Expansion of geoid heights into a spherical harmonic series

Summary An algorithm is derived to compute the coefficients of a spherical harmonic series for the geoid radius vector that are a solution to a system of linear equations, containing Stokes' constants of the geopotential.     

Expansion of a function given over an equipotential surface into spherical harmonics

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Lame surfaces as a generalisation of the triaxial ellipsoid

In modern geodesy the triaxial ellipsoid as a generalisation of the ellipsoid of revolution has a significant position in studying the figure of the Earth. Lame surfaces represent a generalisation of the triaxial ellipsoid. The following paragraphs are devoted to curvatures of the Lame surfaces.     

Two modified algorithms to transform Cartesian to geodetic coordinates on a triaxial ellipsoid

The paper presents modified Lin and Wang??s (1995) and Hedgley??s (1976) algorithms to tackle the problem of transforming Cartesian to geodetic coordinates on a triaxial ellipsoid. Originally, the methods were developed for an ellipsoid of revolution but due to their universality, they may be adapted to the more complicated problem stated on a triaxial ellipsoid what is in fact done in this work. Two modified methods are compared to the vector method recently introduced by Feltens. The modified methods turn out to be more accurate and faster than the algorithm presented by Feltens.     

卫星测高混合边值问题的球谐级数解法

研究了球界面下卫星测高问题的解法,利用有限逼近方法得到了下列结论：若陆地部分是球冠,则卫星测高问题的解可以转换成关于球谐级数位系数的线性方程组,并且位系数的阶和次是以分离形式出现的,从而确保该解法具有实用意义. 利用重力场360阶模型进行模拟计算的结果表明：该解法得到的位系数的相对精度达到了10－１１. 同时证明了常用的Stokes问题、Dirichlet问题、Neumann问题可以看成卫星测高问题的特殊情况.     

An optimized method to transform the Cartesian to geodetic coordinates on a triaxial ellipsoid

Studia Geophysica et Geodaetica - A general triaxial ellipsoid is suitable to represent the reference surface of the celestial bodies. The transformation from the Cartesian to geodetic coordinates...     

Orthogonality of spherical harmonic coefficients

An essentially arbitrary function V(θ, λ) defined on the surface of a sphere can be expressed in terms of spherical harmonics $\text{V(θ, Λ) = a}\text{∑}\text{n=1}\text{∞}\text{∑}\text{m=0}\text{n}\text{p}{}^{\mathrm{m}}{}_{\mathrm{n}}\text{(cos θ) (g}{}^{\mathrm{m}}{}_{\mathrm{n}}\text{cos mΛ + h}{}^{\mathrm{m}}{}_{\mathrm{n}}\text{sin mΛ)}$ where the P_n^m are the seminormalized associated Legendre polynomials used in geomagnetism, normalized so that $\text{〈[P}{}^{\mathrm{m}}{}_{\mathrm{n}}\text{(cos θ) cos mΛ]〉}{}^2\text{=1/}\text{(2n+1)}$ The angular brackets denote an average over the sphere. The class of functions V(θ, λ) under consideration is that normally of interest in physics and engineering. If we consider an ensemble of all possible orientations of our coordinate system relative to the sphere, then the coefficients g_n^m and h_n^m will be functions of the particular coordinate system orientation, but $\text{〈:(g}{}^{\mathrm{m}}{}_{\mathrm{n}}\text{)}{}^2\text{〉) = 〈(h}{}^{\mathrm{m}}{}_{\mathrm{n}}\text{)}{}^2\text{=}\text{S}{}_{\mathrm{n}}\text{/}\text{(2n=1)}$ where $\text{S}{}_{\mathrm{n}}\text{=}\text{∑}\text{m=0}\text{n}\text{[(g}{}^{\mathrm{m}}{}_{\mathrm{n}}\text{)}{}^2\text{+ (h}{}^{\mathrm{m}}{}_{\mathrm{n}}\text{)}{}^2\text{]}$ for any orientation of the coordinate system (S_n is invariant under rotation of the coordinate system). The averages are over all orientations of the system relative to the sphere. It is also shown that 〈g^m_ng^l_p〉 = 〈h^m_nh^l_p〉 = 0 for l ≠ m or p ≠ n and 〈g^m_nh^l_p〉 = 0 fro all n, m, p, l.     

