# Stability Analysis of the Left Bank Slope of Baihetan Hydropower Station Based on the MF-DFA Method

Authors: Haoyu Mao; Min Zhang; Biao Li; Nuwen Xu
Journal: Advances in Civil Engineering
 (2020)

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## Abstract

Based on the left bank slope of Baihetan hydropower station in Southwestern China, a high-precision microseismic monitoring system was established. An early warning model of surrounding rock mass deformation and failure based on MF-DFA was proposed. The results showed that the multifractal characteristics of the microseismic and blasting waveform time series in the left bank slope were obvious, and the multifractal spectrum width of the blasting waveform is much larger than that of microseismic waveform. Before the slope cracks increased, the multifractal time-varying response characteristics of microseismic waveform showed strong regularity, which could be regarded as a precursor of surrounding rock mass deformation. Before the deformation and failure of surrounding rock mass, the multifractal spectrum width Δα showed an increasing trend while the multifractal spectrum of microseismic waveforms Δf(α) presented a decreasing trend, which can be regarded as a precursor of surrounding rock mass deformation; when deformation and failure occurred, Δα showed a decreasing trend and Δf(α) showed an increasing trend, which can be regarded as a deformation failure period; after the occurrence of deformation and failure, both Δα and Δf(α) showed a steady trend, and Δf(α) would approach to the zero line, which can be regarded as a stable period.


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## Body

## 1. Introduction

The monitoring and early warning of high slope instability have always been a research hotspot and difficulty in the field of rock mechanics and engineering [1–3]. The analysis methods of slope stability mainly include the engineering geological analysis method, model testing, numerical simulation methods, limit equilibrium, limit analysis, and reliability analysis method. These methods play an important role in solving the problem of rock slope stability research. Che et al. [4] conducted a series of shaking table physical tests to study the propagation of seismic waves in jointed rock mass and their influence on the stability of rock slope with discontinuous jointed rock slope in high and steep bedding. Griffiths and Fenton [5, 6] first applied the finite element method to the stability analysis of the slope, promoting the development of the finite element method in the slope engineering. Baker and Garber. [7] used variational method to search for the minimum safety factor and its sliding surface. Hungr et al. [8] extended the Janbu method and Bishop method to three dimensions, which improved the calculation accuracy. Sutcliffe et al. [9] discussed the ultimate bearing capacity of jointed rock foundations based on the limit analysis, presented an extensive parametric analysis, and researched the effect of strength properties and joint orientation on the bearing capacity of jointed rock. Xiao-Li and Liu et al. [10] analyzed the reliability of rock slope in the form of stability factor and explored the relationship between probability of failure and mean safety factor.Although some achievements have been made in slope stability analysis, the monitoring and early warning of instability have not been completely solved. The conventional monitoring adopts the ‘point, line’ layout, which has certain spatial limitations. In recent years, microseismic (MS) monitoring technology has been used as a high-precision rock mass fracture and deformation safety monitoring method. By preembedding MS sensors in the monitoring target area, pick up the elastic waves released by microcrack inside the rock mass in real time, automatic inversion calculation quantitative seismic parameter, predict the macroscopic deformation and failure of the rock mass in advance, and then evaluate the overall stability of the engineering rock mass. Xu et al. [11, 12] successfully carried out MS monitoring of the high and steep rock slope on the left bank of Jinping first stage hydropower station and the right bank slope of Dagangshan hydropower station, evaluated the stability of the high rock slope of hydropower station, and achieved a lot of research results in the aspect of rock dynamic disaster.The MS signal collected by the MS monitoring system is a complex nonlinear and nonstationary time series. The fracture of the slope rock mass often has the characteristics of discontinuous multiscale. Compared with the simple fractal dimension, the multifractal method can describe the fluctuation of the rock fracture signal at different levels more accurately. Multifractals, also known as multiscale fractals, represent self-similar fractal systems with different local characteristics. From the perspective of statistical physics, multifractal is an inhomogeneous set which consist of probability subsets with many different singular exponents. At present, the methods to estimate the multifractal spectrum mainly include box counting method, histogram method, partition function method, Wavelet-Based Detrended Fluctuation Analysis method (WB-DFA), Wavelet Transform Modulus Maxima method (WTMM), and Multifractal Detrended Fluctuation Analysis method (MF-DFA). The box counting method adopts a regular gridding method, which does not reflect the distribution of fractal body in the regional space, and the estimation of fractal dimension is very unstable in some cases [13]. The histogram method converges very slowly. Although the partition function method is relatively simple, its calculation results cannot fully reflect the distribution of singularity. Manimaran et al. [14] first proposed the WB-DFA method, which mainly used discrete wavelet transform to decompose signals and extract trends. It did not need to divide intervals in advance, and a certain shape of wavelet is used to approximate trends. However, when the data length is short and the spectrum is narrow, the calculation error is larger. WTMM was proposed by Mallat and Zhong [15], which mainly uses continuous wavelet transform for signal analysis, can process strong unsteady time series, and can estimate the local Hölder exponent, but it requires high-quality data and more detailed parameters to be adjusted and it cannot accurately distinguish between single fractal and multiple fractal. MF-DFA was first proposed by Kentelhardt et al. [16]. Compared with the previous method, its estimation result is better overall.Based on the left bank slope of Baihetan hydropower station in Southwestern China, the MF-DFA method was adopted, the multifractal spectrum was estimated, the MF-DFA preset parameters were determined, the multifractal characteristics of the rock microcrack waveform and the blast vibration waveform were comparatively studied, and the nonlinear dynamic characteristics of the MS waveform were revealed. On this basis, the MS waveform multifractal time-varying response characteristics of rock slope deformation and failure process are discussed, and a rock slope deformation early warning model based on multifractal theory is established.


## 2. Engineering Background

### 2.1. Project Overview

The Baihetan hydropower station is located at the junction of Ningnan County in Sichuan Province and Qiaojia County in Yunnan Province, as shown in Figure1. It is the second cascade hydropower station developed in the lower reach of Jinsha river. It is 182 km away from the upstream Wudongde hydropower station and 195 km away from the downstream Xiluodu hydropower station. The hydropower station adopts an all-underground powerhouse layout, and the left and right bank underground powerhouses adopt a symmetrical layout with a total installed capacity of 16000 MW. It is currently the second largest installed capacity hydropower station over the world. The excavation picture of slope excavation site is shown in Figure 2.Figure 1
Geographical location of the Baihetan hydropower station [17].Figure 2
The pictures of slope excavation site: (a) full view of Baihetan hydropower station excavation; (b) the area of left bank slope spandrel groove.


### 2.2. Geological Condition

The two sides of the dam site are syncline geology, and the river valley has an asymmetric ‘V’ shape with the left bank low and the right bank high. The typical section of the left bank slope along the dam arch axis is shown in Figure3(a). The direction of the abutment on the left bank is approximately north-south and inclines 60° to the east. The stratum lithology of the left dam foundation is mainly composed of laminar basalt (P2β) and a small amount of clastic rock (T3x) and limestone (P1m). The weak structural plane in the study area of the left bank slope mainly includes faults F14 and F17, interlayer staggered zones C3-1 and C3 developed along the rock strata, intraformational disturbed zones LS331 and LS337, and many structural cracks, as shown in Figure 3(a). The geological engineering plane figure of left bank slope is shown in Figure 3(b), and the strike rose diagram of dominant joints is shown in Figure 3(c) The mechanical properties of rock masses and weak structural planes were obtained by the experiment of Hydro-China Huadong Engineering Corporation, as shown in Table 1. The maximum principal stress is σ1 = 8.0–11.0 MPa with orientation of N40W and dip angle of 15°, the medium principal stress is σ2 = 7.0–9.0 MPa with orientation of N12E and dip angle of -48°, and the minimum principal stress is σ3 = 6.0–8.0 MPa with orientation of N74E and dip angle of −29° [18].Figure 3
Arch axis geological section of the left bank slope, (a) geological engineering plane figure of left bank slope, (b) and the strike rose diagram of dominant joints (c) [17].
(a)(b)(c)Table 1
Mechanical properties of the main rock masses and weak structural planes.
Rock mass typeCompressive strength (MPa)Elasticity modulus (GPa)Poisson’s ratioDensity (g/cm3)Shear strength/(°)c (MPa)II100150.232.80361.4III170120.242.68361.1III255110.262.60350.75IV3550.322.50330.5F172310.342.13280.15C3-1220.710.352.05280.04C3180.40.342.13280.1LS331231.30.352.15270.06LS337180.20.352.03250.05


## 2.1. Project Overview

The Baihetan hydropower station is located at the junction of Ningnan County in Sichuan Province and Qiaojia County in Yunnan Province, as shown in Figure1. It is the second cascade hydropower station developed in the lower reach of Jinsha river. It is 182 km away from the upstream Wudongde hydropower station and 195 km away from the downstream Xiluodu hydropower station. The hydropower station adopts an all-underground powerhouse layout, and the left and right bank underground powerhouses adopt a symmetrical layout with a total installed capacity of 16000 MW. It is currently the second largest installed capacity hydropower station over the world. The excavation picture of slope excavation site is shown in Figure 2.Figure 1
Geographical location of the Baihetan hydropower station [17].Figure 2
The pictures of slope excavation site: (a) full view of Baihetan hydropower station excavation; (b) the area of left bank slope spandrel groove.


