Study on the Determination Method of Cast Blasting Stockpile Forms in an Open-Pit Mine

Abstract

Cast blasting–dragline stripping technology is the most advanced mining technology used in open-pit mines. For a long time, however, its precision has been hindered. In this paper, we aim to improve the precision of cast blasting–dragline stripping technology and promote its intelligent design. We present a method to determine cast blasting stockpile forms. First, the 3D point cloud data for the Heidaigou open-pit mine from recent years were collected and counted, and a 3D mathematical model of overcasting stripping steps was constructed. Then, data classification and multivariate statistical analysis were used to establish a cast blasting stockpile characteristic parameter database. Next, locally weighted linear regression was used as the fitting method to achieve shape fitting under different cast blasting step heights. Finally, interval estimation was used as the fitting result test method to verify the morphology of the acquired cast blasting stockpile form. The research results show that the cast blasting stockpile form obtained by fitting can truly reflect the cast blasting effect of the Heidaigou open-pit mine and ensure the reliability and accuracy of the subsequent design of cast blasting–dragline stripping technology.

1. Introduction

As the most mainstream mining technology used in open-pit mines worldwide, cast blasting–dragline stripping technology (hereinafter referred to as the overcasting stripping system) is favored in open-pit mines because of its characteristics of environmental protection, economic efficiency, etc. As a cutting-edge engineering problem, the integrated design of the overcasting stripping system has long been the focus of attention in open-pit mines, especially the determination of the form of a cast blasting stockpile. On the one hand, the cast blasting stockpile form determines the effective throw rate and affects the operation efficiency and production benefit of dragline stripping technology. On the other hand, the cast blasting stockpile form plays a decisive role in the dragline stripping step parameter design, equipment operation area division, field personnel and equipment allocation, and other production organization management issues in dragline stripping technology. Therefore, the determination of the cast blasting stockpile plays a crucial role in the integrated design of the whole overcasting stripping system.

Many researchers have completed a significant amount of scientific research on the prediction of cast blasting stockpile characteristics, including Yang, R.L. et al. [1,2,3], who proposed a 3D motion model to simulate production blasting. Han Liang [4], Xianglong Li [5,6,7], and other researchers simulated the measured blasting pile by introducing the Weibull mathematical model and combining finite group blasting data. Based on ballistic theory and the basic theory of open-pit mines, Zhou Wei [8] and Ma Li [9,10] effectively predicted the effective throwing rate of cast blasting in open-pit mines by scanning the cast blasting stockpile form before and after cast blasting. Feng Chun [11] used the continuous–discontinuous element method (CDEM) to simulate the whole process of cast blasting in an open-pit mine. Weimin Hu, Xiaohua Ding [12,13], and other researchers used general nonlinear theoretical function analysis to gain pertinent information about cast blasting and predict the stockpile form. Weixian Wang, Jiandong Sun, and other researchers [14,15,16,17,18,19,20,21] carried out relevant studies on the prediction and measurement of the cast blasting shape.

However, a previous study found that determination methods for the cast blasting stockpile form and the vast majority of research on blasting data are limited, and there are more rules and special data. By using linear regression and a neural network algorithm for data analysis and processing, it was found that the prediction results and the real blasting effect in open-pit mines have a deviation that can only represent the local blasting effect. However, when considering cast blasting as a large, complex system, it is understandable that the blasting effect is affected by many factors. On the one hand, geological conditions influence the blasting effect, and these factors include the rock properties, fault depression belt, goaf, etc. On the other hand, blasting parameter design also influences the blasting effect; blasting parameter design includes the steps, hole–net parameters, charging structure, the performance of the explosive, and the explosive quantity. If we only rely on a few special blasting parameters’ data as the research object, we can only predict the blasting result for a local area and cannot restore the whole cast blasting stockpile form. The results only reflect the local blasting effect of the blasting parameters for blasting design, so the subsequent overcasting stripping system is affected by high-precision design, further increasing the difficulty of the open-pit mine production and construction.

Therefore, taking the Heidaigou open-pit mine as a case study, a method to determine the characteristics of the cast blasting stockpile is proposed. This method is different from conventional research on the form prediction of cast blasting stockpiles. Based on a large set of on-site blasting data of the Heidaigou open-pit mine, it avoids the uncertainty of geological conditions, such as the probability of the cast blasting system and rocks’ mechanical properties, which may lead to deviation in the prediction effect and affect the overall design of the overcasting stripping system. Therefore, the 3D point cloud data of the Heidaigou open-pit mine from recent years were collected and counted, and a 3D mathematical model of overcasting stripping steps was constructed. Then, data classification and multivariate statistical analysis were used to establish a cast blasting stockpile characteristic parameter database. Next, local weighted linear regression was used as the fitting method to achieve shape fitting under different cast blasting step heights. Finally, interval estimation was used as the fitting result test method to verify the morphology of the acquired cast blasting stockpile form, ensuring the reliability and accuracy of the subsequent design of cast blasting–dragline stripping technology. The research results lay a foundation for popularizing cast blasting technology.

