Abstract

Forward modelling in the space domain is a very important task in geodesy and other geosciences. From topographical or isostatic information in the form of digital terrain model (DTM) and density model, the effects of these parameters or their derivatives on the gravity potential can be evaluated for different applications. In most cases, height or height-layer models are in use, which are gridded with respect to spherical coordinates. This holds for global as well as regional or even local applications. The definition of the spherical gridlines leads immediately to the spherical volume element, that is, the tesseroid. Only in the specific case that the observation point is located on the symmetry axis of the spherical coordinate system does the Newton integral have a closed analytical solution. More specifically, the effect of a tesseroid can be determined by evaluating the analytical solution of a segment of a spherical zonal band. To apply this aspect in practice, the DTM must be transformed into the local spherical azimuthal system of the observation point (UNIPOL approach). In the general case, the Newton integral can be solved, for example, using a Taylor series expansion of the integral kernel and a subsequently applied term-wise integration (GIK approach). Within this contribution, the two fundamentally different tesseroid approaches, namely, the GIK and the UNIPOL approach are compared. This comparison is performed, in particular, with regard to the required computational time and the approximation error under different test scenarios. The numerical studies show that both approaches are equivalent in terms of accuracy for both the gravitational potential and gravity; however, the UNIPOL approach is more time consuming because, for each observation point, the whole DTM must be transformed. Small numerical differences exist between the compared approaches for special constellations regarding the source point and the observation point.

中文翻译：

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比较两种不同计算方法的齿形引力效应

摘要

在大地测量学和其他地球科学中，空间域中的正向建模是一项非常重要的任务。根据数字地形模型（DTM）和密度模型形式的地形或等静线信息，可以针对不同应用评估这些参数或其导数对重力势的影响。在大多数情况下，将使用高度或高度层模型，这些模型相对于球坐标系是网格化的。这适用于全球以及区域甚至本地应用程序。球形网格线的定义立即指向球形体积元素，即tesseroid。仅在观察点位于球坐标系的对称轴上的特定情况下，牛顿积分才具有封闭的解析解。进一步来说，可以通过评估球状带状区段的解析解来确定类畸形的效果。要在实践中应用此方面，必须将DTM转换为观察点的局部球形方位角系统（UNIPOL方法）。在一般情况下，牛顿积分可以例如通过使用积分核的泰勒级数展开和随后应用的按项积分（GIK方法）来求解。在此贡献中，比较了两种根本不同的锯齿状方法，即GIK和UNIPOL方法。尤其是在不同的测试场景下，根据所需的计算时间和近似误差进行比较。数值研究表明，两种方法在重力势和重力精度上都是等效的。但是，UNIPOL方法更耗时，因为对于每个观察点，必须对整个DTM进行转换。对于特殊星座，在源点和观测点的比较方法之间存在很小的数值差异。