# Normality Test for Influence Function of Mining Surface Subsidence Based on Network Observation Station

Authors: Pengyu Li; Zhenqi Hu; Yaokun Fu; Gensheng Li; Jiaxin Guo; Dongzhu Yuan
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

Coal is the primary energy source in China, while underground mining is the mainstream way of coal mining. By triggering surface movement and deformation, underground mining can cause damage to arable land, buildings, roads, and so on, which is detrimental to the ecological environment in the mining area. In order to assess the severity of damage caused by this to the ground buildings and ecological environment in the mining area, it is necessary to predict mining-induced surface subsidence before the mining activities are carried out. Currently, the most used prediction method is the Probability Integral Method. It is based on probabilistic theory to mathematically demonstrate that the surface downwelling caused by underground extraction conforms to normal distribution. However, there is a lack of validation with measured subsidence basin data. Since the 1960s, China has been paying increased attention to the study of mining subsidence. However, there are still few network stations built across China to monitor subsidence basin. Herein, SPSS software is applied to re-analyze these valuable historical data. By analyzing the observation results and comparing them with the calculation results as obtained by using the probability integral method, it can be found out that the surface subsidence in the center of the subsidence basin conforms to normal distribution, not the subsidence at the edge of the subsidence basin. Therefore, it is inevitable for errors to occur at the edge of the subsidence basin when the normal distribution function is used as the mining influence function to calculate the surface subsidence. This conclusion is expected to provide practical reference for the prediction of surface subsidence in coal mines, and this experience can be extended to the mining of other solid minerals.


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## Body

## 1. Introduction

Although China has been putting much efforts into the development and utilization of renewable energy [1, 2], coal remains the dominant source of energy [3]. In 2020, the national raw coal production in China amounted to 3.9 billion tons, accounting for 47.3% of the world's total, to which underground mining contributed as high as 85%. Nowadays, China has the highest proportion of coal production and underground mining in the world [4–6]. In order to evaluate the severity of damage caused by mining to ground buildings and ecological environment in the mining area, it is necessary to assess the risk of surface movement and deformation before mining gets underway. So far, the research on surface subsidence in coal mines has been conducted for more than 100 years [7]. There are a wide variety of empirical function models proposed in plenty of studies for specific geological and mining conditions based on the data collected by linear observation stations, such as negative exponential function, hyperbolic function, trigonometric function, etc. [8–11]. In the 1950s, Knothe proposed a mining influence function that conformed to normal distribution according to the observation data [12, 13] by treating the surface movement process caused by mining as a random process, Litwiniszyn put forward the random medium theory, arguing that the mining influence function followed normal distribution as well [14]; In 1965, Liu Baochen and Liao Guohua adopted the probability theory to mathematically demonstrate that the mining influence function conformed to normal distribution, which led to the Probability Integral Method [15–22]. Among the three calculation methods as mentioned above, the Knothe influence function method assumes that the mining influence function conforms to normal distribution based on the observation data. Differently, Litwiniszyn and Liu Baochen reached their conclusion through the simplified mathematical processing in an ideal state. Despite being based on different assumptions, these three calculation methods lead to the same conclusion, which making people default that the mining influence function obeys the normal distribution. However, there is still a lack of requisite test data.As the Chinese government attaches increasing significance to addressing the environmental damage caused by surface subsidence [23, 24], it is essential for coal mine productions to comply with national laws and regulations. On the one hand, it is necessary to reduce the surface subsidence caused by mining. On the other hand, it is necessary to take remedial actions in those areas where surface subsidence have occurred. At the same time, in order to reduce the cost of damage compensation incurred by mining for buildings, it is necessary to accurately calculate and evaluate the surface subsidence occurring coal mines. To achieve this purpose, it is essential to evaluate the mining impact function, and to verify the rationality of using the normal distribution as the mining influence function.To establish whether the subsidence occurring outside the main section of the settlement basin conforms to normal distribution, it is necessary to obtain the measured data for the entire network of observation stations in the basin. In China, surface movement observation did not start until the early 1960s. Up to now, there have been many linear surface movement observation stations built in the mining areas. However, there are still as few as two networks of observation stations built for the Pingdingshan and Chaili coal mines [25]. In this paper, the historical data collected by Chaili coal mine network observation station are reanalyzed in an advanced way to explore the distribution law of mining influence function. According to the research results, a significant calculation error will arise from the practice of taking the normal distribution as the mining influence function at the edge of the subsidence basin. This result will influence the judgment on mining subsidence basin boundary, which affects production safety, land damage, and building damage compensation. The results of this study provide a practical reference for the prediction of surface subsidence in coal mining areas, which can be extended to the mining of other solid minerals.


## 2. Study Area

The study area is located in Xigang Town, Tengzhou City, which is about 22 km away from the city center of Tengzhou, and its location is shown in Figure1. In terms of climate, the study area shows the characteristics of temperate arid to monsoon continental climate, with an annual average temperature of 13.6°C. The annual precipitation reaches about 793.0 mm, and the average annual evaporation stands at 1791.7 mm [26, 27]. Geographically, the study area features a lakeside alluvial plain with flat terrain, highly developed road network and dense population. The terrain declines slowly from the northeast to the southwest. Ground elevation decreases from +43.73 m to +35.30 m, with a slope of about 1‰. The soil texture in the area is dominated by clay loam and loam, the sandy clay ratio is moderate, the permeability is high, the soil quality is excellent, the level of soil fertility is satisfactory, and the production capacity is high.Figure 1
Location of the Chaili colliery.


### 2.1. General Situation of Working Face

The strike length of working face 331 is 540 m, and the inclined length is 136 m. Strike longwall comprehensive mechanized coal mining is adopted for the working face, and the full caving method is applied for roof management. On average, the ground elevation is about +40.0 m, and the mining depth is 132 m. The first-time mining was conducted in July, 1973, and the last-time mining took place in October, 1976. According to the histogram analysis of a borehole near the working face, the overlying strata is 132 m deep on average. The quaternary system results from the interactive deposition of the clay layer and sandstone layer, with a thickness of 71 m. Besides, the sandstone layer is 30 m and the mudstone layer is 31 m in thickness. The overlying strata show a medium hardness in the comprehensive lithology. The coal seam is simple in structure and stability in occurrence, with an average thickness of 1.8 m and an average dip angle of 4°. The false roof is 2–7 m mudstone, and the coal seam floor is grayish black sandy mudstone, 9–14 m. Figure2 shows the comprehensive histogram of the overlying strata.Figure 2
Comprehensive histogram of overburden of 331 working face.


