's surface have error estimates associated with them that appear realistic. For plausible choices of a priori information the error of the field at the core is unbounded, but integrals over patches of the core surface can have finite errors. The hypothesis that magnetic fields are frozen to the core fluid implies that certain integrals of the secular variation vanish. This idea is tested by computing the integrals and their standard and maximum errors. Most of the integrals are within one standard deviation of zero, but those over the large patches to the north and south of the magnetic equator are many times their standard error, because of the dominating influence of the decaying dipole. All integrals are well within their maximum error, indicating that it will be possible to construct core fields, consistent with frozen flux, that satisfy the observations.     

Maximum entropy regularization of time-dependent geomagnetic field models

We incorporate a maximum entropy image reconstruction technique into the process of modelling the time-dependent geomagnetic field at the core–mantle boundary (CMB). In order to deal with unconstrained small lengthscales in the process of inverting the data, some core field models are regularized using a priori quadratic norms in both space and time. This artificial damping leads to the underestimation of power at large wavenumbers, and to a loss of contrast in the reconstructed picture of the field at the CMB. The entropy norm, recently introduced to regularize magnetic field maps, provides models with better contrast, and involves a minimum of a priori information about the field structure. However, this technique was developed to build only snapshots of the magnetic field. Previously described in the spatial domain, we show here how to implement this technique in the spherical harmonic domain, and we extend it to the time-dependent problem where both spatial and temporal regularizations are required. We apply our method to model the field over the interval 1840–1990 from a compilation of historical observations. Applying the maximum entropy method in space—for a fit to the data similar to that obtained with a quadratic regularization—effectively reorganizes the magnetic field lines in order to have a map with better contrast. This is associated with a less rapidly decaying spectrum at large wavenumbers. Applying the maximum entropy method in time permits us to model sharper temporal changes, associated with larger spatial gradients in the secular variation, without producing spurious fluctuations on short timescales. This method avoids the smearing back in time of field features that are not constrained by the data. Perspectives concerning future applications of the method are also discussed.     

Geomagnetic field analysis—IV. Testing the frozen-flux hypothesis

Summary. We present a model of the magnetic field at the core–mantle boundary, for epoch 1959.5, based on a large set of observatory and survey measurements. Formal error estimates for the radial field at the core are 50 μT, compared with 30 and 40 μT for our previous MAGSAT (1980) and POGO (1970) models.
Current work on the determination of the velocity of the core fluid relies on the assumption that the core behaves as a perfect conductor, so that the field lines remain frozen to the fluid at the core surface. This frozen-flux condition requires that the integrated flux over patches of the core surface bounded by contours of zero radial field remain constant in time. A new method is presented for constructing core fields that satisfy these frozen-flux constraints. The constraints are non-linear when applied to main field data, unlike the case of secular variation which was considered in an earlier paper. The method is applied to datasets from epochs 1969.5 and 1959.5 to produce fields with the same flux integrals as the 1980 model.
The frozen-flux hypothesis is tested by comparing the changes in the flux integrals between 1980/1969.5, 1969.5/1959.5 and 1980/1959.5 with their errors. We find that the hypothesis can be rejected with 95 per cent confidence. The main evidence for flux diffusion is in the South Atlantic region, where a new null flux curve appears between 1960 and 1970, and continues to grow at a rapid rate from 1970 to 1980. However, the statistical result depends critically on our error estimates for the field at the core surface, which are difficult to assess with any certainty; indeed, doubling the error estimates negates the statistical argument. The conclusion is therefore, at this stage, tentative, and requires further evidence, either from older data, if good enough, or from future satellite measurements.