Engineering Application and Prediction of the Influence Area of the Rockfall Hazards

The identiﬁcation of potential rockfall and the accurate prediction of its trajectory are critical in prevention and mitigation of rockfall hazard. It is an important precondition to assess the uncertainty of rockfall motion, study the eﬀective identiﬁcation technology of potential rockfall, predict the rockfall trajectory, and calculate the threatened area by rockfall hazards. In this study, ﬁeld investigations and numerical simulations were carried out to identify potential rockfall on a weathered rock slope. As a case study, our calculations results show that the area of tensile stress concentration and plastic failure is the potential area where the rockmass will fall oﬀ the surface of the weathered rock slope. A mathematical model for calculating the rockfall inﬂuence area of the weathered rock slope was established based on the optimization theory, neural network technology, and genetic optimization algorithm. The rockfall inﬂuence area of the weathered rock slope was determined using maximum horizontal distance of rockfall in the speciﬁed slope cross sections and described on the topographic map using spline curves to form a closed possibly vulnerable area. As a case study, our calculations conﬁrm that the distributions of the plastic failure and tensile stress areas obtained from the numerical simulations are consistent with the dangerous rock masses identiﬁed by ﬁeld investigations at Guanyindong Slope that is a popular tourist scenic spot in Zhejiang Province, China. In this study, it has been indicated that the inﬂuence area can be used as the basis for the design of passive protection methods for rock slopes vulnerable to rockfall hazards.


Introduction
Rockfall is a common geological disaster that occurs during any construction activities in the mountainous region. Because rockfalls are random and destructive, the trajectory of rockfalls has not been effectively predicted. ese disasters can result in enormous losses to any construction projects and tourism sites in the hilly/mountainous areas.
In recent years, many studies have been conducted on the controlling factors and motion characteristics of rockfalls using theoretical analysis [1], field investigations [2][3][4], statistical analysis [5], field tests [6][7][8], and numerical simulation analysis [9]. Previous models can be divided into three groups: empirical models [3,[10][11][12], analysis and calculation models based on the theory of kinematics [5,10,[13][14][15][16][17][18][19], and mathematical models based on Geographic Information System (GIS) [20][21][22]. Empirical models usually require a great deal of field investigation and experiments for statistical analysis and thus consume a great amount of manpower and financial resources. Formulas based on empirical models are often only locally applicable. Both analysis and calculation models based on the theory of kinematics and mathematical models based on GIS require a large number of input parameters related to slope morphology and rolling blocks' characteristics, which are usually not readily available.
All of the above-mentioned studies satisfactorily explain the characteristics of the rockfall motion of the slope and provide a theoretical basis for the prevention and control of rockfall hazards. However, it is impossible to use empirical formulas or specific mathematical expression to accurately predict the potential trajectory of dangerous rockmasses after falling because the trajectory is closely related to the complex and variable slope parameters (e.g., the shape of the slope, the slope's vegetation cover, and the shape and size of the blocks). Due to many factors controlling rockfall, the slope roughness exerts a limiting effect on the lateral motion of the rockfall, especially for steep rock slopes [15,23]. In addition, several complex parameters are required to simulate the three-dimensional rockfall run-out, which increases the level of uncertainty. However, it is feasible to approximate the rockfall motion as a planar motion for a steep rock slope with a relatively smooth surface [24]. Taking the Guanyindong Rock Slope, which is located in a tourist scenic spot in Zhejiang Province, China, as an example, field investigations and numerical simulation analysis were used to identify potential dangerous rockmass. In addition, the neural network technique was used to calculate the influence area of rockfall. Based on field investigation and numerical simulation results, a corresponding mathematical model for determining the influence range was established, and the maximum run-out distance of the potential rockfall was computed using a genetic optimization algorithm.
is study provides a basis for the design of passive preventative methods for preventing rockfall.

Calculation Theory of the Influence Area of Rockfall
Authors have established the model to calculate the run-out distance of rockfall. is model can calculate the maximum run-out distance of rockfall at a specific profile of the rock slope.