## 2.2. Geological Condition

The two sides of the dam site are syncline geology, and the river valley has an asymmetric ‘V’ shape with the left bank low and the right bank high. The typical section of the left bank slope along the dam arch axis is shown in Figure3(a). The direction of the abutment on the left bank is approximately north-south and inclines 60° to the east. The stratum lithology of the left dam foundation is mainly composed of laminar basalt (P2β) and a small amount of clastic rock (T3x) and limestone (P1m). The weak structural plane in the study area of the left bank slope mainly includes faults F14 and F17, interlayer staggered zones C3-1 and C3 developed along the rock strata, intraformational disturbed zones LS331 and LS337, and many structural cracks, as shown in Figure 3(a). The geological engineering plane figure of left bank slope is shown in Figure 3(b), and the strike rose diagram of dominant joints is shown in Figure 3(c) The mechanical properties of rock masses and weak structural planes were obtained by the experiment of Hydro-China Huadong Engineering Corporation, as shown in Table 1. The maximum principal stress is σ1 = 8.0–11.0 MPa with orientation of N40W and dip angle of 15°, the medium principal stress is σ2 = 7.0–9.0 MPa with orientation of N12E and dip angle of -48°, and the minimum principal stress is σ3 = 6.0–8.0 MPa with orientation of N74E and dip angle of −29° [18].Figure 3
Arch axis geological section of the left bank slope, (a) geological engineering plane figure of left bank slope, (b) and the strike rose diagram of dominant joints (c) [17].
(a)(b)(c)Table 1
Mechanical properties of the main rock masses and weak structural planes.
Rock mass typeCompressive strength (MPa)Elasticity modulus (GPa)Poisson’s ratioDensity (g/cm3)Shear strength/(°)c (MPa)II100150.232.80361.4III170120.242.68361.1III255110.262.60350.75IV3550.322.50330.5F172310.342.13280.15C3-1220.710.352.05280.04C3180.40.342.13280.1LS331231.30.352.15270.06LS337180.20.352.03250.05


## 3. MS Monitoring of the Left Bank Slope

The left bank slope of Baihetan hydropower station adopted the MS monitoring system produced by Canadian ESG (Engineering Seismology Group). The MS monitoring system was successfully installed and operated on November 10, 2014. The network topology of the MS monitoring system is shown in Figure4. The MS monitoring network consists of a Hyperion data processing system, 3 Paladin data acquisition substations, and 18 uniaxial acceleration sensors. The sensors are installed in the sidewalls of three tunnels (i.e., grouting tunnels and drainage tunnels) at different elevations (i.e., 610 m, 660 m, and 750 m). Hyperion data processing system mainly includes HANS signal real-time acquisition and recording software, WaveVis waveform processing software, and SeisVis 3-dimensional visualization software. The HANS signal real-time acquisition and recording software can control parameters such as the sampling frequency, signal gain, and signal trigger threshold of Paladin data acquisition substation. The sampling frequency of the MS monitoring system in the left bank slope of Baihetan is 20 kHz, the threshold value of the ratio of short to long Windows is adopted to trigger, with a threshold value of 3, and the response frequency range of the sensor is 50 Hz∼5 kHz. WaveVis waveform processing and analysis software can automatically identify or manually process the collected waveform files and obtain the time, position, moment magnitude, energy release, and other parameters of MS events through inversion calculation. The SeisVis 3-dimensional visualization software can display the processing results of MS events in real time and identify the potential danger areas by analyzing the activity characteristics of MS events. The sensor receives the elastic wave to generate an electrical signal, which is transmitted to Paladin substation via the cable. Paladin data acquisition substations transmit the collected MS data to Hyperion processing system through communication cable (optical fiber cable). Hyperion data processing system automatically filters background noise and completes record storage of MS events, providing users with complete waveform and spectrum information for analysis and research.Figure 4
Network topology of the MS monitoring system.With comprehensive consideration of economic conditions, technical conditions, and engineering conditions, aiming at the problems of excavation deformation and safety construction of the left bank slope of Baihetan hydropower station, the optimal arrangement plan of sensors of the MS monitoring system is proposed as shown in Figure5.Figure 5
Spatial arrangement diagram of the sensor in the left bank slope.


## 4. Multifractal Detrended Fluctuation Analysis

### 4.1. Basic Principle of the MF-DFA Method

Fractal is generally divided into two categories, one is geometric self-similarity or uniform fractal, and the other is statistical self-similarity or nonuniform fractal, i.e., multifractal. Geometric self-similarity is usually described by a simple fractal dimensionD, but the fractal in nature is generally statistically self-similar, which needs to be described by the multifractal spectrum f(α) − α. Multifractal spectrum f(α) − α, also known as singular spectrum, is a commonly used parameter to describe multifractal. The segmented structure of singular measure can be analyzed by multifractal spectrum [19]. In multifractal calculation, the fractal body will be divided into multiple intervals. Among them, α is the singularity exponent, which represents the fractal dimension of a certain interval, the fractal dimension of each interval can be expressed by the corresponding α, and the intervals with the same α value form a fractal subset. Due to the large number of intervals, a multifractal spectrum function f(α) composed of infinite sequences of different α can be obtained.Multifractal Detrended Fluctuation Analysis (MF-DFA) is an extension of Detrended Fluctuation Analysis (DFA), which can effectively reveal the dynamic behavior in nonlinear and nonstationary signals. Compared with the traditional multifractal calculation method, the core advantages of MF-DFA mainly have three points: (1) make full use of the sequence data length and divide the sequence into equal time length in forward and reverse; (2) fitting polynomials are for each interval by least squares to eliminate the impact of nonstationary trends in time series; (3) analyze the behavior of time series at different levels by using volatility functions of different orders and accurately describe the fractal characteristics of time series, revealing the multifractal characteristics hidden in nonstationary time series.


### 4.2. Algorithm of the MF-DFA Method

The MF-DFA calculation program consists of five steps, of which the first three steps are traditional DFA.Let the time series of the MS waveform signal bexk,k=1,2,…,N, which is a nonlinear and nonstationary sequence.Step 1: construct the signal profile:(1)Yi=∑k=1ixk−x,i=1,2,…,N,wherex is the mean of the time series {x(k)}; i.e.,(2)x=1N∑k=1Nxk.Step 2: divide the signal profileY(i) into Ns intervals of equal time length s; i.e.,(3)Ns=intNs.SinceN is not necessarily an integer multiple of s, the signal profile Y(i) will have a residual value during the division process. In order to make full use of the data and retain this part of the residual value, the above division process can be repeated from the tail of the signal profile Y(i); at this time, 2Ns equal length intervals will be obtained.Step 3: the least square method is used to fit the local trend of the data in each interval in Step 2, and then the variance is calculated. This step is the most time-consuming part of the MF-DFA.Whenv=1,2,⋯,Ns,(4)F2s,v=1s∑i=1sYv−1s+i−yvi2.Whenv=Ns+1,Ns+2,…,2Ns,(5)F2s,v=1s∑i=1sYN−v−Nss+i−yvi2.whereyvi represents the m-order (m = 1, 2, 3, …) fitting polynomial in vth interval.In the time series, the elimination of the ‘trend’ is completed by subtracting the fitting polynomialyvi from the signal profile Y(i), so different fitting orders m can reflect the degree of elimination of ‘trend.’Steps 1–3 are the traditional DFA method.Step 4: calculateq-order volatility function Fq(s):(6)Fqs=12Ns∑v=12NsF2s,vq/21/q.Whenq = 2, MF-DFA degenerates into standard DFA.Step 5: makeq-order volatility function Fq(s) − s double logarithmic graph.If {x(k)} has self-similarity characteristics, i.e., {x(k)} is a multifractal time series, then there is a power law relationship between q-order volatility function Fq(s) and s:(7)Fqs∼shq,where h(q) is the generalized Hurst exponent, which represents the correlation of the original sequence and the size of h(q) depends on the value of q. For stationary time series, when q = 2, h(2) is the same as Hurst exponent. Normally, Fq(s) is an increasing function of s.If {x(k)} is a single fractal time series, F2 (s, v) has the same scale among all intervals, and h(q) is a constant, independent of the value of q.In particular, whenq = 0, formula (6) diverges, and then h0 can be determined by the logarithmic averaging process:(8)F0s≡exp14N∑v=12NslnF2v,s≈sh0.


