Space-time modeling of the main magnetic field by combined methods of spherical harmonic analysis and natural orthogonal components

A simple method for obtaining a space-time model of the main magnetic field from the high-precision satellite survey data is described. At the first stage, the CHAMP satellite data for one-day interval are expanded into the spherical harmonics with constant coefficients. This yields a set of daily mean spherical harmonic models (DMSHM) over the survey interval of a few years. At the second stage, the coefficients of this set are used as source data for expansion into the natural orthogonal components (NOC). It is shown that the terms of the NOC series decrease rapidly, and the accuracy of the space-time model of the main geomagnetic field over the time interval under discussion is not worse than the accuracy of the models obtained by traditional methods.     

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.     

Uncertainties in field-line tracing in the magnetosphere. Part I: the axisymmetric part of the internal geomagnetic field

The technique of tracing along magnetic field lines is widely used in magnetospheric physics to provide a magnetic frame of reference that facilitates both the planning of experiments and the interpretation of observations. The precision of any such magnetic frame of reference depends critically on the accurate representation of the various sources of magnetic field in the magnetosphere. In order to consider this important problem systematically, a study is initiated to estimate first the uncertainties in magnetic-field-line tracing in the magnetosphere that arise solely from the published (standard) errors in the specification of the geomagnetic field of internal origin. Because of the complexity in computing these uncertainties for the complete geomagnetic field of internal origin, attention is focused in this preliminary paper on the uncertainties in magnetic-field-line tracing that result from the standard errors in just the axisymmetric part of the internal geomagnetic field. An exact analytic equation exists for the magnetic field lines of an arbitrary linear combination of axisymmetric multipoles. This equation is used to derive numerical estimates of the uncertainties in magnetic-field-line tracing that are due to the published standard errors in the axisymmetric spherical harmonic coefficients (i.e. g_n⁰ ± g_n⁰). Numerical results determined from the analytic equation are compared with computational results based on stepwise numerical integration along magnetic field lines. Excellent agreement is obtained between the analytical and computational methods in the axisymmetric case, which provides great confidence in the accuracy of the computer program used for stepwise numerical integration along magnetic field lines. This computer program is then used in the following paper to estimate the uncertainties in magnetic-field-line tracing in the magnetosphere that arise from the published standard errors in the full set of spherical harmonic coefficients, which define the complete (non-axisymmetric) geomagnetic field of internal origin. Numerical estimates of the uncertainties in magnetic-field-line tracing in the magnetosphere, calculated here for the axisymmetric part of the internal geomagnetic filed, should be regarded as first approximations in the sense that such estimates are only as accurate as the published standard errors in the set of axisymmetric spherical harmonic coefficients. However, all procedures developed in this preliminary paper can be applied to the derivation of more realistic estimates of the uncertainties in magnetic-field-line tracing in the magnetosphere, following further progress in the determination of more accurate standard errors in the spherical harmonic coefficients.Also Visiting Reader in Physics, University of Sussex, Falmer, Brighton, BN1 9QH, UK     

最新国际地磁参考场模型IGRF11研究

详细阐述了国际地磁参考场模型IGRF11的建模原理、 数据来源、 球谐系数及误差来源等. 以NOAA/NGDC候选地磁场模型为例, 分析了数据的选取原则和校正方法. 在MATLAB环境下, 研制了该模型的可视化软件, 为制备地磁仿真基准图提供了最有可能和最为便捷的途径, 能有效使用国际公开数据, 减少外场地磁实测工作量和成本, 对地磁导航工程化具有重要的价值. 对模型检验值和实测值与模型计算值进行比较, 结果表明, 软件计算结果是正确的、 可靠的; 对IGRF11与WMM2010模型的异同点进行比较, 结果表明, WMM2010的模型精度要高于IGRF11, 具有更为广阔的适用范围. 最后, 从模型长期变化估算角度入手, 分析了引起IGRF11模型计算精度低的原因.     

