
==== Front
Heliyon
Heliyon
Heliyon
2405-8440
Elsevier

S2405-8440(24)13370-1
10.1016/j.heliyon.2024.e37339
e37339
Research Article
Improved complete ensemble empirical mode decomposition with adaptive noise and composite multiscale permutation entropy for denoising blast vibration signal
Kang Yi-ze ab
Yao Ying-kang shanxiyao@jhun.edu.cn
ab⁎
Dong Run-long ab
Jia Yong-sheng abc
Xie Quan-min ab
Wang Jian-ning d
a State Key Laboratory of Precision Blasting, Jianghan University, Wuhan, 430056, China
b Hubei Key Laboratory of Blasting Engineering, Jianghan University, Wuhan, 430056, China
c Wuhan Explosion & Blasting Co., Ltd., Wuhan, 430056, China
d China National Machinery Industry Co., Ltd., Beijing, 100080, China
⁎ Corresponding author. State Key Laboratory of Precision Blasting, Jianghan University, Wuhan, 430056, China. shanxiyao@jhun.edu.cn
05 9 2024
30 9 2024
05 9 2024
10 18 e3733926 8 2024
31 8 2024
2 9 2024
© 2024 The Authors
2024
https://creativecommons.org/licenses/by-nc/4.0/ This is an open access article under the CC BY-NC license (http://creativecommons.org/licenses/by-nc/4.0/).
Monitoring the building blast vibration signal is an efficient way to determine the power of blast vibration hazards. Due to the harsh measurement environment, noise is inevitably introduced into the recorded signals. This research presents a denoising approach based on Improved complete ensemble empirical mode decomposition with adaptive noise(ICEEMDAN) and Composite Multiscale Permutation Entropy (CMPE). First, the noisy blast vibration signal is decomposed into different intrinsic mode functions using ICEEMDAN; then multiple intrinsic mode functions (IMFs) are separated into pure and noisy using CMPE, the noisy IMFs are denoised using wavelet thresholding; finally the blast wave is reconstructed using the pure and denoised mixed IMFs. The proposed approach was compared with four other approaches (CEEMDAN-CMPE, VMD-CMPE, SVMD-CMPE, and WST). The results indicate that the proposed approach has better performance and can be considered as an effective denoising method for building blast vibration signals.

Highlights

• The ICEEMDAN-CMPE method is proposed to denoise the building blast vibraiton signal.

• Distinguish between explosive and noise signals in the high frequency signal component using CMPE.

• Compared with other denoising methods, the proposed method can significantly improve the SNR of building blast vibration signals in complex urban environments and noise interference.

Keywords

Demolition blasting
Vibration signal
Improved complete ensemble empirical mode decomposition with adaptive noise(ICEEMDAN)
Signal noise reduction
Composite Multiscale Permutation Entropy (CMPE)
==== Body
pmc1 Introduction

During the blast demolition of urban buildings, the accurate acquisition and analysis of blast vibration signals is critical for the assessment of blast vibration hazards [[1], [2], [3], [4], [5], [6]]. The buildings to be demolished are usually located in urban areas with complex environments. When monitoring blast vibration signals, the high-frequency signals generated by the blast often overlap with the low-frequency signals produced by the collapse of the structure. This overlap, combined with disturbances in the on-site measurement environment, results in the acquisition of blast vibration signals containing many frequency components. It is therefore important to reduce the noise in blast and demolition vibration signals.

In order to effectively suppress the influence of noise on the feature extraction of vibration signals, many noise reduction algorithms have been proposed [[7], [8], [9]], and the traditional noise reduction methods include the Fourier transform spectral filtering methods [10,11], such as low-pass, high-pass and band-pass filters, and the wavelet series noise reduction methods, such as wavelet decomposition, wavelet packet decomposition and wavelet thresholding [[12], [13], [14], [15], [16], [17], [18], [19], [20]]. Traditional orthogonal wavelet bases with fixed waveforms are unable to effectively match the mixed frequency components within the signals, which limits their performance in distinguishing different frequency components. Furthermore, because of their fixed waveforms, these wavelet bases may suffer from insufficient resolution when identifying high-frequency blasting signals and low-frequency ground impact vibration signals. This leads to a decrease in recognition accuracy. Although the pattern-adaptive wavelet construction method aims to enhance adaptability, this approach requires the reconstruction of wavelet bases for different signals, which significantly increases the workload and reduces the efficiency of analysis.

To solve this problem, Huang et al. [21] proposed Empirical Modal Decomposition (EMD) in 1998, which fundamentally eliminates the limitations of Fourier transform theory and establishes a signal analysis method based on instantaneous frequency for the first time. The EMD algorithm can decompose signals into multiple intrinsic modal function (IMF) components, but there may be modal aliasing between the IMFs, which may lead to distortion of the denoised signal. Therefore, Wu [22] proposed the EEMD method to suppress the modal aliasing problem by adding uniformly distributed white noise to the original signal so that the distribution of signal extreme points tends to be uniform. Yeh [23] proposed the complementary ensemble empirical modal decomposition (CEEMD) to improve the decomposition efficiency and reduce the noise by adding positive and negative pairs of white noise. TORRES [24] added the IMF component of white noise at each decomposition, proposed the complete ensemble empirical modal decomposition of adaptive noise (CEEMDAN), which makes the number of screenings and reconstruction noise residuals smaller. However, CEEMDAN still has some aspects that need to be improved. For example, its modes contain some residual noise, and modal aliasing occurs in the early stages of decomposition. To address these issues, Colominas [25] developed an Intrinsic Computing Expressive Empirical Mode Decomposition with Adaptive Noise (ICEEMDAN) to obtain less noisy and more physically meaningful components.

