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.     

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.     

Models of Earth's main magnetic field incorporating flux and radial vorticity constraints

We describe a new technique for implementing the constraints on magnetic fields arising from two hypotheses about the fluid core of the Earth, namely the frozen-flux hypothesis and the hypothesis that the core is in magnetostrophic force balance with negligible leakage of current into the mantle. These hypotheses lead to time-independence of the integrated flux through certain 'null-flux patches' on the core surface, and to time-independence of their radial vorticity. Although the frozen-flux hypothesis has received attention before, constraining the radial vorticity has not previously been attempted. We describe a parametrization and an algorithm for preserving topology of radial magnetic fields at the core surface while allowing morphological changes. The parametrization is a spherical triangle tesselation of the core surface. Topology with respect to a reference model (based on data from the Oersted satellite) is preserved as models at different epochs are perturbed to optimize the fit to the data; the topology preservation is achieved by the imposition of inequality constraints on the model, and the optimization at each iteration is cast as a bounded value least-squares problem. For epochs 2000, 1980, 1945, 1915 and 1882 we are able to produce models of the core field which are consistent with flux and radial vorticity conservation, thus providing no observational evidence for the failure of the underlying assumptions. These models are a step towards the production of models which are optimal for the retrieval of frozen-flux velocity fields at the core surface.     

The Backus–Gilbert theory for piecewise continuous structures with variable discontinuity levels and its application to the magnetotelluric inverse problem

Summary. The Backus–Gilbert theory is extended to the case when the models are piecewise continuous vector functions of depth with variable discontinuity locations. In addition, some of the layers may be represented by linear combinations of known functions. For such layers only a finite number of discrete parameters is to be determined. The iterative process for obtaining a model satisfying the data is convergent, the numerical procedure by which the iterations are performed being equivalent to the method of spectral decomposition for continuous structures. The method of obtaining the Fréchet kernels by using the first perturbation of the differential system satisfied by the corresponding functionals is shown to be valid.
The theory is applied to the magnetotelluric problem, Fréchet kernels being calculated for isotropic and non-isotropic structures.
A few numerical examples are described.     

Finite difference seismograms for laterally varying marine models

Summary. The method of finite differences is applied to the elastic wave equation to generate synthetic seismograms for laterally varying seafloor structures. The results are compared with borehole seismic data from the Gulf of California (Deep Sea Drilling Project Site 485) in which lines were shot over flat and rough topography. The significant new phenomenon observed in both the synthetic seismograms and the field data is the generation of a 'double head wave' due to the interaction of the incident wavefront with the side of a hill and the flat seafoor adjacent to the hill.
In these models the hills are on the order of a seismic wavelength in height and steep velocity gradients occur over distances comparable to wavelengths. Ray theoretical methods would not be suitable for studying such structures. True amplitude record sections are obtained by the finite difference method, which show for these models that the head wave generated at the flat seafloor adjacent to the hill is lower in amplitude than if the hill were not present and is lower in amplitude than the head wave generated at the hill.
A second feature which is important for borehole receivers is the existence of the 'direct wave root' in the upper basement. This energy occurs below the sharp interface when the direct wave impinges on the interface from above. There is no corresponding Snell's law ray path for this energy and the energy is evanescent with depth in the lower medium.
The properties of both the double head wave and the direct wave root are clearly demonstrated in the finite difference 'snapshot' displays.     

Two-dimensional magnetotelluric inversion

Summary. When complex structure is encountered in magnetotelluric surveys, interpretation by locally fitted layered models is of questionable validity. However, when the processed data show two-dimensional structure, numerical inversion schemes for two-dimensional models may be constructed as an aid to regional data interpretation.
The two-dimensional magnetotellurics inversion problem is here formulated in a way that may be applied to many problems. A resulting computer program is analysed carefully in terms of its cost relative to that of simpler layered modelling.
As an example, the method is applied to some field data where the interpretive advantages of the program become evident.     

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.     

A model for energy dissipation at the mantle—core boundary

Summary. An existing experimentally verified model for energy dissipation in a processing spherical cavity filled with liquid assumed to be in a semirigidized state except for a viscous Ekman boundary layer is applied to the Earth's liquid core to assess energy dissipation magnitudes. Application of the model to the best available Earth data occurs at the derived energy dissipation maximum for the model. Other existing research showing that the Earth's atmosphere appears to adjust to a state of maximum dissipation led to generic models for systems of maximum dissipation. The maximum dissipation mantle—core model with core motion driven by Earth precession alone, coupled to the mantle only by viscous shear stresses, and with a spherical mantle—core boundary leads to energy dissipation rates on the order of 10⁴ times those necessary for an Earth dynamo. The maximum dissipation model also leads to excessive magnetic field drift rates and to excessive retardation of the Earth's rotation rate. Effects of the mantle—core ellipticity and of magnetic field coupling are briefly discussed and are used to help develop a less than maximum dissipation model also driven by precession alone but using the additional coupling to yield a model more consistent with observed phenomena.     

