XG Boost Algorithm to Simultaneous Prediction of Rock Fragmentation and Induced Ground Vibration Using Unique Blast Data

Abstract

The two most frequently heard terms in the mining industry are safety and production. These two terms put a lot of pressure on blasting engineers and crew to give more while consuming less. The key to achieving the optimum blasting results is sophisticated bench analysis, which must be combined with design blast parameters for good fragmentation and safe ground vibration. Thus, a unique solution for forecasting both optimum fragmentation and reduced ground vibration using rock mass joint angle and blast design parameters will aid the blasting operations in terms of cost savings. To arrive at a proper understanding and a solution, 152 blasts were carried out in various mines by adjusting blast design parameters concerning the measured joint angle. The XG Boost, K-Nearest Neighbor, and Random Forest algorithms were evaluated, and the XG Boost outputs were shown to be superior in terms of Mean Absolute Percentage Error (MAPE), Root Mean Squared Error (RMSE), and Co-efficient of determination (R²) values. Using XG Boost, the decision-tree-based ensemble Machine Learning algorithm that uses a gradient-boosting framework and a simultaneous formula was developed to predict both fragmentation and ground vibration using joint angle and the same set of parameters.

1. Introduction

The use of explosive energy in blasting affects both rock fragmentation and induced ground vibration. Rock movement may be desired, and manifests in amuck profile suitable for the loading equipment. The complete and proper utilization of explosive energy is the main objective in this process; energy used in achieving proper fragmentation automatically reduces negative aspects such as ground vibration. Positive blast results can be obtained by equalizing the energy of the explosive to the strength of the rock, optimum design parameters, and geospatial positioning of the blast holes.

There are many equations in blasting to make use of explosive energy properly to yield a safe and effective blast, but most formulas are designed based on controllable parameters such as burden, spacing, bench height, hole diameter, stemming, decking, firing pattern, and quantity of explosive, etc. Many researchers have revealed that uncontrollable parameters such as joints and bedding planes, rock compressive and tensile strengths also significantly affect the performance of the blast in terms of fragmentation and ground vibration. The daunting task of any blasting engineer is to ensure that the selected blast design parameters meet all post-blast requirements and the targeted fragmentation of an enterprise. Blast-induced ground vibrations are a major issue to be tackled.

The presence of geological discontinuities in a rock mass can significantly influence both rock fragmentation and ground vibration [1,2,3]. Joints are among the most common defects in the rock mass and are also defined as planes of weakness. Jointed rock gives poorer fragmentation than un-jointed rock, as stress waves dissipate and gases escape from joints and create an imbalance in attenuation and produce uneven fragmentation [4]. Likewise, rock joint angles have a substantial impact on shockwave propagation post blasting [5]. There is a pronounced impact of joint orientation on mean fragmentation size and ground vibration [6,7,8,9,10]. Good fragmentation is obtained when the orientation of the free face is parallel to and on the dip side of the principle joint planes [11,12]. Similarly, the rate of attenuation depends on the incidence angle of the joint face [13,14,15]. Usually, a joint angle of 90° generates the very fast attenuation of stress wave [16]; however, with an increase in joint angle, the attenuation rate of vibration velocity increases and decreases the efficiency of fragmentation. Thus, alteration in joint angle has a potential effect on the blast results [10,17]. Joint intensity also influences blasting results [16].

Blast design parameters such as spacing burden ratio, firing pattern, and explosive quantity pose a substantial impact on both rock fragmentation and ground vibration [17]. Poor blast design results in desensitization of explosives and detonator damage [18]. Geological aspects such as open joints, discontinuities, and voids are the cause behind the premature detonation and detonator damage due to the merging of side-by-side blast holes; in the case of surface, blasting would explain the cause–effect relationship. Similarly, excessive burden results in fly rock; therefore, burden and/or spacing of the decks should be sufficient to avoid the risk of fly rock and over ground vibrations. According to thumb rules, considering a minimum 8 to 25 ms delay between charges in the same row and between rows prevents blast shock overlap and could result in fewer ground vibrations [18].

A firing pattern provides a synchronized opportunity for the explosive charges to exercise their combined effect. Thus, firing pattern provides a free face, to the upcoming blast holes in some order, with the blast progression. A firing pattern determines the movement and direction of the rock throw. The firing pattern reflects a substantial effect on twin products of blasting, i.e., fragmentation and ground vibration. During the blast trials, it was observed that the V pattern had better fragmentation, probably due to in-flight collision between broken rock fragments [18]. Analogously, investigations found that firing patterns can significantly influence and govern both fragmentation and ground vibration results [19].

