Spherical harmonic analysis and Fourier transformation

Summary The paper deals with the computation of spherical harmonic coefficients from surface measurements of the magnetic or gravity field of the Earth when the measurements are distributed regularly. The Fourier representation of associated Legendre functions which this procedure makes use of, then enables the harmonic analysis to be transformed to Fourier analysis which has better numerical properties.

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Spherical harmonic analyses of the palaeomagnetic field

Analytical models of the palaeomagnetic field have been constructed for a number of geological periods (Quaternary Neogene, Jurassic, Triassic, Permian and Permo-Carboniferous) by spherical harmonic analysis using the present-day world map as a basis and (for the earlier periods) using also a palaeogeographic reconstruction. The use of the palaeogeographic chart for the earlier periods simplifies the models, and its use appears to be valid. The low accuracy, small number and uneven distribution of palaeomagnetic data severely limit the conclusions which can be drawn from the analyses. Nevertheless the results for all periods indicate that throughout the past 300 million years the geomagnetic field has maintained its global structure, and has remained similar to the field of a dipole slightly shifted from the Earth's centre. It appears that there have not been any persistent systematic anomalies or variations in the Earth's magnetic field throughout that time, but rather that the field has been oscillating around a mean level not greatly different from that of the present epoch.     

Spherical harmonic analysis of the geomagnetic secular variation — A review of methods

Methods of producing spherical harmonic models of the secular variation of the geomagnetic field are reviewed and classified, more emphasis being given to modern methods. The relative merits of the different techniques are discussed and their relevance to possible procedures for updating the International Geomagnetic Reference Field (IGRF) are considered.     

Meridional transport of sensible heat in contrasting monsoon activity: Spherical harmonic analysis

Spherical harmonic analysis is made of the grid point values of geopotential heights at 700 mb and 300 mb levels for the months April to August for the years 1967 and 1972. The year 1967 is a good monsoon year and 1972 is a bad monsoon year in India. Meridional transport of sensible heat is obtained in wave number domain using spherical harmonic coefficients at 500 mb level form=1 to 10 andn–m=0 to 10, wherem represents the wave number round the globe andn–m gives the numbers of zero points from north pole to south pole excluding the poles themselves.Large northward transports of sensible heat in the month of May and in the monsoon months at the subtropics are characteristic of bad monsoon. Wave 1 transports sensible heat southward (forn–m=0) and wave 2 transports sensible heat northward (forn–m=4). Strengthening of wave 1 is conducive to good monsoon year and strengthening of wave 2 is conducive to bad monsoon year. These are the same features obtained in Fourier analysis. The contrasting features exist in waves 1 and 2 both in good and in bad monsoon and are better defined in the present analysis than in the Fourier analysis of the earlier study. However, waves 1 and 2 reveal clearer contrast in the present analysis than in the Fourier analysis. Bad monsoon activity is associated with large divergence of heart at subtropics and large convergence of heat at extra tropics.     

Spherical cap harmonic analysis of regional magnetic anomalies based on CHAMP satellite data

We used CHAMP satellite vector data and the latest IGRF12 model to investigate the regional magnetic anomalies over mainland China. We assumed satellite points on the same surface (307.69 km) and constructed a spherical cap harmonic model of the satellite magnetic anomalies for elements X, Y, Z, and F over Chinese mainland for 2010.0 (SCH2010) based on selected 498 points. We removed the external field by using the CM4 model. The pole of the spherical cap is 36N° and 104°E, and its half-angle is 30°. After checking and comparing the root mean square (RMS) error of ΔX, ΔY, and ΔZ and X, Y, and Z, we established the truncation level at K ^max = 9. The results suggest that the created China Geomagnetic Referenced Field at the satellite level (CGRF2010) is consistent with the CM4 model. We compared the SCH2010 with other models and found that the intensities and distributions are consistent. In view of the variation of F at different altitudes, the SCH2010 model results obey the basics of the geomagnetic field. Moreover, the change rate of X, Y, and Z for SCH2010 and CM4 are consistent. The proposed model can successfully reproduce the geomagnetic data, as other data-fitting models, but the inherent sources of error have to be considered as well.     

Spherical harmonic expansion of the continents and oceans distribution function

Summary An algorithm is derived to compute the coefficients of a spherical harmonic series for the functionE(, ) representing the distribution of continents and oceans with the least-squares method. Some properties of the system of normal equations, when measuring points are distributed in a regular grid, are discussed. The fully normalized complex coefficients to the ninth degree are given in the table.