The building blast vibration signal is decomposed by ICEEMDAN method to obtain the modal component (IMF), which contains rich feature information of the blast vibration signal, and the IMF feature information is accurately characterized by entropy theory [26]. Various entropy calculation methods have been applied in different fields of vibration signal feature extraction, such as approximate entropy (AE), sample entropy (SE), and multiscale entropy (MSE) [[27], [28], [29]]. The permutation entropy (PE) algorithm [30] can accurately find the mutation information of vibration signals. However, due to the non-smooth time-varying characteristics of building blast vibration signal, its characteristic information is contained in multiple scales of time series, and the information extracted by single-scale permutation entropy is limited. Multi-scale permutation entropy (MPE) [31] for complex vibration signals in accordance with the time series of multi-scale feature extraction, can greatly increase the ability to extract effective information of the blast vibration signal. However, this method only considers the direct interaction of neighboring variables, does not take into account the non-neighboring variables directly interact with each other. This will lead to the extraction of vibration signal features, the omission of important information etc. CMPE [32] solves these problems, and it is more effective in dealing with nonlinear and non-smooth blast vibration signals.

In this paper, a joint ICEEMDAN and CMPE signal noise reduction processing method is proposed. The ICEEMDAN algorithm is used to decompose the demolition blast collapse touchdown vibration signal. For the decomposed intrinsic modal components, the entropy sensitivity analysis of the noise components is carried out to determine the optimal parameters for calculating the composite multiscale entropy value. The CMPE value of signal components is calculated by the multiscale entropy algorithm, and the wavelet thresholding method is selected to perform the noise reduction process on the noise components to obtain the decomposed intrinsic modes. Finally it is reconstructed to achieve the purpose of noise reduction. The method is applied to the blast demolition project of Yinfeng Hotel, and the measured building collapse touchdown vibration signal is processed for noise reduction, and the effectiveness of the method is verified.

2 Methodology

2.1 Intrinsic Computing Expressive Empirical Mode Decomposition with adaptive noise

In contrast to the CEEMDAN approach, which introduces Gaussian white noise directly into the original signal, the ICEEMDAN method applies this noise specifically to the K-th component. The K-th component is identified as the Intrinsic Mode Function (IMF) and is determined by the first step of the Empirical Mode Decomposition.

In the multistage decomposition process, each decomposition step produces a new IMF component and a residual term. These components can be further decomposed until only the residual term remains, which cannot undergo any further decomposition. The decomposition is considered complete when the residual term satisfies the condition of monotonicity, and its amplitude remains within a predefined threshold.

During the multi-level decomposition of the original signal, ICEEMDAN adjusts the addition of white noise in each decomposition round based on the results of the previous round and the characteristics of the current signal. The noise addition mechanism employed by ICEEMDAN enables it to effectively handle the various signal features present in explosive vibration signals, precisely identify signal termination conditions, and avoid situations of over-decomposition or under-decomposition.

The introduction of two new operators is fundamental: M(·) and Ek(·). M(·) is the local mean value of the original signal obtained via EMD, and Ek(·) is the function that constructs the EMD decomposition model. Here's a step-by-step account of the ICEEMDAN decomposition process.(1) Adding Gaussian white noise to the original signal.

(1) X(i)=X+βoE1(W(i))

In the equation, X is the original signal, β0 is the standard deviation of the added Gaussian white noise, W(i) is Gaussian white noise with a mean of zero and variance of zero, and E1(·) is the function for calculating the first Intrinsic Mode Function (IMF1).(2) Obtain the first residual value.

(2) R1=1N∑i=1NM(X(i))

In the equation, N is the number of data points in the original signal X.(3) Let k represent the k-th modal component produced through the ICEEMDAN breakdown process. At k = 1, this results in the extraction of the very first mode.

(3) F1=1N∑i=1NE1(x)=X−R1

(4) Upon setting k to 2, the second residual is derived by computing the local mean of R1 following the addition of Gaussian white noise via the Empirical Mode Decomposition (EMD) process.

(4) R2=1N∑i=1NM(R1+β1E2(W(i)))

The second modal component emerges from determining the discrepancy between R1 and R2.(5) F2=R1−R2=R1−M(R1+β1E2(W(i)))

(5) When k equals 3, 4, …, K, the k-th residual and the k-th mode are calculated.

(6) Rk=1N∑i=1NM(Rk−1+βk−1Ek(W(i)))

(7) Fk=Rk−1−Rk

(6) The procedure is reiterated to extract all Intrinsic Mode Functions (IMFs). This continues until the final residual, Rn, exhibits monotonic behavior and is no longer decomposable. Consequently, the original signal is separated into two distinct components.

(8) X=Rn+∑i=1NFi

2.2 Composite multiscale permutation entropy principle

The Composite Multiscale Permutation Entropy (CMPE) represents an enhanced approach that builds upon the foundation of Multiscale Permutation Entropy (MPE). Like its predecessor, CMPE makes use of the coarse-graining process. The novel aspect, however, is that for a single scale factor, it creates several coarse-grained time sequences. The permutation entropy values calculated from these sequences are then averaged to arrive at the CMPE value for the respective scale factor [33].

The main steps involved in calculating the Composite Multiscale Permutation Entropy can be summarized as follows.(1) Just like with Multiscale Permutation Entropy (MPE), a scale factor S is defined. This is used to coarsen the time series X, resulting in a new sequence, termed Ys.

(2) When dealing with a specific time series X={Xi,i=1,2, …...,N} and a chosen scale factor S, various coarse-grained sequences are created through the use of random sampling to apply different coarse-graining techniques. The procedure is carried out K times, each time picking sample points that either do not overlap or overlap only partially. This approach yields K unique coarse-grained sequences denoted as {Ys,k}k=1K.

In the context of the K-th sequence that has undergone the process of coarse-graining, Ys,k, a subset of S data points is chosen from the entire time series to be part of this sequence.(9) {x(i−1)s+1,x(i−1)s+2,x(i−1)s+3,···,xis}

In the equation, i is a randomly selected integer.and 1≤≤i ≤ N−s+1. A set of K sequences is derived through the process of coarse-graining.(10) Ys,k={ys,k(1),ys,k(2),ys,k(3),···,ys,k(|NS|)}k=,1,2,3,···,K

In the equation, ys,k(j) is the j-th coarse-grained sample at the scale S in the k-th coarse-graining, and its calculation method is as follows:(11) ys,k(j)=1s∑s=1sx(ik+(j−1)s+l−1)

In the equation, ik is the starting position index of the k-th coarse-grained sequence, j is the position of the current coarse-grained sample (ranging from 1 to N/S), l is the local index within the current coarse-grained sample (ranging from 1 to s), used for selecting specific sample points in the original time series.(3) For each coarse-grained sequence Ys, k, we reconstruct the phase space sequence by embedding dimension m and time delay t.