Soft computing approaches such as self-learning, adaptive recognition, and nonlinear dynamic processing can substantially aid in the resolution of intractable and perplexing geotechnical problems [20,21,22,23,24,25,26]. Many researchers have employed artificial intelligence-based algorithms such as ANN, FIS, GEP, Regression, XG Boost, Random Forest, AMC, and K-NN to solve and predict rock fragmentation and resultant ground vibration [27,28,29,30]. Nevertheless, none of these models have delivered a unique formula for predicting fragmentation and ground vibration using joint angle and blast design parameters, so an attempt has been made by the authors in this direction.

Two models—FIS, an artificial intelligence approach, and regression—were developed to forecast rock fragmentation using 415 blast design datasets in an Iranian mine and the research indicated that the FIS model performed well in predicting results [23]. Similarly, various methods showed that models can accurately forecast rock fragmentation based on input factors such as burden, spacing, and explosive quantity, among others [31]. Analogously, the Gene Expression Programming (GEP) model was used to predict Peak Particle Velocity (PPV) in Malaysian mines. A total of 102 datasets of blast design parameters such as burden-to-spacing ratio, stemming, hole-depth, a maximum charge per delay, and powder factor were fed into the trained model, and the GEP produced good R² and Random Mean Square Error (RMSE) metrics and predicted well [32]. On 93 blasts of data, algorithms such as SVM, XG Boost, Random Forest, and K-NN were employed, and the results showed that XG Boost predicted the closest PPV value among all of them [33,34].

Objectives

The main objectives of the research are to predict the ground vibration and fragmentation using tools such as XG Boost, K-Nearest Neighbor, and Random Forest algorithms.

2. Materials and Methods

2.1. Predictability and Assessment of Blast Results

Traditionally, the blast results are assessed mostly after observing the muck profile and the fragments by way of estimation. After the excavator handles the blast material, the time taken for handling the material is noted, too; the rock fragments that are too difficult to handle are kept aside. Thus, the post-blast analysis is not properly quantified, and is prone to error, due to non-standard methods used. Nowadays, in mining, Artificial Intelligence (AI) and Machine Learning (ML) are being used for quick and accurate analysis. STRAYOS software is used for the analysis of the video input of the blast area, producing a joint pattern after analysis. Fragmentation is analyzed through STRAYOS software and O-PITBLAST software is used in designing the blast.

2.2. A Brief of the Mine and Blast Site

Opencast mine I of Ramagundam III Area, Singareni Collieries Company Limited, Telangana, India is the mine where the studies were conducted. Figure 1, Figure 2 and Figure 3 shows the site location on an Indian map, its Google map position, and experimental overburden benches.The mine was previously worked through underground methods but later was converted to an opencast mine. In the study area, overburden benches were 12 m high. Rock strata are comprised of sandstone and alluvium soil. The average density of sandstone was 2.3 g/cc.

a. 

Identification of bench structural geo-property through UAV:

For the reconfiguration of the new blast design, extensive work was under taken to determine the rock condition and joint intensity. DJI Mavic Unmanned Aerial Vehicle (UAV) was used to detect joint planes.

The UAV was capable of capturing and recording photos and videos at 4K resolution. It can run for 25 min in the air with a single battery charge and weighs 258 g. Calibration was performed to fix signal problems, allowing for a better take-off of the drone placed on the helipad shown in Figure 4a, as well as the bench scanning test shown in Figure 3.

The drone operated with mobile application LITCHI in FIRST-PERSON VIEW (FPV) mode with 18 satellite signal poles, as shown in Figure 4b.

The UAV flew through the bench range at a height of 100 feet, with a 75 percent overlap. To get error-free images, the camera angle was set to 45 degrees so as to focus on the bench’s top edge and the beginning of the bench floor. In order to maintain accuracy, at least four to five images were taken at the same time in order to generate multiple point cloud data for use in STRAYOS software to generate a 3D model. In total, 580 photos were captured from 1 de-coaled,2B seam, 3A coal seam, and 3A de-coaled.

b. 

Geo–technical properties of the site:

To design blasts in O-PITBLAST SOFTWARE, geo-rock attributes are required. To complete the task, rock samples were collected from each proposed site and tested in a laboratory.

Table 1. Averaged values of geo-technical properties of the rock samples.

Mine	Name of Bench	Density, g/cm3	Uni-Axial Compressive Strength, MPa	Young Modulus, GPa	Poisson’s Ratio	Joint Angle, Degree	Joint Spacing, m
OC I	1 De-Coaled	2.490	67	22.58	0.290	DIP—48 to 74 STRIKE–110 to 180	0.2–0.49
	2A Coaled	2.28	61	31.00	0.156	DIP—27 to 61 STRIKE–94 to 172	0.15–0.76
	3A Seam	2.35	52	27.55	0.310	DIP—50 to 84 STRIKE–121 to 179	0.2–0.5
OC II	2B OB Seam	2.28	52	20.65	0.250	DIP—34 to 85 STRIKE–120 to 183	0.23–0.49
	1B OB Seam	2.41	71	21.67	0.356	DIP—52 to 73 STRIKE–121 to 175	0.39–0.51

shows the results of various tests, such as uniaxial compressive, Young Modulus, and Poisson’s tests. The average mineralogical composition of sandstone was calcite 43–50%, quartz 14–18%, feldspar 13–15%, and the remaining 20–25% in a combination of clay, mica, and dolomite.

