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| One-dimensional accurate calculation of transient electromagnetic responses based on B-spline interpolation |
XING Tao1( ), WANG Yao2, LI Jian-Hui2,3() |
1. Beijing Tanchuang Resources Technology Co., Ltd., Beijing 100071, China
2. School of Geophysics and Geomatics, China University of Geosciences (Wuhan), Wuhan 430074, China
3. State Key Laboratory of Geological Processes and Mineral Resources, China University of Geosciences (Wuhan), Wuhan 430074, China |
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Abstract In the one-dimensional (1D) forward modeling of transient electromagnetic (TEM) responses based on spectral methods, multiple calculation steps significantly influence the calculation accuracy of TEM responses. To improve the efficiency of 1D forward modeling, the common practice is to directly calculate the frequency-domain electromagnetic responses of dozens of frequency points and then obtain the responses of hundreds of frequency points through cubic spline interpolation. Although the numerical results calculated using the cubic spline interpolation function can meet the requirements of most forward modeling scenarios, their accuracy can be further improved. This study introduced high-order B-spline interpolation into the 1D forward modeling of TEM responses to replace the conventional cubic spline interpolation and verified the accuracy of the method based on magnetic dipole sources and circle-shaped loop sources. The results show that the TEM responses of several geoelectric models calculated based on high-order B-spline interpolation exhibit higher accuracy than those calculated using conventional cubic spline interpolation.
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Received: 09 December 2022
Published: 27 October 2023
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Corresponding Authors:
LI Jian-Hui
E-mail: 156663062@qq.com;ljhiicumt@126.com
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1D forward modeling technology roadmap for transient electromagnetic methods
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The magnetic field impulse response excited by vertical magnetic dipole source on the surface of 1 000 Ω·m homogeneous half-space on the condition of 5 frequencies per decade
a—cubic B-spline numerical solution;b—cubic B-spline relative error curve;c—quintic B-spline relative error curve;d—heptatic B-spline relative error curve
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| 插值算法 | 平均误差率/% | | 4频点 | 5频点 | 6频点 | 7频点 | 8频点 | 9频点 | | 传统三次样条 | 9.08×10-1 | 5.27×10-1 | 3.75×10-1 | 2.60×10-1 | 3.21×10-1 | 2.19×10-1 | | 三次B样条 | 9.11×10-1 | 5.54×10-1 | 3.68×10-1 | 2.70×10-1 | 3.10×10-1 | 2.17×10-1 | | 四次B样条 | 7.20×10-1 | 4.02×103 | 1.50×102 | 1.00×101 | 6.22×10-2 | 3.01×10-2 | | 五次B样条 | 6.53×10-2 | 2.74×10-2 | 1.28×10-2 | 3.70×10-2 | 5.71 | 6.31×101 | | 六次B样条 | 1.03×103 | 5.61×10-1 | 4.46×10-3 | 2.03×10-3 | 1.65×10-3 | 1.29×10-3 | | 七次B样条 | 1.53×10-2 | 4.22×10-3 | 2.65×102 | 1.15 | 3.47×10-4 | 3.64×10-4 | | 八次B样条 | 4.03×101 | 1.26 | 9.43×10-4 | 2.16×10-2 | 1.08×103 | 1.53 | | 九次B样条 | 4.13 | 1.81×10-3 | 2.57×101 | 3.72×10-3 | 2.52×10-4 | 2.89×10-4 |
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Average error rate of different interpolation algorithms for a vertical magnetic dipole source in a 1000 Ω·m uniform half-space