### 4.3. Estimation of the Standard Multifractal Spectrum by the MF-DFA Method

The generalizedHurst exponent h(q) obtained in the previous section is closely related to the mass exponent τ(q) based on the partition method.Now suppose that the time series {x(j), j = 1, 2, …, N} is a stable and normalized time series. {x(j)} does not need to eliminate the “trend.” Detrended Fluctuation Analysis (DFA) can be simplified and replaced by Fluctuation Analysis (FA). Formula (4) can be rewritten as(9)FFA2s,v≡Yvs−Yv−1s2.SubstituteFFA2s,vinto formulas (6) and (7) and get(10)12Ns∑v=12NsYvs−Yv−1sq1/q∼shq.For simplicity, assume that the length of {x(j)} is an integer multiple of equal time length s; i.e.,(11)Ns=Ns,∑v=1N/sYvs−Yv−1sq∼sqhq−1.According to the definition of profileY(i) constructed in formula (1), the box probability measure Ps(v) can be obtained:(12)Psv≡∑j=v−1svsxj=Yvs−Yv−1s.The mass exponentτ(q) is determined by the partition function χq(s):(13)χqs=∑v=1N/sPsvq∼sτq.By comparing formulas (12) and (13), the following can be obtained:(14)τq=qhq−1.The generalized multifractal dimensionD(q) can be expressed as(15)Dq=τqq−1=qhq−1q−1.It is worth noting that, as mentioned in the previous section, for a single fractal time series,h(q) has nothing to do with the value of q, but the generalized multifractal dimension D(q) is still related to the value of q.The singularity exponentα and the multifractal spectral function f(α) can be obtained through the change of Legendre:(16)α=τ′q=α=hq+qh′q,fα=qα−τq=qα−hq+1.According to the above algorithm, the calculation process is shown in Figure6.Figure 6
MF-DFA calculation flow chart.


### 4.4. Key Parameter Setting of the MF-DFA Method

Multifractal theory has been widely applied in many fields such as physics, biomedicine, economics, materials science, and geology [20]. The characteristics of the nonstationary time series obtained under different conditions are obviously different, such as the length of the signal time window and the fluctuation trend. Therefore, it is necessary to trial calculation and preset the key parameters to obtain more reliable estimation results. Key parameters of multifractal mainly include time length s, weight factor q, and fitting order m. The value of the parameter affects the calculation result from different aspects, and, at the same time, we can understand the application of multifractal theory in engineering MS monitoring from a deeper level. The parameters of the MS waveform of the left bank slope of Baihetan are calculated and preset as follows.


#### 4.4.1. Time Lengths

In the calculation program, the divided time lengths is used to calculate the local fluctuation root mean square (RMS). Before determining the length of time s, there are two important premises: (1) the minimum time length s (smin) must be much larger than the polynomial fitting order m, to prevent the overfitting of the polynomial trend and to prevent the fitting trend from being similar to the time series and making the residual fluctuation close to 0. (2) The maximum length of time s (smax) should make the number of intervals involved in the calculation larger than 10 [21]. Since the time series {x(k)} in MF-DFA algorithm is divided by equal time length in both directions, the number of sampling points should be at least 5 times of the time length smax.Taking the time series of the MS event waveform in the left bank slope of Baihetan at 13 : 30 on June 15, 2016, as an example,q-order volatility function Fq(s) − s double logarithmic graph is shown in Figure 7(a) where the fitting order m = 3 and the different color curves represent Fq(s) − s double logarithmic relationship corresponding to different q values. The value of q ranges from −20 to 20, the equal component Δq = 0.04, and a total of 101 curves were obtained. According to formula (7), the slope of the fitting curve is generalized Hurst exponent, which is used for the estimation of multifractal spectrum, as shown in Figure 7(b). It can be clearly seen from Figure 7 that when Log2(s) = 8∼12, the fitting effect is better. In particular, the local Hurst exponent h(q) with a large time length will show a smooth and slow change trend, which is related to a large calculation interval and a small number of intervals.Figure 7
Determining the length of times: (a) Fq(s) − s double logarithmic graph; (b) generalized Hurst exponent.
(a)(b)Based on the above considerations, the value of the multifractal time lengths of the MS signal in the left bank slope of Baihetan is smin = 28 = 256 and smax = 212 = 4096.


#### 4.4.2. Weight Factorq

The value of the weight factorq should include positive and negative values in order to periodically weight the fluctuation changes in the time series. The size of q value represents the proportion of the RMS of local fluctuations in the whole. For a large q value, it means that the big fluctuation occupies the dominant position in the whole time series. h(q) mainly describes the scale behavior of the big fluctuation. On the contrary, for small q value, small fluctuations are dominant, and h(q) mainly describes the scale behavior of small fluctuations. Therefore, the value of q should avoid larger or smaller values to reduce the error they caused at the tail of the multifractal spectrum. In the actual calculation, when the value of q has no significant effect on the calculation result, the range of value of q can be cut off [22].Four typical MS event waveform time series in the left bank slope of Baihetan were selected for MF-DFA calculation. The absolute value ofq ranged from 0.5 to 50. The calculation results are shown in Table 2. It can be clearly seen from Table 2 that when |q| was 0.5∼20, the value of Δα varies greatly, indicating that the value of q has a great effect on the calculation result. Therefore, the value range of q should be increased. However, when |q| was 20∼50, the variation error was controlled within 0.01, indicating that when |q| was 20, the calculation result has tended to be stable, and the range of q can be cut off.Table 2
Calculation of the width of the multifractal spectrum Δα.
Time of MS eventWeight factor |q|0.51351020305006–30 02 : 46 : 240.33020.50700.59380.66860.76140.79910.80520.807506–24 15 : 50 : 310.36970.53000.66000.73180.82170.85210.85680.859106–18 04 : 37 : 290.09070.18720.42480.53250.62410.65240.65550.656606–15 13 : 30 : 010.41210.67210.89870.99271.06791.08901.09201.0930Note. The calculation result retains four decimal places.The same result can also be obtained fromh(q) − q graph shown in Figure 8. The values of |q| are 0.5, 3, 20, and 50, respectively. When |q| was 0.5, h(q) − q graph was approximately a straight line; when |q| was 3, h(q) − q graph had a certain curvature; and when q was 20 and 50, two curves had a certain similarity. From the perspective of ‘global-local,’ h(q) − q graph when |q| was 0.5 and 3 can be regarded as the ‘local distribution graph’ of the central part of h(q) − q when |q| was 20 and 50. Therefore, it cannot fully reflect the overall trend of h(q) − q graph.Figure 8
Determining the weight factorq: (a) q∈[−0.5, 0.5], (b) q∈[−3, 3], (c) q∈[−20, 20], and (d) q∈[−50, 50].
(a)(b)(c)(d)


#### 4.4.3. Fitting Orderm

In the MF-DFA calculation, the larger fitting orderm can ensure that the multifractal spectrum is not affected by the nonstationary trend in the time series. However, a larger value of m may lead to overfitting of a small sample time series, and the calculation time will increase. In the calculation process, in order to ensure the stability of Fq(s), m should also satisfy m + 2 ≤ s [23].Taking the waveform of the MS event in the left bank slope of Baihetan at 13 : 30 on June 15, 2016, as an example,Fq(s) − s relationships at different m orders are calculated, respectively, and the contour projection is shown in Figure 9. It can be seen from Figure 9 that when m = 1∼2, Fq(s) − s relationship fluctuates greatly and the fitting effect is not good, while when m ≥ 3, the fitting effect is better. Considering the huge amount of data and the calculation time, the fitting order in the multifractal calculation of the MS waveform on the left bank slope of Baihetan is m = 3.Figure 9
Different m-orderFq(s) − s relationship contour projection.In summary, the preset parameters of MS multifractal of the left bank slope of Baihetan hydropower station are as follows:(17)smin=28=256,smax=212=4096,q=20,m=3.