Basic Assumptions.
For any profile of a steep slope that weathers easily, the slope's shape, slope angle, the accumulation at the toe of the slope, and the amount of vegetation covering on the slope's surface are all fixed constants. Historical rockfall data also shows that the mass of the rockfall block and its location usually change. erefore, in this study, the following assumptions are made to establish a model for calculating the horizontal distance of the rockfall for any slope profile.
(1) e rockfall motion is a planar motion.
(2) e initial velocity of the dangerous rock mass at the start of the rockfall is zero. (3) For any given profile of the slope, the mass of the dangerous rock mass and the position of the rockfall block are variables.

Establishment of the Local Coordinate
System. Based on the above assumptions, a local coordinate system was established for the slope (see Figure 1). e x axis was defined as the direction of the slope's surface; the z axis was defined as the elevation direction of the slope; and the y axis was defined as the direction perpendicular to the paper.

Basic Equations and Analysis.
It is assumed that rockfall occurs for any section x � x i and that rockfall has a planar motion. Also, it is assumed that, at the beginning of the rockfall motion, the position of the dangerous rockmass v is (x i , y i , z i ), the mass of the potential rockfall is m v (kg), and the position after the rockfall of the dangerous rockmass is (x j , y j , z j ). For section x � x i , the dangerous rockmass v rockfall and the resulting horizontal distance of the motion are (1) In the above formulas, D v,x�x i is the rockfall horizontal distance (m) of the dangerous rockmass v at cross section x � x i of the slope, and f and f x are the rockfall motion functions. e calculation of the maximum horizontal distance of the rockfall at cross profile x � x i can be converted into an optimization problem expressed as follows: where v is the number of the dangerous rockmasses at profile x � x i ; i.e., v � 1, 2, 3, . . . , n 1 ; and i is the number of the cross sections of the slope; i.e., i � 1, 2, 3, . . . , n 2 .
To solve the optimality theory problem expressed by (3), the position of the dangerous rockmass where the weathered slope may be rockfall, i.e., the range of the x and z values, is determined first. For convenience, a finite number of profiles were defined along the x direction of the slope, the maximum rockfall horizontal distance of each profile was calculated and marked on the topographic map, and a spline curve was used to connect all of the points to form an enclosed area; this area represents the influence area of the weathered slope rockfall. (2) and (3), the calculation of the influence area of the rockfall at profile x � x i requires that the position z and mass m (kg) of the rockfall dangerous rock mass, i.e., the constraints of the model in (3), be determined as follows.

Position Constraints for the Rockfall Dangerous
Rockmass.
e rockfall and spalling of a rock slope are the results of the interaction of various external forces. Rockfall occurs due to a combination of the rockmass gravity and external forces, such as fissure water pressure, expansion pressure exerted by plant roots, and weathering, which is caused by high-steep slopes and steeply inclined fissures. Examples of this failure include brittle tensional cracks in the surface of the rockmass under low stress conditions, spalling of the surface rockmass due to the close relationship between the rockmass structure and the stress state of the rockmass, and cantilever failure, which is the formation of tensional cracks caused by the shear failure or plastic failure of the rockmass in the rock cavity area under high stress conditions. erefore, the tensile stress distribution and the plastic failure inside the rockmass of steep weathered slopes are important conditions that can accelerate the occurrence of rockfall. Accordingly, the area where the slope may be rockfall can be identified and used as the position constraints needed to calculate the horizontal distance of the rockfall motion.
FLAC 3D is a 3D numerical simulation software developed by ITASCA that is commonly used in geotechnical engineering. FLAC 3D uses the explicit difference scheme to solve the governing differential equation of the field. erefore, it can satisfactorily simulate characteristics such as the internal stress state, the yield failure, and the plastic flow of the rockmass. In addition, it can be used to analyze the 3D spatial deformation behavior of a rockmass after it reaches the yield limit [25] and to determine the plastic failure zone and the tensile stress concentration of the rockmass. In order to determine the position range of the dangerous rockmass and where it may be rockfall, we used FLAC 3D to calculate the tensile stress zone and plastic failure zone at any position on the slope's surface. Based on this, the detailed slope failure conditions of the site can be used to determine the two zones, and the highly overlapping zone of the two can be used as the position constraints for the dangerous rockmass that may be rockfall.