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| 插值算法 | 最大相对误差/% | | 4频点 | 5频点 | 6频点 | 7频点 | 8频点 | 9频点 | | 传统三次样条 | 1.81 | 1.09 | 7.24×10-1 | 4.57×10-1 | 6.98×10-1 | 8.82×10-1 | | 三次B样条 | 1.93 | 1.06 | 6.89×10-1 | 5.50×10-1 | 6.66×10-1 | 9.21×10-1 | | 四次B样条 | 1.38 | 2.60×104 | 1.59×103 | 1.12×102 | 6.56×10-1 | 1.26×10-1 | | 五次B样条 | 1.31×10-1 | 5.60×10-2 | 2.50×10-2 | 1.01×10-1 | 2.17×101 | 5.25×102 | | 六次B样条 | 5.19×103 | 7.74 | 3.39×10-2 | 1.17×10-2 | 9.84×10-3 | 6.47×10-3 | | 七次B样条 | 9.31×10-2 | 1.08×10-2 | 2.36×103 | 1.35×101 | 1.62×10-3 | 1.81×10-3 | | 八次B样条 | 1.48×102 | 1.82×101 | 1.47×10-2 | 5.29×10-2 | 9.53×103 | 1.36×101 | | 九次B样条 | 4.77×101 | 8.67×10-3 | 1.62×102 | 4.68×10-2 | 5.12×10-3 | 2.08×10-3 |
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Maximum relative error of different interpolation algorithms for a vertical magnetic dipole source in a 1000 Ω·m uniform half-space
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The magnetic field impulse response excited by the vertical magnetic dipole source on the surface of 1 Ω·m homogeneous half-space on the condition of 5 frequencies per decade
a—cubic B-spline numerical solution;b—cubic B-spline relative error curve;c—quintic B-spline relative error curve;d—heptatic B-spline relative error curve
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| 插值算法 | 平均误差率/% | | 4频点 | 5频点 | 6频点 | 7频点 | 8频点 | 9频点 | | 传统三次样条 | 1.71 | 5.59×10-1 | 3.11×10-1 | 1.82×10-1 | 5.64×10-2 | 5.69×10-2 | | 三次B样条 | 1.21×101 | 7.83×10-1 | 3.16×10-1 | 1.49×10-1 | 7.80×10-2 | 5.66×10-2 | | 四次B样条 | 4.76×10-1 | 8.95 | 7.80×10-1 | 1.03×10-1 | 2.40×10-2 | 1.42×10-2 | | 五次B样条 | 1.02 | 7.57×10-2 | 3.21×10-2 | 1.46×10-2 | 1.07×10-2 | 4.93×10-2 | | 六次B样条 | 4.31×101 | 1.08 | 3.12×10-2 | 5.84×10-3 | 1.36×10-3 | 9.76×10-4 | | 七次B样条 | 1.55 | 5.09×10-2 | 1.33 | 2.11×10-2 | 8.84×10-4 | 3.44×10-4 | | 八次B样条 | 3.54 | 8.14×10-1 | 1.09×10-2 | 1.00×10-3 | 6.33×10-2 | 4.56×10-4 | | 九次B样条 | 8.00 | 3.74×10-2 | 2.78×10-1 | 1.13×10-3 | 2.54×10-4 | 4.82×10-5 |
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Average error rate of different interpolation algorithms for a vertical magnetic dipole source in a 1 Ω·m uniform half-space
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| 插值算法 | 最大相对误差/% | | 4频点 | 5频点 | 6频点 | 7频点 | 8频点 | 9频点 | | 传统三次样条 | 4.85 | 3.49 | 2.63 | 1.37 | 2.35×10-1 | 3.78×10-1 | | 三次B样条 | 1.08×102 | 4.13 | 2.32 | 1.54 | 4.11×10-1 | 3.30×10-1 | | 四次B样条 | 3.87 | 1.32×102 | 8.80 | 6.70×10-1 | 1.80×10-1 | 9.19×10-2 | | 五次B样条 | 3.60 | 4.80×10-1 | 2.02×10-1 | 1.45×10-1 | 8.59×10-2 | 5.62×10-1 | | 六次B样条 | 6.46×102 | 8.81 | 1.80×10-1 | 6.33×10-2 | 8.03×10-3 | 9.17×10-3 | | 七次B样条 | 8.70 | 3.74×10-1 | 1.72×101 | 1.62×10-1 | 6.13×10-3 | 3.09×10-3 | | 八次B样条 | 4.44×101 | 6.69 | 3.85×10-2 | 7.31×10-3 | 8.12×10-1 | 3.63×10-3 | | 九次B样条 | 8.02×101 | 1.72×10-1 | 3.32 | 3.87×10-3 | 1.65×10-3 | 2.44×10-4 |
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Maximum relative error of different interpolation algorithms for a vertical magnetic dipole source in a 1 Ω·m uniform half-space
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The impulse response of magnetic field excited by circular loop on the surface of 1 000 Ω·m homogeneous half-space on the condition of 5 frequencied per decade
a—cubic B-spline numerical solution;b—cubic B-spline relative error curve;c—quintic B-spline relative error curve;d—heptatic B-spline relative error curve
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| 插值算法 | 平均误差率/% | | 4频点 | 5频点 | 6频点 | 7频点 | 8频点 | 9频点 | | 传统三次样条 | 9.04×10-1 | 5.41×10-1 | 3.73×10-1 | 3.18×10-1 | 4.41×10-1 | 1.86×10-1 | | 三次B样条 | 1.41 | 9.90×10-1 | 8.59×10-1 | 7.32×10-1 | 8.65×10-1 | 7.36×10-1 | | 四次B样条 | 2.83 | 6.24×103 | 2.21×103 | 1.28×102 | 2.92 | 4.89×10-2 | | 五次B样条 | 5.75×10-2 | 2.77×10-2 | 1.59×10-2 | 4.41×10-1 | 3.23×102 | 7.69×104 | | 六次B样条 | 4.25×103 | 7.34 | 8.54×10-3 | 3.98×10-3 | 1.36×10-2 | 4.44×10-3 | | 七次B样条 | 1.99×10-2 | 1.19×10-2 | 3.45×103 | 1.54×101 | 9.40×10-3 | 4.12×10-3 | | 八次B样条 | 7.38×101 | 1.96×101 | 5.72×10-3 | 2.44×10-1 | 7.07×104 | 3.71×101 | | 九次B样条 | 6.33 | 1.29×10-2 | 4.33×102 | 4.77×10-2 | 1.32×10-2 | 5.25×10-3 |