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Spherical harmonic expansion of the Earth's crustal thickness up to degree and order 30

Summary The knowledge of the depth of the Mohoroviic' discontinuity (Moho) is important for investigating the 3-D structure of the Earth's interior, since the variations in the crustal thickness can significantly affect geophysical observations made at the Earth's surface. In this paper, the information about the topography of the crust-mantle boundary is compiled from various sources and the crustal thickness is represented in a spherical harmonic expansion up to degree and order 30.     

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.

     

A primitive pseudo wave equation formulation for solving the harmonic shallow water equations

A finite element method formulation for solving the harmonic shallow water equations in their primitive or unmodified form is developed and analysed. The scheme, referred to as the Primitive Pseudo Wave Equation Formulation (PPWE), involves developing a weak weighted residual form of the continuity equation and furthermore forming a pseudo wave equation by substituting the discretized form of the momentum equation into the discretized form of the continuity equation. The final set of equations to be solved, the pseudo wave equation and the primitive momentum equations, deceptively resemble the discretized equations of the wave equation formulation of Lynch and Gray. Despite this resemblance, Fourier analysis indicates that the PPWE scheme is still fundamentally primitive.However, application of the PPWE scheme to a set of stringent test problems results in very good solutions with well controlled nodal oscillations. It is shown that this low degree of spurious oscillations is due to the treatment of the boundary conditions such that elevation is taken as an essential condition and normal flux is taken as a natural condition. This particular boundary condition treatment is suggested by the formation of the pseudo wave equation. Furthermore, even though the equation re-arrangement does not in itself effect the solutions, it does make the scheme much more efficient.     

Compressibility,core dynamics and the subseismic wave equation

It is shown that in the dynamics of a deep fluid of planetary scale such as the Earth's core, compressibility, stratification and self-gravitation are all important as well as rotation. The existing proof of Cowling's theorem prohibiting non-stationary axisymmetric dynamos, and the application of the Proudman-Taylor theorem to core flows, both based on the assumption of solenoidal flow, need to be reconsidered. For sufficiently small (subacoustic) frequencies or reciprocal time scales, an approximation which neglects the effect of flow pressure on the density is valid. We call this the “subseismic approximation” and show that it leads to a new second-order partial differential equation in a single scalar variable describing the low frequency dynamical behaviour. The new “subseismic wave equation” allows a direct connection to be made between the various possible physical regimes of core structure and its dynamics.     

Stabilized finite elements for harmonic soil dynamics problems near the undrained-incompressible limit

Simple Finite Element models for soil dynamics and earthquake engineering problems in the frequency domain are a fast and valuable tool providing a first approximation before a full non-linear analysis in the time domain is performed.Quite often the problem concerns saturated soils with very small permeability and pore fluid of neglectable compressibility. In the limit, the permeability is assumed to be zero and the pore fluid incompressible. Here, engineers use standard finite element codes formulated in terms of displacements but incompressibility may result in volumetric locking of the mesh with a severe loss of accuracy.The purpose of this paper is to present a simple mixed finite element formulation in the frequency domain based on displacements and pore pressures as main variables. A suitable stabilization technique allowing for equal order interpolation of displacements and pressures has been introduced for incompressible and zero permeability limits.Of course, the range of application is limited to those problems in which the behaviour of the material can be approximated by linear models, and therefore modelling of phenomena such as liquefaction, cyclic mobility or cavitation occur is excluded.The paper shows as well an extremely simple way of coupling solid and water domains as it occurs for instance in quay walls under dynamic loading.     

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.     

Linearization of rigid body dynamics on frictional interfaces under harmonic loading

The non-linear dynamic response of the rigid block is linearized by means of a friction model that implicitly considers block response through an experimental parameter obtained from shaking table experiments. In view of the great difficulty in carrying out shaking table experiments, in this technical note some recommendations to estimate friction model parameters are given. The selection of parameters considers the sliding response mode of the block–plane-excitation system: stick–slip or continuous sliding. Once all the friction parameters and block response mode were estimated, a methodology was proposed to compute rigid block dynamic response. The numerical results were then compared to actual experimental data for a rigid block sliding on a geotextile–wood interface, along an inclined plane subjected to base harmonic acceleration. Experiments were carried out for both the stick–slip and the continuous sliding modes. Computed and measured responses for both cases showed good agreement, thus indicating that the methodology developed in this research is adequate to capture the physics (of non-linear nature) of rigid blocks sliding on frictional interfaces subjected to complex harmonic loading. The findings encourage the extension of the linearization technique to the more general seismic loading case.     

