Computation of spherical harmonic coefficients and their error estimates using least-squares collocation

 Equations expressing the covariances between spherical harmonic coefficients and linear functionals applied on the anomalous gravity potential, T, are derived. The functionals are the evaluation functionals, and those associated with first- and second-order derivatives of T. These equations form the basis for the prediction of spherical harmonic coefficients using least-squares collocation (LSC). The equations were implemented in the GRAVSOFT program GEOCOL. Initially, tests using EGM96 were performed using global and regional sets of geoid heights, gravity anomalies and second-order vertical gravity gradients at ground level and at altitude. The global tests confirm that coefficients may be estimated consistently using LSC while the error estimates are much too large for the lower-order coefficients. The validity of an error estimate calculated using LSC with an isotropic covariance function is based on a hypothesis that the coefficients of a specific degree all belong to the same normal distribution. However, the coefficients of lower degree do not fulfil this, and this seems to be the reason for the too-pessimistic error estimates. In order to test this the coefficients of EGM96 were perturbed, so that the pertubations for a specific degree all belonged to a normal distribution with the variance equal to the mean error variance of the coefficients. The pertubations were used to generate residual geoid heights, gravity anomalies and second-order vertical gravity gradients. These data were then used to calculate estimates of the perturbed coefficients as well as error estimates of the quantities, which now have a very good agreement with the errors computed from the simulated observed minus calculated coefficients. Tests with regionally distributed data showed that long-wavelength information is lost, but also that it seems to be recovered for specific coefficients depending on where the data are located. Received: 3 February 2000 / Accepted: 23 October 2000     

Spatially restricted data distributions on the sphere: the method of orthonormalized functions and applications

 In many geoscientific applications data are irregularly distributed and not globally available, e.g. caps around the poles which are uncovered due to non-polar satellite orbits, or signals being defined solely on bounded regions on the globe. Starting from a sequence of base functions with global support, which in the present case is composed of spherical harmonics being initially non-orthogonal on a bounded subdomain, a set of functions is generated that constitutes an orthonormal basis. Different approaches to realize this transformation are studied and compared with respect to numerical stability and computational effort, and the corresponding effects on the coefficient recovery are investigated. A number of synthetic tests demonstrate the applicability, the benefit, but also the limitations, of this method. Received: 24 March 2000 / Accepted: 9 October 2000     

A harmonic approach to global ocean tide analysis based on TOPEX/Poseidon satellite

The constant and harmonic parts of the global ocean tide are modeled by up to nine major tidal constituents, namely, S2, M2, N2, K1, P1, O1, Mf, Mm, and Ssa. Our computations start with the Fourier sine and cosine series expansion for the tidal constituents, including the constant Mean Sea Level (MSL). Although the frequencies of the tidal constituents are considered known, the coefficients of the sine and cosine functions are assumed to be unknown. Subsequently, the coefficients of the sine and cosine functions, as well as the constant part of the Fourier expansion, were expanded into spherical harmonics up to degree and order n, where n corresponds to the number of linearly independent spherical harmonic base functions needed to model the tidal constituents, determined via independent columns of the Gram matrix. The unknown coefficients of the spherical harmonic expansions are computed using sea level observations within cycles #1–#350 of the TOPEX/Poseidon satellite altimetry over 11 years of its mission. A set of orthonormal base functions was generated for the marine areas covered by TOPEX/Poseidon observations from the spherical harmonics using a Gram-Schmidt orthogonalization process. These were used for modeling the dominant tidal constituents. The computed models based on orthonormal base functions for the nine tidal constituents and the constant part of the Fourier expansion, were tested numerically for their validity and accuracy, proving centimeter accuracy.     

Earth2014: 1 arc-min shape,topography, bedrock and ice-sheet models – Available as gridded data and degree-10,800 spherical harmonics