Mass Constraints of the Collapsing Dangerous
Rockmass.
e rock mass with potential rockfall was determined using the following methods: (1) A detailed investigation of historical rockfall of the steep weathered slope was conducted. e shape, size, location, and distance of the rockfall block were counted and analyzed in the detailed investigation, so as to determine the volume for the historical rockfall mass.
(2) Unmanned aerial vehicle (UAV) tilt photography modeling and 3D laser scanning techniques were employed to establish a 3D numerical model of the slope mass and to identify and measure the size of the dangerous rockmass that could be rockfall.  Due to the randomness of the rockfall, the function of the horizontal distance of the rockfall motion y � f x (z, m) is uncertain for any cross profile of the weathered slope; i.e., the functional mapping relationship between the horizontal distance of the rockfall y, the rockfall block position coordinate z, and the mass of the rockfall block m cannot be described by an exact expression. However, based on the uncertainty function's mapping relationship, the BP (backpack) neural network can learn the rules of the corresponding function by analyzing the sample data; thus, the result closest to the expected output value can be obtained given an input value [27,28]. As a result, the BP neural network can be used to express the function of the horizontal distance of the rockfall y � f x (z, m); see Figure 2.
(1) Establishment of the Network Model. We propose the fact that the structural model of the BP neural network of the horizontal distance of the rockfall function has the three hidden layer neural network structure shown in Figure 2.
(2) e Input Layer. e input layer includes the position coordinate z of the dangerous rockmass and the mass m of the dangerous rockmass. e input layer can be expressed in vector form as (3) e Hidden Layer. e output vector of each hidden layer is expressed as

e weight matrix from the input layer to the hidden layer is
e column vector v i j is the weight vector corresponding to the jth neuron of the ith hidden layer. If the threshold of the neuron is T i , then (4) e Output Layer. e output layer is the value of the horizontal distance of the rockfall motion, which can be expressed by the vector O � (o 1 ) T . e rockfall horizontal distances from investigations of historical rockfalls or from numerical simulation are taken as the expected output value, which can be expressed by the vector d � (d 1 ) T . is value can be used as the basis for the neural network, which can be used to calculate and adjust the weights of the output layer and the hidden layers. If the weight matrix between hidden layer 3 and the output layer is d � (ω 1 ) T � (ω 11 , ω 21 , . . . , ω m 3 1 ), the output layer can be expressed as (5) e Transfer Function. e unipolar Sigmoid function was used for the transfer functions of both the output layer and the hidden layer [29]: In summary, when using the neural network model, as shown by (4) through (8), as the model for calculating the horizontal distance of the rockfall motion, as shown in (2), it is necessary to use a large number of rockfall motion statistical data points to train the neural network model. As a result, the model can be used to calculate the horizontal distance of the rockfall motion y of any dangerous rockmass. In the calculation, the rockfall motion sample data can be obtained from historical rockfall data from field investigations, the rockfall numerical simulation, and/or from physical model tests.

Solution for the Influence Area of the Rockfall.
e genetic optimization algorithm is a type of mathematical optimization algorithm formed by mimicking the biological evolution process of chromosomes, which undergo reproduction, crossover, and mutation to produce new, fitter chromosomes [30]. Because it is an effective method for solving the optimality problems, the optimality problem shown in (3) can be solved using the genetic optimization algorithm using the following steps: (1) Randomly generate the initial population P 0 and then use the model expressed in (4)  (2) Perform the binary tournament selection, crossover, and mutation of P 0 to obtain a new population Q 0 and let t � 0. (3) Combine P t and Q t to form a new population R t and then perform the nondominated sorting of R t and select the best N individuals to generate a population P t+1 . (4) Reproduce, cross over, and mutate the population P t+1 to generate a population Q t+1 Qt + 1. (5) If the number of iterations reaches the preset maximum number, the optimization ends. Otherwise, go to step 1.
Based on the above-mentioned steps, a genetic algorithm (GA) program was written using MATLAB 2012b to calculate the maximum rockfall horizontal distance of any cross profile of the weathered slope mass.

Case Study
Multiple episodes of rockfall have occurred historically on the Guanyindong Slope, which is a tourist scenic spot in Zhejiang Province, China. Future rockfall poses a threat to the lives and property of tourists. Taking the Guanyindong Slope as an example, we used the above calculation method to determine the influence area of the Guanyindong Slope, providing a basis for determining and designing disaster prevention and reduction measures for the Guanyindong Slope.

