High resolution spherical and ellipsoidal harmonic expansions by Fast Fourier Transform
High resolution spherical and ellipsoidal harmonic expansions by Fast Fourier Transform
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DOI：
10.1007/s11200-013-0578-3
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发表时间：
2014-08
影响因子：
0.9
通讯作者：
C. Gruber;Oleh Abrykosov
中科院分区：
文献类型：
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作者：
C. Gruber;Oleh Abrykosov
常规地球物理数据和谐波模型系数之间的高分辨率变换可以通过快速傅立叶变换 (FFT) 最有效地计算。然而，先决条件是数据网格是在适当的几何域中给出的。例如，如果数据位于等角缩减纬度的椭球上，则可以采用球谐函数分析，并随后通过 Jekeli 变换对系数进行转换。这导致了地心纬度域中的球谐谱。然而，数据很可能是在大地纬度（椭球体）给出的，这意味着 FFT 基础需要通过纬度相关的相位滞后进行移动，以获得正确的球谐谱。这需要通过傅里叶求和对移位纬度进行适当的采样率转换，并且不能通过FFT算法有效地处理。本文将讨论另一种解决方案。由于球面和椭球面之间的可变高度可以通过一系列 Tschebyshev 多项式精确近似，因此可以将它们卷积为球面基础。我们将展示这种新型的椭球变换与 Jekeli 对两个表面之间光谱的转换如何最终实现将采样率转换为偏移纬度。这避免了前面提到的不合适的傅里叶求和。本文讨论了 FFT 在球面和椭球面领域以及使用地心纬度、约化纬度和大地纬度的三种应用。分辨率为 5 角分的地球引力模型 EGM2008 已用于演示数值结果和计算优势。
High resolution transformations between regular geophysical data and harmonic model coefficients can be most efficiently computed by Fast Fourier Transform (FFT). However, a prerequisite is that the data grids are given in the appropriate geometrical domain. For example, if the data are situated on the ellipsoid at equi-angular reduced latitudes, spherical harmonic analysis can be employed and the coefficients subsequently converted by Jekeli’s transformation. This results in the spherical harmonic spectrum in the domain of geocentric latitudes. However, the data are most likely given at geodetic (ellipsoidal) latitudes which means that the FFT base needs to be shifted by latitude dependent phase lags in order to obtain the correct spherical harmonic spectrum. This requires appropriate sample rate conversion about the shifted latitudes by means of Fourier summation and cannot be treated efficiently by an FFT algorithm. In this article another solution is discussed instead. Since the variable heights between the spherical and ellipsoidal surfaces can be accurately approximated by a series of Tschebyshev polynomials, they can be convolved into the spherical basis. It will be shown how this new type of transformation to and from the ellipsoid in combination with Jekeli’s conversion of the spectra between the two surfaces allows eventually the sample rate conversion to shifted latitudes. This avoids the inexpedient Fourier summation mentioned previously. In this paper three applications for FFT in the domain of spherical and ellipsoidal surfaces, and using geocentric, reduced and geodetic latitudes are discussed. The Earth gravitational model EGM2008 of 5 arcminutes resolution has been used to demonstrate numerical results and computational advantages.