Improved Parker's method for topographic models using Chebyshev series and low rank approximation
Improved Parker's method for topographic models using Chebyshev series and low rank approximation
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使用切比雪夫级数和低秩近似改进地形模型的 Parker 方法
DOI：
10.1093/gji/ggx093
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发表时间：
2017-05
影响因子：
2.8
通讯作者：
Lin Qiang
中科院分区：
文献类型：
--
作者：
Wu Leyuan;Lin Qiang
我们提出了一种新的方法，以改善著名的帕克的公式模拟的重力和磁场所造成的源与复杂地形的收敛性。在原有的帕克公式中，有两种近似方法可能会引起相当大的数值误差和不稳定性：（1）用离散的正、逆快速傅里叶变换（FFT）算法来近似正、逆连续傅里叶变换;（2）用泰勒级数展开式来近似指数函数。在我们以前的论文中，我们已经努力解决第一个问题，应用高斯FFT方法，而不是标准的FFT算法。新的基于高斯-FFT的方法显示出更高的数值效率，并与空间域解析或混合解析-数值算法相一致。然而，即使在计算表面是所有地形源上方的水平面的简化假设下，该方法在某些情况下仍然可能失败或变得不准确。当地形的峰太接近观测表面时，达到适当精度所需的泰勒级数展开的项数变大，并且减慢计算。本文证明了这一问题是由上述第二近似引起的，并且由于泰勒级数展开的收敛性质，该算法对于某些具有大振幅的地形模型变得不准确。在此基础上，借助Chebfun软件系统，提出了一种利用指数函数低秩逼近的修正帕克方法。以这种方式，实现了最佳的收敛速度。需要一些预先计算，但不会导致显著的计算开销。合成和真实的模型试验表明，该方法现在几乎适用于任何实际的地形模型，只要假设，整个地形质量位于观测表面以下，是满足的。
We present a new method to improve the convergence of the well-known Parker's formula for the modelling of gravity and magnetic fields caused by sources with complex topography. In the original Parker's formula, two approximations are made, which may cause considerable numerical errors and instabilities: (1) the approximation of the forward and inverse continuous Fourier transforms using their discrete counterparts, the forward and inverse Fast Fourier Transform (FFT) algorithms; (2) the approximation of the exponential function with its Taylor series expansion. In a previous paper of ours, we have made an effort addressing the first problem by applying the Gauss-FFT method instead of the standard FFT algorithm. The new Gauss-FFT based method shows improved numerical efficiency and agrees well with space-domain analytical or hybrid analytical-numerical algorithms. However, even under the simplifying assumption of a calculation surface being a level plane above all topographic sources, the method may still fail or become inaccurate under certain circumstances. When the peaks of the topography approach the observation surface too closely, the number of terms of the Taylor series expansion needed to reach a suitable precision becomes large and slows the calculation. We show in this paper that this problem is caused by the second approximation mentioned above, and it is due to the convergence property of the Taylor series expansion that the algorithm becomes inaccurate for certain topographic models with large amplitudes. Based on this observation, we present a modified Parker's method using low rank approximation of the exponential function in virtue of the Chebfun software system. In this way, the optimal rate of convergence is achieved. Some pre-computation is needed but will not cause significant computational overheads. Synthetic and real model tests show that the method now works well for almost any practical topographic model, provided that the assumption, that the entire topographic mass lies below the observation surface, is met.
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影响因子：
--
作者：
K. Macdonald;S. P. Miller;B. Luyendyk;T. Atwater;L. Shure
通讯作者：
K. Macdonald;S. P. Miller;B. Luyendyk;T. Atwater;L. Shure
DOI：
10.1007/bf02521068
发表时间：
1985-12
期刊：
Bulletin géodésique
影响因子：
--
作者：
R. Forsberg
通讯作者：
R. Forsberg
影响因子：
2
作者：
V. Grigoriadis;I. Tziavos;G. Tsokas;A. Stampolidis
通讯作者：
V. Grigoriadis;I. Tziavos;G. Tsokas;A. Stampolidis
DOI：
10.1016/j.cageo.2004.11.004
发表时间：
2005-05
期刊：
Comput. Geosci.
影响因子：
--
作者：
D. Gómez-Ortiz;B. Agarwal
通讯作者：
D. Gómez-Ortiz;B. Agarwal
影响因子：
2.9
作者：
M. Meijde;J. Julià;M. Assumpção
通讯作者：
M. Meijde;J. Julià;M. Assumpção