A tool for designing digital filters for the Hankel and Fourier transforms in potential, diffusive, and wavefield modeling

Abstract

The open-source code fdesign makes it possible to design digital linear filters for the Hankel and Fourier transforms used in potential, diffusive, and wavefield modeling. Digital filters can be derived for any electromagnetic (EM) method, such as methods in the diffusive limits (direct current, controlled-source EM [CSEM]) as well as methods using higher frequency content (ground-penetrating radar [GPR], acoustic and elastic wavefields). The direct matrix inversion method is used for the derivation of the filter values, and a brute-force minimization search is carried out over the defined spacing and shifting values of the filter basis. Included or user-provided theoretical transform pairs are used for the inversion. Alternatively, one can provide layered subsurface models that will be computed with a precise quadrature method using the EM modeler empymod to generate numerical transform pairs. The comparison of the presented 201 pt filter with previously presented filters indicates that it performs better for some standard CSEM cases. The derivation of a longer 2001 pt filter for a GPR example with a 250 MHz center frequency proves that the filter method works not only for diffusive EM fields but also for wave phenomena. The presented algorithm provides a tool to create problem specific digital filters. Such purpose-built filters can be made shorter and can speed up consecutive potential, diffusive, and wavefield inversions.

Time‐Domain Modeling of Three‐Dimensional Earth's and Planetary Electromagnetic Induction Effect in Ground and Satellite Observations

“…This is possible only if f R (ω) and f I (ω) are even and odd functions of the angular frequency ω, respectively. Therefore, Equation A5 reduces to Using substitutions t = e x and ω = e −y and multiplying by e x we can rewrite (A9) as a convolution integral and approximate it by a N-point digital filter η as (Anderson, 1975) The optimal values for η n , Δ and ν in Equations B1-B2 were found by following the method of Werthmüller et al (2019). In this work, we designed a 50-point filter such that it requires as few values of  The Figure B1 shows the designed filter and its performance for the chosen analytic pair.…”

“…One way to calculate a planet's EM induction effect is through frequency domain (FD) transfer functions, which describe a planet's response due to "elementary" extraneous currents. Modeling three-dimensional EM induction effects with transfer functions was previously applied to analyze daily magnetic field variations (Yamazaki & Maute, 2017) in ground (Guzavina et al, 2019;Koch & Kuvshinov, 2013;Kuvshinov et al, 1999) and satellite measurements (Chulliat et al, 2016;Sabaka et al, 2004Sabaka et al, , 2015Sabaka et al, , 2018. Additionally, it was applied in the analysis of aperiodic geomagnetic variations in ground observations (Honkonen et al, 2018;Munch et al, 2020;Olsen & Kuvshinov, 2004;Sun et al, 2015).…”

Fast Fourier transform of electromagnetic data for computationally expensive kernels

“…Key 2009). A simple tool to design digital linear filters was recently presented by Werthmüller et al (2019a), together with a comprehensive overview of the history and development of DLF in geophysics. FFTLog, introduced by Hamilton (2000), is another transform algorithm which proved to be powerful for the frequency-to-time domain transformation of EM responses (e.g.…”

Efficient Filter Generation Based on Particle Swarm Optimization Algorithm

“…More recently authors develop the matrix inversion method to calculate the filters by directly discretizing the Hankel transformation [29], [30]. This can be considered as a variant of the Wiener-Hopf least square method [27], [31].…”

“…This method needs to find optimal spacing and shift parameter, which are optimized over a sequence of gradually refined grids. The literature shows that the optimal solution of the spacing and shift parameter can be random over a large detailed region (in this region every filter is good), and the refinement needs expertise especially during the last stages of the search process [30]. Automatic determination of optimal spacing and shift parameters is of practical importance and not reported in the literature, which is the primary goal of this paper.…”