2.3. AI Tools for Rock Characterization

STRAYOS, Artificial Intelligence (AI)-based software has been used to analyze joint intensity and joint pattern at the target bench. Images of rock joints are analyzed repeatedly to see if characteristic patterns can be found in the picture data.

Joints in rock mass break the propagation of blast waves; hence, fragmentation may be affected. Under certain circumstances, there may not be any back-break or over-break, and fragmentation may be aided by joints, improving overall performance. There may be excessive sapling towards free faces at joints when rock is weak and explosive gases give a push effect without actually shattering them. Because of this complex relationship, the characterization of bench cracks is important for designing the spatial disposition of blast holes and the placement of decks.

STRAYOS Software requires a minimum of 10 drone photographs to create a 3D model. When data is successfully fed into the software, the interface displays a 3D model of a bench, as shown in Figure 5. After enabling different clusters suggested by software with the aid of AI, planes and lines will be visible on a 3D bench. Lines and planes are two ways in which the software helps the user to analyze discontinuities. When the cluster lines and planes option is activated, the DIP direction, STRIKE direction, and DIP angle of joints on the bench can be identified. A wire-framed bench with lines is shown in Figure 6.

Table 2. Model training and testing datasets.

S.No	Spacing Burden Ratio (Se/Be), m	Total Explosive, kg	Firing Pattern	Joint Angle, Degree	Fragmentation	PPV	MaximumCharge/Delay, Kg
Training Data (80%) 118 Samples
1	1.3	7425	1	86	0.67	5.33	200
2	1.2	10,125	1	83	1.4	5.29	195
3	1.3	8100	1	85	0.91	4.94	187
4	1.3	8230	2	85	1.2	4.56	180
5	1.3	8100	3	80	0.89	5.15	195
6	1.3	8100	1	74	0.55	4.79	186
7	1.3	5670	3	86	0.55	3.91	155
8	1.2	5690	1	31	0.45	3.8	140
9	1.3	8100	2	84	0.97	4.61	180
10	1.3	5130	1	90	0.44	3.21	145
11	1.3	7100	1	39	0.38	2.8	125
12	1.3	5940	3	38	0.31	2.21	122
13	1.2	6110	3	90	0.88	3.52	145
14	1.3	7290	3	82	1.1	3.78	149
15	1.3	4500	3	46	0.91	3.94	146
16	1.3	6210	2	85	1.2	3.96	158
17	1.3	5940	1	86	0.82	4.51	180
18	1.3	6615	3	36	0.47	3.55	141
19	1.3	6210	3	80	0.56	2.9	128
20	1.2	6885	2	41	0.33	4.18	148
21	1.3	7290	1	29	0.97	5.39	210
22	1.3	5400	1	90	0.38	2.55	120
23	1.3	4995	3	80	0.38	2.03	130
24	1.3	6210	3	35	0.34	2.85	110
25	1.3	6480	3	85	0.57	4.11	145
26	1.3	9280	3	45	0.56	3.96	170
27	1.4	7020	3	68	0.91	4.21	195
28	1.2	6000	1	38	0.34	4.9	120
29	1.3	7420	2	78	0.45	4.54	180
30	1.2	7290	3	75	0.65	4.28	185
31	1.3	7105	2	88	0.35	3.7	145
32	1.3	5940	3	35	0.44	3.26	147
33	1.3	7020	2	90	0.97	3.5	142
34	1.3	5265	3	84	0.29	2.11	135
35	1.3	4725	3	86	0.38	1.85	119
36	1.3	5670	1	77	0.33	1.49	121
37	1.3	5240	1	91	0.39	3.11	155
38	1.3	7132	1	37	0.36	2.9	115
39	1.3	5940	3	36	0.333	2.2	150
40	1.2	6110	3	90	0.88	3.52	165
41	1.3	7290	3	82	1.1	3.78	135
42	1.3	4500	3	46	0.91	3.94	125
43	1.3	6220	2	84	1.19	3.86	125
44	1.3	5940	1	86	0.82	4.51	185
45	1.3	6615	3	36	0.47	3.55	130
46	1.3	6210	3	80	0.56	2.9	130
47	1.2	6886	2	42	0.34	4.19	175
48	1.3	7290	1	29	0.97	5.39	195
49	1.3	5401	1	93	0.36	2.32	130
50	1.3	4995	3	80	0.366	2.08	123
