The spherical Slepian basis as a means to obtain spectral consistency between mean sea level and the geoid
The spherical Slepian basis as a means to obtain spectral consistency between mean sea level and the geoid
复制标题
球形斯莱普基作为获得平均海平面和大地水准面之间光谱一致性的手段
作者：
D. C. Slobbe;Frederik J. Simons;R. Klees
平均动态地形（MDT）可以计算为平均海平面（MSL）和重力大地水准面之间的差异。这要求两个数据集在光谱上是一致的。在实际应用中，大地水准面数据的分辨率往往低于MSL数据的分辨率，因此，在计算MDT之前，需要对MSL数据进行低通滤波。为此目的，传统的低通滤波器是不够的，在沿海地区，他们遇到大陆上的不确定的MSL信号失败。在本文中，我们考虑使用一个带限，空间集中Slepian基础，以获得一个低分辨率近似的MSL信号。我们计算Slepian函数的海洋和海洋的一部分，并通过这种方法与其他方法，特别是迭代球谐方法结合高斯低通滤波，和各种修改计算MDT的性能进行比较。基于数值实验，我们得出结论，这些方法都没有提供一个低分辨率的MSL近似在亚分米级。特别是，我们表明Slepian函数不是解决这个问题的合适基函数，并且低分辨率MSL信号的Slepian表示存在宽带泄漏。我们还表明，一个有意义的定义，低分辨率MSL不完整的球形域涉及正交基函数与Slepian函数不具备的其他属性。一个低分辨率的MSL信号，频谱上与一个给定的大地水准面模型一致，通过适当的截断的MSL信号的扩展这些正交基函数。我们使用球面调和函数的Gram-Schmidt正交化来计算这些正交基函数之一。对于海洋，我们可以构造一个正交基，其分辨率仅等于球谐次数36。由于固有的不稳定性，具有较高分辨率的基的计算失败。正则化减少了不稳定性，但破坏了正交性，因此，提供了不切实际的低分辨率MSL近似。需要更多的研究来解决不稳定性问题，也许是通过找到一个不同的正交基来完全避免它。
The mean dynamic topography (MDT) can be computed as the difference between the mean sea level (MSL) and a gravimetric geoid. This requires that both data sets are spectrally consistent. In practice, it is quite common that the resolution of the geoid data is less than the resolution of the MSL data, hence, the latter need to be low-pass filtered before the MDT is computed. For this purpose conventional low-pass filters are inadequate, failing in coastal regions where they run into the undefined MSL signal on the continents. In this paper, we consider the use of a bandlimited, spatially concentrated Slepian basis to obtain a low-resolution approximation of the MSL signal. We compute Slepian functions for the oceans and parts of the oceans and compare the performance of calculating the MDT via this approach with other methods, in particular the iterative spherical harmonic approach in combination with Gaussian low-pass filtering, and various modifications. Based on the numerical experiments, we conclude that none of these methods provide a low-resolution MSL approximation at the sub-decimetre level. In particular, we show that Slepian functions are not appropriate basis functions for this problem, and a Slepian representation of the low-resolution MSL signal suffers from broadband leakage. We also show that a meaningful definition of a low-resolution MSL over incomplete spherical domains involves orthogonal basis functions with additional properties that Slepian functions do not possess. A low-resolution MSL signal, spectrally consistent with a given geoid model, is obtained by a suitable truncation of the expansions of the MSL signal in terms of these orthogonal basis functions. We compute one of these sets of orthogonal basis functions using the Gram–Schmidt orthogonalization for spherical harmonics. For the oceans, we could construct an orthogonal basis only for resolutions equivalent to a spherical harmonic degree 36. The computation of a basis with a higher resolution fails due to inherent instabilities. Regularization reduces the instabilities but destroys the orthogonality and, therefore, provides unrealistic low-resolution MSL approximations. More research is needed to solve the instability problem, perhaps by finding a different orthogonal basis that avoids it altogether.