The fifth order geomagnetic multipole: The duotrigintupole

Summary The notion of a dipole is generalized to the case of the fifth order spherical harmonic coefficients of the geomagnetic potential. The corresponding five axes and fifth order multipole strength are computed for ten epochs in the interval 1845 to 1965.     

Geomagnetic multipoles, 1829 to 1960

Summary Using the results of spherical harmonic analyses of the geomagnetic field for some fourteen different epochs, includingGauss' first analysis for epoch 1835, and theErman-Petersen analysis for epoch 1829, the strength and axes of geomagnetic multipoles have been computed. In particular, a dipole from the three first order spherical harmonic coefficients, a quadrupole from five second order coefficients, and an octupole from seven third order coefficients. The axes of the quadrupole and octupole have moved quite rapidly when compared with movements of the dipole axis, and show a general movement westwards. Although the strength of the dipole has generally diminished, the strengths of the quadrupole and octupole have generally increased.On leave National University of La Plata, Argentina     

Changes in geomagnetic multipole parameters 1942.5–1962.5

Summary From a smooth series of spherical harmonic coefficients for the geomagnetic potential, the corresponding multipole parameters have been calculated for five epochs from 1942.5 until 1962.5, at five year intervals. Changes in multipole parameters are discussed in relation to the secular variation field and to theSchmidt eccentric dipole.     

Geomagnetic multipoles, 1965.0

Summary Following the recent appearance of spherical harmonic coefficients for the potential of the geomagnetic field for epoch 1965.0, the strengths and axes for a dipole, quadrupole and octupole have been computed from first, second and third order spherical harmonic coefficients respectively. An interesting relation existing between the parameters of the dipole, quadrupole and geomagnetic eccentric dipole is also indicated.     

On recurrence relations for multipole coefficients

Summary General recurrence relations between the coefficients in thenth and (n+1)th order spherical harmonic multipole expansions are derived. The particular application presented here is the derivation of the equations concerned with representing the geomagnetic field by magnetic multipoles. The equations up to the 3rd order multipole are given as an example of the method. The main advantage in using these recurrence relations rather than other methods is that the mathematics is reduced to merely a matter of successive substitutions and this allows a fast step by step generation of the required equations, in a form for which there is a simple numerical program for solution.     

Possible configurations of the magnetic field in the outer magnetosphere during geomagnetic polarity reversals

Possible configurations of the magnetic field in the outer magnetosphere during geomagnetic polarity reversals are investigated by considering the idealized problem of a magnetic multipole of order m and degree n located at the centre of a spherical cavity surrounded by a boundless perfect diamagnetic medium. In this illustrative idealization, the fixed spherical (magnetopause) boundary layer behaves as a perfectly conducting surface that shields the external diamagnetic medium from the compressed multipole magnetic field, which is therefore confined within the spherical cavity. For a general magnetic multipole of degree n, the non-radial components of magnetic induction just inside the magnetopause are increased by the factor 1 + [(n + 1)/n] relative to their corresponding values in the absence of the perfectly conducting spherical magnetopause. An exact equation is derived for the magnetic field lines of an individual zonal (m = 0), or axisymmetric, magnetic multipole of arbitrary degree n located at the centre of the magnetospheric cavity. For such a zonal magnetic multipole, there are always two neutral points and n – 1 neutral rings on the spherical magnetopause surface. The two neutral points are located at the poles of the spherical magnetopause. If n is even, one of the neutral rings is coincident with the equator; otherwise, the neutral rings are located symmetrically with respect to the equator. The actual existence of idealized higher-degree (n > 1) axisymmetric magnetospheres would necessarily imply multiple (n + 1) magnetospheric cusps and multiple (n) ring currents. Exact equations are also derived for the magnetic field lines of an individual non-axisymmetric magnetic multipole, confined by a perfectly conducting spherical magnetopause, in two special cases; namely, a symmetric sectorial multipole (m = n) and an antisymmetric sectorial multipole (m = n – 1). For both these non-axisymmetric magnetic multipoles, there exists on the spherical magnetopause surface a set of neutral points linked by a network of magnetic field lines. Novel magnetospheric processes are likely to arise from the existence of magnetic neutral lines that extend from the magnetopause to the surface of the Earth. Finally, magnetic field lines that are confined to, or perpendicular to, either special meridional planes or the equatorial plane, when the multipole is in free space, continue to be confined to, or perpendicular to, these same planes when the perfectly conducting magnetopause is present.Also Honorary Research Associate, Rutherford Appleton Laboratory, Chilton, Didcot, Oxon OX11 0QX, UK and Visiting Reader in Physics. University of Sussex, Falmer, Brighton BN1 9QH, UK     

GOCE Data Processing: The Spherical Cap Regularization Approach

Due to the sun-synchronous orbit of the satellite gravity gradiometry mission GOCE, the measurements will not be globally available. As a consequence, using a set of base functions with global support such as spherical harmonics, the matrix of normal equations tends to be ill-conditioned, leading to weakly determined low-order spherical harmonic coefficients. The corresponding geopotential strongly oscillates at the poles. Considering the special configuration of the GOCE mission, in order to stabilize the normal equations matrix, the Spherical Cap Regularization Approach (SCRA) has been developed. In this approach the geopotential function at the poles is predescribed by an analytical continuous function, which is defined solely in the spatially restricted polar regions. This function could either be based on an existing gravity field model or, alternatively, a low-degree gravity field solution which is adjusted from GOCE observations. Consequently the inversion process is stabilized. The feasibility of the SCRA is evaluated based on a numerical closed-loop simulation, using a realistic GOCE mission scenario. Compared with standard methods such as Kaula and Tikhonov regularization, the SCRA shows a considerably improved performance.     