51	1.3	7420	2	78	0.42	4.57	186
52	1.2	7290	3	74	0.64	4.31	178
53	1.3	7105	2	88	0.35	3.7	160
54	1.3	5940	3	35	0.44	3.43	150
55	1.3	7425	1	86	0.67	5.33	210
56	1.2	10,222	1	80	1.5	5.2	198
57	1.3	8100	1	85	0.91	4.94	190
58	1.3	8230	2	85	1.2	4.56	187
59	1.3	8100	3	80	0.89	5.15	200
60	1.3	8100	1	74	0.55	4.79	195
61	1.3	5120	1	91	0.44	3.3	155
62	1.3	7100	1	39	0.38	2.8	135
63	1.3	5940	3	38	0.31	2.21	128
64	1.2	6110	3	90	0.88	3.52	175
65	1.3	7290	3	82	1.1	3.78	179
66	1.2	6110	3	90	0.88	3.52	160
67	1.3	7260	3	82	1.1	3.78	178
68	1.3	4500	3	45	0.92	3.9	184
69	1.3	6210	2	85	1.2	3.96	185
70	1.3	5940	1	86	0.82	4.51	190
71	1.25	7230	3	79	1.1	4.26	179
72	1.25	7110	3	78	0.92	3.41	160
73	1.25	8222	3	55	1.5	5.2	195
74	1.25	8900	3	57	0.98	3.22	139
75	1.3	8240	3	89	1.6	4.56	185
76	1.25	8500	3	51	0.77	3.78	170
77	1.3	9500	3	54	0.55	4.67	185
78	1.2	8125	1	60	1.5	5.29	210
79	1.3	8100	1	82	0.91	4.94	190
80	1.3	8230	2	59	1.2	4.56	183
81	1.3	7880	3	69	1.2	2.5	125
82	1.3	8100	1	74	0.55	4.79	170
83	1.3	7688	3	72	0.85	4.2	160
84	1.2	5690	1	70	1.3	3.9	145
85	1.3	8100	2	58	0.97	4.61	165
86	1.3	6130	1	90	1.2	4.23	155
87	1.3	7555	1	42	1.1	2.1	122
88	1.3	5988	3	57	0.31	4.21	170
89	1.2	6121	3	94	0.88	2.33	143
90	1.3	7356	3	59	1.1	2.31	137
91	1.3	4200	3	46	0.99	2.67	150
92	1.3	6345	2	73	0.89	5.1	205
93	1.3	5789	1	86	0.99	3.6	138
94	1.3	6618	3	42	0.59	3.22	137
95	1.3	6220	3	81	0.73	2.9	132
96	1.2	6845	2	57	0.99	4.18	176
97	1.3	7256	1	77	0.73	4.5	182
98	1.3	5400	1	55	0.92	2.9	135
99	1.3	4998	3	34	0.67	1.2	110
100	1.3	6240	3	54	0.34	1.66	115
101	1.3	6480	3	67	0.57	1.59	112
102	1.3	8956	3	34	0.69	2.33	129
103	1.4	7455	3	54	0.8	2.41	132
104	1.25	6890	1	60	0.82	4.9	194
105	1.35	4789	2	87	0.45	4.54	185
106	1.25	6800	3	89	1.33	3.55	155
107	1.32	8905	2	87	0.57	3.41	145
108	1.32	6000	3	56	0.78	2.2	132
109	1.25	6800	2	74	0.97	2.78	143
110	1.25	6580	3	35	0.99	2.57	137
111	1.25	5890	3	46	0.78	1.85	125
112	1.25	6001	1	45	0.68	1.56	120
113	1.3	5240	1	67	0.39	3.14	139
114	1.2	7800	1	54	0.79	2.2	120
115	1.3	6570	3	58	1.4	2.2	123
116	1.2	6777	3	43	1.2	3.52	145
117	1.3	7833	3	56	1.45	3.78	155
118	1.3	6544	3	45	1.28	3.33	139
119	1.3	8000	2	423	1.23	3.66	134
Testing Data (20%) 28 Samples
120	1.31	7665	1	32	0.82	3.51	144
121	1.31	6502	3	69	0.61	3.55	149
122	1.35	6300	3	80	0.72	2.9	132
123	1.26	6211	2	42	0.34	3.61	155
124	1.36	6733	1	29	0.97	3.87	145
125	1.36	7500	1	69	0.66	2.32	130
126	1.36	6744	3	80	0.59	2.08	127
127	1.3	6500	2	48	0.71	3.6	145
128	1.2	7580	3	74	0.64	3.91	156
129	1.3	6900	2	56	0.77	3.7	155
130	1.3	5900	3	50	0.53	3.43	140
131	1.3	7425	1	48	0.89	3.74	155
132	1.2	10,222	1	92	1.5	3.88	158
133	1.3	8100	1	60	1.1	4.94	195
134	1.36	8900	2	778	1.2	4.56	190
135	1.3	8700	3	92	1.21	3.61	155
136	1.3	8600	1	74	1.25	2.24	135
137	1.3	5120	1	91	0.44	3.3	148
138	1.3	7300	1	68	0.28	2.8	148
139	1.36	6400	3	89	0.51	2.21	132
140	1.25	6500	3	83	0.88	3.52	151
141	1.35	7300	3	79	1.1	3.78	159
142	1.25	6544	3	69	0.88	3.52	151
143	1.35	7559	3	82	1.1	3.78	155
144	1.3	5200	3	45	0.32	3.9	160
145	1.3	6300	2	85	0.43	3.96	163
146	1.3	6400	1	86	0.82	4.51	192
147	1.25	7111	3	79	0.31	3	142
148	1.25	7390	3	66	0.92	3.22	157
149	1.25	7890	3	56	0.27	2.67	132
150	1.25	8234	3	87	0.98	3.22	142
151	1.3	8240	3	34	0.46	4.56	179
152	1.25	8221	3	45	0.77	3.78	155