### 2.2. Layout of Observation Stations

In general, there are two types of surface movement observation stations: linear observation station and network observation station [25]. At present, linear observation stations are most commonly seen in China. Usually, the observation line of linear observation station is comprised of strike line and inclination line, which are perpendicular to each other. In some cases, it is possible to set only one or half observation line. For example, the strike length of the working face is too large, the workload of setting the full strike observation line is high, and the observation time is long. In this case, a common practice is to set half of the strike observation line. In another example, one side of the dip direction is affected by the mining of the adjacent working face, which means only half of the dip observation line can be set. The observation line setting scheme depends on the exact purpose of observation and on-site conditions in the setting area. Its selection is based on the practicalities and principle of obtaining as much reliable data as possible. The layout of several different linear observation stations is shown in Figure 3(a).Figure 3
Diagrammatic sketch of linear and network observation stations. (a) are linear observation stations, Observation lines are the main section observation line; (b) is a network observation station.
(a)(b)In the network observation station, the observation points are arranged into a network, so as to monitor the whole area, half area or quarter area of surface movement and deformation, as shown in Figure3(b). The layout of surface movement network observation stations is treated as the control network for monitoring the whole subsidence basin to fully show the movement and deformation law of the whole surface movement basin.The network observation station involved in this study was established on June 15,1973. After 4 years of observation, the final data of surface subsidence observation were collected. Figure4 shows the layout of observation station network and embedding of point markers. There are 146 points in total, the distance between two observation stations is 15 m, and there are 20 observation lines, of which 7 are in the strike direction and 13 are in the inclination direction.Figure 4
Layout of the network of observation stations.After all the points of the observation station are marked, dig pits and bury point markers, which are made of concrete. The top surface of the point mark is sized 250∗250 mm, the height is 1000 mm, the mark center is 520 mm iron bar, and the top is engraved with a cross. Theodolite is used to indicate the direction of the observation line at the time of burying stake, and a steel ruler is used to mark the position on the observation line. After all points are buried, the point number is red painted on the top surface of the stake.The altitude is measured at the observation stations by using precision leveling instruments and a total of 18 leveling measurements. Also, data processing is carried out through the subsidence calculation formula. The formula used to calculate subsidence for pointn during the mth time observation is expressed as(1)Wn=Hn0−Hnm,where Wn represents the subsidence value of point n. Hn0,Hnm refer to the altitudes at the first and the m-th observation, respectively.The final subsidence values at each point for the net observation lines are listed in Table1.Table 1
Final subsidence value of network observation station.
Subsidence value/mmLine ALine BLine CLine DLine ELine FLine GLine A1–G1−1318.00−1263.00−1087.00−713.00−366.00−162.00−78.00Line A2–G2−1265.28−1326.15−1065.26−698.74−355.02−160.38−70.98Line A3–G3−1225.74−1212.48−1054.39−698.74−338.34−184.68−68.64Line A4–G4−1212.56−1199.85−1054.39−698.74−344.04−163.62−62.40Line A5–G5−1146.66−1136.7−967.43−641.70−286.47−118.26−53.04Line A6–G6−948.96−896.73−739.16−456.32−208.62−86.33−45.24Line A7–G7−553.56−505.20−391.32−235.29−139.08−58.32−22.62Line A8–G8−276.78−239.97−195.66−114.08−62.22−43.74−16.27Line A9–G9−92.26−88.41−76.09−49.91−36.60−30.78−14.04Line A10–G10−52.72−50.52−40.33−28.52−22.37−19.44−13.66Line A11–G11−26.36−23.73−21.74−18.23−14.64−13.28−10.14Line A12-G12−13.18−12.63−10.22−14.26−12.43−11.34−8.37Line A13–G13−11.23−9.88−21.74−12.67−10.47−9.76−10.14


## 2.1. General Situation of Working Face

The strike length of working face 331 is 540 m, and the inclined length is 136 m. Strike longwall comprehensive mechanized coal mining is adopted for the working face, and the full caving method is applied for roof management. On average, the ground elevation is about +40.0 m, and the mining depth is 132 m. The first-time mining was conducted in July, 1973, and the last-time mining took place in October, 1976. According to the histogram analysis of a borehole near the working face, the overlying strata is 132 m deep on average. The quaternary system results from the interactive deposition of the clay layer and sandstone layer, with a thickness of 71 m. Besides, the sandstone layer is 30 m and the mudstone layer is 31 m in thickness. The overlying strata show a medium hardness in the comprehensive lithology. The coal seam is simple in structure and stability in occurrence, with an average thickness of 1.8 m and an average dip angle of 4°. The false roof is 2–7 m mudstone, and the coal seam floor is grayish black sandy mudstone, 9–14 m. Figure2 shows the comprehensive histogram of the overlying strata.Figure 2
Comprehensive histogram of overburden of 331 working face.


## 2.2. Layout of Observation Stations

In general, there are two types of surface movement observation stations: linear observation station and network observation station [25]. At present, linear observation stations are most commonly seen in China. Usually, the observation line of linear observation station is comprised of strike line and inclination line, which are perpendicular to each other. In some cases, it is possible to set only one or half observation line. For example, the strike length of the working face is too large, the workload of setting the full strike observation line is high, and the observation time is long. In this case, a common practice is to set half of the strike observation line. In another example, one side of the dip direction is affected by the mining of the adjacent working face, which means only half of the dip observation line can be set. The observation line setting scheme depends on the exact purpose of observation and on-site conditions in the setting area. Its selection is based on the practicalities and principle of obtaining as much reliable data as possible. The layout of several different linear observation stations is shown in Figure 3(a).Figure 3
Diagrammatic sketch of linear and network observation stations. (a) are linear observation stations, Observation lines are the main section observation line; (b) is a network observation station.
(a)(b)In the network observation station, the observation points are arranged into a network, so as to monitor the whole area, half area or quarter area of surface movement and deformation, as shown in Figure3(b). The layout of surface movement network observation stations is treated as the control network for monitoring the whole subsidence basin to fully show the movement and deformation law of the whole surface movement basin.The network observation station involved in this study was established on June 15,1973. After 4 years of observation, the final data of surface subsidence observation were collected. Figure4 shows the layout of observation station network and embedding of point markers. There are 146 points in total, the distance between two observation stations is 15 m, and there are 20 observation lines, of which 7 are in the strike direction and 13 are in the inclination direction.Figure 4
Layout of the network of observation stations.After all the points of the observation station are marked, dig pits and bury point markers, which are made of concrete. The top surface of the point mark is sized 250∗250 mm, the height is 1000 mm, the mark center is 520 mm iron bar, and the top is engraved with a cross. Theodolite is used to indicate the direction of the observation line at the time of burying stake, and a steel ruler is used to mark the position on the observation line. After all points are buried, the point number is red painted on the top surface of the stake.The altitude is measured at the observation stations by using precision leveling instruments and a total of 18 leveling measurements. Also, data processing is carried out through the subsidence calculation formula. The formula used to calculate subsidence for pointn during the mth time observation is expressed as(1)Wn=Hn0−Hnm,where Wn represents the subsidence value of point n. Hn0,Hnm refer to the altitudes at the first and the m-th observation, respectively.The final subsidence values at each point for the net observation lines are listed in Table1.Table 1
Final subsidence value of network observation station.
Subsidence value/mmLine ALine BLine CLine DLine ELine FLine GLine A1–G1−1318.00−1263.00−1087.00−713.00−366.00−162.00−78.00Line A2–G2−1265.28−1326.15−1065.26−698.74−355.02−160.38−70.98Line A3–G3−1225.74−1212.48−1054.39−698.74−338.34−184.68−68.64Line A4–G4−1212.56−1199.85−1054.39−698.74−344.04−163.62−62.40Line A5–G5−1146.66−1136.7−967.43−641.70−286.47−118.26−53.04Line A6–G6−948.96−896.73−739.16−456.32−208.62−86.33−45.24Line A7–G7−553.56−505.20−391.32−235.29−139.08−58.32−22.62Line A8–G8−276.78−239.97−195.66−114.08−62.22−43.74−16.27Line A9–G9−92.26−88.41−76.09−49.91−36.60−30.78−14.04Line A10–G10−52.72−50.52−40.33−28.52−22.37−19.44−13.66Line A11–G11−26.36−23.73−21.74−18.23−14.64−13.28−10.14Line A12-G12−13.18−12.63−10.22−14.26−12.43−11.34−8.37Line A13–G13−11.23−9.88−21.74−12.67−10.47−9.76−10.14