## 4.1. Basic Principle of the MF-DFA Method

Fractal is generally divided into two categories, one is geometric self-similarity or uniform fractal, and the other is statistical self-similarity or nonuniform fractal, i.e., multifractal. Geometric self-similarity is usually described by a simple fractal dimensionD, but the fractal in nature is generally statistically self-similar, which needs to be described by the multifractal spectrum f(α) − α. Multifractal spectrum f(α) − α, also known as singular spectrum, is a commonly used parameter to describe multifractal. The segmented structure of singular measure can be analyzed by multifractal spectrum [19]. In multifractal calculation, the fractal body will be divided into multiple intervals. Among them, α is the singularity exponent, which represents the fractal dimension of a certain interval, the fractal dimension of each interval can be expressed by the corresponding α, and the intervals with the same α value form a fractal subset. Due to the large number of intervals, a multifractal spectrum function f(α) composed of infinite sequences of different α can be obtained.Multifractal Detrended Fluctuation Analysis (MF-DFA) is an extension of Detrended Fluctuation Analysis (DFA), which can effectively reveal the dynamic behavior in nonlinear and nonstationary signals. Compared with the traditional multifractal calculation method, the core advantages of MF-DFA mainly have three points: (1) make full use of the sequence data length and divide the sequence into equal time length in forward and reverse; (2) fitting polynomials are for each interval by least squares to eliminate the impact of nonstationary trends in time series; (3) analyze the behavior of time series at different levels by using volatility functions of different orders and accurately describe the fractal characteristics of time series, revealing the multifractal characteristics hidden in nonstationary time series.


## 4.2. Algorithm of the MF-DFA Method

The MF-DFA calculation program consists of five steps, of which the first three steps are traditional DFA.Let the time series of the MS waveform signal bexk,k=1,2,…,N, which is a nonlinear and nonstationary sequence.Step 1: construct the signal profile:(1)Yi=∑k=1ixk−x,i=1,2,…,N,wherex is the mean of the time series {x(k)}; i.e.,(2)x=1N∑k=1Nxk.Step 2: divide the signal profileY(i) into Ns intervals of equal time length s; i.e.,(3)Ns=intNs.SinceN is not necessarily an integer multiple of s, the signal profile Y(i) will have a residual value during the division process. In order to make full use of the data and retain this part of the residual value, the above division process can be repeated from the tail of the signal profile Y(i); at this time, 2Ns equal length intervals will be obtained.Step 3: the least square method is used to fit the local trend of the data in each interval in Step 2, and then the variance is calculated. This step is the most time-consuming part of the MF-DFA.Whenv=1,2,⋯,Ns,(4)F2s,v=1s∑i=1sYv−1s+i−yvi2.Whenv=Ns+1,Ns+2,…,2Ns,(5)F2s,v=1s∑i=1sYN−v−Nss+i−yvi2.whereyvi represents the m-order (m = 1, 2, 3, …) fitting polynomial in vth interval.In the time series, the elimination of the ‘trend’ is completed by subtracting the fitting polynomialyvi from the signal profile Y(i), so different fitting orders m can reflect the degree of elimination of ‘trend.’Steps 1–3 are the traditional DFA method.Step 4: calculateq-order volatility function Fq(s):(6)Fqs=12Ns∑v=12NsF2s,vq/21/q.Whenq = 2, MF-DFA degenerates into standard DFA.Step 5: makeq-order volatility function Fq(s) − s double logarithmic graph.If {x(k)} has self-similarity characteristics, i.e., {x(k)} is a multifractal time series, then there is a power law relationship between q-order volatility function Fq(s) and s:(7)Fqs∼shq,where h(q) is the generalized Hurst exponent, which represents the correlation of the original sequence and the size of h(q) depends on the value of q. For stationary time series, when q = 2, h(2) is the same as Hurst exponent. Normally, Fq(s) is an increasing function of s.If {x(k)} is a single fractal time series, F2 (s, v) has the same scale among all intervals, and h(q) is a constant, independent of the value of q.In particular, whenq = 0, formula (6) diverges, and then h0 can be determined by the logarithmic averaging process:(8)F0s≡exp14N∑v=12NslnF2v,s≈sh0.


## 4.3. Estimation of the Standard Multifractal Spectrum by the MF-DFA Method

The generalizedHurst exponent h(q) obtained in the previous section is closely related to the mass exponent τ(q) based on the partition method.Now suppose that the time series {x(j), j = 1, 2, …, N} is a stable and normalized time series. {x(j)} does not need to eliminate the “trend.” Detrended Fluctuation Analysis (DFA) can be simplified and replaced by Fluctuation Analysis (FA). Formula (4) can be rewritten as(9)FFA2s,v≡Yvs−Yv−1s2.SubstituteFFA2s,vinto formulas (6) and (7) and get(10)12Ns∑v=12NsYvs−Yv−1sq1/q∼shq.For simplicity, assume that the length of {x(j)} is an integer multiple of equal time length s; i.e.,(11)Ns=Ns,∑v=1N/sYvs−Yv−1sq∼sqhq−1.According to the definition of profileY(i) constructed in formula (1), the box probability measure Ps(v) can be obtained:(12)Psv≡∑j=v−1svsxj=Yvs−Yv−1s.The mass exponentτ(q) is determined by the partition function χq(s):(13)χqs=∑v=1N/sPsvq∼sτq.By comparing formulas (12) and (13), the following can be obtained:(14)τq=qhq−1.The generalized multifractal dimensionD(q) can be expressed as(15)Dq=τqq−1=qhq−1q−1.It is worth noting that, as mentioned in the previous section, for a single fractal time series,h(q) has nothing to do with the value of q, but the generalized multifractal dimension D(q) is still related to the value of q.The singularity exponentα and the multifractal spectral function f(α) can be obtained through the change of Legendre:(16)α=τ′q=α=hq+qh′q,fα=qα−τq=qα−hq+1.According to the above algorithm, the calculation process is shown in Figure6.Figure 6
MF-DFA calculation flow chart.


## 4.4. Key Parameter Setting of the MF-DFA Method

Multifractal theory has been widely applied in many fields such as physics, biomedicine, economics, materials science, and geology [20]. The characteristics of the nonstationary time series obtained under different conditions are obviously different, such as the length of the signal time window and the fluctuation trend. Therefore, it is necessary to trial calculation and preset the key parameters to obtain more reliable estimation results. Key parameters of multifractal mainly include time length s, weight factor q, and fitting order m. The value of the parameter affects the calculation result from different aspects, and, at the same time, we can understand the application of multifractal theory in engineering MS monitoring from a deeper level. The parameters of the MS waveform of the left bank slope of Baihetan are calculated and preset as follows.


### 4.4.1. Time Lengths

In the calculation program, the divided time lengths is used to calculate the local fluctuation root mean square (RMS). Before determining the length of time s, there are two important premises: (1) the minimum time length s (smin) must be much larger than the polynomial fitting order m, to prevent the overfitting of the polynomial trend and to prevent the fitting trend from being similar to the time series and making the residual fluctuation close to 0. (2) The maximum length of time s (smax) should make the number of intervals involved in the calculation larger than 10 [21]. Since the time series {x(k)} in MF-DFA algorithm is divided by equal time length in both directions, the number of sampling points should be at least 5 times of the time length smax.Taking the time series of the MS event waveform in the left bank slope of Baihetan at 13 : 30 on June 15, 2016, as an example,q-order volatility function Fq(s) − s double logarithmic graph is shown in Figure 7(a) where the fitting order m = 3 and the different color curves represent Fq(s) − s double logarithmic relationship corresponding to different q values. The value of q ranges from −20 to 20, the equal component Δq = 0.04, and a total of 101 curves were obtained. According to formula (7), the slope of the fitting curve is generalized Hurst exponent, which is used for the estimation of multifractal spectrum, as shown in Figure 7(b). It can be clearly seen from Figure 7 that when Log2(s) = 8∼12, the fitting effect is better. In particular, the local Hurst exponent h(q) with a large time length will show a smooth and slow change trend, which is related to a large calculation interval and a small number of intervals.Figure 7
Determining the length of times: (a) Fq(s) − s double logarithmic graph; (b) generalized Hurst exponent.
(a)(b)Based on the above considerations, the value of the multifractal time lengths of the MS signal in the left bank slope of Baihetan is smin = 28 = 256 and smax = 212 = 4096.