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Average error rate of different interpolation algorithms for circular loops in 1 000 Ω·m uniform half-space
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| 插值算法 | 最大相对误差/% | | 4频点 | 5频点 | 6频点 | 7频点 | 8频点 | 9频点 | | 传统三次样条 | 2.57 | 1.02 | 7.66×10-1 | 5.90×10-1 | 1.75 | 4.51×10-1 | | 三次B样条 | 7.46 | 2.82 | 2.86 | 2.92 | 2.41 | 2.47 | | 四次B样条 | 5.16 | 4.05×104 | 2.36×104 | 1.46×103 | 5.07×101 | 4.04×10-1 | | 五次B样条 | 1.17×10-1 | 1.07×10-1 | 7.81×10-2 | 9.37×10-1 | 1.34×103 | 4.66×105 | | 六次B样条 | 2.06×104 | 7.51×101 | 8.01×10-2 | 7.54×10-2 | 2.28×10-1 | 4.39×10-2 | | 七次B样条 | 1.93×10-1 | 1.35×10-1 | 3.00×104 | 1.92×102 | 1.62×10-1 | 4.46×10-2 | | 八次B样条 | 2.92×102 | 1.98×102 | 7.46×10-2 | 4.25×10-1 | 6.04×105 | 4.61×102 | | 九次B样条 | 7.23×101 | 1.00×10-1 | 5.02×103 | 7.35×10-1 | 2.34×10-1 | 4.21×10-2 |
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Maximum relative error of different interpolation algorithms for circular loops in 1 000 Ω·m uniform half-space
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The impulse response of magnetic field excited by circular loop on the surface of 1 Ω·m homogeneous half-space on the condition of 5 frequencies per decade
a—cubic B-spline numerical solution;b—cubic B-spline relative error curve;c—quintic B-spline relative error curve;d—heptatic B-spline relative error curve
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| 插值算法 | 平均误差率/% | | 4频点 | 5频点 | 6频点 | 7频点 | 8频点 | 9频点 | | 传统三次样条 | 6.14×10-1 | 3.23×10-1 | 2.07×10-1 | 1.26×10-1 | 1.21×10-1 | 5.65×10-2 | | 三次B样条 | 1.02 | 3.55×10-1 | 1.96×10-1 | 1.26×10-1 | 1.17×10-1 | 5.67×10-2 | | 四次B样条 | 1.45×10-1 | 2.81×10-1 | 4.03×10-2 | 2.19×10-2 | 1.70×10-2 | 7.16×10-3 | | 五次B样条 | 1.75×10-1 | 2.46×10-2 | 1.05×10-2 | 4.96×10-3 | 6.48×10-3 | 7.58×10-2 | | 六次B样条 | 2.07 | 4.60×10-2 | 4.76×10-3 | 1.42×10-3 | 7.26×10-4 | 2.12×10-4 | | 七次B样条 | 1.84×10-1 | 6.60×10-3 | 1.54×10-2 | 1.62×10-3 | 2.52×10-4 | 6.78×10-5 | | 八次B样条 | 1.15×10-1 | 3.40×10-2 | 1.22×10-3 | 2.43×10-4 | 5.44×10-2 | 3.00×10-5 | | 九次B样条 | 3.25×10-1 | 5.25×10-3 | 8.17×10-4 | 4.29×10-4 | 4.50×10-5 | 1.08×10-5 |
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Average error rate of different interpolation algorithms for circular loops in 1 Ω·m uniform half-space
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| 插值算法 | 最大相对误差/% | | 4频点 | 5频点 | 6频点 | 7频点 | 8频点 | 9频点 | | 传统三次样条 | 1.88 | 9.94×10-1 | 6.83×10-1 | 4.15×10-1 | 5.90×10-1 | 2.82×10-1 | | 三次B样条 | 5.34 | 1.03 | 7.21×10-1 | 4.86×10-1 | 5.75×10-1 | 2.50×10-1 | | 四次B样条 | 3.57×10-1 | 3.29 | 1.09×10-1 | 5.52×10-2 | 6.24×10-2 | 3.16×10-2 | | 五次B样条 | 6.67×10-1 | 5.16×10-2 | 2.52×10-2 | 1.02×10-2 | 6.00×10-2 | 8.42×10-1 | | 六次B样条 | 2.65×101 | 3.43×10-1 | 1.40×10-2 | 2.55×10-3 | 2.22×10-3 | 9.95×10-4 | | 七次B样条 | 1.16 | 2.72×10-2 | 1.81×10-1 | 1.07×10-2 | 5.10×10-4 | 2.07×10-4 | | 八次B样条 | 1.15 | 2.93×10-1 | 7.41×10-3 | 1.08×10-3 | 6.67×10-1 | 8.09×10-5 | | 九次B样条 | 3.41 | 3.01×10-2 | 5.73×10-3 | 1.91×10-3 | 1.64×10-4 | 3.27×10-5 |
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Maximum relative error of different interpolation algorithms for circular loops in 1 Ω·m uniform half-space
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