## 3. Results

### 3.1. Test Results of the Normal Subsidence of the Observation Line

The commonly used normal test methods can be divided into three categories in general [28]. The first one is graphical representation, which is examined mainly using probability plots (P–P plots) and quantile plots (Q-Q plots). The second one is the kurtosis–skewness joint test. Kurtosis indicates how flattened or steep the top of the frequency distribution curve is for the test sample, and skewness is used to indicate the degree to which a distribution deviates from symmetry. The last one is the Kolmogorov Smirnov test (K–S normality test). The latter two are more reliable than graphical representation. Therefore, the kurtosis–skewness joint test and K–S normality test are adopted in this study to verify the normality of surface subsidence with the assistance of SPSS software.


#### 3.1.1. Kurtosis–Skewness Normality Test

The kurtosis and skewness coefficient values of the data are used to determine whether the samples conform to normal distribution or not. If the absolute values of kurtosis coefficientZK and skewness coefficient ZS fall below 1.96 simultaneously, the distribution of sample data is considered to conform to normal distribution. If either or both the absolute values of ZK or ZS exceed 1.96, the sample data are considered not to conform to normal distribution [29]. The formulas used to calculate ZK and ZS are expressed as follows:(2)ZK=K−0SK,ZS=S−0SS. Where SK and SS represent standard error for kurtosis coefficient and skewness coefficient, respectively.Table2 depicts the calculation results obtained from the normality test on the subsidence kurtosis and the skewness of the observation lines in the inclination direction. It is found out that the absolute kurtosis coefficient and skewness coefficient values are less than 1.96 during 12 observation lines, the sedimentation profiles of which conform to normal distribution. However, the absolute values of kurtosis coefficient and skewness coefficient of the A13-G13 observation line exceed 1.96, which means the subsidence of this observation line does not follow normal distribution.Table 2
Results of the kurtosis–skewness normality test of the observation lines in the inclination direction.
Observation linesNNormal parameterKSKZKSSSSKMean valueStandard deviationLineA1–G17−712.429522.1160.0360.7940.045−2.1711.5871.368LineA2–G27−705.973524.5060.0030.7940.005−2.1391.5871.348Line A3–G37−683.287492.8900.0750.7940.094−2.2161.587−1.396Line A4–G47−676.514492.4840.1010.7940.127−2.2141.587−1.395Line A5–G57−621.466474.3310.0560.7940.071−2.2471.587−1.416Line A6–G67−479.050387.682−0.0730.794−0.092−2.1801.587−1.374Line A7–G77−272.199213.994−0.2110.794−0.266−1.9051.587−1.200Line A8–G87−135.531102.435−0.2820.794−0.355−1.8721.587−1.180Line A9–G97−53.82132.265−0.0030.794−0.004−2.0201.587−1.273Line A10–G107−32.50915.486−0.2860.794−0.360−1.8131.587−1.142Line A11–G117−18.3035.937−0.0040.794−0.005−1.4641.587−0.922Line A12–G127−11.7761.9800.7320.7940.9220.1661.5870.105Line A13–G137−12.2704.296−2.3650.794−2.9795.7671.5873.634Table3 presents the calculation results obtained from the normality test on the subsidence kurtosis and the skewness of the strike parallel observation lines. The absolute values of the kurtosis coefficient and the skewness coefficient of the seven strike parallel observation lines fall below 1.96, which suggests that the subsidence distribution of the strike parallel observation lines conforms to normal distribution.Table 3
Results of the kurtosis–skewness normality test of the observation lines in the strike direction.
Observation linesNNormal parameterKSKZKSSSSKMean valueStandard deviationLine I13−626.407564.244−0.0650.616−0.106−2.0971.191−1.761Line A13−612.712562.983−0.1150.616−0.187−2.0811.191−1.748Line B13−517.287478.136−0.1420.616−0.231−2.1001.191−1.7638Line C13−336.938314.855−0.1900.616−0.308−2.0911.191−1.7568Line D13−168.946150.802−0.2240.616−0.364−1.9651.191−1.6508Line E13−81.68767.439−0.3900.616−0.634−1.6841.191−1.414Line F13−36.42627.040−0.3840.616−0.623−1.7811.191−1.495


#### 3.1.2. K–S Normal Test

The rationale of the K–S normality test is described as follows. To calculate the theoretical cumulative probability valueFx of the sample data, it is assumed that normal distribution is followed. Then, the actual cumulative probability value Sx of the sample is calculated. The D-value Dx between them is calculated as follows:(3)D=maxSxi−Fxi.D must be corrected into(4)D=maxmaxSxi−Fxi,maxSxi−1−Fxi−1.SPSS software is directly applicable to calculate the probability value(p-value). As for whether the sample value conforms to the normal distribution, it is determined by whether the significance level is exceeded (0.05 in general) [30]. If the probability value falls below 0.05, it is considered that the test sample does not conform to normal distribution. If the value reaches above 0.05, the test sample is considered to conform to normal distribution.Table4 lists the results of normality test conducted on the inclination parallel observation lines. It can be seen clearly from the table that there are 12 observation lines with a greater p-value than 0.05. That is to say, the subsidence of 12 inclination direction observation lines conforms to normal distribution. Only the p-value of the Line A13-G13 is less than 0.05, which suggests that the subsidence of this observation line does not follow normal distribution.Table 4
Results of the K–S normality test of each line in the inclination direction.
Observation linesNNormal parameterStatisticpMeanStandard deviationLine A1–G17−712.429522.1160.5080.959Line A2–G27−705.973524.5060.4810.975Line A3–G37−683.287492.8900.5370.936Line A4–G47−676.514492.4840.5480.925Line A5–G57−621.466474.3310.5180.951Line A6–G67−479.050387.6820.4920.969Line A7–G77−272.199213.9940.4280.993Line A8–G87−135.531102.4350.5070.960Line A9–G97−53.82132.2650.4860.972Line A10–G107−32.50915.4860.4580.985Line A11–G117−18.3035.9370.4230.994Line A12–G127−11.7761.9800.5320.940Line A13–G137−12.2704.2960.8470.047Table5 shows the results of normality test on the strike parallel observation lines. Since the p-value of the strike parallel observation lines are significantly higher than 0.05, it is judged that the subsidence distribution of each observation line parallel to the strike of the main section conforms to normal distribution.Table 5
Results of the K–S normality tests of each observation line in the strike direction.
Observation linesNNormal parameterStatisticpMean valueStandard deviationLine I13−626.407564.2440.7670.599Line A13−612.712562.9830.7530.623Line B13−517.287478.1360.7620.607Line C13−336.938314.8550.8000.543Line D13−168.946150.8020.8000.544Line E13−81.68767.4390.6710.759Line F13−36.42627.0400.8420.477There is consistence shown in the conclusions between the normal test of the kurtosis–skewness and the K–S normal test. Among the 20 observation lines tested, only observation line A13–G13, which is the farthest away from the subsidence basin center, does not conform to normal distribution. However, the subsidence of the other strike and inclination parallel observation lines conforms to normal distribution, i.e., the subsidence in the surface of the mining area conforms to normal distribution overall.