各向异性半空间上方垂直磁偶极子电磁场的球面波展开式

文中推导出单轴各向异性媒质半空间上方垂直磁偶极源电磁场的精确解析表达式. 应用圆柱波函数的球面波展开式和超几何函数理论,场分量中的索末菲尔德型积分被表示成快速、绝对收敛的球面波函数系展开式;展开系数是以物性参数为复宗量的勒让德多项式. 该展开式数学物理意义明显,并且不受场点和源点的位置、媒质的物性参数和频率等条件的限制. 利用本文的结果可十分方便地计算和分析任意场点处的电磁场分布.     

Improved determination of heights using a conversion surface by combining gravimetric quasi-geoid/geoid and GPS-levelling height differences

The quasi-geoid/geoid can be determined from the Global Positioning System (GPS) ellipsoidal height and the normal/orthometric heights derived from levelling (GPS-levelling). In this study a gravimetric quasigeoid and GPS-levelling height differences are combined to develop a new surface, suitable for “levelling” by GPS. This new surface provides better conversion of GPS ellipsoidal heights to the national normal heights. Different combining procedures, a four-parameter solution, linear and cubic splines interpolations, as well as the least-squares collocation method were investigated and compared over entire Norway. More than 1700 GPS-levelling stations were used in this study. The combined surface provides significant accuracy improvement for the normal height transformation of GPS height data, as demonstrated by the post-fitting residuals. The best solution, based on the least-squares collocation, provided a conversion surface for the transformation of GPS heights into normal height in Norway with an accuracy of about 5 cm.     

Modelling of neutral-gas releases into the earth's ionosphere

This paper discusses a mathematical model which is aimed at the interpretation of the effects by injections of chemically active gases into the ionosphere. A study is made of neutral-gas expansion processes as well as of processes associated with the formation of decreased plasma density regions, ionospheric holes. The spatial-temporal distribution of particles released during the transition to the diffusion regime is constructed using an approximate, analytic solution of the Boltzmann equation with the collision integral in the form of BGK, with proper allowance for gravitation and atmospheric inhomogeneity. The investigation of processes associated with the formation of ionospheric holes is carried out using a numerical model of ionosphere-plasmasphere coupling, in which the transfer equations for charged components are integrated along geomagnetic field lines. The purpose of the model is to calculate the electron density and temperature, ion composition and emission of a disturbed region in the 6300 Å line at ionospheric and plasmaspheric heights.     

Sector spherical harmonic analysis of the solar magnetic field

The theory of sector harmonic analysis has been developed as applied to the specific conditions of the solar magnetic field. A computer program has been developed. The possibilities of the program are illustrated in an analysis of a large-scale open solar field “singularity” observed on August 11, 2004, using MDI magnetometer data.     

Expansion of a plane wave into Gaussian beams

Summary An integral expansion which expresses a plane monochromatic wave as a superposition of Gaussian beams is found. The expansion can be used to solve many wave propagation problems in complicated structures, including laterally inhomogeneous media with curved interfaces.     

Improved expressions for the ellipsoidal harmonic series representing the Earth gravitational potential and its first and second derivatives on and outside the reference ellipsoid of revolution

The Legendre functions of the second kind, renormalized by Jekeli, are considered in the external space on a set of ellipsoids of revolution which are confocal with respect to the normal ellipsoid. Among these ellipsoids a reference one is chosen which bounds the Earth. New expressions for the first and second order derivatives of the Legendre functions are derived. They depend on two very quickly convergent Gauss hypergeometric series which are obtained by transforming the slowly convergent initial hypergeometric series. The derived expressions are applied for constructing the ellipsoidal harmonic series for the Earth disturbing gravitational potential and its derivatives of the first and second orders. Since outside the chosen reference ellipsoid there are no Earth masses (as compared to the normal ellipsoid) then it is more appropriate for constructing the boundary-value equation and solving it on the basis of surface gravity data reduced to this ellipsoid.     

Expansion of a Plane Wave into Gaussian Beams

Magnetic susceptibility measurements on topsoils have often been used during the last few years to detect anthropogenic pollution. In most cases, a Bartington susceptibility meter for field measurements was used. However, up to now, no standard procedure has been developed for carrying out such investigations. The purpose of our study was to test the compatibility of different set-ups of instruments used for this purpose and the possible influences of subjective (human) factors. Field magnetic susceptibility measurements, carried out with four different Bartington MS2D instruments in strictly defined positions, are very consistent both for low and high values. The correlation coefficient between the magnetic susceptibility values recorded with different Bartington MS2D probes reached 97–98%. A test area was mapped independently by two groups, without any restrictions concerning the choice and distribution of the measured points, but respecting a few standard conditions (e.g., measuring at a distance from tree trunks; on the flattest place possible; recording between 10–30 values per point). The resulting susceptibility maps show the same general features in both cases, suggesting that the measuring strategy applied is suitable for topsoil magnetic screening. The methodology proposed can be used to map magnetic susceptibility on a larger scale—for example Europe—providing large sets of representative data and eliminating border-transition biases and human errors.     