(12) Ys,kt={ys,kt(1),ys,kt(2),ys,kt(3),···,ys,kt(m)}

In the equation, Ys, k is the i-th vector in the reconstructed series, each vector is composed of m consecutive coarse-grained samples, t is the time delay.(4) Calculate the entropy of the permutations Hs,k for each reconstructed phase space sequence Ys,kt. First, sort the elements in the sequence and calculate the probability of occurrence of each permutation P(π).

(13) P(π)=N(π)m!(n−m+1)

In the equation, N (π) arranges the number of times π occurs, and (n-m+1) is the number of all possible subsequences in the sequence. The permutation entropy Hs,k is a weighted average of the probabilities P(π) of all permutations, using natural logarithms as weights.(14) Hs,k=−∑πP(π)lnP(π)

(5) The average of the entropy values of the alignments of the K coarse-grained sequences under the scale factor S.

(15) CS=1K∑k=1KHs,k

Averaging the CMPE values across all scale factors, we obtain.(16) CMPE=1S∑S=1SCS

The CMPE algorithm ensures comprehensive and consistent data analysis. This is achieved by taking into account the interactions between neighboring and non-neighboring variables. The algorithm not only focuses on the interactions between neighboring variables (x1 and x2, ×3 and x4), but also includes the information interactions between non-neighboring variables, (x2 and x3, ×4 and x5), effectively avoiding the omission of important information. Moreover, during the CMPE coarsening process, each scale factor corresponds to a different coarsening sequence. The CMPE method enhances the stability of entropy values by calculating these values for sequence comparisons across various scaling factors and then averaging them. This approach not only minimizes the fluctuation in entropy values but also ensures a more consistent and reliable measurement. By utilizing the average entropy value as the representative arrangement entropy for a given scale factor, CMPE effectively addresses the issue of unstable entropy calculations in MPE as the scale factor increases.

The process of coarse-graining as applied to the specified scale factor is shown in Fig. 1.Fig. 1 Coarse granulation process under scale factor.

Fig. 1

Fig. 2 Time domain waveforms of different modal signals.

Fig. 2

2.3 The definition of CMPE threshold based on randomness detection

To better distinguish the noisy signal components, we generate six sets of signals with various modes using randomness detection and measure their CMPE values independently (see Fig. 2). With the features of each signal and the related entropy value, a threshold value may be calculated to efficiently detect the noise signal component.

Create six sets of signals with different modes.

In this setup, ×1 represents a white noise signal with a length of 2048 samples;

Intermittent signals: ×2 = [zeros(1,300), randn(1,600), zeros(1,300), randn(1,500), zeros(1,348)];

Amplitude modulation signal: ×3 = sin(2π·10t)·in(2π·100t);

Amplitude modulation and frequency modulation signals: ×4 = 0.5sin(2π·5t)sin(2π·50 t + 2π·10t);

High-frequency sinusoidal signal: ×5 = sin(2π·200t);

Low-frequency sinusoidal signal: ×6 = sin(2π·50t); As shown in Fig. 4, the time-domain waveform of x1∼x6 is displayed.

The entropy values of the signals in different modalities are calculated using CMPE, as shown in Table 1.Table 1 Entropy of different modal signals.

Table 1Signal	x1	x2	x3	x4	x5	x6	
CMPE	0.9974	0.6137	0.5683	0.3968	0.4903	0.3683	

Analysis of the Composite Multiscale Permutation Entropy (CMPE) values for six sets of signals reveals the complexity and randomness of each signal. The CMPE value of the Gaussian white noise signal (x1) is 0.9974, indicating that the signal has a high level of randomness and complexity, which is consistent with theoretical expectations. This also suggests a greater likelihood of dynamic mutations in the signal. In contrast, the AM signal (x3), AM-FM signal (x4), high-frequency sinusoidal signal (x5), and low-frequency sinusoidal signal (x6) have CMPE values of 0.5683, 0.3968, 0.4903, and 0.3968, respectively. They are relatively low and exhibit more regular variations. This result aligns with the characteristics of the signals, as sinusoidal and modulated signals typically exhibit some degree of periodicity and regularity, while noise signals display a high degree of randomness.

The intermittent signal (x2) is a signal with abrupt changes. The calculated CMPE value is 0.6137, which has a large entropy value and is more random compared to other FM and sinusoidal signals. It is proved that the signal can simulate the characteristics of blast vibration signal. After many comparative tests, it is found that when the entropy value is greater than or equal to 0.6, the signal generally has a sudden change, and its change rule is in line with the dynamic characteristics.

The comprehensive analysis results show that 0.6 is a reasonable threshold, which can be used to distinguish the noise signal component and other signals.

2.4 Wavelet soft thresholding method

Donoho presented a wavelet value denoising approach based on the wavelet transform that involves three steps: wavelet decomposition, thresholding, and rebuilding. First, the signal is wavelet-decomposed. Then, each layer's high-frequency coefficients are treated using threshold quantization. Finally, wavelet reconstruction is applied to the processed wavelet coefficients to produce the noise reduction signal. Among these three steps, the thresholding approach has the most influence on the denoising output, and the number of thresholds is critical. The thresholding function consists mostly of hard thresholding and soft thresholding functions. In this research, the discovered noise signal components are quantified using the soft thresholding function, which is expressed as follows:(17) wj,k′={sgn(wi,k)(|wj,k|−λ),|wi,k|⩾λ0,|wi,k|<λ

In the equation, wj,k is the wavelet coefficient; λ is the threshold value; wj,k′ is the processed wavelet coefficient.

2.5 Hybrid ICEEMDAN-CMPE denoising approach

The algorithm presented in this study, referred to as ICEEMDAN-CMPE, initially employs the ICEEMDAN method to decompose the original blast vibration signal into a sequence of Intrinsic Mode Functions (IMFs) and residual elements. Subsequently, it computes the entropy value of each IMF by utilizing the Composite Multiscale Permutation Entropy (CMPE) approach. Through randomness detection, a threshold of 0.6 was determined to distinguish between signals containing noise and those not containing noise. A component with a value greater than 0.6 indicates a high level of noise within the signal, while a value less than 0.6 indicates a lower level of noise, potentially indicating a noise-free signal. For IMF components identified as noisy, noise reduction is achieved by applying wavelet thresholding techniques. The Intrinsic Mode Function (IMF) components are subjected to a noise reduction process and then reconstructed with the original IMF components to produce a noise-reduced signal for blast vibration analysis. Fig. 3 illustrates the comprehensive flowchart of the entire noise reduction process.Fig. 3 Flow chart of joint noise reduction.

