﻿ 矿区地表沉降监测的一种组合模型预测方法
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 大地测量与地球动力学  2021, Vol. 41 Issue (3): 308-312  DOI: 10.14075/j.jgg.2021.03.016

### 引用本文

ZHOU Wentao, ZHANG Wenjun, YANG Yuanji, et al. A Combined Model Prediction Method for Surface Subsidence Monitoring in Mining Areas[J]. Journal of Geodesy and Geodynamics, 2021, 41(3): 308-312.

### Foundation support

National Key Research and Development Program of China, No.2018YFC150540202; National Natural Science Foundation of China, No.41871174.

### Corresponding author

ZHANG Wenjun, professor, majors in remote sensing information technology, E-mail: 113999066@qq.com.

### 第一作者简介

ZHOU Wentao, postgraduate, majors in application of InSAR in surface subsidence of mining area, E-mail: 269026354@qq.com.

### 文章历史

1. 西南科技大学环境与资源学院，四川省绵阳市青龙大道中段59号，621010;
2. 国家遥感中心绵阳科技城分部，四川省绵阳市青龙大道中段59号，621010

1 SBAS沉降监测及组合预测模型建立 1.1 SBAS-InSAR技术原理

1.2 基于IOWA算子的ARIMA和Holt-Winters模型的组合预测方法原理

ARIMA模型是针对非平稳时间序列建模的常见模型。ARIMA(p, d, q)称为差分整合移动平均自回归模型，其中AR为自回归项，p为自回归阶数，MA为移动平均项，q为移动平均阶数，d为差分阶数。ARIMA(p, d, q)模型可表示为：

 $\left( {1 - \sum\limits_{i = 1}^p {{\varphi _i}} {L^i}} \right){(1 - L)^d}{X_t} = \left( {1 + \sum\limits_{i = 1}^q {{\theta _i}} {L^i}} \right){\varepsilon _t}$ (1)

 ${y^\prime }_{t + 1} = a{y_t} + (1 - a){y^\prime }_t$ (2)

Holt-Winters线性指数平滑法是在上述一次指数平滑的基础上进行二次指数平滑，其公式为：

 $S_{t}^{(2)}=a S_{t}^{(1)}+(1-a) S_{t-1}^{(2)}$ (3)

 $\sum\limits_{t = 1}^m {{l_t}} = 1,{l_t} \ge 0,t = 1,2, \cdots ,m$ (4)

 ${{\hat x}_i} = \sum\limits_{t = 1}^m {{l_t}} {x_{ti}},i = 1,2, \cdots ,N$ (5)

t种方法在时刻i的预测精度pti可表示为：

 ${p_{ti}} = \left\{ {\begin{array}{*{20}{l}} {1 - \left( {{x_i} - {x_{ti}}} \right)/{x_i},\left( {{x_i} - {x_{ti}}} \right)/{x_i} < 1}\\ {0,\left( {{x_i} - {x_{ti}}} \right)/{x_i} \ge 1} \end{array}} \right.$ (6)

 $\begin{array}{*{20}{c}} {{F_L}\left( {\left( {{p_{1i}},{x_{1i}}} \right),\left( {{p_{2i}},{x_{2i}}} \right), \cdots ,\left( {{p_{mi}},{x_{mi}}} \right)} \right) = }\\ {\sum\limits_{t = 1}^m {{l_t}} {x_{a - {\mathop{\rm index}\nolimits} (ti)}}} \end{array}$ (7)

ea-index(ti)=xi-xa-index(ti)，则以误差平方和为准则的基于IOWA的组合预测最优化模型可表示为：

 $\begin{array}{*{20}{c}} {\min S(L) = \sum\limits_{t = 1}^m {\sum\limits_{g = 1}^M {{l_t}} } {l_g}\left( {\sum\limits_{i = 1}^N {{e_{a - {\mathop{\rm index}\nolimits} (ti)}}} {e_{a - {\mathop{\rm index}\nolimits} (gi)}}} \right)}\\ {{\rm{ s}}{\rm{.t}}{\rm{. }}\left\{ {\begin{array}{*{20}{l}} {\sum\limits_{t = 1}^m {{l_t}} = 1}\\ {l \ge 0} \end{array}} \right.} \end{array}$ (8)
2 实验分析 2.1 研究区概况及数据来源