2. Methods and Steps

2.1. Project Description

Heidaigou open-pit mine is the only open-pit mine in China that adopts cast blasting–dragline stripping technology. After 14 years of technology introduction, absorption, digestion, and innovation, Heidaigou open-pit mine has rich on-site practical experience in applying this technology. At present, the stope has advanced to the second mining area (as shown in Figure 1). With the continuous downward and forward advance of the stope, the coal seam rises and falls steeply, the rock characteristics are changeable and difficult to control, and the parameters of the cast blasting step are complex and changeable, which leads to uncertainty of the form of cast blasting. However, cast blasting stockpile form and the effective rate of blasting, dragline operation efficiency, and the overcasting stripping system are closely related to the overall production cost calculation, the subsequent dragline stripping steps’ parameter design and equipment operation, and the production organization of the personnel and equipment allocation, which makes determining the influence on the overcasting stripping system integration design incredibly difficult. Due to this, the Heidaigou open-pit mine often features a coal mining link and stripping link that cannot be synchronized, production organization and management of the personnel and equipment allocation that cannot be guaranteed in time, and a coal seam reveal speed that is always advanced or delayed (and thus causes the overall development of a seriously disturbed stope), all of which affect the open-pit mine’s production schedule and plan.

2.2. Data Acquisition and Analysis

2.2.1. Data Collection

The strike length of cast blasting steps in Heidaigou open-pit mine is about 1800 m, the width of the steps is about 85 m, and the height of the steps is 38 m to 50 m. The cast blasting steps are divided into 4 areas, including perforation area, blasting area, overcasting area, and mining area, and the length of the single area is 450 m to 550 m.

A Quarryman high-precision laser scanner (equipment parameters are shown in

Table 1. Quarryman high-precision scanner equipment parameters.

Project	Parameter	Describe
The data collection	—	360° scan and panoramic photo collection
Real time together	—	Based on visual tracking technology of automatic point cloud splicing
The laser wavelength	1550 nm	No visible light
The scanning angle	0° to 360°	Horizontal angle
	−45° to 80°	The vertical angle
The scanning distance	0.5 m to 700 m	—
The scanning speed	7000 p/h	—
The transmitter resolution	1 cm	Precision can be up to 10 cm
Scanning angular resolution	0.01°	Accurate to 0.02°
Image acquisition	432 million pixels	Panoramic scope
The sensor	—	IMU obliquity sensor, altimeter, GNSS
Wireless communications	802.11 b/g/n	Integrated wireless communication module of WLAN
Data storage	256GB, USB3.0	External storage
Length of equipment	209 mm × 243 mm × 419 mm	Length, width, and height
Equipment weight	10.7 kg	After adding leveling equipment, weighing 11.2 kg
Appropriate temperature	−10 °C to 45 °C	Working temperature

) was used in this study for the selection of the data acquisition location (as shown in Figure 2). Positions 1 and 6 mainly assist in scanning goaf and cast blasting stockpile; positions 3 and 4 mainly scan high steps adjacent to goaf and cast blasting stockpile; and positions 2 and 5 mainly scan goaf and cast blasting stockpile to obtain 3D point cloud data for each scanned area.

2.2.2. Construction of 3D Model of Cast Blasting Stockpile

After being collected by the Quarryman scanner, the point cloud data for each area of high steps, goaf, and cast blasting stockpile are first imported into blasting digital comprehensive processing system [22], and then the data are automatically classified by the system. Then, the data are output in the form of a 3D model (as shown in Figure 3). The 3D model data for the cast blasting stockpile include key parameters such as blasting pile slope surface, coal seam top board surface, goaf, and high step slope surface.

In the table: “_spline” is a kind of computer language used in computer automation fixed-point drawing.

To ensure that reconstruction after cast blasting stockpile profiling is consistent with the real cast blasting stockpile profile, first, the cast blasting stockpile profile is removed according to the integration of the relationship of space and time and position of cast blasting; then, automatic proofreading occurs for each group of cast blasting stockpile profile coordinates. Next, proofreading occurs for the cast blasting stockpile slope top line and bottom line, goaf, and high step slope line point cloud data, such as the implementation code (as shown in

Table 2. Partial point cloud data for cast blasting stockpile.