传染病多维度聚集性探测方法

及早发现异常健康事件的苗头是有效进行传染病早期预警的关键.现有的传染病聚集性探测仅限于时间、空间或时空维度,往往容易忽略病例个人情况从其他方面反映的信息,从而造成过度预警.论文结合蚁群聚类算法和Bayesian Gamma-Poisson 模型,提出一种全新的传染病多维度聚类探测技术.研究区麻疹爆发案例证明该技术在继承以往时空聚集性探测技术思想的基础上,考虑了病例的属性信息,能更为灵敏、准确地找出传染病聚集区域.此方法在实际工作中具有潜在的重要应用价值.     

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.     

Geography and math: a technique for finding population centers

In the absence of core requirements that secondary and/or college-level students fulfill specific geography proficiency standards prior to graduation, geography will continue to be taught informally by teachers as a corollary to other major disciplines. These teachers, however, generally lack formal training in the concepts and sheer information of the geography field. Moreover, available texts at these levels tend to simply present statistical data on countries and maybe compare them to those for other countries. The duty of explaining spatial distribution and density implications falls upon the teacher. The basic mathematical concepts of mean and media are easy measures which can be used by teachers to teach the concepts of mean center and median of a population distribution. The processes by which these central points are determined are described and applied respectively to province and state data for China and the US as suggested student activities for practice and discussion.     

Evaluating the Potential of an Airborne Laser-scanning System for Measuring Volume Changes of Glaciers

Airborne laser scanning (ALS) is well suited for the production of digital elevation models (DEM), and can, in contrast to photographic methods, be used to acquire a DEM independently of surface texture and external light sources. ALS thus serves as a tool to generate DEMs of firn areas where photogram- metric methods often fail.
The potential of an integrated ALS system – comprising a laser scanner, precise differential global positioning system, and a gyro platform – for DEM generation of firn areas is currently being assessed. The Unteraargletscher, Bernese Alps, Switzerland, has been chosen as a test site. As part of a pilot project aimed at determining the mass balance distribution of that glacier without the use of in situ information, ALS measurements were conducted in autumn 1997. The DEM derived from laser measurements is extremely sensitive to the position and attitude of the aircraft. Currently the main work focuses on assessing and improving the system's accuracy by error modelling and by the development of error-correction algorithms.
Preliminary results from Unteraargletscher are presented, and the potential of this method for the generation of DEMs of firn areas is discussed.     

Measurements and Models of the Mass Balance of Hintereisferner

This paper summarizes the methods applied to determine the mass balance of Hintereisferner and several other glaciers in the Tyrolean Alps since 1952. On an annual basis the direct glaciological method was applied with fixed date measurements on 10–15 accumulation pits and 30–90 ablation stakes on 9 km².
Indirect mass balance determination from equilibrium line altitude, accumulation area ratios or representative stakes, yield fair results and some exceptions could be related to anomalous meteorological conditions.
Monthly or more frequent stake readings supplied time series of ablation at various altitudes and slope aspects that served as basis for the calibration of energy and mass balance models. Of various models developed, two are presented in this paper. Both are based on degree days, one using daily values from a valley station to predict the mean annual balance of the entire glacier, while the other calculates day-to-day changes at 50-m grid points on the glacier.
The geodetic method has been applied for longer periods and yields results consistent with those of the glaciological method. The balance velocity calculated from recent ice thickness soundings and accumulation measurements is significantly less than observed velocity.     

Secular trend of the Earth's rotation pole: consideration of motion of the latitude observatories

Secular polar motion has been recorded in ILS data over the past 75 years, an amount greater by a factor of ten than the 'true polar wandering' deduced from paleomagnetic data. In this work, the possibility that the secular trend is an observational artifact of the continental drift of the ILS stations is directly examined by consideration of several absolute plate velocity models earlier proposed by Minster et al. (1974), Kaula (1975), and Solomon, Sleep & Richardson (1975). The assumptions underlying those models are discussed; in general, the absolute velocity models are more likely to be valid when geologically short timescales are considered.
The corrections to the ILS data due to the stations' motion fail by an order of magnitude to explain the ILS trend; even by taking into account possible plate hyperactivity and non-rigidity, the corrections could explain no more than 30 per cent of the trend. The corrections are small because the absolute plate velocities of North America and Eurasia are small and primarily east—west. Consequently, the rotation pole is undergoing significant motion of its own relative to the surface of the Earth.
The Kimura z term found by the ILS observations provides an independent means of estimating the relative motion between Eurasia and North America. It also contains other geophysical information; the 7.5-yr periodicity discovered by Naito & Ishii (1974) may be widespread.
Lastly, tectonically induced changes in the zenith direction, such as at Mizusawa, are probably too small to be detected, contrary to earlier proposals.     