Fig. 3

Fig. 4 Simulated signal and time domain waveforms of each component.

Fig. 4

3 Simulation signal denoising analysis

To further verify the effectiveness of the improved denoising algorithm proposed in this paper for denoising demolition blast vibration signals, a set of noisy simulation signals consisting of four signals with different frequencies and amplitudes is designed. The expressions of the signals are as follows:(18) {x1=0.25sin(2π30t);x2=0.35sin(2π60t)(1+1.5sin(2π35t))x3=0.1e(−0.01t)sin(2π120t)x4=sin2π10tX=x1+x2+x3+x4

X1 is a mid-low frequency signal with a frequency of 30Hz and an amplitude of 0.25, used to test the algorithm's ability to decompose mid-low frequency signals. X2 is a slightly higher frequency signal with a frequency of 60Hz and an amplitude of 0.35. X2 also contains a mid-low frequency modulation signal with a frequency of 35Hz and an amplitude of 1.5, used to evaluate the algorithm's ability to process slightly higher frequencies and modulation signals. X3 is a high frequency signal with attenuation effects, with a frequency of 120Hz and an initial amplitude of 0.1, with the amplitude decreasing over time, used to test the algorithm's ability to detect high frequency signals. X4 is a low frequency signal at 10Hz with an amplitude of 1, used to test the algorithm's effectiveness in decomposing low frequency signals. These four signals are combined with noise N with a signal-to-noise ratio of 10 to simulate the noise interference in real signals, forming the mixed signal.

As can be seen from Fig. 4, after adding noise to the signal, a large number of spikes appear in the signal, indicating that the noise has a significant effect on the simulation signal. The simulation signals were decomposed using CEEMDAN and ICEEMDAN, respectively, and the spectrum of each component was calculated, with the results shown in Fig. 5(a and b).Fig. 5 ICEEMDAN and CEEMDAN decomposition of simulated signals and corresponding spectra.

Fig. 5

Fig. 5 shows that the noise is concentrated in the high frequency part of the signal. Additionally, there is a difference in the number of modal components generated by the ICEEMDAN and CEEMDAN methods during the modal decomposition process. Specifically, the ICEEMDAN method generates 8 modal components, while the CEEMDAN method generates 13 modal components. The CEEMDAN method produced a greater number of modal components compared to the ICEEMDAN method, resulting in an increase in the number of pseudo-components. For instance, the IMF7 and IMF8, IMF11 and IMF12 components in the CEEMDAN decomposition are typical examples of modal aliasing.

The corresponding spectrograms reveal that the ICEEMDAN algorithm outperforms in separating the frequency components of the signal. The spectra of each modal component are clearer and more concentrated, indicating that the ICEEMDAN algorithm effectively distributes the signal's energy in different frequency bands, revealing the signal's frequency characteristics better.

The ICEEMDAN algorithm effectively avoids generating a large number of pseudo-components during signal decomposition and successfully solves the modal mixing problem present in the CEEMDAN method. It outperforms the CEEMDAN method in terms of the number of decomposition results, computational efficiency, and ability to avoid the appearance of pseudo-components. The ICEEMDAN algorithm can accurately identify the true components of a signal and is more computationally efficient and stable for larger amounts of data.

The components with distinct modes decomposed by the ICEEMDAN and CEEMDAN algorithms were selected and their entropy values were calculated using CMPE and MPE, respectively, as shown in Fig. 6.Fig. 6 Entropy calculation for different algorithms.

Fig. 6

As shown in Fig. 6, the composite multiscale permutation entropy value is higher than the multiscale permutation entropy value. This suggests that the composite multiscale permutation entropy algorithm is more effective in identifying noise in high-frequency signals with multiple noise components, resulting in more accurate noise reduction and feature extraction.

During the calculation process for CMPE, there are four parameters that need to be set: the delay factor t, the sample length N, the embedding dimension M, and the scale factor S. Appropriate parameter settings can improve the frequency resolution, enabling the algorithm to discriminate the subtle changes in the vibration signals at different stages, which is crucial for identifying the shock wave and vibration characteristics in the blast vibration signals.

The selection of parameters such as delay factor t, sample length N, embedding dimension M, and scale factor S directly affects the feature extraction ability of the CMPE algorithm, and optimizing the parameters can avoid overfitting or underfitting problems in the feature extraction process. For example, too high a setting of the embedding dimension M may capture the noise rather than the physical properties of the signal, while too low a setting may lead to the loss of important information; the size of the delay factor t affects the temporal resolution; the choice of the scale factor S affects the distribution of the entropy value of the composite multiscale array, which in turn affects the identification and differentiation of different frequency components.

To analyze the sensitivity of the entropy value, we will focus on the IMF1 component, which contains the most noise. First, we will keep M, N, and S unchanged, and calculate the CMPE value under different delay factors t, as shown in Fig. 7.Fig. 7 Comparison results of CMPE with different time delay.

Fig. 7

Fig. 7 shows that the composite multiscale entropy initially increases and then decreases as the scale factor increases when t = 1. However, when t > 1, the composite multiscale entropy decreases with the increase of the scale factor, and different delay factors have no significant effect on the composite multiscale entropy. Therefore, to reduce the algorithm's operation time, set t = 3 as the optimal value.

The sample length N was varied as 128, 256, 512, 1024, 2048, and 4000 while keeping the M, t, and S parameters constant for comparative analysis. The results are presented in Fig. 8.Fig. 8 Comparison results of CMPE with different sample lengths.

Fig. 8

Fig. 8 shows that the entropy value becomes more stable as the sample length increases, therefore N is set to 4000. Keeping N, t, and S parameters unchanged, a comparative study was conducted by setting the range of embedding dimension M as [2,8], as shown in Fig. 9.Fig. 9 Comparison results of CMPE with different embedding dimensions.