### 3.2. Calculation Error of the Surface Subsidence at Any Point

Based on the Knothe influence function method [13], Figure 5 is built to show the working face and the calculation coordinate system of the horizontal coal seam, where l represents the calculation length of the trend; L indicates the calculation width of the tendency; and Ax,y denotes any point on the surface.Figure 5
Calculation coordinate system.Since the probable directions ofx and y are independent of each other, the subsidence at Ax,y is expressed as follows:(5)Wx,yA=Wmax∬Ffx,ydF=Wmax∬Ffxfydxdy,fx=1re−πx2r2,fy=1re−πy2r2,where Wmax represents the maximum surface subsidence; r indicates the main mining influence radius; fx, fy denotes the Knothe influence function, which follows a N0,σ2 distribution, and σ=r/2π.O represents the coordinate origin in Figure 5, so that the subsidence at Ax,y is expressed as follows:(6)Wx,yA=Wmaxr2∫−xl−xe−πxr2dx∫−yl−ye−πyr2dy=Wmax×12erfπxr+1−erfπx−lr+1×12erfπyr+1−erfπy−Lr+1=WmaxCxCy,where Cx,Cy refer to the subsidence distribution coefficients of the trend and tendency, respectively.The working face is a rectangular mining area, the mining length along the strike isl=540m , the mining width along the strike is L=136m , and the main influence radius of the mining is r=98m , The coordinate system is established with the lower left corner of the mining area as the coordinate origin as of the coordinate system (Figure 5): The maximum settlement value of the main section is measured to be W0=1318mm.


#### 3.2.1. Prediction Error of the Distribution of the Surface Settlement along the Inclined Observation Lines

According to the observation data, the settlement coefficient is determined asq=0.73 , the tangent of the main influence angle is determined as tanβ=2.2 , and the subsidence at any point on the surface is calculated using (6). Figure 6 shows all the predicted and measured values of the inclination parallel observation lines. It can be found out that the predictions of each observation line in the inclined direction deviate clearly from the actual subsidence curve, with similarity shown only around the maximum subsidence values.Figure 6
The measured and predicted distribution of the surface subsidence along the inclination parallel lines.
(a)(b)(c)(d)(e)(f)(g)(h)(i)(j)(k)(l)(m)The Root Mean Square Error (RMSE) [31]and the Percent Relative Error (PRE) [32]are adopted to examine how subsidence is accurately and reliably predicted by using the formula for each observation line at any point in the surface. As a numerical measure of prediction accuracy, RMSE refers to the square root of the sum of the squares of the difference between predicted and measured values and the ratio of the number of observations n. A smaller RMSE value indicates a higher accuracy of prediction [33]. The PRE is advantageous in reflecting the reliability of the predicted formula, with a lower absolute value of PRE indicating a greater reliability.(7)RMSE=±∑ΔΔnPRE=Δ/L′×100%,where Δ is the true error between predicted and measured values and L′ is the measured value.Table6 shows the calculation of the RMSE and PRE between the predicted value and the measured value of the surface subsidence along the inclination parallel observation lines. In general, the RMSE of the predicted values of the subsidence for observation lines is high in the inclined direction. However, according to the PRE, the prediction accuracy of the inclined direction observation lines shows a gradual trend of increasing from the center of the subsidence basin to the edge.Table 6
Predicted errors of each line in the inclined direction.
Observation linesMeasured mean value/mmRMSE of predicted/(mm)PRE%Line A1–G1−712.43380.3443.26Line A2–G2−705.97380.7143.94Line A3–G3−673.48363.1444.00Line A4–G4−676.51333.5438.74Line A5–G5−621.47320.1541.74Line A6–G6−479.05314.9753.71Line A7–G7−272.20335.76117.67Line A8–G8−135.53315.56224.74Line A9–G9−53.82254.37452.33Line A10–G10−32.51149.32436.30Line A11–G11−18.3028.95147.53Line A12–G12−11.7840.40317.56Line A13–G13−12.278.2760.00


#### 3.2.2. Prediction Error of the Distribution of the Surface Subsidence along the Strike Parallel Observation Lines

Figure7 shows the comparison between the predicted and measured values of all the strike parallel observation lines. It can be discovered that the predictions of observation lines A and B at the center of the subsidence basin are relatively close to the measured subsidence curves, despite a significant difference in the observation line subsidence curves toward the edge of the basin.Figure 7
The measured and predicted distribution of the surface subsidence along the strike parallel observation lines.
(a)(b)(c)(d)(e)(f)(g)Table7 shows the measured and predicted distribution of the surface subsidence along the strike parallel observation lines as well as the RMSE and PRE. To be specific, RMSE is predicted to be the minimum for the A-line subsidence toward the main section. When shifting away from the center line of the subsidence basin, the RMSE for the other observations increases first and then decreases. However, judging from the PRE, the prediction accuracy of the strike line reaches the highest in the center of the basin and it increases gradually when moving away from the center of the subsidence basin.Table 7
Predicted relative errors of each line in the strike direction.
Observation lineMeasured mean value (mm)RMSE of predicted (mm)PRE%Line A−626.41139.5515.37Line B−612.71153.4617.72Line C−517.29207.2834.58Line D−336.94354.1692.78Line E−168.95436.50219.78Line F−81.69346.23353.46Line G−36.43192.14436.92According to the comparison in subsidence prediction and RMSE between the actual measurement data and the prediction result for each observation line, it can be known that the subsidence prediction formula at any point on the surface has lower RMSE and higher prediction accuracy in the central part of the subsidence basin. In the region which is more distant from the center of the subsidence basin, however, the RMSE of the predicted values is larger and the level of prediction accuracy is lower.