shows the results of using 580-point cloud data to detect joints in four benches.

2.4. Blast Modeling Using Software

A 12 m bench with SME explosive was designed with alternative decking at 0.5 m, 2.5 m, and 1 m and was fired with a V pattern in the presence of joints, as shown in Figure 7. By a manual method it is difficult to include all of the parameters; therefore, computer software called O-PITBLAST SOFTWARE was used for designing the blast round. Similarly, blast check and charging-checking are shown in Figure 8a,b.

2.5. Blast Experimentation

1.

The four OB benches chosen for experimentation were 1 de-coaled, 3A coal seam, and 3A de-coaled with bench heights of 12 m, 10.5 m, 11 m, and 9.5 m, respectively. Because the mine was previously worked underground, there was a high risk of disturbances in strata and induction of cracks. To address this issue, benches were initially cleaned to a depth of 0.3 m for improved visibility and identification of cracks and joint planes, as shown in Figure 9a,b.

Figure 9. (a,b) Bench leveling and preparation for drilling.

Figure 9. (a,b) Bench leveling and preparation for drilling.

2.

The joint planes of the benches were identified by STRAYOS software and marked with white powder on the bench top surface to avoid the seizing-up of drilling bits in joints and to have a reference in deciding blast pattern and connections, as shown in Figure 10. Table 1 shows the design burden and spacing values in the O-PITBLAST that yielded good predicted results. The drill bit diameter was 150 mm, which was adequate for the existing bench height, burden, and spacing, and the drilled holes are shown in Figure 11.

Figure 10. Bench–joint planes marking.

Figure 10. Bench–joint planes marking.

Figure 11. Bench photograph after drilling.

Figure 11. Bench photograph after drilling.

A total of 152 blasts were performed to determine the effect of Se/Be ratio, total explosive quantity, and firing pattern on rock fragmentation and ground vibration. In all these blasts, site mixed emulsion (SME) explosive was used with a booster-nonel combination, and Down hole delays of 425 and 450 ms were chosen, with hole-to-hole delays of 17 milliseconds and row-to-row delays of 25 and 42 ms, adopted, respectively, according to rock conditions. The average explosive quantity was between 45–55 kg and 2 boosters per 1 m of the hole were used.

152 blasts were performed in four phases in 27 benches. The phases are A, B, and C, as shown below.

Phase A: All blast design parameters were maintained the same, but the firing pattern was altered with respect to joint angle.

Phase B: All blast design parameters were maintained the same, but the Spacing Burden Ratio was altered with respect to joint angle.

Phase C: All blast design parameters were maintained the same, but the Explosive Quantity was altered with respect to joint angle.

2.6. Assessment of Fragmentation

STRAYOS software, with the help of AI, can identify from the orthophotograph the entire dataset for all the rock sizes. When the border is identified, the rocks within the border are sorted by diameters. D10, D20…D80, D90 values are generated based on the concepts of KUZ-RAM and SWEBREC.

The drone flew at a height of 90 feet, with a 90° camera angle, i.e., perpendicular to the ground and with 80% capturing overlap, immediately after the blast at noon over the blasted muck pile to avoid the settling of dust layer on rock and the shadows, one over the other.

AI automatically detects the boundary of a muck pile and allows the user to change the edge points as desired; different rock fragmentation sizes will be marked with different colors in the configured boundary, as shown in Figure 12a,b. All blasts produced fragmentation graphs similar to the one shown in Figure 13.

2.7. Ground Vibration

As shown in Figure 14a, the ground vibration was monitored using NOMIS, an engineering seismograph. The transducer was attached to spikes and firmly pressed into the ground surface to maintain direct contact with the earth. The maximum charge per delay was between 110 and 210 kg, since the distance between the blast location and the monitoring station remained constant throughout the research. The measuring distance was 500 m. To establish a precise distance between the blasting site and the monitoring spot, drone geo-coordinates were used, as shown in Figure 14b. Vibration data produced by each blast is exported and saved. PPV for longitudinal (R), vertical (V), and transverse (T) components, vector sum velocity (VS) were reported by the seismograph during the blasts, as shown in Table 2.