Fig. 9

Fig. 9 shows that for 1<M < 6 the CMPE values are aliased and the changes in entropy values are too small to effectively discriminate feature information. For 6 <M < 9, the decreasing trend of the CMPE values does not correspond to the intrinsic modal characteristics of the IMF1 component. In addition, increasing the embedding dimension M leads to longer running times, which significantly affects the efficiency of signal detection. When M is equal to 6, the CMPE values calculated under different scaling factors are consistent with the intrinsic modal characteristics of IMF.

Keeping N, t, S, and M unchanged, it is evident from Fig. 4 that the first seven of the eight IMF components obtained from signal decomposition exhibit distinct spectral features. These seven components form a feature matrix from which features are extracted using the CMPE entropy calculation method to form feature vectors, and the CMPE value of each component is calculated separately.

Fig. 10 shows that the entropy value of the composite multiscale arrangement of IMF1, IMF2, IMF3, and IMF4 gradually decreases when 1<S < 4, which is consistent with the characteristics of the intrinsic modal components of the signal. However, when 3<S < 10, IMF1, IMF2, IMF3, and IMF4 exhibit obvious modal mixing. Notably, the composite multiscale arrangement entropy values of IMF1∼IMF7 are most efficiently detected when the scale factor S = 2.Fig. 10 Comparison results of CMPE with different signal components.

Fig. 10

The scale factor must be carefully chosen to extract the feature information of the blast demolition vibration signal effectively. If the scale factor is too small, the feature information cannot be fully extracted. On the other hand, if the scale factor is too large, the information of the CMPE feature vector becomes redundant, which negatively affects the algorithm's detection rate. Simultaneously, if the sample length and embedding dimension are too small, the reconstructed vectors will contain insufficient states, causing the algorithm to lose its significance. Conversely, if the embedding dimension is too large, it will fail to reflect the subtle changes in the time series.

The optimal values for each parameter were obtained after conducting a sensitivity analysis of the composite multi-scale arrangement of entropy values. The values obtained were N = 4000, M = 6, S = 3, and t = 2. Table 2 shows the CMPE values of each IMF component.Table 2 MPE values of modal components after ICEEMDAN decomposition.

Table 2IMFs	IMF1	IMF2	IMF3	IMF4	IMF5	IMF6	IMF7	IMF8	
CMPE	0.9752	0.9579	0.8923	0.7518	0.3411	0.3036	0.2219	0.1871	

The larger the entropy value, the greater the random volatility of the components, among which the noise contained in IMF1∼IMF4 is most obvious, which corresponds to the waveform description shown in Fig. 4. The noise-containing components IMF1∼IMF4 are noise-reduced using the wavelet thresholding method, and the processed signal components are reconstructed with the noise-free ones to obtain the final noise-reduced signal.

Fig. 11 shows the results of noise reduction of the simulated signal using different methods, where most of the added white noise is removed after denoising the processing using different methods. However, the denoised signal corresponding to CEEMDAN-MPE has a large number of burrs. The denoised signals corresponding to CEEMDAN-CMPE and ICEEMDAN-MPE have small amplitude burrs and significant information loss occurs near the peak. The denoised signal corresponding to ICEEMDAN-CMPE is smooth, and the signal reconstructed by ICEEMDAN-CMPE has better smoothness and preserves effective information well compared to the other three methods. In addition, the signal has the highest similarity to the original simulated signal, indicating that the ICEEMDAN-CMPE method has the best denoising performance.Fig. 11 Comparison of the time-domain plots of the simulated signals at each stage.

Fig. 11

4 Engineering background

To test the method's denoising capability in engineering practice, we denoised the observed vibration signals induced by building blasting. The specifics are as follows.

4.1 Summary of works

This paper is based on the blasting and demolition project of the Hanzheng Street group of buildings in Qiaokou District of Wuhan City, including the Yinfeng Hotel Building (20-story frame structure, hereinafter referred to as Building 1) and the two residential buildings of Zhongzizhi Subdistrict (9-story frame structure, hereinafter referred to as Buildings 2 and 3), with a total floor area of about 35,000 square meters (Fig. 12.). The Project is located in the Hanzheng Street Commercial Circle between Zhongshan Avenue, Youyi South Road, Changdi Street and Duo Fu Road in Qiaokou District of Wuhan City, and its north side is adjacent to Zhongshan Avenue with busy traffic and high-rise buildings such as Meiji Shopping Plaza with dense flow of people, and the west, south and east sides of the Project are vacant land after demolition and relocation, with a large number of residential houses and commercial buildings distributed outside the vacant land. Among them, there is an underground structure at 6.9m on the north side, which is "combined with peace and war", as an underground commercial street in peacetime and a human defense fortification in wartime, and is distributed along Zhongshan Road, with a length of about 1,000m.Fig. 12 Demolition site map.

Fig. 12

4.2 Monitoring program and signal acquisition

The measurement of this project adopts 941B low-frequency sensor, which is a multi-functional vibration sensor jointly developed by the Institute of Engineering Mechanics of China Earthquake Administration and Beijing Tengsheng Qiaokang Science and Technology Co. It adopts the principle of differential magnet-induced vibration and utilizes passive closed-loop servo technology to achieve good low-frequency characteristics and good impedance matching characteristics, and is mainly used for industrial vibration measurement of general structures. In blast vibration monitoring, the measurement point is located near the foundation of the protected building.

According to the field blast monitoring program, 1#, 2# building to the south of the directional dumping, 3# building in three parts, respectively, to the east, south, and west of the directional dumping, the detonation sequence for the 3# building →1# building →2# building, the building between the detonation of the time difference of 1. 0 s. All of the instrumentation is located in the Zhongshan Road, "Ground Avenue" beneath business street, near the source of the explosion on the side of the entryway of the commercial tenants channel on the ground. For sensor layout specifics, see Fig. 13.Fig. 13 Layout of measurement points.

Fig. 13

In order to analyze the time-frequency and energy characteristics of the blast vibration signals during the demolition process, we select the most typical longitudinal blast vibration signals from the D1 measurement point for noise reduction analysis. As shown in Fig. 14.Fig. 14 Time-domain graph of raw signal of monitoring data.