### 4.4.2. Weight Factorq

The value of the weight factorq should include positive and negative values in order to periodically weight the fluctuation changes in the time series. The size of q value represents the proportion of the RMS of local fluctuations in the whole. For a large q value, it means that the big fluctuation occupies the dominant position in the whole time series. h(q) mainly describes the scale behavior of the big fluctuation. On the contrary, for small q value, small fluctuations are dominant, and h(q) mainly describes the scale behavior of small fluctuations. Therefore, the value of q should avoid larger or smaller values to reduce the error they caused at the tail of the multifractal spectrum. In the actual calculation, when the value of q has no significant effect on the calculation result, the range of value of q can be cut off [22].Four typical MS event waveform time series in the left bank slope of Baihetan were selected for MF-DFA calculation. The absolute value ofq ranged from 0.5 to 50. The calculation results are shown in Table 2. It can be clearly seen from Table 2 that when |q| was 0.5∼20, the value of Δα varies greatly, indicating that the value of q has a great effect on the calculation result. Therefore, the value range of q should be increased. However, when |q| was 20∼50, the variation error was controlled within 0.01, indicating that when |q| was 20, the calculation result has tended to be stable, and the range of q can be cut off.Table 2
Calculation of the width of the multifractal spectrum Δα.
Time of MS eventWeight factor |q|0.51351020305006–30 02 : 46 : 240.33020.50700.59380.66860.76140.79910.80520.807506–24 15 : 50 : 310.36970.53000.66000.73180.82170.85210.85680.859106–18 04 : 37 : 290.09070.18720.42480.53250.62410.65240.65550.656606–15 13 : 30 : 010.41210.67210.89870.99271.06791.08901.09201.0930Note. The calculation result retains four decimal places.The same result can also be obtained fromh(q) − q graph shown in Figure 8. The values of |q| are 0.5, 3, 20, and 50, respectively. When |q| was 0.5, h(q) − q graph was approximately a straight line; when |q| was 3, h(q) − q graph had a certain curvature; and when q was 20 and 50, two curves had a certain similarity. From the perspective of ‘global-local,’ h(q) − q graph when |q| was 0.5 and 3 can be regarded as the ‘local distribution graph’ of the central part of h(q) − q when |q| was 20 and 50. Therefore, it cannot fully reflect the overall trend of h(q) − q graph.Figure 8
Determining the weight factorq: (a) q∈[−0.5, 0.5], (b) q∈[−3, 3], (c) q∈[−20, 20], and (d) q∈[−50, 50].
(a)(b)(c)(d)


### 4.4.3. Fitting Orderm

In the MF-DFA calculation, the larger fitting orderm can ensure that the multifractal spectrum is not affected by the nonstationary trend in the time series. However, a larger value of m may lead to overfitting of a small sample time series, and the calculation time will increase. In the calculation process, in order to ensure the stability of Fq(s), m should also satisfy m + 2 ≤ s [23].Taking the waveform of the MS event in the left bank slope of Baihetan at 13 : 30 on June 15, 2016, as an example,Fq(s) − s relationships at different m orders are calculated, respectively, and the contour projection is shown in Figure 9. It can be seen from Figure 9 that when m = 1∼2, Fq(s) − s relationship fluctuates greatly and the fitting effect is not good, while when m ≥ 3, the fitting effect is better. Considering the huge amount of data and the calculation time, the fitting order in the multifractal calculation of the MS waveform on the left bank slope of Baihetan is m = 3.Figure 9
Different m-orderFq(s) − s relationship contour projection.In summary, the preset parameters of MS multifractal of the left bank slope of Baihetan hydropower station are as follows:(17)smin=28=256,smax=212=4096,q=20,m=3.


## 4.4.1. Time Lengths

In the calculation program, the divided time lengths is used to calculate the local fluctuation root mean square (RMS). Before determining the length of time s, there are two important premises: (1) the minimum time length s (smin) must be much larger than the polynomial fitting order m, to prevent the overfitting of the polynomial trend and to prevent the fitting trend from being similar to the time series and making the residual fluctuation close to 0. (2) The maximum length of time s (smax) should make the number of intervals involved in the calculation larger than 10 [21]. Since the time series {x(k)} in MF-DFA algorithm is divided by equal time length in both directions, the number of sampling points should be at least 5 times of the time length smax.Taking the time series of the MS event waveform in the left bank slope of Baihetan at 13 : 30 on June 15, 2016, as an example,q-order volatility function Fq(s) − s double logarithmic graph is shown in Figure 7(a) where the fitting order m = 3 and the different color curves represent Fq(s) − s double logarithmic relationship corresponding to different q values. The value of q ranges from −20 to 20, the equal component Δq = 0.04, and a total of 101 curves were obtained. According to formula (7), the slope of the fitting curve is generalized Hurst exponent, which is used for the estimation of multifractal spectrum, as shown in Figure 7(b). It can be clearly seen from Figure 7 that when Log2(s) = 8∼12, the fitting effect is better. In particular, the local Hurst exponent h(q) with a large time length will show a smooth and slow change trend, which is related to a large calculation interval and a small number of intervals.Figure 7
Determining the length of times: (a) Fq(s) − s double logarithmic graph; (b) generalized Hurst exponent.
(a)(b)Based on the above considerations, the value of the multifractal time lengths of the MS signal in the left bank slope of Baihetan is smin = 28 = 256 and smax = 212 = 4096.


## 4.4.2. Weight Factorq

The value of the weight factorq should include positive and negative values in order to periodically weight the fluctuation changes in the time series. The size of q value represents the proportion of the RMS of local fluctuations in the whole. For a large q value, it means that the big fluctuation occupies the dominant position in the whole time series. h(q) mainly describes the scale behavior of the big fluctuation. On the contrary, for small q value, small fluctuations are dominant, and h(q) mainly describes the scale behavior of small fluctuations. Therefore, the value of q should avoid larger or smaller values to reduce the error they caused at the tail of the multifractal spectrum. In the actual calculation, when the value of q has no significant effect on the calculation result, the range of value of q can be cut off [22].Four typical MS event waveform time series in the left bank slope of Baihetan were selected for MF-DFA calculation. The absolute value ofq ranged from 0.5 to 50. The calculation results are shown in Table 2. It can be clearly seen from Table 2 that when |q| was 0.5∼20, the value of Δα varies greatly, indicating that the value of q has a great effect on the calculation result. Therefore, the value range of q should be increased. However, when |q| was 20∼50, the variation error was controlled within 0.01, indicating that when |q| was 20, the calculation result has tended to be stable, and the range of q can be cut off.Table 2
Calculation of the width of the multifractal spectrum Δα.
Time of MS eventWeight factor |q|0.51351020305006–30 02 : 46 : 240.33020.50700.59380.66860.76140.79910.80520.807506–24 15 : 50 : 310.36970.53000.66000.73180.82170.85210.85680.859106–18 04 : 37 : 290.09070.18720.42480.53250.62410.65240.65550.656606–15 13 : 30 : 010.41210.67210.89870.99271.06791.08901.09201.0930Note. The calculation result retains four decimal places.The same result can also be obtained fromh(q) − q graph shown in Figure 8. The values of |q| are 0.5, 3, 20, and 50, respectively. When |q| was 0.5, h(q) − q graph was approximately a straight line; when |q| was 3, h(q) − q graph had a certain curvature; and when q was 20 and 50, two curves had a certain similarity. From the perspective of ‘global-local,’ h(q) − q graph when |q| was 0.5 and 3 can be regarded as the ‘local distribution graph’ of the central part of h(q) − q when |q| was 20 and 50. Therefore, it cannot fully reflect the overall trend of h(q) − q graph.Figure 8
Determining the weight factorq: (a) q∈[−0.5, 0.5], (b) q∈[−3, 3], (c) q∈[−20, 20], and (d) q∈[−50, 50].
(a)(b)(c)(d)


## 4.4.3. Fitting Orderm

In the MF-DFA calculation, the larger fitting orderm can ensure that the multifractal spectrum is not affected by the nonstationary trend in the time series. However, a larger value of m may lead to overfitting of a small sample time series, and the calculation time will increase. In the calculation process, in order to ensure the stability of Fq(s), m should also satisfy m + 2 ≤ s [23].Taking the waveform of the MS event in the left bank slope of Baihetan at 13 : 30 on June 15, 2016, as an example,Fq(s) − s relationships at different m orders are calculated, respectively, and the contour projection is shown in Figure 9. It can be seen from Figure 9 that when m = 1∼2, Fq(s) − s relationship fluctuates greatly and the fitting effect is not good, while when m ≥ 3, the fitting effect is better. Considering the huge amount of data and the calculation time, the fitting order in the multifractal calculation of the MS waveform on the left bank slope of Baihetan is m = 3.Figure 9
Different m-orderFq(s) − s relationship contour projection.In summary, the preset parameters of MS multifractal of the left bank slope of Baihetan hydropower station are as follows:(17)smin=28=256,smax=212=4096,q=20,m=3.