2.8. XG Boost Regression Algorithm

XG Boost provides a scalable, portable, and Distributed Gradient Boosting Library, and thus an enhanced method based on gradient boosting decisions [35,36,37]. XG Boost model can effectively generate boosted trees, work in parallel, and tackle classification and regression problems [38]. The optimization of the value of the objective function lies at the heart of the method. It uses the gradient boosting framework to construct machine learning algorithms. With parallel tree boosting, XG Boost can handle various engineering issues quickly and accurately.

$$\mathrm{Obj} (\mathsf{\theta })=\frac{1}{\mathrm{n}}{{\displaystyle \sum }}_{\mathrm{i} }^{\mathrm{n}}L(y_i-Y_i)+{{\displaystyle \sum }}_{\mathrm{j}=1}^{\mathrm{j}}\Omega \left(\mathrm{fj}\right)$$

(1)

L is the training loss function, while $\Omega$ denotes the regularization term. The training loss is used to assess the performance of the model on training data. The regularization term seeks to regulate the model’s complexity by preventing over-fitting [39]. In the formula, fj means a prediction coming from the jth tree.

It makes use of gradient (the error term) and hessian to create the trees. Hessian is a second-order derivative of the loss at the present estimate, which is provided as:

$${ \mathrm{h}}_{\mathrm{m}}\left(x\right)=\frac{{\partial }^{2}L\left(Y, f\left(x\right)\right)}{\partial f{\left(x\right)}^2}$$

(2)

where f(x) = f^(m−1)(x) and L is Loss of function

$$\mathrm{Similarity} \mathrm{Score} =\frac{{\left(Sumofresiduals\right)}^2}{\left(N+\lambda \right)}$$

(3)

where $\lambda$ is the L2 regularization term of weights.

Gain of the root node:

$$Gain=Left similarity+Right similarity-Root similarity$$

(4)

$$\mathrm{Output} \mathrm{Value}=\frac{\left({\mathsf{\Sigma }\mathrm{Residual}}_{\mathrm{i}}\right)}{\mathsf{\Sigma }\left[{ \mathrm{Previous} \mathrm{Probability}}_{\mathrm{i}}\times \left(1-{\mathrm{Previous} \mathrm{Probability}}_{\mathrm{i}}\right)\right]+\lambda }$$

(5)

2.9. Random Forest Algorithm

The decision tree method was initially introduced by Breiman [40]. It is popular for being a reliable non-parametric statistical approach for both regression and classification issues. It was an ensemble approach for achieving prediction accuracy based on the outcomes of various trees [41]. RF combines the projected values from each tree in the forest to provide the best result for each new observation. Each tree in the forest serves as an important member of the RF’s ultimate decision [42].

Three phases summarize the crux of the RF model for regression:

• Based on the dataset, generate bootstrap samples that are the number of trees in the forest (ntree).
• Create an unpruned regression tree for each bootstrap sample by picking predictors at random (mtry). Choose the optimal split among those factors.
• Assemble the anticipated values of the trees to forecast fresh observations (ntree). The average value of the projected values by each tree in the forest was utilized to solve the regression problem as well as forecast fragmentation and blast-induced PPV.

The error rate can be obtained from the training dataset in two ways, using out-of-bag (OOB) and aggregate of OBB.

K-NN Algorithm

It is a popular approach in machine learning for tackling regression and classification problems. It was developed by Altman NS [43]. The K-NN method finds the testing point and classifies it based on its nearest neighbors (k-neighbors). From the training data, the algorithm teaches nothing. It solely retains the weights of its functional space neighbors. When anticipating a new observation, it looks for comparable findings and computes the closeness to those neighbors.

In geosciences, the K-NN algorithm has been widely utilized to predict rock fragmentation, back break, and ground vibration [33]. KNN primarily employs the weighted average value of the k-nearest neighbors; the computation begins by calculating the distance between the uncertain and labeled neighbors using Equation (6), and then the neighbors’ numbers are re-arranged by raising the distance and RMSE using the cross-validation method. Finally, the average inverse distance between K-Nearest Neighbors will be computed.

$$\mathrm{D}({\mathrm{x}}_{\mathrm{tr}}, {\mathrm{x}}_{\mathrm{t}})={{\displaystyle \sum }}_{\mathrm{i}=1}^{\mathrm{n}}{\mathrm{w}}_{\mathrm{n}}({\mathrm{x}}_{\mathrm{tr},\mathrm{n}}-{\mathrm{x}}_{\mathrm{t},\mathrm{n}})$$

(6)

Above n denotes the number of features, x_tr,n & x_t,n represent the nth feature of training and testing data, and W_n presents the weight of the nth feature.

