分数算子应用于地球电磁学
Fractional Operators Applied to Geophysical Electromagnetics
AI总结：
本文研究了分数算子在地球电磁学中的应用，提出了一种基于谱方法和有限元离散化的求解亥姆霍兹方程的实用算法，并揭示了分数阶数对电磁场长程相关性、阻抗幅度与波数关系以及视电阻率谱的影响。
摘要：
近年来，应用数学文献中越来越多地将分数微积分应用于异常传输问题的研究中。在这些分析中，假设异常传输（电荷、示踪剂、流体等）归因于复杂且在某些情况下具有自相似性的导体介质中材料属性的长程相关性。而不是考虑极其离散化（且计算不可行）的介质表示，通过重新表述支配相关传输物理的偏微分方程，使得其系数成为光滑但配以分数阶空间导数。本文将这些概念应用于标量亥姆霍兹方程及其在磁电法中用于探测地球内部的应用。我们概述了使用谱方法与有限元离散化相结合的实用算法来求解亥姆霍兹方程。执行该算法以解决磁电问题揭示了几个在实地数据中可观测到的有趣特征：预测电磁场的长程相关性；平方阻抗幅度与平方波数之间的幂律关系，其斜率是支配亥姆霍兹方程的分数阶数的函数；以及一个非常数的视电阻率谱，其变化仅来源于分数阶数。在具有自相似性（例如：裂隙系统；厚且富含纹理的沉积层等）的地质环境中，我们提出这些诊断对于地质特征的远距离表征是有用的，这些特征远低于地球物理学中电磁方法的典型分辨率极限。
英文摘要：
A growing body of applied mathematics literature in recent years has focussed on the application of fractional calculus to problems of anomalous transport. In these analyses, the anomalous transport (of charge, tracers, fluid, etc.) is presumed attributable to long-range correlations of material properties within an inherently complex, and in some cases self-similar, conducting medium. Rather than considering an exquisitely discretized (and computationally intractable) representation of the medium, the complex and spatially correlated heterogeneity is represented through reformulation of the PDE governing the relevant transport physics such that its coefficients are, instead, smooth but paired with fractional-order space derivatives. Here we apply these concepts to the scalar Helmholtz equation and its use in electromagnetic interrogation of Earth's interior through the magnetotelluric method. We outline a practical algorithm for solving the Helmholtz equation using spectral methods coupled with finite element discretization. Execution of this algorithm for the magnetotelluric problem reveals several interesting features observable in field data: long--range correlation of the predicted electromagnetic fields; a power-law relationship between the squared impedance amplitude and squared wavenumber whose slope is a function of the fractional exponent within the governing Helmholtz equation; and, a non-constant apparent resistivity spectrum whose variability arises solely from the fractional exponent. In geologic settings characterized by self--similarity (e.g. fracture systems; thick and richly-textured sedimentary sequences, etc.) we posit that diagnostics are useful for geologic characterization of features far below the typical resolution limit of electromagnetic methods in geophysics.
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