基于贝叶斯证据的全球地磁场模型主磁场阶数分析
doi: 10.11728/cjss2023.04.2022-0009 cstr: 32142.14.cjss2023.04.2022-0009
Analysis of Global Geomagnetic Main Field Model Order Based on Bayesian Evidence
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摘要: 全球主磁场模型描述了主磁场的时空分布特征。模型的主磁场阶数是构建主磁场模型的关键问题之一。使用贝叶斯推理分析全球主磁场模型，依据贝叶斯证据比较模型阶数，为主磁场的阶数分析提供一种统计依据。利用Swarm卫星磁测数据，估计不同阶数的主磁场模型的证据。结果表明，在1~20的模型阶数中，阶数N = 12具有全局最佳证据。参照Jefrrey’s scale的阈值区间，数据对阶数N = 12的偏好显著优于其他阶数。实验表明，主磁场阶数的证据推理可用于研究主磁场贡献，结果与14阶球谐函数的功率谱分析相匹配。Abstract: The global main magnetic field model describes the space-time distribution characteristics of the main magnetic field. The order of the main magnetic field in the model is one of the key issues to build the main magnetic field model. This paper used Bayesian inference to analyze the global geomagnetic main field model and compares model orders based on Bayesian evidence. It provided a statistical perspective for the main field order selection. Using magnetic observations from Swarm satellites, the evidence for different orders of the main field model was estimated. The results show that order N = 12 has the global best evidence in model from 1 to 20. Referring to the threshold interval given by Jefrrey’s scale, the data preference for order N = 12 is significantly better than other orders. The experiment shows that the evidence reasoning of the main magnetic field order can be used to study the main magnetic field contribution, and the results match the power spectrum analysis of spherical harmonic of order N = 14.
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Key words:
- Geomagnetic field /
- Main field model /
- Bayesian evidence /
- Model comparison /
- Geomagnetic power spectrum
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图 2 数据直方分布与密度曲线。（a）地磁标量强度$ {F} $，（b）观测噪声方差，（c）地壳场贡献模拟修正量，（d）外源场贡献模拟修正量
Figure 2. Data histogram distribution and density curve. (a) Geomagnetic scalar intensity $ {F} $, (b) observed noise variance, (c) simulation correction of crustal field, (d) simulation correction of external field
图 3 地壳场与外源场的主磁场修正值的KDE等高图
Figure 3. KDE contour map of the main field correction value of the crustal field and the external field
图 5 阶数$ N=3 $ 的球谐系数的后验分布
Figure 5. Posterior distribution of spherical harmonic coefficients of order N=3
图 6 利用14阶球谐函数计算的功率谱
Figure 6. Power spectrum calculated using spherical harmonics of order 14
表 1 基于SHA方法的全球地磁场模型及其主磁场对应阶数
Table 1. Geomagnetic field model and its corresponding main field order based on SHA method
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表 2 证据因子差异度量表
Table 2. Evidence factor difference scale
$ {Z}_{i}/{Z}_{j} $ $ \mathrm{lb}{(Z}_{i}/{Z}_{j}) $ $ \mathrm{ln}({Z}_{i}/{Z}_{j}) $ $ \mathrm{lg}({Z}_{i}/{Z}_{j}) $ Evidence
against $ {H}_{i} $$1\sim3.2$ $0\sim1.7$ $0\sim1.2$ $0\sim0.5$ Weak $3.2\sim10$ $ 1.7 \sim 3.3 $ $ 1.2 \sim 2.3 $ $ 0.5 \sim 1 $ Substantial $ 10 \sim 100 $ $3.3\sim6.6$ $2.3\sim4.6$ $1\sim2$ Srtong $ > 100 $ $ > 6.6 $ $ > 4.6 $ $ > 2 $ Decisive $ > 1000 $ $ > 10 $ $ > 7 $ $ > 3 $ Beyond reasonable doubt
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表 3 1~20阶主磁场模型的证据
Table 3. Evidence of main field model of order 1~20
$ N $ $ \mathrm{lg}{Z}_{i} $ $ N $ $ \mathrm{lg}{Z}_{i} $ $ 1 $ $ -2236994722 $ $ 11 $ $ -20514 $ $ 2 $ $ -37342005 $ $ 12 $ $ -19973 $ $ 3 $ $ -37552523 $ $ 13 $ $ -20250 $ $ 4 $ $ -1053325 $ $ 14 $ $ -20411 $ $ 5 $ $ -96379 $ $ 15 $ $ -24028 $ $ 6 $ $ -35091 $ $ 16 $ $ -25807 $ $ 7 $ $ -22232 $ $ 17 $ $ -26269 $ $ 8 $ $ -20626 $ $ 18 $ $ -29179 $ $ 9 $ $ -20102 $ $ 19 $ $ -724596 $ $ 10 $ $ -20023 $ $ 20 $ $ -1372094 $
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表 4 证据因子度量结果
Table 4. Result of the evidence factor measurement
Order i $ \mathrm{lg}{(Z}_{12}/{Z}_{i}) $ Evidence against $ {H}_{i} $ 10 725.2 Beyond reasonable doubt 11 496.7 Beyond reasonable doubt 13 277.2 Beyond reasonable doubt 14 438.0 Beyond reasonable doubt
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