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• CN 62-1112/TF
• ISSN 1005-2518
• 创刊于1988年

## Prediction of Mine Subsidence Area Based on Chaotic Time Series Analysis

ZENG Junhui,, LI Xibing,

School of Resources and Safety Engineering，Central South University，Changsha 410083，Hunan，China

 基金资助: 国家自然科学基金重点项目“深部资源开采诱发岩体动力灾害机理与防控方法研究”.  编号：41630642

Received: 2018-01-23   Revised: 2018-03-28   Online: 2019-04-29

Abstract

With the advance of mining to deep in domestic underground metal mines,surface subsidence caused by mining has seriously threatened the industrial production of the mine and the ecological environment of the mining area.Therefore，accurately predicting the extent of mine subsidence area is of great significance to mine safety production and ecological protection of mining areas.Because the mine subsidence area is a complex system，it contains various random factors，and the evolution process of the system is often accompanied by the exchange of energy，showing nonlinear characteristics.This paper demonstrates the mine subsidence area as a nonlinear dissipative dynamic system.Combined with the time series analysis method in chaos theory，the range of mine subsidence area is predicted.The specific method studied in this paper is based on the phase space reconstruction theory，and the phase space reconstruction of the deformation time series of the mine subsidence area is carried out.A small data amount algorithm is used to calculate the key index of the time series-the largest Lyapunov exponent.According to the calculation results，the chaos of the mine subsidence zone system is identified.The evolution law and energy variation law of the phase distance of the collapse zone in the phase space are studied.The prediction model of the boundary of the subsidence zone is established.The model is used to analyze and predict the subsidence zone of the Hongling lead-zinc mine.The results show that the formation and expansion of the subsidence area of the Hongling lead-zinc mine is the result of the comprehensive effects of all kinds of factors such as geological conditions，mining engineering operations and inherent nonlinear characteristics.The maximum Lyapunov exponent and associated dimension of the time series of the subsidence area of the Hongling lead-zinc mine are calculated.It is verified that the variation of the subsidence area of Hongling lead-zinc mine has chaotic characteristics.The time series analysis method can well reflect the inherent law of the range change of the subsidence area.The chaotic characteristics of the different locations in the collapsed zone are different in the phase space due to different geological conditions and mechanical properties.The established mine subsidence area prediction system can better predict the change of the subsidence area.The predicted value is basically consistent with the actual value，and the error size does not exceed 0.1%，which verifies the reliability of the method.It provides a new idea for the prediction of mine subsidence area and guides mine safety production.

Keywords： underground mining ; mine monitoring ; chaotic time series ; phase space reconstruction ; Lyapunov index ; surface deformation law ; subsidence area prediction

ZENG Junhui, LI Xibing. Prediction of Mine Subsidence Area Based on Chaotic Time Series Analysis[J]. Gold Science and Technology, 2019, 27(2): 249-256 doi:10.11872/j.issn.1005-2518.2019.02.249

### 1.2 最大 Lyapunov指数计算

Lyapunov指数是判断系统是否具有混沌特性的指标，若其计算结果大于0则说明系统具有混沌性。同时，Lyapunov指数还能说明系统轨道的幅散程度，系统本身的可预测性问题则与最大Lyapunov 指数有关，它决定着初始信息的影响大小、邻近轨道的辐散程度和吸引子的发散快慢等，因此准确得到最大Lyapunov指数会有一个很好的预测结果。计算Lyapunov指数的方法有多种，由于本文数据量较小，对结果精度的要求较高，因此选取小数据量方法来计算最大Lyapunov指数。

（1）塌陷区变形序列重构相空间。由于塌陷区本身是个闭合的不规则曲线，将其近似为一个矩形，那么对于水平轴线来说上下边的位移属于单变量移动，选取了一些能够确定塌陷区范围的关键点，即x1，x2，…，x16。对于x1来说，设单变量的位移为{x1（ti），i=1，2，…，n}，通过G-P算法（遗传编程算法）得到关联维数d，嵌入维数m则可以通过m≥2d+1来确定。选择自相关函数达到极小值时确定时间间隔为λ=kd，可得到时间序列{x1（ti），i=1，2，…，n}的相空间表示为