## 5. Multifractal Characteristics of MS Signals

The signals collected by MS monitoring of rock slope mainly include rock microfracture signals (MS signals), blasting vibration signals (blasting), mechanical vibration signals, current interference signals, car whistle, and unknown signals. The multifractal characteristics of different signals are obviously different. The following mainly analyzes the multifractal characteristics of rock microfracture and blasting vibration waveform time series by MF-DFA method.


### 5.1. Multifractal Spectrum of MS Signals

Figure10 shows the typical rock microfracture signal and rock blasting vibration signal, and Figure 11 shows the multifractal spectrum corresponding to the two types of typical signals.Figure 10
Typical signal waveform: (a) rock microfracture signals; (b) blasting vibration signals.
(a)(b)Figure 11
Multifractal spectrum of typical signal: (a) MS signal; (b) blasting signal.
(a)(b)In Figure11, Δα is the width of the multifractal spectrum, which represents the multifractal strength of the waveform and the complexity of the fluctuation. The larger the Δα is, the greater the multifractal strength of the waveform is, and the more intense and complex the fluctuation is. And vice versa, the calculation can be expressed as Δα = αmax − αmin. It can be seen from Figure 11 that the multifractal spectrum width of rock microfracture waveform Δα1 = 0.99 is much smaller than the multifractal spectrum width of blasting vibration waveform Δα2 = 3.14, indicating that the multifractal strength of blasting vibration waveform is larger and the fluctuation is more intense and complex. Δf(α) represents the proportion of large fluctuations and small fluctuations in the waveform. The larger the Δf(α) is, the larger the proportion of small fluctuations in the waveform will be and vice versa. The calculation can be expressed as Δf(α) = f(αmax) − f(αmin). It can be seen from Figure 11 that the multifractal spectrum of rock microfracture waveform Δf(α1) = −0.06, which is larger than the multifractal spectrum of blasting Δf(α2) = -0.21, indicating that small fluctuations in the MS waveform account for a large proportion.


### 5.2. Multifractal Characteristics of Rock Microfracture Waveform with Background Noise

In the previous section, the multifractal characteristic of a typical MS waveforms is described. However, in actual MS monitoring of rock slope, the collected MS signals were often mixed with various types of noise due to the complexity of the engineering field construction and the rock mass itself. The following studied the effect of background noise on multifractal spectrum estimation by MF-DFA calculation.The MS waveformsA∼J with background noise collected by different channels at the same time were selected for MF-DFA calculation, and the waveform is shown in Figure 12. It can be seen from Figure 12 that the amplitudes of the waveforms from A∼J are in descending order, including ADHJ with a large background noise and BCEF with a small background noise. The mass exponent τ(q) − q graph and the multifractal spectrum f(α − α) graph were made for A∼J, as shown in Figure 13. Figures 13(a) and 13(b) show the multifractal characteristics of waveform corresponding to the window size of 1500 ms, and Figures 13(c) and 13(d) show the multifractal characteristics of the waveform corresponding to the window size of 750 ms (i.e., 375 ms to 1125 ms in Figure 12). There are two main advantages of focusing the multifractal analysis window on the effective waveform area: (1) speeding up calculation efficiency, suitable for a large number of waveform multifractal calculations, and (2) better reflecting the multifractal characteristics of effective waveforms, which is beneficial to distinguish the multifractal difference of waveform produced by different inducements. In general, different waveforms for multifractal difference analysis should ensure the same time window size.Figure 12
MS waveforms with background noises.Figure 13
Multifractal characteristics of MS waveform with background noise: (a)τ(q) − q graph with window size of 1500 ms; (b) f(α) − α graph with window size of 1500 ms; (c) τ(q) − q graph with window size of 750 ms; (d) f(α) − α graph with window size of 750 ms.
(a)(b)(c)(d)According to the mass exponentτ(q) − q graph shown in Figure 13(a), the curve corresponding to the rock microfracture waveform ADHJ with background noise is located above the left end and below the right end in the mass exponent τ(q) − q graph. Meanwhile, the waveform ADHJ has a smaller αmax in the multifractal spectrum f(α) − α diagram shown in Figure 13(b). The time window was narrowed to make the rock microfracture waveform fill the window as much as possible, to improve the effectiveness of waveform multifractal analysis, and MF-DFA was calculated again. The results of the recalculation are shown in Figures 13(c) and 13(d). The calculation results are more obvious: the width of the multifractal spectrum Δα of the rock microfracture waveform ADHJ with large background noise is small, and the microfracture waveform BC with low background noise and large amplitude has a larger spectrum width Δα.This result reflects the effectiveness of MF-DFA method in removing nonstationary trend; that is, MF-DFA method puts more emphasis on the multifractal characteristics of rock microfracture waveform after removing noise. Therefore, when the MF-DFA is used to analyze the MS waveform data collected at different channels without sufficient noise filtering, the waveform with small background noise and large amplitude should be selected for analysis as much as possible to improve the reliability of multifractal spectrum estimation.


## 5.1. Multifractal Spectrum of MS Signals

Figure10 shows the typical rock microfracture signal and rock blasting vibration signal, and Figure 11 shows the multifractal spectrum corresponding to the two types of typical signals.Figure 10
Typical signal waveform: (a) rock microfracture signals; (b) blasting vibration signals.
(a)(b)Figure 11
Multifractal spectrum of typical signal: (a) MS signal; (b) blasting signal.
(a)(b)In Figure11, Δα is the width of the multifractal spectrum, which represents the multifractal strength of the waveform and the complexity of the fluctuation. The larger the Δα is, the greater the multifractal strength of the waveform is, and the more intense and complex the fluctuation is. And vice versa, the calculation can be expressed as Δα = αmax − αmin. It can be seen from Figure 11 that the multifractal spectrum width of rock microfracture waveform Δα1 = 0.99 is much smaller than the multifractal spectrum width of blasting vibration waveform Δα2 = 3.14, indicating that the multifractal strength of blasting vibration waveform is larger and the fluctuation is more intense and complex. Δf(α) represents the proportion of large fluctuations and small fluctuations in the waveform. The larger the Δf(α) is, the larger the proportion of small fluctuations in the waveform will be and vice versa. The calculation can be expressed as Δf(α) = f(αmax) − f(αmin). It can be seen from Figure 11 that the multifractal spectrum of rock microfracture waveform Δf(α1) = −0.06, which is larger than the multifractal spectrum of blasting Δf(α2) = -0.21, indicating that small fluctuations in the MS waveform account for a large proportion.