## 3.1. Test Results of the Normal Subsidence of the Observation Line

The commonly used normal test methods can be divided into three categories in general [28]. The first one is graphical representation, which is examined mainly using probability plots (P–P plots) and quantile plots (Q-Q plots). The second one is the kurtosis–skewness joint test. Kurtosis indicates how flattened or steep the top of the frequency distribution curve is for the test sample, and skewness is used to indicate the degree to which a distribution deviates from symmetry. The last one is the Kolmogorov Smirnov test (K–S normality test). The latter two are more reliable than graphical representation. Therefore, the kurtosis–skewness joint test and K–S normality test are adopted in this study to verify the normality of surface subsidence with the assistance of SPSS software.


### 3.1.1. Kurtosis–Skewness Normality Test

The kurtosis and skewness coefficient values of the data are used to determine whether the samples conform to normal distribution or not. If the absolute values of kurtosis coefficientZK and skewness coefficient ZS fall below 1.96 simultaneously, the distribution of sample data is considered to conform to normal distribution. If either or both the absolute values of ZK or ZS exceed 1.96, the sample data are considered not to conform to normal distribution [29]. The formulas used to calculate ZK and ZS are expressed as follows:(2)ZK=K−0SK,ZS=S−0SS. Where SK and SS represent standard error for kurtosis coefficient and skewness coefficient, respectively.Table2 depicts the calculation results obtained from the normality test on the subsidence kurtosis and the skewness of the observation lines in the inclination direction. It is found out that the absolute kurtosis coefficient and skewness coefficient values are less than 1.96 during 12 observation lines, the sedimentation profiles of which conform to normal distribution. However, the absolute values of kurtosis coefficient and skewness coefficient of the A13-G13 observation line exceed 1.96, which means the subsidence of this observation line does not follow normal distribution.Table 2
Results of the kurtosis–skewness normality test of the observation lines in the inclination direction.
Observation linesNNormal parameterKSKZKSSSSKMean valueStandard deviationLineA1–G17−712.429522.1160.0360.7940.045−2.1711.5871.368LineA2–G27−705.973524.5060.0030.7940.005−2.1391.5871.348Line A3–G37−683.287492.8900.0750.7940.094−2.2161.587−1.396Line A4–G47−676.514492.4840.1010.7940.127−2.2141.587−1.395Line A5–G57−621.466474.3310.0560.7940.071−2.2471.587−1.416Line A6–G67−479.050387.682−0.0730.794−0.092−2.1801.587−1.374Line A7–G77−272.199213.994−0.2110.794−0.266−1.9051.587−1.200Line A8–G87−135.531102.435−0.2820.794−0.355−1.8721.587−1.180Line A9–G97−53.82132.265−0.0030.794−0.004−2.0201.587−1.273Line A10–G107−32.50915.486−0.2860.794−0.360−1.8131.587−1.142Line A11–G117−18.3035.937−0.0040.794−0.005−1.4641.587−0.922Line A12–G127−11.7761.9800.7320.7940.9220.1661.5870.105Line A13–G137−12.2704.296−2.3650.794−2.9795.7671.5873.634Table3 presents the calculation results obtained from the normality test on the subsidence kurtosis and the skewness of the strike parallel observation lines. The absolute values of the kurtosis coefficient and the skewness coefficient of the seven strike parallel observation lines fall below 1.96, which suggests that the subsidence distribution of the strike parallel observation lines conforms to normal distribution.Table 3
Results of the kurtosis–skewness normality test of the observation lines in the strike direction.
Observation linesNNormal parameterKSKZKSSSSKMean valueStandard deviationLine I13−626.407564.244−0.0650.616−0.106−2.0971.191−1.761Line A13−612.712562.983−0.1150.616−0.187−2.0811.191−1.748Line B13−517.287478.136−0.1420.616−0.231−2.1001.191−1.7638Line C13−336.938314.855−0.1900.616−0.308−2.0911.191−1.7568Line D13−168.946150.802−0.2240.616−0.364−1.9651.191−1.6508Line E13−81.68767.439−0.3900.616−0.634−1.6841.191−1.414Line F13−36.42627.040−0.3840.616−0.623−1.7811.191−1.495


### 3.1.2. K–S Normal Test

The rationale of the K–S normality test is described as follows. To calculate the theoretical cumulative probability valueFx of the sample data, it is assumed that normal distribution is followed. Then, the actual cumulative probability value Sx of the sample is calculated. The D-value Dx between them is calculated as follows:(3)D=maxSxi−Fxi.D must be corrected into(4)D=maxmaxSxi−Fxi,maxSxi−1−Fxi−1.SPSS software is directly applicable to calculate the probability value(p-value). As for whether the sample value conforms to the normal distribution, it is determined by whether the significance level is exceeded (0.05 in general) [30]. If the probability value falls below 0.05, it is considered that the test sample does not conform to normal distribution. If the value reaches above 0.05, the test sample is considered to conform to normal distribution.Table4 lists the results of normality test conducted on the inclination parallel observation lines. It can be seen clearly from the table that there are 12 observation lines with a greater p-value than 0.05. That is to say, the subsidence of 12 inclination direction observation lines conforms to normal distribution. Only the p-value of the Line A13-G13 is less than 0.05, which suggests that the subsidence of this observation line does not follow normal distribution.Table 4
Results of the K–S normality test of each line in the inclination direction.
Observation linesNNormal parameterStatisticpMeanStandard deviationLine A1–G17−712.429522.1160.5080.959Line A2–G27−705.973524.5060.4810.975Line A3–G37−683.287492.8900.5370.936Line A4–G47−676.514492.4840.5480.925Line A5–G57−621.466474.3310.5180.951Line A6–G67−479.050387.6820.4920.969Line A7–G77−272.199213.9940.4280.993Line A8–G87−135.531102.4350.5070.960Line A9–G97−53.82132.2650.4860.972Line A10–G107−32.50915.4860.4580.985Line A11–G117−18.3035.9370.4230.994Line A12–G127−11.7761.9800.5320.940Line A13–G137−12.2704.2960.8470.047Table5 shows the results of normality test on the strike parallel observation lines. Since the p-value of the strike parallel observation lines are significantly higher than 0.05, it is judged that the subsidence distribution of each observation line parallel to the strike of the main section conforms to normal distribution.Table 5
Results of the K–S normality tests of each observation line in the strike direction.
Observation linesNNormal parameterStatisticpMean valueStandard deviationLine I13−626.407564.2440.7670.599Line A13−612.712562.9830.7530.623Line B13−517.287478.1360.7620.607Line C13−336.938314.8550.8000.543Line D13−168.946150.8020.8000.544Line E13−81.68767.4390.6710.759Line F13−36.42627.0400.8420.477There is consistence shown in the conclusions between the normal test of the kurtosis–skewness and the K–S normal test. Among the 20 observation lines tested, only observation line A13–G13, which is the farthest away from the subsidence basin center, does not conform to normal distribution. However, the subsidence of the other strike and inclination parallel observation lines conforms to normal distribution, i.e., the subsidence in the surface of the mining area conforms to normal distribution overall.