Fig. 14

5 Vibration signal noise reduction

5.1 Decomposition of vibration signals in group building demolition

The demolition of a building by blasting composes of two distinct phases, the explosive blast phase and the collapse ground shock phase. During the collapse phase, a series of ground shocks typically occur, including the initial shock (formation of the blast notch), the secondary shock (closure of the blast notch), and the final shock (overall collapse of the structure). The vibration signals generated by the blast of the explosives are generally high frequency signals, while the vibrations generated during the collapse process are mainly low frequency signals.

During the demolition blasting process of a group of buildings, there is a certain delay in the blasting timing of each building, where the signals are mixed with different frequency components from different stages of different buildings. For example, the high-frequency signals generated by one building during the explosive detonation phase may overlap with the low-frequency signals produced by another building during its collapse phase.

Additionally, environmental noise and interference from mechanical equipment further compound the complexity of the demolition blasting vibration signals. In this case, distinguishing high-frequency signals from different sources becomes a key challenge in noise reduction.

As shown in Fig. 15, this is the signal decomposition diagram of the LMD, VMD, SVMD, and CEEMDAN algorithms. The following analysis will be conducted based on the results of the signal decomposition.Fig. 15 Comparison of different algorithms for signal decomposition.

Fig. 15

As shown in Fig. 15(a), the LMD may encounter issues with mode mixing during signal decomposition (e.g. confusing PF4 and PF5, and PF9 and PF10). When dealing with complex blast vibration signals, the signals of different frequencies may be incorrectly grouped into the same PF, leading to incorrect results. Particularly, during the blasting process, the LMD may have difficulty to effectively distinguish between high-frequency and low-frequency signals, thus affecting the accuracy of signal interpretation.

From Fig. 15(b) and (c), it can be seen that, among the five high-frequency signal components separated by the VMD algorithm, all of them are noise signals (IMF1∼IMF5), and there is no explosion-generated high-frequency signal. Obviously, VMD has confused the high-frequency signals produced by the explosive detonation with noise signals. Conversely, SVMD, as shown in Fig. 5(c), successfully separates the high-frequency signals generated by the explosive detonation (IMF1∼IMF4), but does not identify the noise signal components.

In contrast, the ICEEMDAN algorithm does not require a predefined number of modes. The algorithm emphasizes weak high-frequency signals under noise flooding by adding white noise of uniform frequency to each decomposition process. When dealing with nonlinear and unstable blast vibration signals, this adaptive noise addition mechanism helps to better identify and retain the true signal components accurately.

As shown in Fig. 15(a), after using ICEEMDAN to decompose the blast vibration signals of the building, the high-frequency signals generated by the explosive detonation (IMF2 and IMF3) have been decomposed. Moreover, the signal components with a high amount of noise (IMF1) has been separated. The time domain plot and corresponding spectrum of each component are shown in Fig. 16.Fig. 16 Time-domain and spectral plots of each modal component of the rupture vibration signal.

Fig. 16

5.2 CMPE feature extraction

From Fig. 16, it can be seen that IMF1 is the signal component which is mainly concentrated above 100 HZ. IMF2, IMF3 and IMF4 are the signal components which are mainly concentrated at 50–100 HZ, and the rest are low frequency components and residual terms. To accurately discriminate between noise-free and noise-containing signal components, it is necessary to calculate the multi-scale permutation entropy (CMPE) of each component.

In order to determine the optimal parameters to effectively extract and distinguish the useful features and noise components in the explosion vibration signal. We need to analyze the effects of the four parameters, delay factor t, sample length N, embedding dimension M and scale factor S, on the entropy value. Taking the noisiest IMF1 component as an example, we first calculate the changes of CMPE values under different delay factors t with M, N and S kept constant, as shown Fig. 17.Fig. 17 Comparison results of CMPE with different time delay.

Fig. 17

As seen in Fig. 17, when t = 1, increasing the scale factor causes the trend of composite multiscale entropy changes to first increase and subsequently drop. When t is more than 2, the composite multiscale entropy drops as the scale factor increases.The varied delay factor t has no visible influence on the composite multiscale entropy, thus t = 2 is used to minimize the algorithm's operating time.

Keeping the M, t, and S parameters constant, the sample length N is set to 128, 256, 512, 1024, 2048, 4096 and 8000, respectively, for comparative analysis, with the results displayed in Fig. 18.Fig. 18 CMPE comparison results with different sample lengths.

Fig. 18

As seen in the image, the entropy value becomes more stable as the sample length increases, assuming that the sample data should include data from the whole demolition and blasting cycle, with N = 8000. Keeping the parameters N, t, and S constant, we conduct a comparative investigation with a sample length N = 8000 and an embedding size M range of 2 to 8, and the results are presented in Fig. 19.Fig. 19 CMPE comparison results with different embedding dimension.

Fig. 19

As shown in Fig. 19, when 2 ≤ M ≤ 5, the CMPE value appears heteroscedastic, and the entropy value changes are too small to discriminate the effective feature information. When 7≤M ≤ 8, the decreasing trend of the CMPE value is inconsistent with the inherent modal characteristics of the IMF1 component, and with the increase of the embedding dimension M, it will lead to the prolongation of the operation time, which seriously affects the operation efficiency. When M = 6, the estimated CMPE values with different scale factors are consistent with the fundamental modal characteristics of IMF.

Keeping N, t, S and M unchanged, it can be seen from Fig. 16 that among the 10 IMF components obtained from the signal decomposition, the feature mode components of imf1∼im7 have clear spectra. These 7 IMF components are formed into a feature matrix, and the features are extracted by the CMPE method to form feature vectors, and their CMPE values are calculated respectively.

From Fig. 20, it can be seen that when 1 < S < 4, the entropy value of the composite multiscale arrangement of IMF1, IMF2, IMF3 and IMF4 gradually decreases, which is consistent with the intrinsic modal component characteristics of the vibration signals of building demolition blasting. When 3 < S < 10, IMF1, IMF2, IMF3, IMF4 began to appear obvious modal mixing phenomenon. It can be seen that when the scale factor S = 2, the entropy value of the composite multi-scale arrangement of IMF1∼IMF7 is most efficiently detected.Fig. 20 CMPE comparison results with different IMF components.