Local Ionospheric Modeling Using the Localized Global Ionospheric Map and Terrestrial GPS

global ionosphere maps are generated on a daily basis at CODE using data from about 200 GPS/GLONASS sites of the IGS and other institutions. The vertical total electron content is modeled in a solargeomagnetic reference frame using a spherical harmonics expansion up to degree and order 15. The spherical Slepian basis is a set of bandlimited functions which have the majority of their energy concentrated by optimization inside an arbitrarily defined region, yet remain orthogonal within the spatial region of interest. Hence, they are suitable for decomposing the spherical harmonic models into the portions that have significant strength only in the selected areas. In this study, the converted spherical harmonics to the Slepian bases were updated by the terrestrial GPS observations by use of the least-squares estimation with weighted parameters for local ionospheric modeling. Validations show that the approach adopted in this study is highly capable of yielding reliable results.     

Motion of the Earth’s magnetic poles in the last decade

Based on vector magnetic data from the CHAMP German satellite, average daily spherical harmonic models of the main geomagnetic field to n = m = 10 have been constructed for the period from May 2001 to the end of 2009 at an interval of 4 days. The obtained 16 models, which were averaged over half a year, have been used to calculate the coordinates of the north and south magnetic poles (the points where magnetic field lines are vertical). The changes in these coordinates during these eight and a half years have been traced. Both poles continue moving northward and westward. The north magnetic pole has traveled 400 km during this period. The velocity of its motion has increased up to the year 2003, reaching 62.5 km yr⁻¹, and then started decreasing and reached 45 km yr⁻¹ by the end of 2009. In addition, the direction of motion changed from north-northwestward to northwestward; i.e., the pole started turning slightly towards Canada. The south magnetic pole moved slower by an order of magnitude and has traveled 42 km during this period. The coordinates of the geomagnetic (dipole) poles and the eccentric dipole parameters have also been calculated. The dynamics of these poles has been traced.     

Comparison of the sector and conventional spherical harmonic analyses of the solar magnetic field on the photosphere,source surface,and in the Earth’s orbit on July 10–20, 2004

The results of a spherical harmonic analysis and a sector spherical harmonic analysis of the solar magnetic field on the photosphere, source surface, and in the Earth’s orbit on July 10–20, 2004, were compared. It was found that the field values according to a sector harmonic analysis are an order of magnitude as large as the same values according to a spherical harmonic analysis and differ in the configuration. A twocomponent magnetic field structure was revealed: short-range sources are better described by a sector spherical harmonic analysis; long-range sources are better described by a spherical harmonic analysis. This is caused by the different depths of the occurrence of sources below the photosphere.     

On the significance of multipole axes

Summary The force experienced by a multipole when placed in an external magnetic field (which may be the field of another multipole) is formulated in terms of its dependence on the axes of the multipole.     

关于全球地震层析成像反演方法的探讨

以全球地震面波相速度变化问题为例，使用约３００００道高质量的面波记录数据集，在数据空间和模型空间的误差服从零平均Ｇａｕｓｓ随机分布的假设下，研究反演中的阻尼和参量化问题．发现最佳分块尺度随频率增高而减小；分块参量化方法比球谐展开方法引入的人为因素少，但难以分辨甚低阶横向非均匀性，除非施加极端的空间平滑；球谐函数展开则具备恢复长波结构的优点．为此提出一种混合参量反演方法：首先用球谐函数作为全球基函数，恢复相速度的低阶球谐分量，然后用此作为进一步反演的初始模型，用分块模型迭代反演，得到最终结果．     

On a spectral method of solving the Stokes equation

A method of solving the Stokes equation for a spherical mantle model by expansion in spherical harmonics was developed by Hager and O’Connell [1979]. However, this method is applicable only if the viscosity depends solely on depth. In this case, the Stokes equation reduces to a system of independent equations for each harmonic. Given lateral variations in viscosity, the Stokes equation contains terms in the form of products of harmonics, which invalidates all advantages of harmonic expansion. Zhang and Christensen [1993] developed a perturbation method for the case when terms containing products of lateral viscosity variations are small. These terms are first calculated from the preceding iteration and are then expanded in a series of harmonic functions. As a result, equations for harmonics remain independent. An evident advantage of the spectral method is the simplicity of the technique of incorporating the self-gravitation and compressibility effects. Moreover, this method partially eliminates difficulties related to the singularities at poles. As yet, it has not been applied in practice, possibly because the equations presented in [Zhang and Christensen, 1993] contain misprints that have not been elucidated in the literature. In the present work, a system of equations is derived for the spectral-iterative method of solving the Stokes equation and the errata present in formulas of Zhang and Christensen [1993] and significantly affecting results of calculations are analyzed.     