## 5.2. Multifractal Characteristics of Rock Microfracture Waveform with Background Noise

In the previous section, the multifractal characteristic of a typical MS waveforms is described. However, in actual MS monitoring of rock slope, the collected MS signals were often mixed with various types of noise due to the complexity of the engineering field construction and the rock mass itself. The following studied the effect of background noise on multifractal spectrum estimation by MF-DFA calculation.The MS waveformsA∼J with background noise collected by different channels at the same time were selected for MF-DFA calculation, and the waveform is shown in Figure 12. It can be seen from Figure 12 that the amplitudes of the waveforms from A∼J are in descending order, including ADHJ with a large background noise and BCEF with a small background noise. The mass exponent τ(q) − q graph and the multifractal spectrum f(α − α) graph were made for A∼J, as shown in Figure 13. Figures 13(a) and 13(b) show the multifractal characteristics of waveform corresponding to the window size of 1500 ms, and Figures 13(c) and 13(d) show the multifractal characteristics of the waveform corresponding to the window size of 750 ms (i.e., 375 ms to 1125 ms in Figure 12). There are two main advantages of focusing the multifractal analysis window on the effective waveform area: (1) speeding up calculation efficiency, suitable for a large number of waveform multifractal calculations, and (2) better reflecting the multifractal characteristics of effective waveforms, which is beneficial to distinguish the multifractal difference of waveform produced by different inducements. In general, different waveforms for multifractal difference analysis should ensure the same time window size.Figure 12
MS waveforms with background noises.Figure 13
Multifractal characteristics of MS waveform with background noise: (a)τ(q) − q graph with window size of 1500 ms; (b) f(α) − α graph with window size of 1500 ms; (c) τ(q) − q graph with window size of 750 ms; (d) f(α) − α graph with window size of 750 ms.
(a)(b)(c)(d)According to the mass exponentτ(q) − q graph shown in Figure 13(a), the curve corresponding to the rock microfracture waveform ADHJ with background noise is located above the left end and below the right end in the mass exponent τ(q) − q graph. Meanwhile, the waveform ADHJ has a smaller αmax in the multifractal spectrum f(α) − α diagram shown in Figure 13(b). The time window was narrowed to make the rock microfracture waveform fill the window as much as possible, to improve the effectiveness of waveform multifractal analysis, and MF-DFA was calculated again. The results of the recalculation are shown in Figures 13(c) and 13(d). The calculation results are more obvious: the width of the multifractal spectrum Δα of the rock microfracture waveform ADHJ with large background noise is small, and the microfracture waveform BC with low background noise and large amplitude has a larger spectrum width Δα.This result reflects the effectiveness of MF-DFA method in removing nonstationary trend; that is, MF-DFA method puts more emphasis on the multifractal characteristics of rock microfracture waveform after removing noise. Therefore, when the MF-DFA is used to analyze the MS waveform data collected at different channels without sufficient noise filtering, the waveform with small background noise and large amplitude should be selected for analysis as much as possible to improve the reliability of multifractal spectrum estimation.


## 6. Multifractal Time-Varying Response Characteristics of MS Waveform and Deformation Warning of Rock Slope

### 6.1. Multifractal Time-Varying Response Characteristics of MS Waveform

Multifractal analysis was conducted on the MS waveforms of the rock mass near the crack T3301 (rock mass within 570 m∼630 m elevation) of the left bank slope of Baihetan on June 1 to July 10, 2016. The crack T3301 is shown in Figure14. The multifractal spectrum parameter Δα reflects the multifractal strength and the degree of fluctuation of the waveform time series, while Δf(α) reflects the proportion of large and small fluctuations in the waveform time series. The multifractal spectrum parameters Δα and Δf(α) represent the multifractal characteristics of the time series of rock microfracture waveform. The deformation and instability failure of rock slope is caused by the initiation, development, expansion, interaction, and connection of microfractures in rock mass, which is a nonstationary dynamic evolution process in time series. Therefore, it is necessary to focus on the analysis of the time-varying multifractal characteristics of the rock microfracture waveform during the monitoring period. Combined with the field macrofailure law, a rock slope deformation early warning model based on the MS waveform multifractal time-varying response characteristics was established.Figure 14
The location and photograph of T3301 crack.Figure15 shows the time-varying law of multifractal spectrum parameters Δα and Δf(α) of the MS waveform of the rock mass near the crack T3301. In Figure 15, the upper limit of Δα and the lower limit of Δf(α) have very obvious time series characteristics, so the evolutionary trend of Δα takes the upper limit of the MS event, and the evolutionary trend of Δf(α) takes the lower limit. As can be seen from Figure 15, since June 1, the number of MS events had not increased rapidly, but the overall Δα showed a sharp increase trend, reaching a maximum of 1.53, indicating that the multifractal strength of rock microfracture waveform increased and the fluctuations became complex and intense. The corresponding Δf(α) showed a sharp decrease trend; the minimum reached −0.58, indicating that the proportion of large fluctuations in the waveform time series increased. This was due to the strong unloading effect of blasting excavation, which hinders the further expansion of microcracks in hard rock mass. The number of MS events induced was small, but the local stress was highly concentrated and the strain energy was increased, that is, the ‘quiet period’ before deformation and failure [24]. After June 15, Δα briefly fell back to the valley and then increased to 1.3 again and the corresponding Δf(α) increased from the valley. In this process, the local stress continued to increase until the rock bearing capacity was exceeded, resulting in the connection of microcracks at some locations, the stress concentration position began to shift, and the strain energy began to be released. When the crack expansion was blocked again, the local stress reconcentrated, and the strain energy accumulated again. After June 29, Δα decreased as a whole and was in a stable state, indicating that the fluctuation of the microfracture waveform time series was relatively smooth. And the value of Δf(α) also stabilizes near the zero line after a small increase. At this time, the microcracks almost were completely connected under the action of high stress, forming cracks, causing macroscopic failure, and the stress and strain energy were released.Figure 15
Evolution characteristics of multifractal spectrum parameters vs. time: (a) Δα characteristics with time; (b) Δf(α) characteristics with time.
(a)(b)


### 6.2. Comparison with Conventional Monitoring

The time-varying response characteristics of multifractal spectrum parameters Δα and Δf(α) were closely related to the initiation, development, expansion, and penetration of rock microfractures. Before the deformation and failure of the rock mass, Δα showed an increasing trend and Δf(α) showed a decreasing trend, which can be regarded as a precursor signal of deformation warning; when deformation and failure occurred, Δα showed a decreasing trend and Δf(α) showed an increasing trend, which can be regarded as a deformation failure period; after deformation and failure, both Δα and Δf(α) showed a steady trend, and Δf(α) as a whole would be near the zero line, which can be regarded as a stable period. The division result is shown in Figure 16. In particular, when Δα and Δf(α) increase and decrease for several times, this indicates that the stress concentration degree is getting higher and higher, and the strain energy is accumulating more and more, which indicates that larger deformation and failure will occur, and it is also regarded as the deformation period. Therefore, reinforcement measures should be taken immediately for the slope to timely control the continuous growth of cracks and prevent the failure of slope deformation and instability.Figure 16
Stage division based on MF-DFA.Compared with conventional monitoring, the monitoring results of displacement meter CX04 (Figure17(a)) installed at the height of 615 m are shown in Figure 17(b). Since the installation date of CX04 in June 17, the cumulative deformation of the T3301 crack was 1.35 mm, indicating that the deformation of T3301 was in a slow-growing state, that is, in the ‘precursor period.’ From June 17 to July 8, the cumulative deformation displacement of the T3301 crack was about 6.30 mm. And it was the largest deformation region within the monitoring range of CX04 during the period; the deformation rate reached about 0.3 mm/d; this period was in the ‘deformation period.’ From July 8 to July 15, the cumulative deformation displacement of the T3301 crack was 0.87 mm, and the average deformation rate was reduced to 0.11 mm/d. And, after July 12, the rock mass had almost no deformation and reached a stable state, which corresponded to the ‘stable period.’ Combined with field construction, at the end of June and early July, the prestressed anchor cable was installed at the 605–600 m elevation of the dam foundation slope. It enhanced the integrity of the slope rock mass, increased the bearing capacity of the slope rock mass to a certain extent, increased the antisliding friction resistance of the unstable surface, improved the stress adjustment path, effectively controlled the further development of crack, and made the slope rock mass temporarily stable. The early warning analysis of rock slope deformation and failure based on the multifractal time-varying response characteristics of MS signals had a good correspondence with conventional monitoring in time and space. It can accurately describe the mechanical response characteristics of rock slope under the action of excavation unloading. It proves the feasibility of the early warning method of rock slope deformation and failure in this study, which can provide an important reference for rock slope design and safe construction.Figure 17
Conventional monitoring result: (a) the location of the CX04, and (b) cumulative displacement data of CX04.
(a)(b)


### 6.3. Early Warning Model of Rock Slope Deformation Based on the MF-DFA Method

In actual situations, it is impossible to know when Δα and Δf(α) reach the peak or valley before large deformation occurs, so the ‘precursor period’ and ‘deformation period’ should not be divided at the peak or valley. The warning model is optimized according to the actual situation, as shown in Figure 18. After a period of ‘stable period,’ Δα shows an increasing trend for the first time and Δf(α) shows a decreasing trend for the first time, then entering the ‘precursory period.’ Δα reaches the peak and shows a decreasing trend, Δf(α) drops to the valley and shows an increasing trend, then the ‘precursor period’ ends, and the ‘deformation period’ starts. If Δα and Δf(α) increase and decrease for several times afterwards, it is still considered to be in the ‘deformation period’ until the arrival of the next ‘stable period.’Figure 18
MF-DFA multifractal warning model.
(a)(b)