Position		(x,y)	Position		(x,y)	Position		(x,y)
Cast blasting stockpile		_spline	Cast blasting stockpile		_spline	Cast blasting stockpile		_spline
0	1092.8	0, 1092.8	0	1092.8	0, 1092.8	0	1092.8	0, 1092.8
……	……	……	……	……	……	……	……	……
Slope crest		_spline	Slope crest		_spline	Slope crest		_spline
0	1111.1	0, 1111.1	0	1111.5	0, 1111.5	0	1112.0	0, 1112.0
……	……	……	……	……	……	……	……	……
Goaf		_spline	Goaf		_spline	Goaf		_spline
120	1079.1	1, 201,079	120	1079.1	1, 201,079	120	1079.0	1, 201,079
……	……	……	……	……	……	……	……	……
240	1068.7	2, 401,068	240	1068.7	2, 401,068	240	1068.7	2, 401,068.7
Coal roof		_spline	Coal roof		_spline	Coal roof		_spline
0	1080	0, 1080	0	1080	0, 1080	0	1080	0, 1080
240	1080	2, 401,080	240	1080	2, 401,080	240	1080	2, 401,080

). The information for the explosion stack with code is imported into AutoCAD and then transformed into a cast blasting stockpile profile that is easy to edit, operate, and calculate.

2.2.3. Sample Analysis Based on Cast Blasting Stockpile Form

The cast blasting of Heidaigou open-pit mine is usually carried out at 5:00 p.m. on the day of blasting.

Considering the harm caused by toxic and harmful gases in the air after blasting, the scanning work of cast blasting stockpile form is generally carried out on the next day, during which some areas of cast blasting stockpile form have been in production. As a result, the partial area of the cast blasting stockpile form has been leveled by auxiliary equipment such as bulldozers (as shown in Figure 4), and the scanning operation lags behind. Therefore, there are errors in some data regarding the characteristics of the cast blasting stockpile form, which cannot truly restore the morphological characteristics of the cast blasting stockpile. Therefore, it is necessary to process the acquired sample data of the cast blasting stockpile form.

The cast blasting and the following dragline operation are the two working production procedures of the overcasting stripping system; after the completion of cast blasting, the dragline operation begins to work, and they are closely connected. To ensure that the real form of the cast blasting stockpile form can be obtained, it is necessary to closely combine cast blasting with the subsequent production process of dragline stripping, carry out research and calculations on the task allocation of each area of the cast blasting stockpile, and then screen and classify the reconstructed cast blasting stockpile form samples.

2.3. Research Ideas and Steps

Through field data acquired for the real cast blasting stockpile form, in this study, to simplify the difficulty of feature research, the 3D features were reduced to 2D; namely, using the cast blasting stockpile form section to create a cast blasting stockpile profile. Then, to obtain different bench heights for the precision of typical characteristics, we need to obtain the cast blasting stockpile profile. An analysis and filtering method based on the overcasting stripping system of an internal business process system was the breakthrough point through which we reached an analysis of the overcasting stripping system business relationship and the business mission. Then, a comparative analysis was used to determine the volume of the cast blasting stockpile form, and the overcasting stripping system of the internal business process system was used to filter forms that were not in conformity with the samples of the open-pit mine site. Finally, the selected sample data were used to fit the cast blasting stockpile form. We obtained a confidence interval of more than 95% of the typical profile of cast blasting stockpile form, which provided a decision-making basis for the overcasting stripping system of high-precision design.

Therefore, to obtain an overcasting stripping system for internal business processes, we also deconstructed an overcasting stripping system for the internal business relationship and logical relationship. This study is presented based on an overcasting stripping system of the digital model of the internal business relationship; the core of the research train of thought and main steps are as follows:

Step 1: Deconstruction of the core elements: Building a digital model, deconstructing the cast blasting stockpile profile of various parts based on the cast blasting stockpile form for the division of radiation, upper delamination of the cast blasting stockpile form, lower delamination of cast blasting stockpile form, coal ditch, and effective throw area;

Step 2: Deconstruction of the overcasting stripping system of internal business elements: Detailed analysis of overcasting stripping system elements, access to overcasting stripping system business relationships, each business process determined before overcasting stripping system; namely, the total quantity of the cast blasting stockpile, the total quantity of the upper delamination of cast blasting stockpile, the total quantity of lower delamination of cast blasting stockpile, the total quantity of coal ditch, and the total quantity of lower delamination of the effective throw area;

Step 3: Sample data analysis and selection: According to the actual operation in open-pit mine geological conditions, such as the dip angle of the coal seam, the effective rate of casting, etc., based on the theory of ballistics, conduct a preliminary screening of the data sample. Obtain effective casting rate, total quantity of upper delamination of cast blasting stockpile, and total quantity of coal ditch for the cast blasting stockpile form’s basic indexes such as business and various business data index;

Step 4: Characteristic fitting of cast blasting stockpile: According to the basic indicators, the portfolio can reach typical configuration and eventually obtain the sample database of profile data. Then, the profile can make use of locally weighted linear regression fitting and obtain the typical morphology of the cast blasting stockpile profile under different bench heights;

Step 5: Inspection of cast blasting stockpile: Adopting the method of interval estimation for validation with confidence and confidence intervals of the data fitting.

