﻿ 利用全球地磁场模型研究Δ<i>T</i>与<i>T</i><sub>ap</sub>之间的差异性特征
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 大地测量与地球动力学  2021, Vol. 41 Issue (11): 1141-1145, 1151  DOI: 10.14075/j.jgg.2021.11.008

### 引用本文

CHEN Kang, HU Zhengwang, DU Jinsong. Research on the Differential Characteristics between ΔT and Tap Using the Global Geomagnetic Field Model[J]. Journal of Geodesy and Geodynamics, 2021, 41(11): 1141-1145, 1151.

### Foundation support

National Natural Science Foundation of China, No.41604060; Independent Research Project of the State Key Laboratory of Geological Processes and Mineral Resources, China University of Geosciences, No.MSFGPMR01-4.

### Corresponding author

HU Zhengwang, lecturer, majors in geophysical teaching and research, E-mail: hzw@cug.edu.cn.

### 第一作者简介

CHEN Kang, senior engineer, majors in geophysical exploration, E-mail: chenkang_gsi@163.com.

### 文章历史

1. 广西壮族自治区第七地质队，广西壮族自治区柳州市柳堡路3号，545100;
2. 中国地质大学(武汉)地球物理与空间信息学院地球内部多尺度成像湖北省重点实验室，武汉市鲁磨路388号，430074;
3. 中国地质大学(武汉)地质过程与矿产资源国家重点实验室，武汉市鲁磨路388号，430074

1 ΔTTap差异性的理论分析

 $\begin{gathered} \Delta T=|\boldsymbol{T}|-\left|\boldsymbol{T}_{0}\right| \end{gathered}$ (1)
 $T_{\mathrm{ap}}=\left|\boldsymbol{T}_{a}\right| \cos \theta$ (2)
 图 1 ΔT、Ta与Tap之间的关系示意图 Fig. 1 Relationship diagram between ΔT, Ta and Tap

 $\theta=\arccos \frac{\boldsymbol{T}_{0} \cdot \boldsymbol{T}_{a}}{\left|\boldsymbol{T}_{0}\right|\left|\boldsymbol{T}_{a}\right|}$ (3)

 $E=\Delta T-T_{\mathrm{ap}}=\frac{\left|\boldsymbol{T}_{a}\right|^{2}-(\Delta T)^{2}}{2\left|\boldsymbol{T}_{0}\right|}$ (4)

 $E_{\max }=\left|\boldsymbol{T}_{a}\right|^{2} /\left(2\left|\boldsymbol{T}_{0}\right|\right)$ (5)

 $\theta^{\prime}=\pi-\arccos \frac{\left|\boldsymbol{T}_{a}\right|}{2\left|\boldsymbol{T}_{0}\right|}$ (6)

2 基于全球岩石圈磁场模型的ΔTTap及其差异计算方法

 $\begin{gathered} |\boldsymbol{T}|=\sqrt{X^{2}+Y^{2}+Z^{2}} \end{gathered}$ (7a)
 $\left|\boldsymbol{T}_{0}\right|=\sqrt{X_{0}^{2}+Y_{0}^{2}+Z_{0}^{2}}$ (7b)
 $\Delta T=\sqrt{X^{2}+Y^{2}+Z^{2}}-\sqrt{X_{0}^{2}+Y_{0}^{2}+Z_{0}^{2}}$ (7c)
 $\begin{gathered} \left|\boldsymbol{T}_{a}\right|=\sqrt{\Delta X^{2}+\Delta Y^{2}+\Delta Z^{2}} \end{gathered}$ (7d)
 $T_{\mathrm{ap}}=\Delta X \frac{X_{0}}{T_{0}}+\Delta Y \frac{Y_{0}}{T_{0}}+\Delta Z \frac{Z_{0}}{T_{0}}$ (7e)

 $\theta=\arccos \frac{X \cdot \Delta X+Y \cdot \Delta Y+Z \cdot \Delta Z}{\left|\boldsymbol{T}_{0} \boldsymbol{T}_{a}\right|}$ (8)
3 结果与分析