Forward computation of the magnetic field of a 3D body with arbitrary boundary and continually varying magnetization

The forward computation of the gravitational and magnetic fields due to a 3D body with an arbitrary boundary and continually varying density or magnetization is an important problem in gravitational and magnetic prospecting. In order to solve the inverse problem for the arbitrary components of the gravitational and magnetic anomalies due to an arbitrary 3D body under complex conditions, including an uneven observation surface, the existence of background anomalies and very little or no a priori information, we used a spherical coordinate system to systematically investigate forward methods for such anomalies and developed a series of universal spherical harmonic expansions of gravitational and magnetic fields. For the case of a 3D body with an arbitrary boundary and continually varying magnetization, we have also given the surface integral expressions for the common spherical harmonic coefficients in the expansion of the magnetic field due to the body, and a very precise numerical integral algorithm to calculate them. Thus a simple and effective method of solving the forward problem for magnetic fields due to 3D bodies of this kind has been found, and in this way a foundation is laid for solving the inverse problem of these magnetic fields. In addition, by replacing the parameters and unit vectors in the spherical harmonic expansion of a magnetic field by gravitational parameters and a downward unit vector, we have also derived a forward method for the gravitational field (similar to that for the magnetic case) of a 3D body with an arbitrary boundary and continually varying density.     

Topographic and atmospheric effects on goce gradiometric data in a local north-oriented frame: A case study in Fennoscandia and Iran

Satellite gradiometry is an observation technique providing data that allow for evaluation of Stokes’ (geopotential) coefficients. This technique is capable of determining higher degrees/orders of the geopotential coefficients than can be achieved by traditional dynamic satellite geodesy. The satellite gradiometry data include topographic and atmospheric effects. By removing those effects, the satellite data becomes smoother and harmonic outside sea level and therefore more suitable for downward continuation to the Earth’s surface. For example, in this way one may determine a set of spherical harmonics of the gravity field that is harmonic in the exterior to sea level. This article deals with the above effects on the satellite gravity gradients in the local north-oriented frame. The conventional expressions of the gradients in this frame have a rather complicated form, depending on the first-and second-order derivatives of the associated Legendre functions, which contain singular factors when approaching the poles. On the contrary, we express the harmonic series of atmospheric and topographic effects as non-singular expressions. The theory is applied to the regions of Fennoscandia and Iran, where maps of such effects and their statistics are presented and discussed.     

Computation and analysis of the geomagnetic field model in China and its adjacent area for 2003

Based on the geomagnetic data at 135 stations and 35 observatories in China in 2003, the Taylor polynomial model and the spherical cap harmonic model in China and its adjacent area for 2003 were established. In the model calculation, the truncation order of the model and the influences of the boundary restriction on the model calculation were carefully analyzed. The results show that the geomagnetic data used are precise and reliable, and the selection of the truncation order is reasonable. The Taylor polynomial model and the spherical cap harmonic model in China and its adjacent area established in this paper are consistent very well.     

2003年中国及邻区地磁场模型的计算与分析

根据2003年中国地区的135个测点和35个台站的地磁数据, 建立了2003年中国及邻区地磁场泰勒多项式模型和球冠谐模型. 在模型计算过程中, 细致地分析了模型的截断阶数和边界约束对模型计算的影响. 结果表明, 所使用的地磁观测资料是准确可靠的, 模型截断阶数的选取是合理的. 本文所建立的中国区域地磁场球冠谐模型与泰勒多项式模型具有良好的一致性.      

欧洲及其邻区MAGSAT卫星磁异常冠谐模型

通过对欧洲及其邻近地区 MAGSAT 卫星黎明资料进行了处理,得到1°×1°卫星磁异常网格值。本文使用冠谐分析方法,计算该地区卫星矢量磁异常（ΔX,ΔY,ΔZ）冠谐模型。球冠极点位于33°N和26°E,球冠半角为40°.冠谐模型的截断指数为18。根据卫星磁异常冠谐模型和地磁场球谐模型 DGRF1980,计算卫星总强度磁异常（ΔF）的冠谐一球谐模型。根据卫星磁异常的理论模型,计算并绘制不同高度（300,400,500km）的理论卫星磁异常图。对冠谐模型和大磁异常进行了分析和讨论。     

On the estimation of accuracy for global models of gravitational field of the earth

The methods and techniques for estimating the accuracy of global models of the Earth’s gravity field in the form of spherical harmonic expansion of the geopotential are analyzed. Various methods for obtaining the a priori and a posteriori estimates for the accuracy are considered and classified. The application of different approaches is illustrated by numerical examples for nine models, including those recently developed using the modern methods of space geodesy. The basic requirements for the database and software for estimating the accuracy are formulated.