2.4. Construction of Mathematical Model for Cast Blasting Stockpile

In this paper, a plane mathematical model of the parameters of the cast blasting stockpile profile based on the morphological characteristics of the cast blasting stockpile form is constructed. Additionally, the plane coordinate system is constructed by taking the cast blasting stockpile profile as the research object. It is assumed that y = y_a (x) represents the curve equation of the cast blasting stockpile profile, and y = y_b (x) represents the characteristic equation of broken line ABCDEF [23]. The position of each area in the cast blasting stockpile profile is shown in Figure 5 below.

(1) Sectional area of cast blasting stockpile profile is AMFEDCBA:

$$S_{Bu}={\displaystyle {\mathsf{\int }}_o^f\left[y_a(x)-y_b(x)\right]}dx$$

(1)

$$={\displaystyle {\mathsf{\int }}_o^f{y}_a(x)}dx-\left[\frac{H_r{}^2}{2\tan \alpha }+\frac{1}{2}(\frac{2H_r}{\tan \alpha }+\frac{H_m}{\tan \beta })H_m+H_{m}b+\frac{1}{2}(x_F-x_E)\tan r\right]$$

(2)

$$x_E=\frac{H_r\tan \beta +H_m\tan \alpha }{\tan \alpha \tan \beta }+2b x_D=\frac{H_r\tan \beta +H_m\tan \alpha }{\tan \alpha \tan \beta }+b$$

(3)

where S_Bu is the sectional area of cast blasting stockpile profile, m²; α represents the slope angle of the rock step, °; B represents the width of the cast blasting step, m; β represents the slope angle of the coal seam step, °; H_m is the height of the rock step, m; H_m is the height of the coal seam step, m; R represents the angle of rest about materials; and x_E and x_D represent the abscissae of E and D, respectively.

(2) The sectional area of effective throwing amount of cast blasting DMFMD:

If the correlation equation of line EF is y_EF = tan γ(x − x_E), then y_a(x) = tan γ(x − x_E) can find x_F. Similarly, if the correlation equation of the line EM is y_DH = tan γ(x − x_D), then y_a(x) = tan γ(x − x_E) can be calculated as x_M. The sectional area of effective throwing is:

$$S_{Ef}={\displaystyle {\mathsf{\int }}_f^m{y}_{DM}(x)}dx+{\displaystyle {\mathsf{\int }}_m^f\left[y_a(x)-y_b(x)\right]}dx$$

(4)

$$={\displaystyle {\mathsf{\int }}_m^f{y}_a(x)}dx+\frac{1}{2}\tan \gamma \left[{(x_M-x_D)}^2-{(x_F-x_E)}^2\right]$$

(5)

The effective throwing rate is:

$$k_{Ef}=\raisebox{1ex}{$S_{{}_{Ef}}$}\!\left/ \!\raisebox{-1ex}{$S_{{}_{Bu}}$}\right.$$

(6)

(3) The sectional area of layers on the cast blasting stockpile is ANPA:

$$S_{Up}={\displaystyle {\mathsf{\int }}_0^p\left[y_a(x)-(H_r+H_m)\right]}dx-\frac{{(H_r-H_s)}^2}{2\tan \alpha }$$

(7)

where Hs represents the height of the step of the dragline, m.

(4) The sectional area of secondary dragline stripping is PMJKP:

$$S_{Se}={\displaystyle {\mathsf{\int }}_p^k{y}_{KP}(x)}dx+{\displaystyle {\int }_k^j{y}_{KJ}(x)}dx-{\displaystyle {\mathsf{\int }}_p^m{y}_{PM}(x)}dx-{\displaystyle {\mathsf{\int }}_m^j{y}_{JM}(x)}dx$$

(8)

(5) The sectional area of layers under the cast blasting stockpile is NBRYN:

Assuming that the abscissa of point R at the bottom of the cast blasting stockpile is x_R and the angle of rest R of the material is the angle of inclination RY, the abscissa of point Y is x_R + H_stanr.

$$S_{Do}={\displaystyle {\mathsf{\int }}_n^y{H}_s}dx-\frac{H_s{}^2}{2\tan \alpha }-\frac{H_s{}^2}{2\tan r}$$

(9)

(6) The sectional area of the coal ditch is YRCDMY:

The abscissa of point P on the dragline working face is x_P; thus:

$$S_{Di}={\displaystyle {\mathsf{\int }}_y^p{H}_s}dx+{\displaystyle {\mathsf{\int }}_p^m{y}_a(x)}dx-{\displaystyle {\mathsf{\int }}_r^m{y}_b(x)}dx$$

$$={\displaystyle {\mathsf{\int }}_m^p\left[y_a(x)-y_b(x)\right]}dx+\frac{1}{2}(x_P-x_Y)(1+H_s\tan r)H_s-\frac{1}{2}{(x_M-x_D)}^2\tan r$$

(10)

The dragline stripping working face parameter model is based on building. The height of the cast blasting step, thickness of the coal seam, and dip angle of the coal seam are taken as classification indexes. The dragline of the Heidaigou open-pit mine step height of 13 m is the initial condition, and we can calculate the cast blasting stockpile form section samples of dragline stripping effective quantity, dragline stripping secondary quantity, and single-bucket truck auxiliary work as the key indicators. Some of these databases are shown in

Table 3. Partial database of critical operation index of cast blasting stockpile.