3.1 ΔTTap之间差异的大小与分布特征

 图 2 ΔT及其与Tap之间的差异分布 Fig. 2 Distribution of ΔT and its difference with Tap

3.2 ΔTTap之间差异的影响因素

 图 3 |Ta|与1/|T0|分布 Fig. 3 Distribution of |Ta| and 1/|T0|

 图 4 误差E与|Ta|2、|Ta|2/2|T0|的统计关系 Fig. 4 Statistical correlations between the error E and |Ta|2, | Ta|2/2|T0|
3.3 ΔTTap之间的差异估计与压制方法

 图 5 Emax及其与实际误差E之间的差异分布 Fig. 5 Distribution of Emax and its difference with E

 图 6 T0与Ta之间的夹角θ分布以及当ΔT为0时T0与Ta之间的夹角θ′分布 Fig. 6 Distribution of angle θ between T0 and Ta and angle θ′ between T0 and Ta when ΔT is zero

Lesur等[13]在利用全球总磁场强度异常汇编ΔT数据反演岩石圈磁场球谐模型系数时采用了上述磁力异常矢量的方向约束，并且认为引入该约束是为压制Backus效应[14]。但根据上述分析，本文认为该约束应该是为减小将Tap近似为ΔT的误差，而非压制Backus效应。实际上，所谓Backus效应，即由标量磁场值反演磁场矢量具有多解性；而Tap方向即主磁场方向，为磁力异常矢量在主磁场方向的投影。因此，Backus效应和Tap与ΔT的差异性为两个不同的问题。现今根据ΔT可以精确计算Tap[4-7]，因此在利用总磁场强度异常ΔT数据反演构建岩石圈磁场模型时，无需引入磁力异常矢量的大小或方向约束。

4 结语

1) 对不同波长范围与不同高度情况下误差E的幅值大小与空间分布特征进行分析，可为构建与应用岩石圈磁场模型时是否需要考虑ΔTTap之间的差异性提供判断依据。

2) 在实际情况下，误差E受|Ta|影响最大、ΔT次之、|T0|最弱，并且误差E未表现出随纬度而变化的分布特征，与T0Ta之间的夹角也无相关性。

3) 为压制ΔTTap之间的差异性，可以引入|Ta|趋近于ΔT的约束条件，也可以将磁力异常矢量与主磁场垂直分量的幅度最小化作为约束，但目前研究表明，无论是等效源方法、最优化方法还是迭代方法均能够基于ΔT精确计算Tap，因此在利用总磁场强度异常ΔT数据反演构建岩石圈磁场模型时，无需对磁力异常矢量的大小或方向进行约束。

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Research on the Differential Characteristics between ΔT and Tap Using the Global Geomagnetic Field Model
CHEN Kang1     HU Zhengwang2     DU Jinsong2,3
1. The 7th Geological Team, Guangxi Zhuang Autonomous Region, 3 Liubao Road, Liuzhou 545100, China;
2. Hubei Subsurface Multi-Scale Imaging Key Laboratory, Institute of Geophysics and Geomatics, China University of Geosciences, 388 Lumo Road, Wuhan 430074, China;
3. State Key Laboratory of Geological Processes and Mineral Resources, China University of Geosciences, 388 Lumo Road, Wuhan 430074, China
Abstract: The global geomagnetic field model EMM2017 is used to calculate the global distribution of the difference between the total magnetic field intensity anomaly (ΔT) and the projection component (Tap) of the magnetic anomaly vector (Ta) in the direction of the main magnetic field, and the amplitude, spatial distribution characteristics and influencing factors of the difference are analyzed. This study can provide reference for the construction and application of the global lithospheric magnetic field model.
Key words: global geomagnetic field model; spherical harmonic analysis; total magnetic intensity anomaly (ΔT); magnetic anomaly component (Tap); lithospheric magnetic field