Optimized formulas for the gravitational field of a tesseroid
Optimized formulas for the gravitational field of a tesseroid
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DOI：
10.1007/s00190-013-0636-1
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发表时间：
2013-07-01
影响因子：
4.4
通讯作者：
Heck, Bernhard
中科院分区：
文献类型：
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作者：
Grombein, Thomas;Seitz, Kurt;Heck, Bernhard
大地测量学、地球物理学和相关地球科学中的各种任务需要关于质量分布对重力场相关量（如重力位及其偏导数）的影响的精确信息。使用基于牛顿积分的正演模拟，质量分布通常被分解成规则的基本体。在经典的方法中，主要利用棱柱或点质量近似。考虑到地球球形的影响，应考虑到基于镶嵌体（球面棱柱）的替代质量建模方法，特别是在区域和全球应用中。在笛卡尔坐标系中，质点引力场的表达式相对简单。在积分的情况下，由地心球坐标为界的超曲面体积，它将被证明，它也是有益的，以笛卡尔坐标表示的积分核。这大大简化了确定的超曲面的潜在的衍生物相比，以前公布的方法，利用积分内核表示在球坐标系。基于这一思想，本文给出了齐次超曲面引力势及其二阶导数的优化公式。这些新公式不受球坐标系的极奇异性的影响，因此，可以对地球仪上的任何位置进行计算。由于积分超曲面体积不能解析求解，数值计算是通过扩展积分核的泰勒级数的积分点的空间坐标中的四阶误差。由于笛卡尔积分核的结构被大大简化，泰勒系数可以以紧凑且计算上有吸引力的形式表示。因此，与先前使用的算法相比，优化的镶嵌曲面公式的使用特别受益于计算时间显著减少约45%。为了显示的计算效率和验证的数学推导，新的tesseroid公式应用到两个现实的数值实验，并比较以前发表的tesseroid方法和传统的棱镜的方法。
Various tasks in geodesy, geophysics, and related geosciences require precise information on the impact of mass distributions on gravity field-related quantities, such as the gravitational potential and its partial derivatives. Using forward modeling based on Newton's integral, mass distributions are generally decomposed into regular elementary bodies. In classical approaches, prisms or point mass approximations are mostly utilized. Considering the effect of the sphericity of the Earth, alternative mass modeling methods based on tesseroid bodies (spherical prisms) should be taken into account, particularly in regional and global applications. Expressions for the gravitational field of a point mass are relatively simple when formulated in Cartesian coordinates. In the case of integrating over a tesseroid volume bounded by geocentric spherical coordinates, it will be shown that it is also beneficial to represent the integral kernel in terms of Cartesian coordinates. This considerably simplifies the determination of the tesseroid's potential derivatives in comparison with previously published methodologies that make use of integral kernels expressed in spherical coordinates. Based on this idea, optimized formulas for the gravitational potential of a homogeneous tesseroid and its derivatives up to second-order are elaborated in this paper. These new formulas do not suffer from the polar singularity of the spherical coordinate system and can, therefore, be evaluated for any position on the globe. Since integrals over tesseroid volumes cannot be solved analytically, the numerical evaluation is achieved by means of expanding the integral kernel in a Taylor series with fourth-order error in the spatial coordinates of the integration point. As the structure of the Cartesian integral kernel is substantially simplified, Taylor coefficients can be represented in a compact and computationally attractive form. Thus, the use of the optimized tesseroid formulas particularly benefits from a significant decrease in computation time by about 45 % compared to previously used algorithms. In order to show the computational efficiency and to validate the mathematical derivations, the new tesseroid formulas are applied to two realistic numerical experiments and are compared to previously published tesseroid methods and the conventional prism approach.