## 3.1.1. Kurtosis–Skewness Normality Test

The kurtosis and skewness coefficient values of the data are used to determine whether the samples conform to normal distribution or not. If the absolute values of kurtosis coefficientZK and skewness coefficient ZS fall below 1.96 simultaneously, the distribution of sample data is considered to conform to normal distribution. If either or both the absolute values of ZK or ZS exceed 1.96, the sample data are considered not to conform to normal distribution [29]. The formulas used to calculate ZK and ZS are expressed as follows:(2)ZK=K−0SK,ZS=S−0SS. Where SK and SS represent standard error for kurtosis coefficient and skewness coefficient, respectively.Table2 depicts the calculation results obtained from the normality test on the subsidence kurtosis and the skewness of the observation lines in the inclination direction. It is found out that the absolute kurtosis coefficient and skewness coefficient values are less than 1.96 during 12 observation lines, the sedimentation profiles of which conform to normal distribution. However, the absolute values of kurtosis coefficient and skewness coefficient of the A13-G13 observation line exceed 1.96, which means the subsidence of this observation line does not follow normal distribution.Table 2
Results of the kurtosis–skewness normality test of the observation lines in the inclination direction.
Observation linesNNormal parameterKSKZKSSSSKMean valueStandard deviationLineA1–G17−712.429522.1160.0360.7940.045−2.1711.5871.368LineA2–G27−705.973524.5060.0030.7940.005−2.1391.5871.348Line A3–G37−683.287492.8900.0750.7940.094−2.2161.587−1.396Line A4–G47−676.514492.4840.1010.7940.127−2.2141.587−1.395Line A5–G57−621.466474.3310.0560.7940.071−2.2471.587−1.416Line A6–G67−479.050387.682−0.0730.794−0.092−2.1801.587−1.374Line A7–G77−272.199213.994−0.2110.794−0.266−1.9051.587−1.200Line A8–G87−135.531102.435−0.2820.794−0.355−1.8721.587−1.180Line A9–G97−53.82132.265−0.0030.794−0.004−2.0201.587−1.273Line A10–G107−32.50915.486−0.2860.794−0.360−1.8131.587−1.142Line A11–G117−18.3035.937−0.0040.794−0.005−1.4641.587−0.922Line A12–G127−11.7761.9800.7320.7940.9220.1661.5870.105Line A13–G137−12.2704.296−2.3650.794−2.9795.7671.5873.634Table3 presents the calculation results obtained from the normality test on the subsidence kurtosis and the skewness of the strike parallel observation lines. The absolute values of the kurtosis coefficient and the skewness coefficient of the seven strike parallel observation lines fall below 1.96, which suggests that the subsidence distribution of the strike parallel observation lines conforms to normal distribution.Table 3
Results of the kurtosis–skewness normality test of the observation lines in the strike direction.
Observation linesNNormal parameterKSKZKSSSSKMean valueStandard deviationLine I13−626.407564.244−0.0650.616−0.106−2.0971.191−1.761Line A13−612.712562.983−0.1150.616−0.187−2.0811.191−1.748Line B13−517.287478.136−0.1420.616−0.231−2.1001.191−1.7638Line C13−336.938314.855−0.1900.616−0.308−2.0911.191−1.7568Line D13−168.946150.802−0.2240.616−0.364−1.9651.191−1.6508Line E13−81.68767.439−0.3900.616−0.634−1.6841.191−1.414Line F13−36.42627.040−0.3840.616−0.623−1.7811.191−1.495


## 3.1.2. K–S Normal Test

The rationale of the K–S normality test is described as follows. To calculate the theoretical cumulative probability valueFx of the sample data, it is assumed that normal distribution is followed. Then, the actual cumulative probability value Sx of the sample is calculated. The D-value Dx between them is calculated as follows:(3)D=maxSxi−Fxi.D must be corrected into(4)D=maxmaxSxi−Fxi,maxSxi−1−Fxi−1.SPSS software is directly applicable to calculate the probability value(p-value). As for whether the sample value conforms to the normal distribution, it is determined by whether the significance level is exceeded (0.05 in general) [30]. If the probability value falls below 0.05, it is considered that the test sample does not conform to normal distribution. If the value reaches above 0.05, the test sample is considered to conform to normal distribution.Table4 lists the results of normality test conducted on the inclination parallel observation lines. It can be seen clearly from the table that there are 12 observation lines with a greater p-value than 0.05. That is to say, the subsidence of 12 inclination direction observation lines conforms to normal distribution. Only the p-value of the Line A13-G13 is less than 0.05, which suggests that the subsidence of this observation line does not follow normal distribution.Table 4
Results of the K–S normality test of each line in the inclination direction.
Observation linesNNormal parameterStatisticpMeanStandard deviationLine A1–G17−712.429522.1160.5080.959Line A2–G27−705.973524.5060.4810.975Line A3–G37−683.287492.8900.5370.936Line A4–G47−676.514492.4840.5480.925Line A5–G57−621.466474.3310.5180.951Line A6–G67−479.050387.6820.4920.969Line A7–G77−272.199213.9940.4280.993Line A8–G87−135.531102.4350.5070.960Line A9–G97−53.82132.2650.4860.972Line A10–G107−32.50915.4860.4580.985Line A11–G117−18.3035.9370.4230.994Line A12–G127−11.7761.9800.5320.940Line A13–G137−12.2704.2960.8470.047Table5 shows the results of normality test on the strike parallel observation lines. Since the p-value of the strike parallel observation lines are significantly higher than 0.05, it is judged that the subsidence distribution of each observation line parallel to the strike of the main section conforms to normal distribution.Table 5
Results of the K–S normality tests of each observation line in the strike direction.
Observation linesNNormal parameterStatisticpMean valueStandard deviationLine I13−626.407564.2440.7670.599Line A13−612.712562.9830.7530.623Line B13−517.287478.1360.7620.607Line C13−336.938314.8550.8000.543Line D13−168.946150.8020.8000.544Line E13−81.68767.4390.6710.759Line F13−36.42627.0400.8420.477There is consistence shown in the conclusions between the normal test of the kurtosis–skewness and the K–S normal test. Among the 20 observation lines tested, only observation line A13–G13, which is the farthest away from the subsidence basin center, does not conform to normal distribution. However, the subsidence of the other strike and inclination parallel observation lines conforms to normal distribution, i.e., the subsidence in the surface of the mining area conforms to normal distribution overall.


## 3.2. Calculation Error of the Surface Subsidence at Any Point

Based on the Knothe influence function method [13], Figure 5 is built to show the working face and the calculation coordinate system of the horizontal coal seam, where l represents the calculation length of the trend; L indicates the calculation width of the tendency; and Ax,y denotes any point on the surface.Figure 5
Calculation coordinate system.Since the probable directions ofx and y are independent of each other, the subsidence at Ax,y is expressed as follows:(5)Wx,yA=Wmax∬Ffx,ydF=Wmax∬Ffxfydxdy,fx=1re−πx2r2,fy=1re−πy2r2,where Wmax represents the maximum surface subsidence; r indicates the main mining influence radius; fx, fy denotes the Knothe influence function, which follows a N0,σ2 distribution, and σ=r/2π.O represents the coordinate origin in Figure 5, so that the subsidence at Ax,y is expressed as follows:(6)Wx,yA=Wmaxr2∫−xl−xe−πxr2dx∫−yl−ye−πyr2dy=Wmax×12erfπxr+1−erfπx−lr+1×12erfπyr+1−erfπy−Lr+1=WmaxCxCy,where Cx,Cy refer to the subsidence distribution coefficients of the trend and tendency, respectively.The working face is a rectangular mining area, the mining length along the strike isl=540m , the mining width along the strike is L=136m , and the main influence radius of the mining is r=98m , The coordinate system is established with the lower left corner of the mining area as the coordinate origin as of the coordinate system (Figure 5): The maximum settlement value of the main section is measured to be W0=1318mm.