Overall Characteristics of the Guanyindong Slope.
e geographical coordinates of the Guanyindong Slope are 120°36′31.2″ E and 28°42′11.6″ N. e high-steep weathered slope mass has an elevation of approximately 126 m, a slope of 70-88°, and slope aspect towards 310°.
ere is a 19 × 17 × 10 m 3 rock cave (the Guanyin cavity) at the bottom of the slope (Figure 3(a)). Inside this rock cave there is a Guanyin statue, which integrates the natural and cultural landscapes and merges the religious and cultural characteristics of the region, making it one of the most important tourist attractions. In the middle of the Guanyindong Slope, there is a weak layer (with an elevation of 230-245 m) that contains a number of cavities formed by weathering (W1-W4 in Figure 3(b)). e lithology of the slope is composed of Upper Jurassic Jiuliping Formation altered tuff (J3j1) (Figure 4). To further illustrate the microscopic characteristics of the rock, one of the microscopic images of rock samples is shown in Figure 4(b). In addition, due to weathering and flowing water, part of the surface the rock block is grayish black; however, the fresh surface of the rock is grayish white. e bedrock on the surface of the slope is exposed with less vegetation cover (Figure 3(a)). In general, the rockmass is severely weathered with well-developed structural planes (Figure 3(b)). e main controlling structural planes include regional crack structural plane F3 in the steep slope at the Guanyindong entrance (orientation of N40E/80 NW) and the oblique normal fault bedding F8 at the cavity's entrance (orientation of N58W/60 SW). In addition, the slope has gently inclined bedding with the orientation of N85W/10 NE, and a weak intercalated layer has formed between the planes, with steeply inclined cracks with orientation s of N32E/88 NW and N33E/72 NW. We also observed that these structural planes cut each other within the rockmass to form large, outwardly inclined blocks, such as block GYD1 (62 m long × 8 m wide × 2 m thick) (Figures 3(c) and 3(d)).
When exposed to external factors such as heavy rainfall, the cracks continue to expand and gradually expand the plastic failure zone of the rockmass and ultimately decrease its strength. Eventually, this will trigger failure phenomena. Under the continuous weathering, the rockfall and falling blocks from the surface of the weathered rock slope are uncertain. us, it is difficult to predict the position and size of the rockfall block and to track its movement accurately.

Rockfall Mathematical Model of the Weathered Slope.
e rockfall mathematical model of the Guanyindong Slope can be expressed in (3). Based on the field investigations, a local coordinate system was established for the Guanyindong Slope mass (Figures 6(a) and 6(b)).

Position and Numerical Analysis of Dangerous Rockmass that May Be Rockfall.
Based on the description in Section 2.2, to determine the position of the dangerous rockmass where rockfall may occur on the Guanyindong Slope, i.e., to determine the ranges of the x and z values, we used both FLAC 3D numerical simulation analysis and geological field investigations. e upper boundary of the model is the surface, which is set as the free boundary, and the lower boundary is located approximately 45 m below the bottom of the Guanyindong Slope. e model is fixed in the z direction and around the boundary in the one-way horizontal axial direction (Figure 7(a)). Based on the field investigation (Figure 7(d)), we used the orthogonal test method [31] to determine the distribution of the plastic failure of the rockmass using the theory of displacement back analysis [32] (Figure 7(c)). en, the calculated parameters were used as the physical and mechanical parameters of the Guanyindong rockmass, as shown in Table 1.  (3) Results of the Numerical Simulation Analysis. By numerical simulation, we obtained the results as follows (Figure 7). First, the tensile stress concentration region is mainly distributed in the filling body of the weak intercalated layer between the rockmass above the Guanyindong Slope and the gently inclined bedding (5°∠10°). Second, the plastic failure zone is mainly concentrated in the vicinity of the filling body of the weak intercalated layer and in the cracks (orientation of 302°∠88°) in the gently inclined bedding.