Fig. 20

If the scale factor is too small, the characteristic information of the demolition blasting vibration signal may not be fully extracted. Conversely, if the scale factor is too large, it can lead to information redundancy in the CMPE feature vector, thereby affecting the detection rate of the algorithm.

In summary, the parameters of CMPE are set to N = 8000, M = 6, S = 3, t = 2. The CMPE values of each IMF component are obtained as shown in Table 3.Table 3 CMPE values of modal components after ICEEMDAN decomposition.

Table 3IMFs	IMF1	IMF2	IMF3	IMF4	IMF5	IMF6	IMF7	IMF8	IMF9	IMF10	IMF11	
CMPE	0.9821	0.9571	0.9544	0.7872	0.4960	0.3026	0.2122	0.2166	0.1868	0.1334	0.1292	

5.3 Signal denoising

The data in Table 3 shows that the CMPE values of the earlier components are closer to 1. Components with higher MPE values have greater randomness in their fluctuations, which is consistent with the waveforms described in Fig. 14. The CMPE values of IMF1 to IMF4 are all greater than 0.6, indicating that the noise is mainly concentrated in IMF1 to IMF4. Therefore, wavelet threshold denoising is required for IMF1 to IMF4. The db8 wavelet basis function in the db series is the same as the blast vibration waveform and can be used for wavelet threshold denoising. The default soft threshold function is used for denoising, and the pure signal components (IMF5 through IMF11) are reconstructed with the denoised signal components to form the final signal, as shown in Fig. 21.Fig. 21 ICEEMDAN joint multiscale aligned entropy after noise reduction time domain map.

Fig. 21

To demonstrate the universality of the improved noise reduction algorithm, two collapsed touchdown vibration signals, D1 and D3, are introduced for the noise reduction process, and the comparison of the waveforms before and after the noise reduction is shown in Fig. 22(a and b), and the ICEEMDAN with the multiscale aligned entropy algorithm can effectively eliminate the noise.Fig. 22 Time domain graph before and after noise reduction.

Fig. 22

5.4 Power spectral density analysis

Welch's overlapping segment averaging method, which divides a continuous vibration signal into multiple overlapping segments, applies the Fourier transform to each segment, and calculates the average power spectral density, is a widely used technique for estimating the power spectrum of transient signals [[34], [35], [36], [37]]. The primary frequency components, noise levels and other significant features can be found by examining the power spectral density map.

First, the continuous signal x(t) is segmented into a number of segments of length work. These segments can be partially overlapped with an overlap length of knives. The kth segment starts at kD and ends at (k+1)D-1. D is usually taken as L/2 for maximum overlap.

For each segment, a windowing function w(j) is applied in order to reduce edge effects and spectral leakage. The needs of the analysis will determine which window function is used; common window functions are the Hanning window, Hamming window, etc. After the window is added, the signal segment is called.(19) xk(j)=x(kD+j)w(j)

In the equation, j is indexed from 0 to L-1.

The frequency domain representation Xk(n) is obtained from the windowed signal segment k(j) Fast Fourier Transform (FFT). The square of the FFT result's mode is the periodogram.(20) Pk(f)=|Xk(f)|2

In the equation, f is the frequency and Xk(f) is the Kth segment at frequency f FFT result.

To calculate the power spectral density, the periodogram must be normalized. Normalization removes the effect of the windowing function and accounts for segment length and overlap. The normalized periodogram is calculated as:(21) p‾k(f)=1L|Xk(f)|2

In the equation, 1/L is the normalization factor based on the energy of the window function and the length of the segment.

To determine the power spectral density, add all normalized periodograms and divide by the number of segments.(22) P‾(f)=1K∑K=1KP‾k(f)

In the equation, P‾(f) is the final power spectral density estimate.

The CMPE, MPE, VMD-CMPE, SVMD-CMPE and WST algorithms are used to reduce the noise of the signal and then their Welch's power spectral density estimation maps are calculated as shown in Fig. 23.Fig. 23 Welch power spectral density estimation before and after D3 noise reduction.

Fig. 23

From Fig. 23,the frequency of the blast vibration signal before noise reduction is relatively dispersed, especially in the high-frequency stage, without obvious peaks. Although the blast vibration signal shows a certain peak after WST noise reduction, the energy is not concentrated and the main frequency is not highlighted.

The power spectrum of the blast vibration signal after VMD and SVMD noise reduction has three obvious peaks, but the arc shape between peaks and valleys is not uniform. This is because the VMD algorithm has a different bandwidth for each IMF when decomposing the signal. If some frequency components have wider bandwidths, they may appear as higher peaks and wider baselines in the power spectrum. SVMD introduces a sparsity penalty term in the choice of basis functions, which causes the impulse response function (IMF) obtained from the decomposition to have a more sparse distribution in the power spectrum. Each mode contains only the most important components of the signal, and although the noise is effectively suppressed, the high-frequency energy is not concentrated.

The CEEMDAN algorithm enhances the high-frequency components in the signal by adding white noise. The enhanced signal is decomposed to highlight the high-frequency components of the blast vibration signal. As shown in Fig. 23, the arcs are uniform between the peaks and valleys. However, the bias cannot be completely eliminated if the noise or initial conditions for each decomposition are not properly chosen. The peaks in the power spectrogram will flatten out and the dominant frequencies will not be apparent.

Instead of simply adding randomly generated white noise, ICEEMDAN first performs an EMD decomposition of the white noise, from which an IMF is selected as the noise component that controls the amount of noise introduced and avoids affecting the characteristics of the signal itself by adding too much noise. Therefore, the denoised power spectrum of the ICEEMDAN algorithm shows that the distance between the peaks and valleys is relatively uniform, with no significant bends or arcs in the CEEMDAN algorithm. It indicates that the energy distribution of the signal is more balanced in the frequency domain.

In summary, the power spectra before noise reduction have no significant peaks, indicating that the signal contains multiple frequency components and noise. After noise reduction using the ICEEMDAN-CMPE method, three peaks appear in the power spectrogram curve. In the high-frequency range, the energy distribution between peaks and troughs is average. This indicates that the noise level has been reduced and the important frequency components of the signal have been enhanced, effectively eliminating some of the noisy or unimportant frequency components.