第11代国际地磁参考场

2009年12月,国际地磁学与高空物理学协会（IAGA）发布了第11代国际地磁参考场（IGRF-11）。第11代IGRF包括1900．0-2010．0年代（间隔为5年）共23个地磁模型与2010．0---2015．0年代地磁长期变化的预测模型,其中1900．0-1995．0年代模型的阶次为N=M=10,相应球谐系数的精度为lnT;2000．0—2010．0年代模型的阶次为N=M=13,其球谐系数的精度为0．1nT;而2010．0—2015．0年代地磁长期变化预测模型的阶次为N=M=8,其球谐系数的精度为0．1nT。本文概述了第11代国际地磁参考场及其2010．0年代地磁模型与2010．0--2015．0年代地磁长期变化的预测模型。     

The new approach to global deep sounding

The new approach to global geomagnetic sounding is developed to overcome difficulties of spherical harmonical analysis and subsequent transfer function determination. The approach is based on minimizing the discrepancy between experimental and simulated magnetic fields. The discrepancy is considered as a function of the medium model parameters and the coefficients of external fields. The method can be used for laterally inhomogeneous as well as homogeneous earth models. An example of its application to a radially symmetric model is demonstrated.     

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。     

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     

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。     

地球扁率在其历史上的变化

地球的形状变化是地球演化研究的基本问题之一.为了确定地球形状的变化需要了解各个地质历史时期其扁率的变化.根据地球的扁率公式,得到扁率与地球的半径、质量和角速度关系.通过推测地球的半径、质量和角速度的变化量,计算出地球各个时期的扁率.地球的扁率自形成以来总体趋势在减小.地球是个固液混合的椭球体,在各个地质历史时期中地球扁率的真实值应该小于计算的扁率值.     

地磁正常场的选取与地磁异常场的计算

根据2003年中国地磁观测数据（包括135年地磁测点和35个地磁台）以及我国邻近地区38个IGRF计算点的地磁数据,计算中国地磁异常场的分布。选取两种地磁场模型作为地磁正常场,一是国际参考地磁场的球谐模型,二是中国地磁场泰勒多项式模型。根据各个测点的地磁异常值（观测值减去模型计算值）,用球冠谐分析方法计算地磁异常场的球冠谐模型,并绘制2003年中国地磁异常（△D,△I,△F,△X,△Y,△Z）。分析和讨论了中国地磁异常场。     

Perpendicular error effect in the DGRF model proposals

It is known that when spherical harmonic models of the geomagnetic field are derived from data sets having insufficient component (vector) data these models are liable to pseudo-systematic errors, particularly large for the sectorial coefficients. These coefficient errors correspond to error fields having foci on and near the dip equator, and which are everywhere roughly perpendicular to the main (approximately dipole) field—the perpendicular error effect.This paper demonstrates the existence of such an effect, with foci of up to 300 nT, in the differences between the IGS, NASA, and USGS spherical harmonic models proposed for the DGRF 1965–1975, and deduces that it probably comes almost entirely from the NASA models.It is shown that, although the NASA data set contained a large amount of observatory component data, the method of analysis used by NASA meant that this data was given little weight in the main field model.It appears that ∼ 15% of component data is needed to avoid significant perpendicular error effect.     

青藏高原地磁场模型的研究

根据青藏高原地磁三分量绝对测量资料,使用泰勒多项式方法和冠谐分析方法,计 算了青藏高原地磁场（Ｘ, Ｙ,Ｚ ）的泰勒多项式模型和青藏高原地磁剩余场（△Ｘ,△Ｙ,△Ｚ）的冠谐 模型,并绘制了相应的理论地磁图．分析了磁异常点对地磁场模型的影响,对比分析了地磁 场的多项式模型和冠谐模型,讨论了地磁场模型的边界效应问题．     

青藏高原地磁场模型的研究

根据青藏高原地磁三分量绝对测量资料,使用泰勒多项式方法和冠谐分析方法,计 算了青藏高原地磁场（Ｘ, Ｙ,Ｚ ）的泰勒多项式模型和青藏高原地磁剩余场（△Ｘ,△Ｙ,△Ｚ）的冠谐 模型,并绘制了相应的理论地磁图．分析了磁异常点对地磁场模型的影响,对比分析了地磁 场的多项式模型和冠谐模型,讨论了地磁场模型的边界效应问题．     