Serial Number	Average Height/m	Coal Seam Height/m	Coal Seam Dip/°	Effective Throwing Rate	Upper Layered Material Quantity/m2	Material Quantity of Coal Ditch/m2	Effective Throwing Quantity/m2	Dragline Stripping Secondary Quantity/m2
1	36.3	31.1	1.1	0.232	1301.2	202.3	735.2	994.7
2	36.2	31.2	1.2	0.231	1346.3	227.3	768.8	966.5
3	36.1	31.4	1.1	0.272	1369.5	254.6	963.7	981.3
4	37.2	31.1	0.2	0.321	1368.5	259.9	1158.4	972.5
5	37.5	30.7	0.1	0.312	1286.8	320.9	1216.2	899.2
6	37.6	31.5	0.3	0.344	1247.1	375.2	1387.6	942.9
7	37.2	31.3	0.2	0.352	1237.4	513.3	1508.9	911.7
8	37.7	31.2	0.1	0.367	1296.9	448.5	1575.4	877.6
9	38.5	31.4	0.1	0.355	1308.8	478.2	1531.2	858.8
10	38.2	30.2	0.2	0.323	1378.3	433.1	1409.4	911.3

(in Figure 6, the red area is completed by single-bucket truck operation, and the blue area is completed by the dragline operation).

2.5. Data Processing Methods

In this section, we consider the probability, uncertainty, and difference in the section information of cast blasting. In this paper, data processing principles and methods based on ballistics, such as projectile blasting theory, multivariate statistical analysis, and interval estimation, are used to improve the accuracy of projectile blasting samples.

(1)

Cast blasting stockpile data sample processing based on coal seam dip characteristic parameters

Based on the theory of ballistics [24,25], the cast blasting trajectory study found that the deposit in the blasting and center of mass of the location and the initial velocity are constant when the horizontal ground beta is increased and the landing position projectile reaches the B side near the high steps; namely, from point B to point B’ (as shown in Figure 7). Conversely, with the increase in the ground gamma horizontal angle, the farther away point B of the landing position of the projectile is from the side of the high step; that is, point B moves to point B”. Moreover, when β increases, it is not conducive to rock-throwing. When γ increases, it is beneficial to rock-throwing.

The characteristic parameters of coal seam dip angle were used as indicators to screen the data of the explosion heap shape sample, and the frequency diagram of the cast blasting stockpile form parameter sample with the change in coal seam dip angle was obtained (as shown in Figure 8).

Results: it was found that the dip angle of the coal seam was mainly distributed between 0° and 3°, and the dip angle variation was small. After comprehensively considering the geological conditions and the impact of the dip angle of the coal seam on the throwing effect, all the exploded pile samples were retained.

(2)

Sample processing of cast blasting stockpile data based on the height of blasting steps

The research results for an open-pit mine company in South Africa show that [26] whether the optimal throwing effect is achieved largely depends on the ratio ε between step height H and step width B, and the effective throwing rate γ is positively correlated with ε. After entering the second mining area, the width of each cast blasting step basically remains the same, and the width of the cast blasting step floats at 85 m. Therefore, this paper only focuses on the step height as the feature object. According to the statistics on the step height of cast blasting (as shown in Figure 9), the step height is mainly distributed in the range 33 m to 48 m. To avoid erroneous research results due to the difference in samples with different step heights, we divided the step heights into three characteristic indexes: 35 m (35 ± 2.5 m), 40 m (40 ± 2.5 m), and 45 m (45 ± 2 m).

(3)

Sample processing of cast blasting stockpile data based on operation parameters of dragline stripping process system

The effective throwing rate, the amount of layered material on the cast blasting stockpile, and the amount of material in the coal ditch are the basic indexes to describe the shape of the cast blasting stockpile form. The accuracy of the index data plays a decisive role in the fitting result of the shape of the cast blasting stockpile form. Therefore, we selected cast blasting step heights of 35 m, 40 m, and 45 m as the research objects, and the effective distribution law was statistically analyzed.

Statistical analysis of 35 m step height samples: By analyzing and processing the sample data under the condition of a 35 m step height for cast blasting, the overall data distribution of the three basic indicators was based on the effective throwing rate, and the material quantity of the upper layer and material quantity of the coal trench were obtained (as shown in Figure 10).