在地磁学与地球重力学中的球谐分析

球谐分析在地磁学与地球重力学中得到广泛的应用.由于地球磁场与地球引力场的不同,球谐表述有所不同.地磁场的高斯分析（球谐分析）的结果表明,地磁场没有单极子,而引力场有,地磁场有内外源场之分,而地球引力场只有内源场,地磁场的球谐级数收敛快,地磁场高斯级数所用的蒂合勒让德函数是Schmidt半标准化的,而地球引力场中用的是全标准化的,地磁场的高斯系数随时间变化快,每5年产生一个IGRF（国际地磁参考场）, 而引力场的变化是与地质变化有关,相对于地磁场来说,是缓慢的. 地磁场的高斯分析还存在一个唯一性问题.     

阿尔奇公式的适用性分析及其拓展

阿尔奇年代的储层孔隙结构简单,岩石孔隙可以被实验电解质完全充填,随着油气田勘探开发的逐步深入,对储层孔隙结构复杂程度的认识更加深入.发现岩石孔隙不仅有有效孔隙,也有无效孔隙.不仅有效孔隙导电,无效孔隙也导电,某些岩石骨架同样也导电.如何去除无效孔隙和其他各类岩石附加导电现象的影响,正确评价有效孔隙的贡献往往令人困惑.笔者发现：通过岩石电阻率计算地层因数只是在没有岩石附加导电环境下的特殊应用,在更为普适的条件下,地层因数实质是借助岩石电阻变化率反映孔隙连通性的重要参数.与物理学位移、速度的关系类似,电阻率和电阻变化率之间同样有着密切关系.这一观点的提出为在复杂孔隙结构条件下评价储层孔隙有效性提供了手段.

     

Comparative error analysis in finite element formulations of the advection-dispersion equation

The behaviour of numerical solutions of the one-dimensional advection-dispersion equation is investigated and comparisons between the consistent and the lumped formulations of Galerkin finite element schemes are made. Well-known criteria for the control of accuracy in the lumped (finite difference) formulation are reviewed. It is found that, because the numerical error produced by the consistent formulation is generally less than that produced by the lumped formulation, these criteria can also be used for the control of numerical dispersion in the consistent formulation. However, because the error in both types of solutions decreases in time when the discretization is invariant, the criteria can be relaxed with advancing simulation time. For the consistent formulation it is found that beyond some initial time period, the numerical error depends only on the temporal discretization. This suggests that constant accuracy can be maintained throughout the simulation period while allowing the time step length to grow.     

A perspective on numerical analysis of the diffusion equation

The equation of transient groundwater motion is founded on the principle of mass conservation and can be mathematically described by the diffusion equation. Recently, powerful integral formulations have been developed for numerically solving the diffusion equation under complex conditions. In the literature, it is customary to formulate the integral equations by integrating point differential equations. Instead, in this paper, we shall employ a direct method of formulation, starting from the concepts of set and measure, the notion of partitions and the definition of set-averages.When the direct approach is applied to formulate the well-known finite element (FEM) equations, it is seen that the ‘Galerkin’ weighting function, which is mathematically treated as an artifice for weighting residuals, is but an appropriate spatial partition function. The logical framework of the direct approach is then applied to study the properties of ‘lumped’ and ‘consistent’ matrices arising in the use of the FEM. The lumped matrix, stemming naturally from the direct approach, seeks to conserve mass locally as well as globally, while the consistent matrix, which results only when the differential equation is integrated in a specific fashion, attempts only to preserve global mass balance.It is concluded that the direct approach is simple and complete and, in so far as the integral formulation is concerned, there is little to be gained in starting with the differential equation. Further, in formulating integral equations, it is common practice to evaluate only the time-dependent changes in the mass content of the system and ignore the evaluation of the mass content of the system at any given instant of time. In order to be complete in itself, a true integral approach should evaluate both the time-dependent changes in the mass content of the system as well as the instantaneous mass content at any given time.     

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.