Modelling of electromagnetic fields in thin heterogeneous layers with application to field generation by volcanoes—theory and example

When interpreting electromagnetic fields observed at the Earth's surface in a realistic geophysical environment it is often necessary to pay special attention to the effects caused by inhomogeneities of the subsurface sedimentary and/or water layer and by inhomogeneities of the Earth's crust. The inhomogeneities of the Earth's crust are expected to be especially important when the electromagnetic field is generated by a source located in a magma chamber of a volcano. The simulation of such effects can be carried out using generalized thin-sheet models, which were independently introduced by Dmitriev (1969 ) and Ranganayaki & Madden (1980 ). In the first part of the paper, a system of integral equations is derived for the horizontal current that flows in the subsurface inhomogeneous conductive layer and for the vertical current crossing the inhomogeneous resistive layer representing the Earth's mantle. The terms relating to the finite thickness of the laterally inhomogeneous part of the model are retained in the equations. This only marginally complicates the equations, whilst allowing for a significant expansion of the approximation limits.
  The system of integral equations is solved using the iterative dissipative method developed by the authors in the period from 1978 to 1988. The method can be applied to the simulation of the electromagnetic field in an arbitrary inhomogeneous medium that dissipates the electromagnetic energy. When considered on a finite numerical grid, the integral equations are reduced to a system of linear equations that possess the same contraction properties as the original equations. As a result, the rate at which the iterative-perturbation sequence converges to the solution remains independent of the numerical grid used for the calculations. In contrast to previous publications on the method, aspects of the algorithm implementation that guarantee its effectiveness and robustness are discussed here.     

Time series analysis of the 10 000 yr geomagnetic secular variation record from SE Australia

Summary. A 10 000 yr continuous secular variation record from intensively dated lake sediments in SE Australia has been subjected to periodogram and maximum entropy method analysis. Tests on synthetic data reveal some of the limitations of the latter method, particularly when applied to complex number series. Anticlockwise precession of the magnetic vector at a period of 5000 ± 1000 yr is tentatively ascribed to dipole precession, and clockwise precession at a period of about 2800 yr is probably due to westward drift of features of the non-dipole field.
The effect of calibrating the radiocarbon time-scale is important and results in periodicity shifts of up to 25 per cent. Even for well-dated lacustrine sequences power spectra are poorly constrained: it is thus possible that the geomagnetic secular variation on a time-scale of thousands of years is more uniform than often supposed. Mismatches between declination and inclination spectra can arise as a natural consequence of certain types of source mechanism and should not be simply attributed to figments of the analysis employed.     

Stabilization of Bottom Hole Temperature With Finite Circulation Time and Fluid Flow

As an alternative to finite element or finite difference modelling, analytical solutions are derived by the method of Laplace transformation and numerical results obtained for several models of the bottom hole temperature stabilization. Included in the models are the finite circulation time, the thermal property contrast between the borehole mud and the surrounding formation, and the presence of radial or lateral fluid flow in the formation, all of which are found to have significant effects on the dissipation of the thermal disturbance induced by drilling.
The mud circulation is considered to have the effect of either maintaining the borehole mud at a constant temperature or supplying a constant amount of heat per unit length per unit time to the borehole. For small circulation times, the former reduces to the 'zero circulation' model in which the mud circulation creates an instantaneous temperature anomaly at the hole bottom; for small borehole radii, the latter reduces to the line source model and the traditional 'Homer plot'.
For typical drilling operations in which the bottom hole temperatures are measured several hours to several tens of hours after the hole is shut in, the new models generally predict higher equilibrium formation temperatures than does the Horner plot. However, predictions from the various models converge if the BHTs are taken after the hole has been shut in for a period which is greater than about five times the circulation period.     

Some terms of nutation derived from the BIH data

Summary. The z -term of latitude and the w -term of UT, as computed by the BIH, are suitable for investigating imperfections in the representation of the real nutation in space. They have been previously used for deriving only the amplitude of the principal term of nutation (Feissel & Guinot). In this study they are used more generally for deriving the amplitudes of two terms of nutation and searching for a possible nutation in space due to the nearly-diurnal wobble.
Two kinds of data are used for this purpose: the values of z and w from 1962.0 to 1980.0 at 0.05 yr intervals and the values of z from 1967.0 to 1980.0 at 5 day intervals. The first ones have been used previously by Feissel & Guinot (using the values up to 1974.0 and 1976.95 respectively). The second ones have not previously been used for such studies.
The derived amplitudes of the principal and the fortnightly terms of nutation are in good agreement with other values deduced from observations, and with the theoretical ones corresponding to the best models of a non-rigid earth. The values obtained are not yet sufficiently precise to distinguish between these models, but could become so in the near future.
The least-squares fits, performed among the z and w data in order to look for any error in the representation of nutation, show the existence of a retrograde term, of a 434 day period for z and 444 day for w and an amplitude of the order of 0.01 arcsec. It may be due to the nearly-diurnal wobble.     

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.