Since the release of the ETOPO1 global Earth topography model through the US NOAA in 2009, new or significantly improved topographic data sets have become available over Antarctica, Greenland and parts of the oceans. Here, we present a suite of new 1′ (arc-min) models of Earth’s topography, bedrock and ice-sheets constructed as a composite from up-to-date topography models: Earth2014. Our model suite relies on SRTM30_PLUS v9 bathymetry for the base layer, merged with SRTM v4.1 topography over the continents, Bedmap2 over Antarctica and the new Greenland bedrock topography (GBT v3). As such, Earth2014 provides substantially improved information of bedrock and topography over Earth’s major ice sheets, and more recent bathymetric depth data over the oceans, all merged into readily usable global grids. To satisfy multiple applications of global elevation data, Earth2014 provides different representations of Earth’s relief. These are grids of (1) the physical surface, (2) bedrock (Earth’s relief without water and ice masses), (3) bedrock and ice (Earth without water masses), (4) ice sheet thicknesses, (5) rock-equivalent topography (ice and water masses condensed to layers of rock) as mass representation. These models have been transformed into ultra-high degree spherical harmonics, yielding degree 10,800 series expansions of the Earth2014 grids as input for spectral modelling techniques. As further variants, planetary shape models were constructed, providing distances between relief points and the geocenter. The paper describes the input data sets, the development procedures applied, the resulting gridded and spectral representations of Earth2014, external validation results and possible applications. The Earth2014 model suite is freely available via http://ddfe.curtin.edu.au/models/Earth2014/.     

一种基于球体分片射线的大数据体渲染方法

针对大数据量、大范围对象的体渲染效果不够精细美观的问题,文章提出了一种基于球体分片射线的体渲染方法:通过将立方体形式的数据组织转变为环绕球体的弧形数据组织方式,并对体数据依次分片,逐片进行渲染。详细论述了该方法的原理及流程,并在某四维气象演示系统的台风登陆模拟中得到了很好的验证。实验表明,该方法能有效地渲染大数据量大范围对象的细节,提高了可视化效果。     

1m红外太阳望远镜光电导行系统中像场旋转的分析

云南天文台1m红外太阳望远镜(YNST)光电导行系统是基于检测太阳像在面阵CMOS图像传感器上的偏移量作为反馈控制信号的高精度闭环跟踪系统[1]。由于YNST是一个地平式装置,在太阳像偏移检测量中存在着像场旋转量,为了获取稳定清晰的太阳像,必须对检测量进行消旋[2]才能达到光电导行系统的跟踪要求。根据球面天文学及天体测量学的知识并结合导行镜的光学系统,推导了光电导行系统中像场旋转的变化规律并对其进行模拟分析。     

球体内点位错产生的球型位移场：II．位错Love数

本文进一步研究了ＳＮＲＥＩ地球模型内位错所产生的球型位移场问题．作为特例，讨论了位错位于地表时的解和计算技巧．对于１０６６Ａ地球模型进行了具体计算．给出了深度分别为５，１０，３２，１００，３００和６３７ｋｍ的４种类型位错的位错Ｌｏｖｅ数的结果．     

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.     

Asymptotic theory for calculating deformations caused by dislocations buried in a spherical earth: geoid change

An asymptotic theory is presented for calculating co-seismic potential and geoid changes, as an approximation of the dislocation theory for a spherical Earth. This theory is given by a closed-form mathematical expression, so that it is mathematically simple and can be applied easily. Moreover, since the asymptotic theory includes sphericity and vertical structure effects, it is physically more reasonable than the flat-Earth theory. A comparison between results calculated by three dislocation theories (the flat-Earth theory, the theory for a spherical Earth and its asymptotic solution) shows that the true co-seismic geoid changes are approximated better by the asymptotic results than by those of a flat Earth. Numerical results indicate that the sphericity effect is obvious large, especially for a tensile source on a vertical fault plane. AcknowledgementsThe author is grateful to Dr S. Okubo for his helpful suggestions and discussions. Comments by anonymous reviewers are also greatly acknowledged. This research was financially supported by JSPS research grants (C13640420) and Basic design and feasibility studies for the future missions for monitoring Earths environment.     

500 hPa球函数谱异常用于我国东部汛期降水预测的尝试

用北、南半球500hPa月平均位势高度场球函数系数资料代替格点资料，用区域降水指数表示华北、江淮、华南五区汛期(6～8月)降雨。通过引进复相关系数和构造复相关系数场模．分析了半球500hPa环流与我国东部汛雨异常的同时和时滞相关联系。在此基础上．选取5月份北半球、1月份南半球球函数系数与东部汛期降水作时滞的奇异值分解。由此得到的预测关系在1998～2001年汛期降水预报试验中取得了一定效果，从而对500hPa环流的球函数系数资料在区域月、季长期天气预报中的应用作了有益的尝试。