Vertical mixing of gravel over a long flood series

Vertical sediment exchange is a fundamental component of bedload transport in gravel‐bed channels. This paper describes the characteristic depth of exchange achieved over a long flood series. Analysis is based on 11 recoveries of magnetically tagged gravels deployed in Carnation Creek, Canada, completed between 1990 and 2008. Vertical grain exchange mixes gravels throughout the streambed relatively rapidly. Within one to eight floods the mean burial depth approaches two times the surface layer thickness, quantified by the 90th percentile of the size distribution. Finer gravels are mixed more rapidly into the bed than coarser gravels. Both active and passive grain exchanges throughout most of the bed produce the overall vertical distribution of marked grains. Gravel exchanges exhibit fairly consistent patterns once tracers are well mixed by large floods. Results highlight the role of flood sequence in determining exchange depths, support the notion of an upper limit to exchange, and underscore the importance of passive grain exchange. Copyright © 2011 John Wiley & Sons, Ltd.     

Limitations in the use of spherical harmonic methods for deep conductivity determinations

Diurnal variations supply only four frequencies with sufficient power to be useful, giving an overall frequency ratio of only 4. Also only permanent observatories supply sufficient accurate data, and their distribution on the Earth is very irregular. More importantly, the assumption implicit in deep conductivity determinations, that the conductivity is a function of depth only, breaks down because of near surface conductivity anomalies.     

Least squares and integral methods for the spherical harmonic analysis of theSq-field

Spherical harmonic coefficients (SHCs) for the daily magnetic variation fields (solar and lunar) and the main field of the earth are usually estimated by the method of least squares applied to a truncated spherical harmonic series. In this paper, an integral method for computing the SHCs for the solar quiet daily magnetic variation fieldSq is described and applied toSq data for May and June 1965. TheSq SHCs thus derived are then compared with the results obtained using both unweighted and weighted versions of the least squares method. The weighting used tends to orthogonalize the least squares terms. The integral and weighted least squares results agree closely for terms up to order 4 and degree 30, but both disagree considerably for the higher degree terms with the results of the unweighted least squares. Errors introduced by the numerical integration can be shown to be small, hence the disagreement between integral and unweighted least squares coefficient sets arises from improper weighting. Also, it is concluded that discrepancies between the geomagnetic northward and eastward component-derived coefficient sets arise from either time-dependent external sources that produce non-local-time, based fields or nonpotential sources and not from truncation of the spherical harmonic series as has previously been suggested.Deceased.     

球冠谐分析法与直接解拉普拉斯方程法重力异常转换效果对比

在重力异常转换计算过程中,当观测区域面积巨大时,须考虑地球曲率的影响.本文作者以球冠谐分析法和直接解拉普拉斯方程法为例,通过模型试验和实测数据处理详细对比了球面位场转换和平面位场转换的效果,并取得以下认识:(1)对于简单模型,球冠谐分析法的转换结果优于直接解拉普拉斯方程法,而后者具有明显的边界效应;(2)对于稍复杂的模...     

球谐旋转变换结合非全次Legendre方法的局部六边形网格重力场球谐综合

本文首次提出基于六边形网格剖分的全球重力场结构,并解决了局部六边形网格点模型重力异常快速计算问题.首先,采用全新的方法给出缔合Legendre函数值从稳定振荡区到快速衰减区分界线的理论表达式,并基于该公式提出一种基于跨阶次递推的非全次Legendre方法,实现了高纬度地区点的快速球谐综合.其次,引入球谐旋转（Spherical Harmonic Rotation）理论,实现了2160阶次的球谐系数在坐标系旋转下的变换,结合非全次Legendre方法,解决了中低纬度地区点的快速球谐综合.通过计算南极洲（高纬）低分辨率和加里曼丹岛（低纬）高分辨率六边形网格重力异常表明,非全次Legendre方法以10^-19m·s^-2精度水平与传统全阶次方法计算结果吻合,且计算效率提升1倍多,旋转变换结合非全次Legendre方法的计算精度在10^-16m·s^-2,效率提升近5倍.本文提出的方法不仅提升了球谐综合的计算效率,凡是有高纬度的缔合Legendre函数计算的问题,都可利用该方法提升效率,同时,超高阶次球谐旋转变量变换的实现将在地磁场模型构建、计算机视觉、量子物理等领域发挥重要作用.