Field Investigation Results.
e field investigations provided the following information. Because of continuous weathering, the filling body of the weak intercalated layer was between the gently inclined bedding (orientation of N85W/10 NE) of the Guanyindong Slope experienced rockfall and spalling in multiple locations (see spalling trace locations A and D in Figure 7(d)), forming multiple cavities. e steeply inclined cracks (orientation of N32E/88 NW) and bedding (orientation of N85W/10 NE) cut the rockmass, forming a number of dangerous rockmasses, leading to multiple historical episodes of rockfall and spalling (see spalling trace locations B in Figure 7(d)). Multiple historical events of rockfall and spalling also occurred in the upper part of the Guanyindong Slope (see spalling trace locations C in Figure 7(d)). Authors observed that because areas where the spalling of steep rocks occurred often exhibit fresh grayishwhite surfaces, their traces tend to be obvious and are easy to be identified (Figure 7(d)).

Delineation of Potential Rockfall Areas.
e rockmass failure distribution obtained from the aforementioned field investigations (Figure 7(d)) is in a good agreement with the plastic failure zone and tensile stress distribution zone (Figures 7(a)-7(c)) determined from the numerical simulation. According to Section 2.2.1, we can delineate the potential rockfall area of the Guanyindong Slope mass, as shown in Figure 7(d). e value range is e procedure for solving the calculation model is as follows:
(2) e objective function y � f x (z, m v ) in (11)fd11 is an uncertainty function, which was represented using the neural network model described in Section 2.3.1. (3) Using the historical rockfall data, the normal and tangential coefficients of restitution [33][34][35] and the friction angle of the Guanyindong Slope were determined using the theory of displacement back analysis (Figure 8) [32]. Based on the parameters, RocFall 2D software was used to simulate possible rockfall events for each cross profile and to obtain sample data for these rockfall events. In order to reduce the error of slope geometry to numerical simulation, according to the principle of aerial photogrammetry [36], the slope image data are acquired by using the tilt photography of five routes including East, South, West, North, and orthoimage with the help of PHANTOM 4 RTK Unmanned Aerial Vehicle (UAV). Among them, the UAV camera of the East, South, West, and North routes forms a 45°angle with the horizontal plane, and the UAV camera of the orthoimage route forms a 90°angle with the horizontal plane. Using context Capture Center Master and other software applications, combined with the principle of reverse modeling [37], the collected image data is generated into a three-dimensional model and 3D point cloud, and then the position, shape, size, and other data of the slope section are identified (Figures 1 and 8). e computer program for the neural network model, which was constructed in training step (2), was written using MATLAB. en, the trained neural network was used to predict possible rockfall events.
Using the GA described in Section 2.3.2, a relevant program was prepared using MATLAB 2012b.
en, the calculation model represented by (11) was solved to obtain the maximum possible rockfall horizontal distance of each cross profile.

Calculation Model Solution.
According to the model solving procedure described in Sections 3.5.1 and 2.3.1, the RocFall ™ was used to obtain neural network training data for each cross section of the Guanyindong Slope. Because the historical rockfall data for the Guanyindong Slope obtained from the field investigations was not sufficient to satisfy the requirements of the neural network training, we obtained the normal and tangential restitution coefficients and the friction angle of the rockmass based on the theory of displacement back analysis [32] using historical rockfall data (see Figure 8). en, numerical simulations of the possible rockfall events of each cross profile of the slope were carried out to obtain the necessary corresponding sample training data.
(1) Determination of the Simulation Parameters of the Rockfall Motion. e historical rockfall events were backcalculated as shown in Figures 8(a) and 7(b). e size of the rockfall rock was 1.8 m × 1.5 m × 0.75 m, and the obtained parameters for the rockfall of the rockmass included a normal restitution coefficient of 0.53, a tangential restitution coefficient of 0.75, and a friction angle of 46.5°. e rockfall motion was verified using back-calculated the historical rockfall events shown in Figures 8(c) and 7(d).
e size of the rockfall rock was 2 m × 1.5 m × 1 m. e simulation results by rockfall are consistent with the field investigations and confirm the validity of the above parameters.
(2) Acquisition of Sample Data. According to the rockfall parameters obtained from the above, we used the uniform design table to design a rockfall simulation test for each cross profile and to obtain training sample data for the neural network of each cross profile. Table 2 shows the orthogonal factor level table of the cross profile x � 12 m designed using the uniform design table U * 30 (30 12 ). Table 3 presents the experimental arrangement and results of the numerical simulation test, i.e., the training sample data of cross profile x � 12 m (the coordinate system is shown in Figure 6(b)). Due to space limitations, only data for cross profile x � 12 m is reported. e experimental arrangement and results are shown in Table 3.
(3) Calculation of the Influence Area of the Rockfall of Each Cross Section. e GA calculation program was created using MATLAB 2012b according to the method described in Section 2.3.2. en, it was used to solve the maximum influence area of each profile and the height Z and mass M of the corresponding rockfall rockmass. e calculation results are shown in Table 4.

