A stable regularization parameter estimation method and its application to gravity anomaly inversion of basin basements
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摘要:
正则化方法通过带有正则化参数的约束项，将不适定问题转换为一个适定问题.如何选取最优正则化参数一直以来都是正则化研究的难点和热点.本文通过定义解的不稳定性度量来直接估算正则化参数μ的最优值，并将这种正则化参数估计方法应用到二维沉积盆地基底重力反演中.测试该方法在通过对一次野外测量的数据加不同噪声得到的多组数据与多次野外测量中得到的多组数据这两种情况中的反演效果.最后将该方法应用到非洲西海岸的北加蓬次盆进行盆地基底反演，测试该方法的实用性.模型测试的结果显示，在这两种情况下获得的反演解非常接近且能够反演得到较为准确的模型基底深度，故该方法适用于一般情况下只进行一次野外测量的实际重力勘探情况且能得到稳定的最优反演解；实际资料的最优反演结果稳定且符合当地的地质构造背景.在模型测试与实际资料测试中，都能够确定最优正则化参数并得到最优反演结果，证明了该方法在重力反演中的正确性和实用性.
Abstract:In the regularization method, the ill posed problem is transformed into a well posed problem by a constraint term with regularization parameters. How to select the optimal regularization parameters has always been a difficult issue in the research of regularization. In this paper, the optimal value of regularization parameter μ is directly estimated by defining the instability measure of the solution, and this regularization parameter estimation method is applied to gravity data inversion of two-dimensional sedimentary basins. The inversion effect of this method is tested in two cases: one is the multiple groups of data obtained by adding different noises to the data of one field survey and the other is the multiple groups of data obtained from multiple field measurements. Finally, the method is applied to basement inversion of the North Gabon sub-basin on the west coast of Africa and the practicability of the method is tested. The results of the model test show that the inversion solutions obtained in these two cases are very close and can obtain accurate model base depth. Therefore, this method is suitable for the actual gravity exploration with only one field survey in general, and can obtain stable optimal inversion solution. The optimal inversion result of the actual data is stable and consistent with the local geological structure background. In both model and actual data tests, the optimal regularization parameters can be determined and the optimal inversion results can be obtained, which proves the correctness and practicability of this method in gravity inversion.
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图 4
沉积盆地基底起伏正演重力异常图
Figure 4.
Forward modeling of gravity anomalies by basement relief
图 5
(a) 不稳定性曲线和数据误差均方根曲线；(b) Δρk/Δμ曲线
Figure 5.
(a) Curves of ρ against μ and RMS of data misfit versus μ; (b) Δρk/Δμ versus μ
图 6
模拟实测数据的25组扰动重力解
Figure 6.
25 groups of perturbed gravity solutions for simulated measured data
图 8
加蓬盆地构造划分与剖面位置示意图(据赵红岩等，2017)
Figure 8.
Structural division and profile position of Gabon basin (modified from Zhao et al., 2017)
图 9
北加蓬次盆CD剖面构造划分(据刘亚雷等，2019)
Figure 9.
CD-section structure of North Gabon sub-basin (modified from Liu et al., 2019)
图 10
(a) 测线布格重力异常；(b) 剩余布格重力异常
Figure 10.
(a) Bouguer gravity anomalies of survey line; (b) Residual Bouguer gravity anomalies of survey line
图 11
(a) 解不稳定定量度量随正则化参数单调递减；(b) Δρk/Δμ曲线
Figure 11.
(a) ρ decreasing monotonically with μ; (b) Curve of Δρk/Δμ versus μ
图 12
北加蓬次盆AB剖面基底反演结果与地质剖面对比图
Figure 12.
Comparison of inversion results and geological section AB profile in North Gabon sub-basin
表 1
反演二维模拟沉积盆地基底深度时的先验信息
Table 1.
Prior information of 2D inversion for basement depth of basin
水平坐标位置x/km 已知深度/km 6.75 2.88 30.75 3.41 40.75 2.86 
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表 2
密度资料(单位：g·cm-3)(据纪晓琳等, 2019)
Table 2.
Density data (unit: g·cm-3) (modified from Ji et al., 2019)
海水密度 盐上层密度 盐层密度 盐下层密度 基底密度 1.03 2.41 2.21 2.45 2.8
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表 3
正则化反演的深度先验信息
Table 3.
Depth prior information of regularized inversion
水平坐标位置x/km 已知深度/km 37.14 6.8 73.75 6.5
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