It can be seen from Figure 10 that the sample data of the morphology characteristics of the blast heap under the condition of 35 m height of the cast blasting step basically follow a normal distribution, and the specific parameters of the standard deviation σ and mean μ of the above three basic indicators were obtained through calculation when the sample n = 113 (as shown in

Table 4. Three basic indexes under the condition of 35 m blasting step height.

Project	Effective Throwing Rate	Upper Layer Material Quantity/m2	Coal Ditch Material Quantity/m2
Average (μ)	0.354	1118.2	423.3
Standard deviation (σ)	0.044	215.837	161.927

).

Statistical analysis of 40 m step height samples: Similarly, the sample data with a height of 40 m for cast blasting were analyzed and processed to obtain the overall data distribution of the three basic index parameters (as shown in Figure 11).

It can be seen from Figure 11 that the sample data of the morphology characteristics of the blast heap under the condition of 40 m height of the cast blasting step basically follow a normal distribution. The specific parameters of the standard deviation σ and mean μ of the above three basic indicators were obtained through calculation when the sample n = 120 (as shown in

Table 5. Three basic indexes under the condition of 40 m blasting step height.

Project	Effective Throwing Rate	Upper Layer Material Quantity/m2	Coal Ditch Material Quantity/m2
Average (μ)	0.333	1408.1	488.1
Standard deviation (σ)	0.033	217.6	107.5

).

Statistical analysis of 45 m step height samples: Similarly, the sample data with a height of 40 m for cast blasting were analyzed and processed to obtain the overall data distribution of the three basic index parameters (as shown in Figure 12).

It can be seen from Figure 12 that the sample data of the morphology characteristics of the blast heap under the condition of 40 m height of the cast blasting step basically follow a normal distribution. The specific parameters of the standard deviation σ and mean μ of the above three basic indicators were obtained through calculation when the sample n = 38 (as shown in

Table 6. Three basic indexes under the condition of 45 m blasting step height.

Project	Effective Throwing Rate	Upper Layer Material Quantity/m2	Coal Ditch Material Quantity/m2
Average (μ)	0.346	1420.8	566.8
Standard deviation (σ)	0.042	190.7	100.5

).

Based on the scientific analysis and rigorous screening of the sample database of the cast blasting stockpile form, the characteristic data of the cast blasting stockpile that can restore the throwing effect were obtained. The screened partial sample data of the cast blasting stockpile are shown in

Table 7. Sample database of final cast blasting stockpile profile data.

Position	1	2	3	4	5	6	7	···
0	1095.0	1094.9	1097.2	1106.8	1094.3	1094.0	1095.4	···
10	1094.5	1093.2	1087.6	1106.8	1094.3	1094.0	1095.4	···
20	1094.1	1092.9	1086.6	1106.3	1094.3	1094.5	1095.4	···
30	1094.0	1093.6	1088.3	1105.8	1098.1	1098.0	1098.	···
···	···	···	···	···	···	···	···	···
180	1047.6	1047.6	1049.9	1063.1	1060.9	1061.5	1062.1	···
190	1047.6	1047.7	1051.0	1063.7	1060.5	1062.7	1064.4	···
200	1047.7	1047.9	1050.4	1063.0	1059.3	1061.9	1063.5	···

below.

2.6. Data Fitting Method

To improve the precision of local detail fitting of the cast blasting stockpile, the cast blasting stockpile under different working conditions must be obtained accurately. In this study, the concept of weight was introduced into the traditional regression method. Through the weight distribution of the kernel function, local weighted linear regression was used to fit the cast blasting stockpile profile. Compared with conventional linear fitting, local linear regression gives greater weight to the data points in the local area, and the weight away from the local area tends to be zero. In addition, the close local samples of each data point were calculated as regression coefficients to achieve effective local regression [27,28]. The formula is as follows:

$$J(\theta )={\displaystyle \sum _{}{w}^{\left(i\right)}}{(y^{\left(i\right)}-{\theta }^T{x}^{\left(i\right)})}^2$$

(11)

where J(θ) is the loss function and represents the optimal value of regression in locally weighted linear regression. x⁽ⁱ⁾ is the sample point; θ^Tx⁽ⁱ⁾ is the linear regression of the current x value; and w⁽ⁱ⁾ is the weight function; namely, the kernel function.

The algebraic function, which is weighted by the w function, ensures that the closer the value is to the point being measured, the larger its weight is, and the further away it is from the point to be measured, the smaller its weight is.

$$w^{(i)}=\exp \left(-\frac{{\left(x^{\left(i\right)}-x\right)}^2}{2{\tau }^2}\right)$$

(12)

Here, x is the point being measured/the predicted point and τ is the rate of weight change.