## 6.1. Multifractal Time-Varying Response Characteristics of MS Waveform

Multifractal analysis was conducted on the MS waveforms of the rock mass near the crack T3301 (rock mass within 570 m∼630 m elevation) of the left bank slope of Baihetan on June 1 to July 10, 2016. The crack T3301 is shown in Figure14. The multifractal spectrum parameter Δα reflects the multifractal strength and the degree of fluctuation of the waveform time series, while Δf(α) reflects the proportion of large and small fluctuations in the waveform time series. The multifractal spectrum parameters Δα and Δf(α) represent the multifractal characteristics of the time series of rock microfracture waveform. The deformation and instability failure of rock slope is caused by the initiation, development, expansion, interaction, and connection of microfractures in rock mass, which is a nonstationary dynamic evolution process in time series. Therefore, it is necessary to focus on the analysis of the time-varying multifractal characteristics of the rock microfracture waveform during the monitoring period. Combined with the field macrofailure law, a rock slope deformation early warning model based on the MS waveform multifractal time-varying response characteristics was established.Figure 14
The location and photograph of T3301 crack.Figure15 shows the time-varying law of multifractal spectrum parameters Δα and Δf(α) of the MS waveform of the rock mass near the crack T3301. In Figure 15, the upper limit of Δα and the lower limit of Δf(α) have very obvious time series characteristics, so the evolutionary trend of Δα takes the upper limit of the MS event, and the evolutionary trend of Δf(α) takes the lower limit. As can be seen from Figure 15, since June 1, the number of MS events had not increased rapidly, but the overall Δα showed a sharp increase trend, reaching a maximum of 1.53, indicating that the multifractal strength of rock microfracture waveform increased and the fluctuations became complex and intense. The corresponding Δf(α) showed a sharp decrease trend; the minimum reached −0.58, indicating that the proportion of large fluctuations in the waveform time series increased. This was due to the strong unloading effect of blasting excavation, which hinders the further expansion of microcracks in hard rock mass. The number of MS events induced was small, but the local stress was highly concentrated and the strain energy was increased, that is, the ‘quiet period’ before deformation and failure [24]. After June 15, Δα briefly fell back to the valley and then increased to 1.3 again and the corresponding Δf(α) increased from the valley. In this process, the local stress continued to increase until the rock bearing capacity was exceeded, resulting in the connection of microcracks at some locations, the stress concentration position began to shift, and the strain energy began to be released. When the crack expansion was blocked again, the local stress reconcentrated, and the strain energy accumulated again. After June 29, Δα decreased as a whole and was in a stable state, indicating that the fluctuation of the microfracture waveform time series was relatively smooth. And the value of Δf(α) also stabilizes near the zero line after a small increase. At this time, the microcracks almost were completely connected under the action of high stress, forming cracks, causing macroscopic failure, and the stress and strain energy were released.Figure 15
Evolution characteristics of multifractal spectrum parameters vs. time: (a) Δα characteristics with time; (b) Δf(α) characteristics with time.
(a)(b)


## 6.2. Comparison with Conventional Monitoring

The time-varying response characteristics of multifractal spectrum parameters Δα and Δf(α) were closely related to the initiation, development, expansion, and penetration of rock microfractures. Before the deformation and failure of the rock mass, Δα showed an increasing trend and Δf(α) showed a decreasing trend, which can be regarded as a precursor signal of deformation warning; when deformation and failure occurred, Δα showed a decreasing trend and Δf(α) showed an increasing trend, which can be regarded as a deformation failure period; after deformation and failure, both Δα and Δf(α) showed a steady trend, and Δf(α) as a whole would be near the zero line, which can be regarded as a stable period. The division result is shown in Figure 16. In particular, when Δα and Δf(α) increase and decrease for several times, this indicates that the stress concentration degree is getting higher and higher, and the strain energy is accumulating more and more, which indicates that larger deformation and failure will occur, and it is also regarded as the deformation period. Therefore, reinforcement measures should be taken immediately for the slope to timely control the continuous growth of cracks and prevent the failure of slope deformation and instability.Figure 16
Stage division based on MF-DFA.Compared with conventional monitoring, the monitoring results of displacement meter CX04 (Figure17(a)) installed at the height of 615 m are shown in Figure 17(b). Since the installation date of CX04 in June 17, the cumulative deformation of the T3301 crack was 1.35 mm, indicating that the deformation of T3301 was in a slow-growing state, that is, in the ‘precursor period.’ From June 17 to July 8, the cumulative deformation displacement of the T3301 crack was about 6.30 mm. And it was the largest deformation region within the monitoring range of CX04 during the period; the deformation rate reached about 0.3 mm/d; this period was in the ‘deformation period.’ From July 8 to July 15, the cumulative deformation displacement of the T3301 crack was 0.87 mm, and the average deformation rate was reduced to 0.11 mm/d. And, after July 12, the rock mass had almost no deformation and reached a stable state, which corresponded to the ‘stable period.’ Combined with field construction, at the end of June and early July, the prestressed anchor cable was installed at the 605–600 m elevation of the dam foundation slope. It enhanced the integrity of the slope rock mass, increased the bearing capacity of the slope rock mass to a certain extent, increased the antisliding friction resistance of the unstable surface, improved the stress adjustment path, effectively controlled the further development of crack, and made the slope rock mass temporarily stable. The early warning analysis of rock slope deformation and failure based on the multifractal time-varying response characteristics of MS signals had a good correspondence with conventional monitoring in time and space. It can accurately describe the mechanical response characteristics of rock slope under the action of excavation unloading. It proves the feasibility of the early warning method of rock slope deformation and failure in this study, which can provide an important reference for rock slope design and safe construction.Figure 17
Conventional monitoring result: (a) the location of the CX04, and (b) cumulative displacement data of CX04.
(a)(b)


## 6.3. Early Warning Model of Rock Slope Deformation Based on the MF-DFA Method

In actual situations, it is impossible to know when Δα and Δf(α) reach the peak or valley before large deformation occurs, so the ‘precursor period’ and ‘deformation period’ should not be divided at the peak or valley. The warning model is optimized according to the actual situation, as shown in Figure 18. After a period of ‘stable period,’ Δα shows an increasing trend for the first time and Δf(α) shows a decreasing trend for the first time, then entering the ‘precursory period.’ Δα reaches the peak and shows a decreasing trend, Δf(α) drops to the valley and shows an increasing trend, then the ‘precursor period’ ends, and the ‘deformation period’ starts. If Δα and Δf(α) increase and decrease for several times afterwards, it is still considered to be in the ‘deformation period’ until the arrival of the next ‘stable period.’Figure 18
MF-DFA multifractal warning model.
(a)(b)


## 7. Conclusion

The MS monitoring technique was adopted to stability analysis of the left bank slope of Baihetan hydropower station. The multifractal characteristics of MS waveforms were carried out based on the MF-DFA method. The following conclusions were drawn.Based on the MF-DFA method, the multifractal preset parameters of the MS waveform of the left bank slope of Baihetan hydropower station were determined as follows:smin = 28 = 256, smax = 212 = 4096, |q| = 20, and m = 3.The difference between the multifractal spectrum of typical MS waveform and blasting waveform was obvious: the width of the multifractal spectrum Δα of the MS waveform was much smaller than that of the blasting waveform, indicating that the multifractal strength of the blasting waveform was larger and the fluctuation was more intense and complex. The multifractal spectrum Δf(α) of the MS waveform was larger than that of the blasting waveform, indicating that small fluctuations in the MS waveform account for a relatively large amount. Furthermore, selecting a waveform with small background noise and large amplitude for analysis can improve the reliability of the multifractal spectrum estimation results.An early warning model of deformation and failure of rock slope based on MF-DFA was proposed. Before the deformation and failure of surrounding rock mass, Δα showed an increasing trend and Δf(α) showed a decreasing trend, which can be regarded as ‘precursor period;’ when deformation and failure occurred, Δα showed a decreasing trend and Δf(α) showed an increasing trend, which can be regarded as ‘deformation period;’ after deformation and failure, both Δα and Δf(α) showed a steady trend, and Δf(α) as a whole would be near the zero line, which can be regarded as ‘stable period.’ Compared to the conventional monitoring data, the early warning model was verified to be feasible. The MF-DFA-based early warning method of rock slope deformation and failure can accurately describe the mechanical response characteristics of rock slope under excavation and unloading.



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*Source: 8898318-2020-07-31.xml*