1-D inversion of C-response data from geomagnetic depth sounding with shallow resistivity constraint
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摘要:
地磁测深研究的周期范围通常为105~107s，缺少反映浅部电性结构的短周期信息，而C-响应受浅部电阻率影响明显，因此本文提出在反演中增加浅部（约200 km）电阻率约束以提高深部反演的稳定性和可靠性.在磁层环状电流满足P10假设的条件下，球坐标系中一维导电薄球层状地球的C-响应和电导率分布关系由边界条件通过递推的方法计算得到.反演采用有限内存拟牛顿（L-BFGS）法；浅部电阻率约束通过将目标函数对模型参数的梯度设为零来实现；通过置信区间分析评价约束反演结果的可靠性.合成数据的无约束反演虽然最终的拟合效果很好，但浅部电阻率受初始模型影响，差异较大；采用浅部约束后，反演结果对初始模型依赖性明显减小，同时还能显著提高200~600 km范围内反演结果的准确性.对全球近地轨道卫星观测的C-响应数据约束反演后结果与前人一致，表现为地幔电导率整体上随着深度的增加而增加.参数置信区间分析表明，由于约束反演加入了浅部信息，电阻率的变化范围更加紧致，说明反演结果更加可靠.因此，有必要通过其他地球物理方法，如长周期大地电磁测深等获得浅部电阻率分布，作为先验信息参加反演，进行浅部约束的C-响应反演，获得更可靠的一维全深度电性结构，为地磁测深数据解释奠定基础.
Abstract:C-response of geomagnetic depth sounding (GDS) usually ranges from 105 to 107 seconds, lacking short period information which reflects the electrical conductivity structure of the topmost Earth. However, the shallow portion has obvious effect on C-response. In this paper, a new method incorporating shallow resistivity constraint has been developed to improve the stability and reliability of deep mantle electrical structure inversion. Assuming in the Earth magnetosphere a ring current of P10 as source, the C-response from a spherical thin-sheet conductive Earth model has been estimated by a recurrence method based on the continuity boundary condition of magnetic field. Limited-memory BFGS method was introduced to C-response inversion. Shallow resistivity constraint was implemented by zeroing the gradient of penalty function with respect to the constrained resistivity. Confidence interval was used to measure the reliability of inversion result. Numerical results show that without constraint, the shallow part of inversion profile resembles initial models; and even though these models differ dramatically from each other, the misfit of C-response does not change. However, under shallow constraint, not only the inversion model coincides well with true model, but also is irrelevant to the initial model. The proposed inversion method has been applied to invert the C-response from low Earth orbit satellites. The conductivity profile reveals an increase of the Earth mantle conductivity with depth, which agrees well with previous study. Further analysis by confidence interval method indicates that the inversion with shallow resistivity constraint is capable of narrow resistivity estimation, and hence leading to a more reliable conductivity profile. Therefore, in order to obtain reliable 1-D electrical conductivity profile of the Earth's mantle by GDS inversion, it is necessary to get appropriate resistivity of the shallow Earth by the means such as long period magnetotellurics, and use it to constrain the conductivity in this part during inversion.
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Key words:
- Geomagnetic depth sounding /
- C-response /
- 1-D inversion /
- Model constrained /
- Confidence interval
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图 1
地球的薄层球状模型及球坐标系下的磁场三分量
Figure 1.
Spherical thin-shell Earth model and three components of magnetic field decomposed in spherical coordinates
图 2
浅部 (0~200 km) 电阻率不同时的C-响应
Figure 2.
The C-response of models with different resistivity in shallow layers (0~200 km)
图 4
理论模型有无浅部约束的反演结果
Figure 4.
Inversion results with and without shallow resistivity constraint
图 5
近地轨道卫星观测数据的反演结果
Figure 5.
The 1-D inversion results of low Earth orbit satellite′s data
表 1
用于C-响应数值模拟的理论模型
Table 1.
Model used to calculate C-response
Resistivity/Ωm Thickness/km 76 200 76 320 4.5 126 5.9 145 0.662 2099 Core 3481 
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表 2
有限内存拟牛顿反演法的基本流程
Table 2.
Inversion diagram of L-BFGS method
置k=0 初始化模型mk、正则化参数λ和对称正定矩阵Hk, 指定迭代终止条件 循环 计算导数向量gk, 并计算修正梯度方向dk=-Hkgk 搜索最优步长αk 更新模型mk+1=mk+αkdk
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表 3
用于合成反演数据的模型
Table 3.
Model for inversion test
Resistivity/Ωm Thickness/km 100 200 50 300 5 300 0.5 2090 Core 3481
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