2.7. Inspection Method

To closely fit the actual situation of an open-pit mine, the fitting results of the cast blasting stockpile form need to be further verified with the field blasting effect. Therefore, we chose interval estimation as the test method of fitting results. Interval estimation uses samples to estimate the interval (range) of the global location parameters and gives the confidence level (confidence level) of this interval containing the global location parameters. The solution principle is as follows:

We assume that E is the representative characteristic of the cast blasting stockpile in the general basic index of the cast blasting stockpile form, X₁, X₂…; X_n is a sample extracted from the population to construct two statistics, Ê₁(X₁, X₂, …, X_n) and Ê₂(X₁, X₂, …, X_n); and there is a relation between the statistics Ê₁ < Ê₂. Assuming that there is a parameter α (0 < α < 1), which makes P(Ê₁ < EÊ₂) valid, then the random interval of the sample size of the basic parameters of the cast blasting stockpile form (Ê₁, Ê₂) is the (1 − α) confidence interval of the representative characteristic parameter E of the cast blasting stockpile form. (1 − α) represents the confidence level (confidence) of the confidence interval of the basic index of the cast blasting stockpile form, and the interval length L = Ê₂ − Ê₁ represents the confidence interval accuracy of the basic index of the cast blasting stockpile form.

According to the interval estimation theorem of the normal population mean N (μ, σ₂) [29,30,31], if the sample population standard deviation σ₂—the basic indicator of the cast blasting stockpile form—has been determined, the confidence interval of the sample population mean μ corresponding to the confidence level (1 − α) is:

$$\left({\widehat{E}}_1,{\widehat{E}}_2\right)=\left\{\overline{x}-\frac{\sigma }{\sqrt[2]{n}}{z}_{\raisebox{1ex}{$\alpha $}\!\left/ \!\raisebox{-1ex}{$2$}\right.},\overline{x}+\frac{\sigma }{\sqrt[2]{n}}{z}_{\raisebox{1ex}{$\alpha $}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\right\}$$

(13)

Additionally, the corresponding length of the interval is:

$$L={\widehat{E}}_2-{\widehat{E}}_1=\frac{2\sigma }{\sqrt[2]{n}}{z}_{\raisebox{1ex}{$\alpha $}\!\left/ \!\raisebox{-1ex}{$2$}\right.}$$

(14)

Considering that $\overline{x}$ is an unbiased estimator of μ, $z=\frac{\overline{x}-\mu }{\sigma /\sqrt{n}}~N(0,1)$, and $z=\frac{\overline{x}-\mu }{\sigma /\sqrt{n}}~N(0,1)$, and is independent of all unknown parameters, the α loci can be expressed as follows after analyzing and deducing the corresponding situation of the standard normal distribution:

$$P=\left\{\left|\frac{\overline{x}-\mu }{\sigma /\sqrt{n}}\right|\le z_{\raisebox{1ex}{$\alpha $}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\right\}=1-\alpha$$

(15)

The corresponding confidence level of μ is (1 − α), and the confidence interval is:

$$\left({\widehat{E}}_1,{\widehat{E}}_2\right)=\left\{\overline{x}-\frac{\sigma }{\sqrt{n}}{z}_{\raisebox{1ex}{$\alpha $}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\le \mu \le \overline{x}+\frac{\sigma }{\sqrt{n}}{z}_{\raisebox{1ex}{$\alpha $}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\right\}$$

(16)

3. Result Analysis

After data collection, reconstruction analysis, and strict screening of the cast blasting stockpile profile, we obtained the sample database of cast blasting stockpile profile data (as shown in

Table 8. Confidence intervals of three basic indicators.

Confidence Interval	Effective Throwing Rate/%	Material Quantity of Upper Layer/m2	Material Quantity of Coal Trench/m2
35 m step height	(34.6, 36.2)	(1078.4, 1158.0)	(393.4, 453.2)
40 m step height	(32.7, 33.9)	(1369.2, 1447.1)	(468.8, 507.2)
45 m step height	(33.1, 35.8)	(1360.3, 1481.5)	(534.9, 598.7)

). Then, local weighted linear regression was used to fit the cast blasting stockpile profile, and the cast blasting results were obtained for heights of 35 m, 40 m, and 45 m for the cast blasting stockpile step.

(1)

Standard cast blasting stockpile profile for the 35 m cast blasting step

As shown in Figure 13, the slope of the upper slope (corresponding to the abscissa from 0 to 50) of the cast blasting stockpile profile basically stays at 15°.

The slope of the middle descending slope (corresponding to the abscissa from 50 to 150) is low, basically floating at 45°, while the slope of the lower slope (corresponding to the abscissa from 150 to 200) is gentle. It basically stayed between 5° and 8°.

(2)

Standard cast blasting stockpile profile for the 40 m cast blasting step

As shown in Figure 14, the upper section of the cast blasting stockpile retains the shape of the blasting funnel (corresponding to the abscissa of 0 to 50) and the middle section (corresponding to the abscissa of 50 to 150) has a higher slope. The slope basically stays at 60°, while the slope of the lower section (corresponding to the abscissa of 150 to 200) has a gentle slope. The slope has an upward sloping trend.