$X1i(t)=x1(t),x1(t+λ),⋯,x1t+(m-1)λi=1,2,⋯,M$

（2）小数据量算法计算最大Lyapunov指数。小数据量算法具有计算量少且快速准确等优点，十分适合本研究，其计算步骤如下：

①将该系统的时间序列嵌入欧式空间$R$中，得到一个向量集：$X1i(t)=x1(t),x1(t+λ),⋯,x1t+(m-1)λ$，运用快速傅立叶变换方法，计算得到该组数据的时间延迟$λ$和平均周期t

②关联维数d由G-P方法计算得出，根据Takens嵌入定律计算系统的嵌入维数m，根据时间延迟λ和嵌入维数m重构相空间[11]，即$X1j,j=1,2,⋯,M$

③找到相空间中与$X1j$相对应的最近邻点$X1j⌢$，将其分离开来[12]，即：

$d1j(0)=minjX1j-X1j⌢,j-j⌢>T$

④相空间中的每个点$X1j$，计算出该邻点对的第i个离散时间后的距离为$dJ(i)=Xj+i-Xj⌢+i$

$i=1,2,⋅⋅⋅,minM-j,M-j⌢$

$y(i)=1ph∑j=1plndj(i)$

（3）Lyapunov指数预测模式及预测时间 。设$XM$为预测模式的交点，可得到其在相空间中的最邻近点为$Xk$，两点的距离为

$dM(0)=minXM-Xj=XM-Xk$

$XM-XM+1=Xk-Xk+1eλ1$

$Tm=1λ1$

$Tm$能够说明系统误差增加一倍时系统所需的时间，因此可成为短期系统的可靠性预测指标之一[13]

### 图1

Fig.1   Partial subsidence area of Hongling lead zinc mine

### 图2

Fig.2   Pull cracks in the subsidence area of Hongling lead zinc mine

### 2.4 不同监测点塌陷区变形在相空间中的演变规律

$Z=minX1-Xi$$i=1,2,⋯,8$

$Zk=xtk-xtb2+xtk-1-xtb-12+⋯+xtk-4-xtb-4212$

### 图3

Fig.3   Phase space distance evolution curve of deformation in subsidence area

### 2.5 塌陷区变形时频分析

$Zj=1n-m+1∑j=1n-m+1Zke2πjkn-m+1$

### 图4

Fig.4   Time power spectrum curve of deformation in subsidence area

### 图5

Fig.5   Measured displacement curves

### 图7

Fig.7   Comparison between prediction and actual scope in subsidence area

Table 1  Prediction results of displacement at A2 monitoring point

39029.4229.610.65
42031.4331.680.80
45035.0635.010.14
48040.8541.090.59
51049.8850.210.66
54057.0256.720.53
57066.1466.570.65
60075.0574.450.80

Table 2  Prediction results of displacement at A9 monitoring point

39025.0625.250.76
42026.0826.320.92
45028.7128.650.21
48033.4933.730.72
51041.5541.860.75
54048.6248.360.53
57059.8359.310.87
60071.6971.110.81

### 图6

Fig.6   Comparison curves of measured and predicted displacement

## 3 结论

（1）在矿山生产中，塌陷区的形成及扩大是其地质条件、采矿工程作业及内在非线性特性等多种因素综合作用的结果，研究发现该系统具有混沌特征，在相空间重构理论基础上，综合分析得到系统混沌特性的内在基本规律。

（2）通过研究不同位置、不同地质条件下的监测点变形情况，发现其表现出不同的力学特征，同时在相空间中也表现出不同的混沌特性。

（3）研究结果表明，重构相空间方法能够对塌陷区范围大小进行短期预测，在红岭铅锌矿塌陷区范围预测中取得了良好的结果，计算得出红岭铅锌矿塌陷区范围时间序列的最大Lyapunov指数大于0，该塌陷区范围的预测值与实际值基本吻合，误差大小不超过0.1%。这是研究塌陷区混沌现象内在规律的一次有益尝试，为地表变形范围的圈定和矿山安全生产提供了指导。

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