3. Results and Discussions

The data in this study were split into two sections: a training dataset and a testing dataset. The majority of the datasets (about 118 blasting events) are utilized for training, and the remaining (28 observations) are used for testing. The training dataset is utilized to create the aforementioned models, as shown in Figure 15. The testing dataset is used to evaluate the performance of the built models.

Two statistical metrics, MAPE and RMSE were used in this research to evaluate the performance of XG Boost, KNN, and Random Forest constructed the models shown in Figure 15. The most often-used measure for determining the accuracy is the Mean Absolute Percentage Error. This falls within the category of scale-independent percentage errors, which may be used to compare series on different scales. Likewise, MSE is used to measure the degree of inaccuracy in the statistical models. The average squared difference between observed and expected values is calculated. The MSE equals zero when a model has no errors and the value rises as the model inaccuracy rises.

The following equations were used to calculate MAPE and MSE in this study.

$$\mathrm{MSE}=\frac{1}{\mathrm{n}}{{\displaystyle \sum }}_{\mathrm{i}=1}^{\mathrm{n}}{(y_i-Y_i)}^2$$

(7)

Here, n represents a number of data points; y_i denotes observed values and Y_i are the predicted values. Hence, MSE is the average squared difference between the actual and predicted value.

$$\mathrm{MAPE}=\frac{100\%}{\mathrm{n}}{{\displaystyle \sum }}_{\mathrm{t}=1}^{\mathrm{n}} | \frac{A_t-F_t}{A_t} |$$

(8)

where A_t denotes the current value and Ft denotes the predicted value. The difference between them is divided by the actual value of A_t. This ratio’s absolute value is added for each projected point in time and divided by the number of fitted points (n).

$${\mathrm{R}}^2 =1-\frac{{{\displaystyle \sum }}^{\hspace{0.17em}}\mathrm{i}{\left({\mathrm{y}}_{\mathrm{i}}-\widehat{{\mathrm{y}}_{\mathrm{1}}}\right)}^2}{{{\displaystyle \sum }}^{\hspace{0.17em}} \mathrm{i}{\left({\mathrm{y}}_{\mathrm{i}}-\stackrel{-}{\mathrm{y}}\right)}^2}$$

(9)

A number of data here represented by n, yi & $\widehat{{\mathrm{y}}_{\mathrm{i}}}$ denote the actual and forecasted vales and $\stackrel{-}{\mathrm{y}}$ denotes the mean. Metrics such as MAPE, RMSE, and R² were computed on both training and testing datasets to evaluate the performance and accuracy of the model to decide the final algorithm and to develop a formula to predict fragmentation and ground vibration. The values are presented in

Table 3. MAPE, RMSE, and R² results.

Metric Type	XG Boost Regression		Random Forest		KN Nearest
	Fragmentation
	Training	Testing	Training	Testing	Training	Testing
MAPE	24	22.5	21.2	20.1	19.2	18.1
RMSE	0.0854	3.873	0.0021	4.221	0.0031	5.321
R2	0.953	0.9125	0.89	0.67	0.9	0.87
Peak Particle Velocity
MAPE	19	18.4	17.5	15.8	16.9	14.32
RMSE	0.0056	2.8890	0.0041	3.78	0.0886	4.88
R2	0.96	0.932	0.79	0.65	0.83	0.72

.

According to Table 3 and Figure 16, XG Boost algorithms perform well in both fragmentation and peak particle velocity based on metric values of MAPE, RMSE, and R² in both 80% training and 20% testing data. Fragmentation and peak particle velocity were measured in the field, and the XG Boost, Random Forest, and KNN models predicted values using the testing data shown in Figure 17. The predicted values of the XG Boost model were very close to the measured values of both fragmentation and peak particle velocity. In this study, the XG Boost regression method was finalized and utilized to forecast the simultaneous formula for fragmentation and PPV.

XG Boost Regression

To avoid modeling complexity, two stopping methods, maximum tree depth and n rounds, were selected in the current XG Boost Regression Model. Choosing significant numbers for maximum tree depth and n rounds would result in excessive development of the tree and an over-fitting issue. Therefore, to avoid this problem, the maximum tree depth was set to 1–3 and n rounds to 50, 100, and 150. A trial-and-error approach was used with a range of two values to get the best combination of these two parameters. The dataset was randomly sorted into training (80%) and testing (20%). For the training set, K-fold cross-validation was used to determine the optimum model parameters. The training set was divided into 10 parts of about equal size for 10-fold cross-validation, of which 9 parts were used for training and 1 part utilized for validation. This method was repeated 10 times repeatedly, and the average of these values was used to get the predicted forecast accuracy.