(3)

Standard cast blasting stockpile profile for the 45 m cast blasting step

As shown in Figure 15, the upper section of the cast blasting stockpile profile basically retains the shape of the blasting funnel (corresponding to the abscissa of 0 to 50) and the middle section (corresponding to the abscissa of 50 to 150) has a higher slope. The slope is basically maintained at 65° to 70°, and the slope of the lower section (corresponding to the abscissa of 150 to 200) is relatively gentle. The corresponding abscissa has an upward sloping trend from 175 to 200.

4. Discussion

To verify the reliability of the fitting results of the cast blasting stockpile profile, we adopted interval estimation as a verification method and verified the cast blasting stockpile profile under the conditions of 35 m, 40 m, and 45 m cast blasting steps. The effective throwing rate, the material quantity of the upper layer, and the material quantity of the coal trench were taken as verification indexes. The fitting results were verified by testing whether the three satisfied the confidence of the confidence interval. The data error should be reduced as much as possible, so the interval length L should be maximized as much as possible to improve the selection accuracy of the interval of the detonation morphology. In this study, α = 0.05 ($z_{\raisebox{1ex}{$\alpha $}\!\left/ \!\raisebox{-1ex}{$2$}\right.}=z_{0.025}=1.96$ from the table) was selected, and the confidence of the basic index was 95%.

(1)

Statistical analysis of samples with a 35 m step height

By referring to the calculation principle and the process of interval estimation, the confidence interval of the fitting results of the effective throwing rate, material quantity of the upper layer, and the material quantity of the coal trench at a 35 m high cast blasting step were obtained, as shown in Table 8.

According to the fitting results of the cast blasting stockpile profile with the height step of 35 m, the effective throwing rate of the cast blasting stockpile profile is 34.9%, the material quantity of the upper layer is 1098.2 m², and the material volume of the coal trench is 422.8 m², satisfying the above confidence interval. Therefore, the shape-fitting result of the cast blasting stockpile form is basically consistent with the actual cast blasting stockpile form.

(2)

Statistical analysis of samples with a 40 m step height

Similarly, the confidence interval of the fitting results of the effective throwing rate, the material quantity of the upper layer, and the material quantity of the coal trench with a 40 m high cast blasting step were obtained, as shown in Table 8.

According to the fitting results of the cast blasting stockpile profile with the height step of 40 m, the effective throwing rate of the cast blasting stockpile profile is 33.1%, the material quantity of the upper layer is 1408.3 m², and the material volume of the coal trench is 482.8 m², satisfying the above confidence interval. Therefore, the shape-fitting result of the cast blasting stockpile form is basically consistent with the actual cast blasting stockpile form.

(3)

Statistical analysis of samples with a 45 m step height

Similarly, the confidence interval of fitting results of effective throwing rate, the material quantity of the upper layer, and the material quantity of the coal trench with a 45 m high cast blasting step were obtained, as shown in Table 8.

According to the fitting results of the cast blasting stockpile profile with the height step of 45 m, the effective throwing rate of the cast blasting stockpile profile is 33.8%, the material quantity of the upper layer is 1438.7 m², and the material volume of the coal trench is 554.2 m², satisfying the above confidence interval. Therefore, the shape-fitting result of the cast blasting stockpile form is basically consistent with the actual cast blasting stockpile form.

5. Conclusions

Unlike the conventional prediction of the cast blasting stockpile form, we have presented a method to determine the cast blasting stockpile form based on field production practice data. Using research with the Heidaigou open-pit mine as an example, through an analysis of field data acquisition, reconstruction of form, and combining the building of digital model dragline stripping steps, strict screening can reflect the real blasting effect of the sample database and then use the locally weighted regression method for current screening after fitting the sample data to the cast blasting stockpile form. Finally, the interval estimation method was used to verify the cast blasting stockpile form, and the fitting result was basically consistent with the actual cast blasting effect. The results show that the following:

(1) In the Heidaigou open-pit mine, the distribution of the confidence interval of the effective throwing rate is (34.6%, 36.2%), (32.7%, 33.9%), and (33.1%, 35.8%) at cast blasting step heights of 35 m, 40 m, and 45 m, respectively. According to ballistics theory, the higher the step height is, the more obvious the throwing body motion is. The effective throwing rate should be higher. Therefore, the design of cast blasting parameters above 40 m in the Heidaigou open-pit coal mine needs further optimization and modification.

(2) Three basic indexes, comprising the effective throwing rate, the material quantity of the upper layer, and the material quantity at the coal trench, obtained by data fitting step heights of 35 m, 40 m, and 45 m, all reached 95% confidence. The fitting results were verified, and the cast blasting stockpile form is basically consistent with the actual blasting effect on site.