### 3.2.1. Prediction Error of the Distribution of the Surface Settlement along the Inclined Observation Lines

According to the observation data, the settlement coefficient is determined asq=0.73 , the tangent of the main influence angle is determined as tanβ=2.2 , and the subsidence at any point on the surface is calculated using (6). Figure 6 shows all the predicted and measured values of the inclination parallel observation lines. It can be found out that the predictions of each observation line in the inclined direction deviate clearly from the actual subsidence curve, with similarity shown only around the maximum subsidence values.Figure 6
The measured and predicted distribution of the surface subsidence along the inclination parallel lines.
(a)(b)(c)(d)(e)(f)(g)(h)(i)(j)(k)(l)(m)The Root Mean Square Error (RMSE) [31]and the Percent Relative Error (PRE) [32]are adopted to examine how subsidence is accurately and reliably predicted by using the formula for each observation line at any point in the surface. As a numerical measure of prediction accuracy, RMSE refers to the square root of the sum of the squares of the difference between predicted and measured values and the ratio of the number of observations n. A smaller RMSE value indicates a higher accuracy of prediction [33]. The PRE is advantageous in reflecting the reliability of the predicted formula, with a lower absolute value of PRE indicating a greater reliability.(7)RMSE=±∑ΔΔnPRE=Δ/L′×100%,where Δ is the true error between predicted and measured values and L′ is the measured value.Table6 shows the calculation of the RMSE and PRE between the predicted value and the measured value of the surface subsidence along the inclination parallel observation lines. In general, the RMSE of the predicted values of the subsidence for observation lines is high in the inclined direction. However, according to the PRE, the prediction accuracy of the inclined direction observation lines shows a gradual trend of increasing from the center of the subsidence basin to the edge.Table 6
Predicted errors of each line in the inclined direction.
Observation linesMeasured mean value/mmRMSE of predicted/(mm)PRE%Line A1–G1−712.43380.3443.26Line A2–G2−705.97380.7143.94Line A3–G3−673.48363.1444.00Line A4–G4−676.51333.5438.74Line A5–G5−621.47320.1541.74Line A6–G6−479.05314.9753.71Line A7–G7−272.20335.76117.67Line A8–G8−135.53315.56224.74Line A9–G9−53.82254.37452.33Line A10–G10−32.51149.32436.30Line A11–G11−18.3028.95147.53Line A12–G12−11.7840.40317.56Line A13–G13−12.278.2760.00


### 3.2.2. Prediction Error of the Distribution of the Surface Subsidence along the Strike Parallel Observation Lines

Figure7 shows the comparison between the predicted and measured values of all the strike parallel observation lines. It can be discovered that the predictions of observation lines A and B at the center of the subsidence basin are relatively close to the measured subsidence curves, despite a significant difference in the observation line subsidence curves toward the edge of the basin.Figure 7
The measured and predicted distribution of the surface subsidence along the strike parallel observation lines.
(a)(b)(c)(d)(e)(f)(g)Table7 shows the measured and predicted distribution of the surface subsidence along the strike parallel observation lines as well as the RMSE and PRE. To be specific, RMSE is predicted to be the minimum for the A-line subsidence toward the main section. When shifting away from the center line of the subsidence basin, the RMSE for the other observations increases first and then decreases. However, judging from the PRE, the prediction accuracy of the strike line reaches the highest in the center of the basin and it increases gradually when moving away from the center of the subsidence basin.Table 7
Predicted relative errors of each line in the strike direction.
Observation lineMeasured mean value (mm)RMSE of predicted (mm)PRE%Line A−626.41139.5515.37Line B−612.71153.4617.72Line C−517.29207.2834.58Line D−336.94354.1692.78Line E−168.95436.50219.78Line F−81.69346.23353.46Line G−36.43192.14436.92According to the comparison in subsidence prediction and RMSE between the actual measurement data and the prediction result for each observation line, it can be known that the subsidence prediction formula at any point on the surface has lower RMSE and higher prediction accuracy in the central part of the subsidence basin. In the region which is more distant from the center of the subsidence basin, however, the RMSE of the predicted values is larger and the level of prediction accuracy is lower.


## 3.2.1. Prediction Error of the Distribution of the Surface Settlement along the Inclined Observation Lines

According to the observation data, the settlement coefficient is determined asq=0.73 , the tangent of the main influence angle is determined as tanβ=2.2 , and the subsidence at any point on the surface is calculated using (6). Figure 6 shows all the predicted and measured values of the inclination parallel observation lines. It can be found out that the predictions of each observation line in the inclined direction deviate clearly from the actual subsidence curve, with similarity shown only around the maximum subsidence values.Figure 6
The measured and predicted distribution of the surface subsidence along the inclination parallel lines.
(a)(b)(c)(d)(e)(f)(g)(h)(i)(j)(k)(l)(m)The Root Mean Square Error (RMSE) [31]and the Percent Relative Error (PRE) [32]are adopted to examine how subsidence is accurately and reliably predicted by using the formula for each observation line at any point in the surface. As a numerical measure of prediction accuracy, RMSE refers to the square root of the sum of the squares of the difference between predicted and measured values and the ratio of the number of observations n. A smaller RMSE value indicates a higher accuracy of prediction [33]. The PRE is advantageous in reflecting the reliability of the predicted formula, with a lower absolute value of PRE indicating a greater reliability.(7)RMSE=±∑ΔΔnPRE=Δ/L′×100%,where Δ is the true error between predicted and measured values and L′ is the measured value.Table6 shows the calculation of the RMSE and PRE between the predicted value and the measured value of the surface subsidence along the inclination parallel observation lines. In general, the RMSE of the predicted values of the subsidence for observation lines is high in the inclined direction. However, according to the PRE, the prediction accuracy of the inclined direction observation lines shows a gradual trend of increasing from the center of the subsidence basin to the edge.Table 6
Predicted errors of each line in the inclined direction.
Observation linesMeasured mean value/mmRMSE of predicted/(mm)PRE%Line A1–G1−712.43380.3443.26Line A2–G2−705.97380.7143.94Line A3–G3−673.48363.1444.00Line A4–G4−676.51333.5438.74Line A5–G5−621.47320.1541.74Line A6–G6−479.05314.9753.71Line A7–G7−272.20335.76117.67Line A8–G8−135.53315.56224.74Line A9–G9−53.82254.37452.33Line A10–G10−32.51149.32436.30Line A11–G11−18.3028.95147.53Line A12–G12−11.7840.40317.56Line A13–G13−12.278.2760.00


