﻿ 重力数据探测地下目标的贝叶斯方法研究
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 大地测量与地球动力学  2024, Vol. 44 Issue (2): 154-159, 176  DOI: 10.14075/j.jgg.2023.06.152

### 引用本文

HUANG Jiaxi, BIAN Shaofeng, JI Bing. Bayesian Method for Detecting Subsurface Targets Using Gravity Data[J]. Journal of Geodesy and Geodynamics, 2024, 44(2): 154-159, 176.

### Foundation support

National Natural Science Foundation of China, No.41971416, 42122025, 42074010.

### About the first author

HUANG Jiaxi, PhD candidate, majors in geophysical detection and navigation, E-mail: hgarcia@163.com.

### 文章历史

1. 海军工程大学电气工程学院，武汉市解放大道717号，430033;
2. 中国地质大学(武汉)地质探测与评估教育部重点实验室，武汉市鲁磨路388号，430074

1 研究方法 1.1 规则形体重力场正演

 \begin{aligned} g_z= & G \Delta \rho[x \ln (y+r)+y \ln (x+r)- \\ & \left.z \arctan \left(\frac{x y}{z r}\right)\right] \left.\left|\begin{array}{l} x_b \\ x_a \end{array}\right| \begin{array}{l} y_b \\ y_a \end{array} \right| \begin{array}{l} z_b \\ z_a \end{array} \end{aligned} (1)
 \left.g_{z z}=G \Delta \rho \arctan \left(\frac{x y}{z r}\right)\left|\begin{array}{l} x_b \\ x_a \end{array}\right| \begin{gathered} y_b \\ y_a \end{gathered} \right| \begin{aligned} & z_b \\ & z_a \end{aligned} (2)

tatb(t=xyz)代表积分上限和下限，则式(1)和式(2)中各参量的积分上、下限可定义为ta=t1tptb=t2tp。当目标体的走向长度比截面的尺度和埋藏深度大得多，且观测剖面位于目标长轴中部时，可将其视为二度体，无需考虑长轴方向的重力场变化，此时可对式(1)和式(2)进行进一步简化[10]

 $\boldsymbol{R}=\left[\begin{array}{cc} \cos \beta & \sin \beta \\ -\sin \beta & \cos \beta \end{array}\right]$ (3)
 $\left[\begin{array}{l} x \\ y \end{array}\right]=\boldsymbol{R}(90-\alpha)\left[\begin{array}{l} u \\ v \end{array}\right]$ (4)
 图 1 测量坐标系与长方体坐标系示意图 Fig. 1 Schematic diagram of measurement coordinate andcuboid coordinate
1.2 贝叶斯推断基础 1.2.1 贝叶斯公式

 $\pi(\theta \mid \boldsymbol{q})=\frac{f(\boldsymbol{q} \mid \theta) \pi(\theta)}{\int_\mathit{Θ} f(\boldsymbol{q} \mid \theta) \pi(\theta) \mathrm{d} \theta}$ (5)

 $\begin{gathered} \pi(\theta \mid \boldsymbol{q}, M)=\frac{f(\boldsymbol{q} \mid \theta, M) f(\theta \mid M)}{f(\boldsymbol{q} \mid M)}= \\ \frac{f(\boldsymbol{q} \mid \theta, M) f(\theta \mid M)}{\int_\mathit{Θ} f(\boldsymbol{q} \mid \theta, M) f(\theta \mid M) \mathrm{d} \theta_m} \end{gathered}$ (6)

1.2.2 似然函数

 $q_i=s_i+n_b+n_{\text {inst }}$ (7)

 $\begin{gathered} p(\boldsymbol{q} \mid \theta, M)=\frac{1}{\sqrt{(2 \pi)^k|\mathit{\Sigma}|}} \\ \exp \left(-\frac{1}{2}(\boldsymbol{q}-\boldsymbol{s})^{\mathrm{T}} \mathit{\Sigma}^{-1}(\boldsymbol{q}-\boldsymbol{s})\right) \end{gathered}$ (8)

1.2.3 先验信息

 图 2 贝叶斯推断基本流程 Fig. 2 Flowchart of Bayesian inference
2 实验结果分析 2.1 基于区域测量数据的目标探测实验

 图 3 模拟的区域重力异常和垂直重力梯度异常 Fig. 3 Simulated gravity anomalies and vertical gravity gradients anomalies

 图 4 未知参数的先验概率密度分布 Fig. 4 Prior probability density distributions for unknown parameters

 图 5 模型参数的贝叶斯后验概率分布 Fig. 5 Bayesian posterior probability distributions for model parameters

2.2 基于少量测线重力梯度数据的目标探测实验

Romaides等[15]曾对一处地下地铁停车场进行地面微重力测量，该地下设施长150 m、宽18.3 m、高8.5 m，顶部距地表 3.7 m。因剩余密度未知，本文取值为-1 900 kg/m3，并在目标中部生成2条与目标体长轴呈35°方向的测线(图 1)。测线长90 m，以长方体长轴为中心，测点间距分别设为2 m、4 m、8 m和12 m，并加入不同程度的白噪声(表 3)。图 6(a)为地面(虚线)及30 m高度上(实线)重力和垂直重力梯度正演结果。尽管地面重力异常(蓝色虚线)量级明显低于文献[15]中的实测值，但较小的剩余密度更有助于验证本文方法的可行性。图 6(b)为30 m高度上的重力梯度正演结果(蓝色曲线)及加入6 E白噪声的结果(红色曲线)。

 图 6 不同高度上的重力异常和重力梯度测线数据 Fig. 6 Survey line data of gravityanomalies and gravity gradient anomalies at different altitudes

 图 7 模型参数的贝叶斯后验概率分布 Fig. 7 Bayesian posterior probability distributions for model parameters

3 结语

1) 地面垂直重力梯度测量较微重力测量具有更高的信噪比，使用贝叶斯推断能对地下目标的位置、形状、剩余密度等参数作出合理估计。结合测区的坐标范围合理设定目标位置、形状等参数的先验信息，可使反演结果更趋于真值。

2) 对于未知走向的地下目标，至少需要2条与目标相交的测线才能确定目标的走向参数。

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Bayesian Method for Detecting Subsurface Targets Using Gravity Data
HUANG Jiaxi1     BIAN Shaofeng2     JI Bing1
1. Department of Electrical Engineering, Naval University of Engineering, 717 Jiefang Road, Wuhan 430033, China;
2. Key Laboratory of Geological Survey and Evaluation of Ministry of Education, China University of Geosciences, 388 Lumo Road, Wuhan 430074, China
Abstract: Based on gravity data, with aims to evaluate the effectiveness of Bayesian inference for subsurface target detection in situations where the spatial position and shape of underground targets are unknown, we perform modeling and analysis in terms of regional measurement and line measurement scenarios. The results show that Bayesian method for gravity gradient detection outperforms microgravity detection and can lead to a good parameter prediction quality.
Key words: subsurface target detection; gravity anomaly; gravity gradient; Bayesian inference; highest density interval (HDI)