 图 1 研究区概况 Fig. 1 Overview of the study area

2.2 矿区SBAS-InSAR监测及分析

 图 2 累积沉降值 Fig. 2 Cumulative settlement

 图 3 各点沉降变化曲线 Fig. 3 Settlement curve of each point
2.3 地表沉降预测及结果分析

 $\begin{array}{c} {f_L}\left[ {\left( {{p_{1,24}},{x_{1,24}}} \right),\left( {{p_{2,24}},{x_{2,24}}} \right)} \right] = \\ {l_1}{x_{a - {\mathop{\rm index}\nolimits} (1,24)}} + {l_2}{x_{a - {\mathop{\rm index}\nolimits} (2,24)}} \end{array}$
 $\begin{array}{c} {f_L}\left[ {\left( {{p_{1,25}},{x_{1,25}}} \right),\left( {{p_{2,25}},{x_{2,25}}} \right)} \right] = \\ {l_1}{x_{a - {\mathop{\rm index}\nolimits} (1,25)}} + {l_2}{x_{a - {\mathop{\rm index}\nolimits} (2,25)}} \end{array}$
 $\begin{array}{c} {f_L}\left[ {\left( {{p_{1,26}},{x_{1,26}}} \right),\left( {{p_{2,26}},{x_{2,26}}} \right)} \right] = \\ {l_1}{x_{a - {\mathop{\rm index}\nolimits} (1,26)}} + {l_2}{x_{a - {\mathop{\rm index}\nolimits} (2,26)}} \end{array}$

 $\begin{array}{c} {f_L}[(0.808, - 17.93),(0.959, - 21.28)] = \\ - 21.28{l_1} - 17.93{l_2} \end{array}$
 $\begin{array}{c} {f_L}[(0.999, - 20.99),(0.936, - 22.30)] = \\ - 20.99{l_1} - 22.30{l_2} \end{array}$
 $\begin{array}{c} {f_L}[(0.814, - 23.87),(0.794, - 23.31)] = \\ - 23.87{l_1} - 23.31{l_2} \end{array}$

 $\begin{array}{*{20}{c}} {\min S\left( {{l_1},{l_2}} \right) = 1{\kern 1pt} {\kern 1pt} {\kern 1pt} 463.2l_1^2 + 1{\kern 1pt} {\kern 1pt} {\kern 1pt} 361.66l_2^2 + }\\ {2{\kern 1pt} {\kern 1pt} {\kern 1pt} 811.6{l_1}{l_2} - 3{\kern 1pt} {\kern 1pt} {\kern 1pt} 225.42{l_1} - 3{\kern 1pt} {\kern 1pt} {\kern 1pt} 098.15{l_2} + 1{\kern 1pt} {\kern 1pt} {\kern 1pt} 793.00} \end{array}$
 ${{\rm{ s}}{\rm{.t}}{\rm{. }}\left\{ {\begin{array}{*{20}{l}} {{l_1} + {l_2} = 1}\\ {{l_1} \ge 0,{l_2} \ge 0} \end{array}} \right.}$

 ${l_1} = 1,{l_2} = 0$

3 结语

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A Combined Model Prediction Method for Surface Subsidence Monitoring in Mining Areas
ZHOU Wentao1,2     ZHANG Wenjun1,2     YANG Yuanji1,2     MA Xudong1,2     RAN Maoying1,2
1. School of Environment and Resource, Southwest University of Science and Technology, 59 Mid-Qinglong Avenue, Mianyang 621010, China;
2. Mianyang Science and Technology City Division, National Remote Sensing Center of China, 59 Mid-Qinglong Avenue, Mianyang 621010, China
Abstract: Based on induced ordered weighted averaging (IOWA) operator, we combine the difference autoregressive integrated moving average (ARIMA) model and Holt-Winters exponential smoothing model. We use the SBAS-InSAR monitoring value to predict mining area surface subsidence, and compare this prediction with the results of each single model. The results show that the prediction accuracy of the combined model based on IOWA operator is significantly improved compared with that of the single model. For the combined model, the mean square error (MSE) and the mean absolute error (MAE) of each point reaches 1.458 mm and 2.175 mm respectively, which can be used for the monitoring and prediction of mining area surface subsidence.
Key words: SBAS-InSAR; ARIMA model; Holt-Winters model; IOWA operator; subsidence prediction