## 3.2.2. Prediction Error of the Distribution of the Surface Subsidence along the Strike Parallel Observation Lines

Figure7 shows the comparison between the predicted and measured values of all the strike parallel observation lines. It can be discovered that the predictions of observation lines A and B at the center of the subsidence basin are relatively close to the measured subsidence curves, despite a significant difference in the observation line subsidence curves toward the edge of the basin.Figure 7
The measured and predicted distribution of the surface subsidence along the strike parallel observation lines.
(a)(b)(c)(d)(e)(f)(g)Table7 shows the measured and predicted distribution of the surface subsidence along the strike parallel observation lines as well as the RMSE and PRE. To be specific, RMSE is predicted to be the minimum for the A-line subsidence toward the main section. When shifting away from the center line of the subsidence basin, the RMSE for the other observations increases first and then decreases. However, judging from the PRE, the prediction accuracy of the strike line reaches the highest in the center of the basin and it increases gradually when moving away from the center of the subsidence basin.Table 7
Predicted relative errors of each line in the strike direction.
Observation lineMeasured mean value (mm)RMSE of predicted (mm)PRE%Line A−626.41139.5515.37Line B−612.71153.4617.72Line C−517.29207.2834.58Line D−336.94354.1692.78Line E−168.95436.50219.78Line F−81.69346.23353.46Line G−36.43192.14436.92According to the comparison in subsidence prediction and RMSE between the actual measurement data and the prediction result for each observation line, it can be known that the subsidence prediction formula at any point on the surface has lower RMSE and higher prediction accuracy in the central part of the subsidence basin. In the region which is more distant from the center of the subsidence basin, however, the RMSE of the predicted values is larger and the level of prediction accuracy is lower.


## 4. Discussion and Conclusion

### 4.1. Discussion

At present, most of the research focuses on linear observation stations, analyzing only the law of subsidence for the main section of the subsidence basin. Consequently, there is little research on the law of subsidence in other areas of the subsidence basin. In this study, the historical subsidence monitoring data of the network observation station is reanalyzed. According to the research results, the subsidence in the central area of the subsidence basin conforms to normal distribution law, not that on the edge of the subsidence basin. Such a deviation is related to the exact ground structure. The mining depth of the working face in the study area is 132 m, but the thickness of the Quaternary loose layer in the overlying strata is 71 m, which is relatively large. To some extent, this will contribute to the occurrence of surface subsidence [34–36].As a highly complex mechanical and spatiotemporal process, the surface subsidence caused by mining is attributed to many factors, which in turn affects the distribution law of subsidence. According to the superposition principle of mining influence, mining surface subsidence can be regarded as the superposition of multiple independent unit mining, with each unit mining having limited impact on the overall subsidence. Therefore, the stratum subsidence caused by underground coal seam mining basically conforms to normal distribution. Therefore, it is universally reasonable to apply the normal distribution function as the mining surface subsidence influence function [37, 38].Due to the exact geological conditions of mining, which affects how surface subsidence develops, the surface subsidence may not fully conform to normal distribution. Therefore, it is necessary to evaluate the geological mining conditions in the first place when the probability integral method based on the normal distribution is adopted. Under exceptional geological conditions, such as the stratum that contains thick topsoil or thick and hard, it is necessary to modify the model parameters of the probability integral method, or introduce other distribution functions and probability integral method into the construction of a combined model for the improvement of prediction accuracy.


## 4.1. Discussion

At present, most of the research focuses on linear observation stations, analyzing only the law of subsidence for the main section of the subsidence basin. Consequently, there is little research on the law of subsidence in other areas of the subsidence basin. In this study, the historical subsidence monitoring data of the network observation station is reanalyzed. According to the research results, the subsidence in the central area of the subsidence basin conforms to normal distribution law, not that on the edge of the subsidence basin. Such a deviation is related to the exact ground structure. The mining depth of the working face in the study area is 132 m, but the thickness of the Quaternary loose layer in the overlying strata is 71 m, which is relatively large. To some extent, this will contribute to the occurrence of surface subsidence [34–36].As a highly complex mechanical and spatiotemporal process, the surface subsidence caused by mining is attributed to many factors, which in turn affects the distribution law of subsidence. According to the superposition principle of mining influence, mining surface subsidence can be regarded as the superposition of multiple independent unit mining, with each unit mining having limited impact on the overall subsidence. Therefore, the stratum subsidence caused by underground coal seam mining basically conforms to normal distribution. Therefore, it is universally reasonable to apply the normal distribution function as the mining surface subsidence influence function [37, 38].Due to the exact geological conditions of mining, which affects how surface subsidence develops, the surface subsidence may not fully conform to normal distribution. Therefore, it is necessary to evaluate the geological mining conditions in the first place when the probability integral method based on the normal distribution is adopted. Under exceptional geological conditions, such as the stratum that contains thick topsoil or thick and hard, it is necessary to modify the model parameters of the probability integral method, or introduce other distribution functions and probability integral method into the construction of a combined model for the improvement of prediction accuracy.


## 5. Conclusion

In this paper, the field measurement data of the network observation station are analyzed through statistical methods, so as to explore the law of subsidence in the whole subsidence basin. Furthermore, the following conclusions are drawn.(1)
By conducting Kurtosis–Skewness normality test and K–S normality test, the subsidence data of each observation line of network observation are analyzed. On this basis, it is concluded that the surface subsidence caused by coal mining usually conforms to normal distribution, rather than the subsidence of observation line A13-G13 at the edge of subsidence basin. Besides, the subsidence of all other observation lines conforms to normal distribution.(2)
The prediction equation intended for the calculation of surface settlement at any point is used to predict the settlement along the observation lines of the observation station network. According to the calculation results, the relative error is the least significant for the equation used to predict the surface settlement at any point through the probability integral method. In the strike direction, the PRE is 15.37% for the settlement prediction of the observation line a, and the PRE reaches the maximum of 436.92% for the observation linef. As the observation line shifts away from the center of the sinking basin, the PRE increases progressively. In the tilt direction, there is a small PRE of the four observation lines A1–G1 to A5–G5 close to the center of the basin, reaching about 40%, which indicates high reliability. Then, as the observation line shifts away from the center of the basin, it gradually increases, with the PRE of the observation line A9–G9 reaching a maximum of 452.33%. Judging from the change of PRE in the comprehensive strike and tilt direction, PRE gradually increases from the center of the basin to the edge of the basin. That is to say, the reliability of probability integral method in predicting the subsidence at any point on the surface diminishes gradually from the center of the basin to the edge of the basin.(3)
When the surface subsidence prediction method is adopted that is based on normal distribution as the mining influence function, it is necessary to take into account the particularity of geological mining conditions and the possibility of prediction deviation. Due to the limited observation data involved, it is necessary to further explore the law of surface subsidence under extraordinary geological mining conditions.



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*Source: 2201996-2022-09-05.xml*