The initial prediction began by taking into account the mean of the dependent variables, MFS and PPV, from the dataset. The residual values from the previous prediction points were then computed. Furthermore, utilizing (80%) of the data, the model was trained to build an XG Boost tree using Equations (1)–(3). As illustrated in Figure 18, Figure 19, Figure 20 and Figure 21 the tree divided the data into two portions. The similarity score and gain were calculated by taking the average of the two nearest leaves and moving the residue to the leaf with the highest score and gain. To avoid model complexity, the learning rate and maximum depth were set to 1.0 and 3. After acquiring predictions (residuals) from model 1, all data points were run through model 1 to obtain h1(x) and the F1(x) prediction and residuals of model 2. Similarly, the technique for obtaining h2(x) for model 3 was carried out.

The final empirical formulas for predicting fragmentation and peak particle velocity are provided in Equations (10) and (11), which were generated from the trained model using Figure 18. The flow chart created utilizing the weight–age of the Se/Be ratio, Total explosive quantity (TE), Firing Pattern (FP), and joint angle degree (JAD) in terms of squared errors, samples, and prediction values from the trees shown in Figure 18, Figure 19, Figure 20 and Figure 21. Maximum Charge per Delay was not considered since the output iterations did not balance accuracy in anticipating fragmentation and PPV as the numbers ranged from 110 to 210 kg with little fluctuation, as shown in Table 2.

Fragmentation = 0.77 + 0.1(Tree 1) + 0.1(Tree 2)

(10)

PeakParticleVelocity = 3.60 + 0.1(Tree 1) + 0.1(Tree 2)

(11)

where tree 1 and 2 are final prediction values of models.

Using Figure 22, one can predict the simultaneous results of fragmentation and PPV. The process depicted in Figure 18 can be used to run the predetermined values of Se/Be, TE, FP, and JAD based on the geo-mechanical condition of the rock mass. Both fragmentation and PPV tree models produced TE as the root node. If the predetermined value of TE is less than or equal to 7181, the pattern continued to the left (Yes); otherwise, it proceeded to the right (No). Furthermore, the final prediction value of tree 1 was chosen based on the comparison values of pre-decided blast parameters and actual parameters derived by the models shown in Figure 22. Tree 2 of fragmentation and PPV trees 1 and 2 were predicted using similar step.

Figure 23 and Figure 24 demonstrate the relationship between the characteristics Se/Be, FP, TE, and JAD and target both fragmentation and peak particle velocity. The PYTHON Seaborn module was used to build correlation heat maps. The association between the dependent and independent variables is strong. Se/Be ratio, TE, and JAD are coefficients mapped with fragmentation, while FP, TE, and JAD are coefficients mapped with PPV, the same characteristics generated from both trees in fragmentation and PPV segments.

A Taylor diagram, shown in Figure 25, was used for rigorous evolution, which allows for more robust comparisons between models. Taylor multi-metrics outperform a single model in terms of accuracy. It may exhibit several criterion outputs in a single figure, which is a great way to grasp the findings. An XG Boost model depicts a good coefficient value among the other two models.

Using Python’s seaborn function, a pair grid was utilized to depict pair-wise connections in datasets. Pair grid provides greater versatility than the pair plot shown in Figure 26 and Figure 27. Pair grid assigned each variable in a dataset to a column and row on a multi-axes grid. A variety of axes level-plotting functions were employed to create a bivariate plot in the upper and lower triangles, with the marginal distribution of each variable in the diagonals. Furthermore, the hue parameter was utilized to express an extra level of conditions that plots various subplots in different hues.

4. Conclusions

The study’s goal was to develop a unique empirical formula using algorithms to simulate the simultaneous prediction of fragmentation and ground vibration using the same set of blast design parameters, including joint angle. There was no fruitful blast design parameters database accessible that included the proper joint angle to direct input to the algorithm, thus achieving the purpose. One hundred and fifty-two blasts were conducted in various opencast mines in the Singareni coal mines of Telangana, measuring the Se/Be ratio, firing pattern, and total explosive, and recording the joint angle. The data was used to train three models: XG Boost, Random Forest, and KNN, which were then evaluated using three metrics: MAPE, RMSE, and R². XG Boost model was selected.

• Use of O-PITBLAST SOFTWARE aided in the design of blasting and provided preliminary warnings for iterations.
• Available technical tools such as STRAYOS SOFTWARE are helpful in identifying the joint angle for rock mass characterization as well as fragmentation analysis
• A Correlation matrix was used to understand relationships between dependent and independent variables, and it proved to be quite useful.
• The XG Boost Regression Algorithm was found to be useful for creating an empirical formula to forecast simultaneous fragmentation and peak particle velocity utilizing joint angle and other blast design parameters such as Se/Be ratio, Total Explosive, and Firing Pattern while keeping the charge per delay constant.
• Empirical formulas of fragmentation and ground vibration are substantial in predicting results.

Fragmentation = 0.77 + 0.1(Tree 1) + 0.1 (Tree 2)

Peak Particle Velocity = 3.60 +0.1(Tree 1) +0.1(Tree 2)