Inverse Multiscale Geoid Computation

从地球实际表面的重力值出发，介绍了一种确定大地水准面外部和地面扰动势的多尺度方法。与标准方法不同的是，提出了用旋转椭球体代替球体作为参考面。此外，根据真实地球表面上给定的电位值，电势本身应由大地水准面外的椭球谐和小波近似。求解过程采用椭球调和正则化小波的多分辨率分析。其基本思想是通过求值泛函，即实际地球表面上的点值，以适当的方式逼近(正则化)小波变换大地球面(即参考椭球)上定义的ALS积分。对于正则化，必须区分带限(即截断奇异值分解)和非带限(即Tikhonov)情况，对于带限和非带限应用，必须分别实现不同类型的金字塔模式的快速小波变换。最后，利用著名的物理大地测量学中的布朗斯公式，将扰动势用于大地水准面的确定。由于大地水准面上越来越多的空间局部化小波的放大特性，理论和实践方面都可以这样表述，从而使从全球到局部的高精度大地水准面模拟变得高效和经济。

Starting from geopotential values as data on the actual Earth's surface a multiscale method of determining the disturbing potential outside and on the telluroid is introduced. In contrast to the standard procedure an ellipsoid of revolution is proposed to act as reference surface instead of a sphere. Furthermore, the potential itself is intended to be approximated by ellipsoidal harmonic wavelets outside the telluroid from given potential values on the real Earth's surface. The solution process is performed by multiresolution analysis in terms of ellipsoidal harmonic regularization wavelets. The essential idea ist to approximate in suitable way the (regularization) wavelet transforms defined als integrals over the telluroid, i.e. the reference ellipsoid, by evaluation functionals, i.e. point values on the actual Earth's surface. The bandlimited (i.e. truncated singular value decomposition) case and the non-bandlimited (i.e. Tikhonov) case must be distinguished for regularization, and different types of pyramid schemata must be implemented for fast wavelets transform for the bandlimited and non-bandlimited appraoch, respectively. Finally, the disturbing potential is used for geoidal determination by using the well-known Bruns formula of physical geodesy. Both the theoretical and practical aspects can be formulated in such a way that high-precision geoidal modelling from global to local scale becomes efficiently and economically available by virtue of the zooming-in property of more and more space-localizing wavelets on the telluroid.