Geomagnetic evolution: 400 years of change on planet Earth

Spherical harmonic coefficients of the geomagnetic field, calculated from historical observations of declination, inclination and intensity, and from archaeomagnetic inclination results, have been used to produce a film of geomagnetic change since 1600 A.D. The non-dipole geomagnetic field is found to be constantly changing: no fixed or standing non-dipole features are observed. Non-dipole foci are seen to have lifetimes of a few hundred years. The westward drift, which was an important feature of the 18th and early 19th century geomagnetic field, was less pronounced in the 17th century. The growth, evolution, decay and replacement of non-dipole foci, but not their movement are found to have been the major features producing century-long secular directional magnetic variation. Most of the low degree and order spherical harmonic coefficients have changed significantly over the last few hundred years. In particular the change in sign of the axisymmetric quadrupole around 1837 A.D. is noted. Sustained, century-long, intensity changes, however, appear to have been dominated by variations in the intensity of the centred dipole, rather than by non-dipole field fluctuations.     

The effect of observational errors on geomagnetic field models based solely on total-intensity measurements

An investigation is made of the changes produced in the spherical harmonic coefficients of a geomagnetic main field model when values of the total intensity synthesized from the model are systematically perturbed. Corresponding differences between the values of the field components from the original model and those from the perturbed model are also derived. The effects of varying the altitude of the perturbed point and of including a small proportion of component data are discussed. It is shown that significant errors can arise in the region of the dip equator when only scalar data are used, but that these errors can be eliminated if the data set contains 10% of well-distributed component data, some of which refers to the equatorial zone.     

甘肃大地电场的变化特征分析

对甘肃省平凉、松山、古丰、山丹、高台、嘉峪关、瓜州7个台站的地电场分钟值、小时值的日变特征进行了对比,对2012年3月地电场小时值进行了频谱分析,同时结合地磁场H分量、分量应变(固体潮)进行了对比分析。结果表明:1地电场静日的日变化都很明显,呈明显"双峰"和"单谷"形态,并且NS向与EW向变化形态基本相同;各台站同测向及相互之间的自相关系数基本在0.85;不同台站之间同测向的相关系数多数在0.75以上;这是地磁场H分量与固体潮共同影响的结果。2通过频谱分析各台站的优势周期主要以12h和8h为主,但存在差异性,谱值最大相差20倍。3各台站的日变幅也不尽相同,其中,差异性主要表现为与地磁场和固体潮影响量的差异性,并与台站的位置、地质构造、电性结构、浅层电阻率等有很大关系。     

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.     

中国地区地磁场的球冠谐和分析

本文根据中国及其邻近地区的地磁三分量绝对测量资料,利用球冠谐和分析方法,计算出中国地区地磁剩余场的冠谐模型,地磁剩余场△X,△Y,△Z的模型均方根偏差分别为106.9,89.7,137.6 nT.提出由地磁场的国际参考地磁场和地磁剩余场的冠谐模型组成的联合模型作为中国参考地磁场的模型,它能较好地表示中国地磁场的分布.以地磁场的联合模型为正常背景场,计算出三分量磁异常的冠谐模型,并分析了磁异常的基本特征,它将为深入研究中国岩石层结构提供新证据.     

On the estimation of a multi-resolution representation of the gravity field based on spherical harmonics and wavelets

The gravitational potential of the Earth is usually modeled by means of a series expansion in terms of spherical harmonics. However, the computation of the series coefficients requires preferably homogeneous distributed global data sets. Since one of the most important features of wavelet functions is the ability to localize both in the spatial and in the frequency domain, regional and local structures may be modeled by means of a spherical wavelet expansion. In general, applying wavelet theory a given input data set is decomposed into a certain number of frequency-dependent detail signals, which can be interpreted as the building blocks of a multi-resolution representation. On the other hand, there is no doubt that the low-frequency part of the geopotential can be modeled appropriately by means of spherical harmonics. Hence, the main idea of this paper is to derive a combined model consisting of an expansion in spherical harmonics for the low-frequency part and an expansion in spherical wavelets for the remaining medium and high-frequency parts of the gravity field. Furthermore, an appropriate parameter estimation procedure is outlined to solve for the unknown model coefficients.