Discussion of the Calculation Results
For comparison, we used Rocscience RocFall 6.0 to conduct a numerical simulation of the calculation results again. e results indicate that the deviation between the results of the rockfall numerical simulation and those of the GA No. 14's location calculations is only 0.096-2.491 m ( Figure 9). In particular, the smallest and largest deviations occur for cross profiles x � 72 m and x � 120 m, respectively ( Figure 9). erefore, the calculation results are believed to be reliable.

Application of the Calculation
Results. According to the above calculation and analysis, we obtained a safe horizontal distance for the engineering application of the Guanyindong Slope. is is shown as follows: Next, the maximum horizontal distance values of the potential rockfall of the different cross profiles were connected to form a spline curve in order to obtain the rockfall influence area on the Guanyindong Slope ( Figure 10). e recorded historical rockfall events (labeled as historical rockfalls A to M in Figures 5 and 10) were identified in the field investigations and the positions of the rockfall of the dangerous rockmasses all fall within the calculated influence area of the rockfall. It should be noted that the department in charge of this scenic spot now closes the Guanyindong scenic spot due to the potential rockfalls that would threaten the safety of tourists. en, the optimized design was carried out based on the above research results. Figure 10, to ensure the safety of tourists in the Guanyindong scenic spot, the following disaster prevention and mitigation measures should be considered; see Figure 11(a).

Countermeasures. According to
(1) Abandon the original 151 m long trail into the Guanyindong scenic spot and replace it with shrubs and other vegetations to keep people out of the influence area of potential rockfall. en, construct a new tourist route (the improved tourist route (South) or the improved tourist route (North)) (as shown in Figure 10). (2) e improved tourist route (North) leads to the cobblestone path at Wanxian Bridge. First, it extends to the northeast, and then it turns and extends to the southeast, which avoids the potential rockfall influence area. In addition, a tunnel can be constructed in this location at an elevation of approximately 200 m to provide access to the northern end of the Guanyindong scenic area. e improved tourist   [38] can be adopted for reinforcement. e prestressed bolt was implanted into the risk rockmass (GYD1). e initial elevation of the bolt layout is 210 m, and a total of 7 layers were arranged with each layer having a height of 5 m. e horizontal spacing of the bolt arrangement was 5 m. e dip angle was 30°, and the prestress was 400 KN. e length of bolt was 15 m, the length of free section was 5 m, and the length of anchorage section was 8 m. e bolt was made of 4 × 7(ϕ5 mm) steel strand (Figure 11(a)). Numerical   simulation results show that the displacements of monitoring point were improved obviously after reinforcement (Figure 11(b)).

Conclusions
Since predicting the rockfall trajectory and calculating the influence area of rockfall are of profound importance in studying the uncertainty of rockfall and the active protection against rockfall hazards, a reliable estimate of the influence area of rockfall is crucial to the design of passive preventative methods for preventing rockfall. We studied the calculation method of the influence area of rockfall, by taking an engineering case in a tourist scenic spot in Zhejiang Province, China.
(1) e results of field investigations and numerical simulation indicate that the area of tensile stress concentration and plastic failure is the potential rockfall area where the rockmass will fall from the surface of weathered rock slope. (2) e normal and tangential restitution coefficients and the friction angle of the weathered rock slope for the rockfall numerical simulation were obtained based on back analysis of historical rockfall events. e numerical simulation results based on these parameters can be used as learning samples for the BP neural network. en, the motion and influence area of the potential rockfall can be effectively predicted. However, the RocFall 2D numerical simulation results still deviated slightly from the historical data.
(3) e prediction of the influence area based on optimization theory, neural network technology, and the genetic optimization algorithm provides a new way to eliminate rockfall uncertainties and can be used as a guidance in the prevention of rockfall hazards.

Data Availability
e authors do not have permission to share data.

Conflicts of Interest
e authors declare that there are no conflicts of interest regarding the publication of this paper.