ABSTRACT
The Python-code empymod computes the 3D electromagnetic field in a layered earth with vertical transverse isotropy by combining and extending two earlier presented algorithms in this journal. The bottleneck in frequency- and time-domain calculations of electromagnetic responses derived in the wavenumber-frequency domain is the transformations from the wavenumber to the space domain and from the frequency to the time domain, the so-called Hankel and Fourier transforms. Three different Hankel transform methods (quadrature, quadrature-with-extrapolation [QWE], and filters) and four different Fourier transform methods (fast Fourier transform [FFT], FFTLog, QWE, and filters) are included in empymod, which allows us to compare these different methods in terms of speed and precision. The best transform in terms of speed and precision depends on the modeled frequencies. Published digital filters for the Hankel transform are very fast and precise for frequencies in the range of controlled-source electromagnetic data, but they fail in the frequency range of ground-penetrating radar. Conventional quadrature, on the other hand, is in comparison very slow but can model any frequency. Examples comparing empymod with analytical solutions and with existing electromagnetic modelers illustrate the capabilities of empymod.
REFERENCES
- 1975, Improved digital filters for evaluating Fourier and Hankel transform integrals: Technical report: U.S. Geological Survey, https://pubs.er.usgs.gov/publication/70045426.Google Scholar ,
- 1979, Numerical integration of related Hankel transforms of orders 0 and 1 by adaptive digital filtering: Geophysics, 44,
1287–1305 , doi:10.1190/1.1441007 .GPYSA7 0016-8033 AbstractWeb of ScienceGoogle Scholar , - 1982, Fast Hankel transforms using related and lagged convolutions: ACM Transactions on Mathematical Software Impact Factor, 8,
344–368 , doi:10.1145/356012.356014 .CrossrefWeb of ScienceGoogle Scholar , - 1984, On: Numerical integration of related Hankel transforms by quadrature and continued fraction expansion by (Chave, 1983): Geophysics, 49,
1811–1812 , doi:10.1190/1.1441595 .GPYSA7 0016-8033 AbstractWeb of ScienceGoogle Scholar , - 1989, A hybrid fast Hankel transform algorithm for electromagnetic modeling: Geophysics, 54,
263–266 , doi:10.1190/1.1442650 .GPYSA7 0016-8033 AbstractWeb of ScienceGoogle Scholar , - 1983, Numerical integration of related Hankel transforms by quadrature and continued fraction expansion: Geophysics, 48,
1671–1686 , doi:10.1190/1.1441448 .GPYSA7 0016-8033 AbstractWeb of ScienceGoogle Scholar , - 2009, On the electromagnetic fields produced by marine frequency domain controlled sources: Geophysical Journal International, 179,
1429–1457 , doi:10.1111/j.1365-246X.2009.04367 .GJINEA 0956-540X CrossrefWeb of ScienceGoogle Scholar , - 1991,
Electrical exploration methods for the seafloor , in N. M. Nabghian, ed., Electromagnetic methods in applied geophysics: SEG,931–966 , doi:10.1190/1.9781560802686 .AbstractGoogle Scholar , - 2005, Marine controlled source electromagnetics: principles, methodologies, future commercial applications: Surveys in Geophysics, 26,
675–700 , doi:10.1007/s10712-005-1830-3 .SUGEEC 0169-3298 CrossrefWeb of ScienceGoogle Scholar , - 1971, The application of linear filter theory to the direct interpretation of geoelectrical resistivity sounding measurements: Geophysical Prospecting, 19,
192–217 , doi:10.1111/j.1365-2478.1971.tb00593 .GPPRAR 0016-8025 CrossrefGoogle Scholar , - 2000, Uncorrelated modes of the non-linear power spectrum: Monthly Notices of the Royal Astronomical Society, 312,
257–284 , doi:10.1046/j.1365-8711.2000.03071 .MNRAA4 0035-8711 CrossrefWeb of ScienceGoogle Scholar , - 2016, Inversion of controlled-source electromagnetic reflection responses: Geophysics, 81, no. 5,
F49–F57 , doi:10.1190/geo2015-0320.1 .GPYSA7 0016-8033 AbstractWeb of ScienceGoogle Scholar , - 2015, The electromagnetic response in a layered vertical transverse isotropic medium: A new look at an old problem: Geophysics, 80, no. 1,
F1–F18 , doi:10.1190/geo2013-0411.1 .GPYSA7 0016-8033 AbstractWeb of ScienceGoogle Scholar , - 2009, 1D inversion of multicomponent, multifrequency marine CSEM data: Methodology and synthetic studies for resolving thin resistive layers: Geophysics, 74, no. 2,
F9–F20 , doi:10.1190/1.3058434 .GPYSA7 0016-8033 AbstractWeb of ScienceGoogle Scholar , - 2012, Is the fast Hankel transform faster than quadrature?: Geophysics, 77, no. 3,
F21–F30 , doi:10.1190/GEO2011-0237.1 .GPYSA7 0016-8033 AbstractWeb of ScienceGoogle Scholar , - 2007, Hankel transform filters for dipole antenna radiation in a conductive medium: Geophysical Prospecting, 55,
83–89 , doi:10.1111/j.1365-2478.2006.00585 .GPPRAR 0016-8025 CrossrefWeb of ScienceGoogle Scholar , - 2007, Electromagnetic fields in planarly layered anisotropic media: Geophysical Journal International, 170,
44–80 , doi:10.1111/j.1365-246X.2007.03390 .GJINEA 0956-540X CrossrefWeb of ScienceGoogle Scholar , - 1989, Petroleum exploration using controlled-source electromagnetic methods: Proceedings of the IEEE, 77,
338–362 , doi:10.1109/5.18630 .IEEPAD 0018-9219 CrossrefWeb of ScienceGoogle Scholar , - 1955, Non-linear transformations of divergent and slowly convergent sequences: Journal of Mathematics and Physics, 34,
1–42 , doi:10.1002/sapm19553411 .CrossrefGoogle Scholar , - 2010, Green’s tensors for the diffusive electric field in a VTI half-space: PIER, 107,
1–20 , doi:10.2528/PIER10052807 .CrossrefGoogle Scholar , - 1982, Geo-electromagnetism: Academic Press Inc.Google Scholar ,
- 2016, empymod: An open-source full 3D electromagnetic modeller for 1D VTI media in Python, https://empymod.github.io/, accessed 30 May 2017.Google Scholar ,
- 2017, Getting started with controlled-source electromagnetic 1D modeling: The Leading Edge, 36,
352–355 , doi:10.1190/tle36040352.1 .AbstractGoogle Scholar , - 1997, The equivalent wavefield concept in multichannel transient electromagnetic surveying: Ph.D., University of Edinburgh.Google Scholar ,
- 1956, On a device for computing the transformation: Mathematics of Computation, 10,
91–96 , doi:10.1090/S0025-5718-1956-0084056-6 .CrossrefGoogle Scholar , - 2012, The potential of the controlled source electromagnetic method: A powerful tool for hydrocarbon exploration, appraisal, and reservoir characterization: IEEE Signal Processing Magazine, 29,
36–52 , doi:10.1109/MSP.2012.2192529 .CrossrefGoogle Scholar ,