Is the fast Hankel transform faster than quadrature?
Is the fast Hankel transform faster than quadrature?
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DOI：
10.1190/geo2011-0237.1
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发表时间：
2012-05-01
期刊：
影响因子：
3.3
通讯作者：
Key, Kerry
中科院分区：
文献类型：
--
作者：
Key, Kerry
几十年来，用数字滤波器实现的快速汉克尔变换(FHT)一直是电磁地球物理中的首选算法。然而，其他学科主要依赖于将汉克尔变换积分分解为部分积分之和的方法，每个部分积分都用求积来计算。然后通过非线性序列变换加速部分和的收敛。虽然这种方法在近30年前就被提出用于地球物理学，但它被证明比FHT慢得多。本文提出了一种新的外推求积算法(QWE)来解决这个问题。QWE方法通过使用定点求积规则将求和重新定义为概念上类似于FHT方法的形式。利用Wynn算法计算的Shanks变换，有效地加快了部分积分和的计算速度。在一系列相关的建模问题上，包括频域可控源电磁问题、时间域电磁问题和大回路磁源问题，比较了QWE算法和FHT方法的实验室实现在精度和速度方面的差异。令人惊讶的是，对于所有这三个问题，QWE方法都比FHT方法快。然而，当积分需要在许多偏移量上计算并且FHT的滞后卷积变量适用时，FHT比QWE方法要快得多。对于像大回路问题中遇到的发散积分，当FHT方法失效时，QWE方法可以提供准确的解。
The fast Hankel transform (FHT) implemented with digital filters has been the algorithm of choice in EM geophysics for a few decades. However, other disciplines have predominantly relied on methods that break up the Hankel transform integral into a sum of partial integrals that are each evaluated with quadrature. The convergence of the partial sums is then accelerated through a nonlinear sequence transformation. While such a method was proposed for geophysics nearly three decades ago, it was demonstrated to be much slower than the FHT. This work revisits this problem by presenting a new algorithm named quadrature-with-extrapolation (QWE). The QWE method recasts the quadrature sum into a form conceptually similar to the FHT approach by using a fixed-point quadrature rule. The sum of partial integrals is efficiently accelerated using the Shanks transformation computed with Wynn's e algorithm. A Mat lab implementation of the QWE algorithm is compared with the FHT method for accuracy and speed on a suite of relevant modeling problems including frequency-domain controlled-source EM, time-domain EM, and a large-loop magnetic source problem. Surprisingly, the QWE method is faster than the FHT for all three problems. However, when the integral needs to be evaluated at many offsets and the lagged convolution variant of the FHT is applicable, the FHT is significantly faster than the QWE method. For divergent integrals such as those encountered in the large loop problem, the QWE method can provide an accurate answer when the FHT method fails.