Forward gravity field modelling requires high accurate determination of the topographic potential on the surface of the Earth’s ellipsoid. Since the topography is recently known with high resolution, the determination of the harmonic analysis of the topographic ellipsoidal potential should be carried-out to an ultra-high degree. The paper presents a direct technique to determine the harmonic analysis of the ellipsoidal potential to an ultra-high degree. The detailed derivation of the developed expressions is given. The proposed technique can be used to compute the harmonic analysis of the topographic potential of the whole Earth or for a local/regional window. This has a great advantage when using the proposed technique for local/regional geoid determination within the window remove–restore technique (Abd-Elmotaal and Kühtreiber, Phys Chem Earth Part A 24(1):53–59, 1999, J Geod 77(1–2):77–85, 2003). The developed technique has been compared to existing global solutions for the ellipsoidal topographic potential and it shows quite good agreement. Local tests for the developed technique proved that it works very well.

中文翻译：

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具有全局和局部验证的椭球地形势的直接谐波分析

正向重力场建模需要高精度地确定地球椭球表面的地形势。由于地形是最近以高分辨率已知的，因此地形椭球势的谐波分析的确定应该进行到超高程度。本文提出了一种直接的技术，可以在超高程度上确定椭球势的谐波分析。给出了所开发表达式的详细推导。所提出的技术可用于计算整个地球或局部/区域窗口的地形势的谐波分析。当在窗口移除-恢复技术（Abd-Elmotaal 和 Kühtreiber，物理化学地球 A 部分 24(1):53–59, 1999, J Geod 77(1–2):77–85, 2003)。已将开发的技术与现有的椭球地形势全局解决方案进行了比较，结果显示出非常好的一致性。对所开发技术的本地测试证明它非常有效。