5.5 Evaluation of denoising effect

To assess the performance of the ICEEMDAN-CMPE algorithm, this paper uses three indexes: signal-to-noise ratio (SNR), root mean square error (RMSE), and normalized cross-correlation loss (NCC) [38,39] to measure the effect of noise reduction. The SNR might indicate the energy ratio between signal and noise, but the RMSE measures the amount of the deviation between the original signal and the reconstructed signal; the lower the RMSE, the smaller the departure between the denoised signal and the true signal. The normalized inter-correlation loss evaluates the degree of correlation between the denoised signal and the original signal; r > 0.8 indicates extremely strong correlation. The higher the signal-to-noise ratio, the lower the root-mean-square error and the more effective the noise reduction.

The signal-to-noise ratio, mean square error, and normalized cross-correlation loss correlation coefficients are calculated as follows:(23) SNR=10log10∑i=1Nx2(i)∑i=1N[x(i)−xˆ(i)]2

(24) RMSE=1N∑i=1N[x(i)−xˆ(i)]2

(25) NCC=∑i=1Nx(i)xˆ(i)[∑i=1Nx(i)2][∑i=1Nxˆ(i)2]

In the equation, N is the signal length, x(i) is the original signal, x(i) is the signal after noise reduction.

The above metrics are used to comprehensively compare the effects of noise reduction on the blast vibration signals of D1, D2, and D3 using four algorithms: ICEEMDAN-CMPE, CEEMDAN-MPE, SVMD-CMPE, and VMD-MPE with CMD-MPE. Through these three quantitative indicators, the advantages, disadvantages, and applicability of each algorithm in the noise reduction process become clearly evident.

As can be seen from Table 4, the blast vibration signal denoised by the ICEEMDAN-CMPE method has the best indicators. Among them, the signal-to-noise ratio is the highest, the root mean square error is the lowest and the correlation coefficient is the highest. The signal-to-noise ratios of D1, D2, and D3 are 23.5744, 25.4664, and 23.3491 respectively; the root mean square errors are 0.059, 0.032, and 0.046 respectively; and the Normalized cross-correlation values are 0.9391, 0.9787, and 0.9896 respectively.Table 4 Comparison of denoising metrics for various algorithms.

Table 4Monitoring Points	Evaluation Indicators	ICEEMDAN-CMPE	CEEMDAN-CMPE	SVMD -CMPE	VMD -MPE	WST	
D1	SNR	23.5744	20.3589	17.3094	16.4363	14.3734	
RMSE	0.059	0.35	0.481	0.568	0.753	
NCC	0.9391	0.8957	0.8518	0.8196	0.6923	
D2	SNR	25.4664	19.2536	18.2036	18.3733	15.2675	
RMSE	0.032	0.456	0.496	0.607	0.812	
NCC	0.9787	0.8725	0.8585	0.8443	0.7248	
D3	SNR	23.3491	19.6728	16.6219	16.1335	13.3597	
RMSE	0.046	0.572	0.647	0.618	0.781	
NCC	0.9896	0.8960	0.7906	0.7913	0.6157	

Comparing the noise reduction effects of five decomposition approaches, ICEEMDAN-CMPE, CEEMDAN-MPE, SVMD-CMPE, VMD-CMPE, and WST, the computational findings demonstrate that all five methods reduce noise to some amount. ICEEMDAN-CMPE and CEEMDAN-CMPE have similar noise reduction effects, However, the ICEEMDAN-CMPE algorithm has a higher NCC and SNR and a smaller RMSE. Its waveform not only keeps the information from the original blast vibration waveform, but it also increases noise removal efficiency and has the greatest noise reduction effect.

6 Conclusions

A large amount of noise is mixed in the vibration signals measured from the demolition of group buildings by blasting, which poses a great challenge in extracting the high-frequency signals generated by the explosion and the low-frequency signals generated by the vibration of the collapse touchdown. In this study, a hybrid ICEEMDAN-CMPE method is proposed to suppress these noises. The blast demolition of Yinfeng Hotel is used as a technical background for the noise reduction of blast vibration. The main results are as follows.(1) The decomposition of building blast vibration signals using the ICEEMDAN algorithm allows the extraction of high frequency signals generated by the explosive blast and low frequency signals generated by the three impacts during the collapse phase. The vibration signals at different stages can be used to effectively evaluate the effect of the blast demolition.

(2) In this paper we have systematically investigated the relationship between the choice of CMPE parameters and the different components of the blast vibration signal. The optimal parameter settings (N = 8000, M = 6, S = 2, t = 2) were determined by sensitivity analysis. These parameter settings significantly increase the frequency resolution. This allows a more accurate extraction of the relevant features of the blast vibration signal while reducing the risk of overfitting and underfitting.

(3) The Welch method is used to calculate the power distribution of the blast vibration signal at different normalized frequencies before and after noise reduction. It is found that before noise reduction, the power spectrogram curve fluctuates greatly, contains a certain amount of noise, and the frequency is relatively scattered. After noise reduction, the peaks of the power spectrogram curve become clearer and the smoothness of the curve in the high frequency band is also improved, indicating that the noise has been effectively removed. In addition, after noise reduction, the main frequency components of the signal become more apparent and the energy distribution is more concentrated.

(4) Five algorithms, ICEEMDAN-CMPE, CEEMDAN-CMPE, SVMD-CMPE, VMD-CMPE, and WST, were used to examine the monitoring signals for noise reduction, and it was found that all five algorithms have a certain level of noise reduction effect and fidelity. Among them, the noise reduction effect of ICEEMDAN-CMPE is particularly outstanding. After using this algorithm, the SNR of measurement points D1, D2, and D3 are 23.5744, 25.4664, and 23.3491 respectively; the RMSE are 0.059, 0.032, and 0.046 respectively; and the NCC values are 0.9391, 0.9787, and 0.9896 respectively. The results show that the signal after noise reduction using the ICEEMDAN-MPE algorithm has the highest correlation with the original signal, the highest NCR and NCC, and the smallest RMSE, which proves the effectiveness of the noise reduction.

The effectiveness of the joint noise reduction algorithm based on ICEEMDAN-CMPE is demonstrated by a thorough evaluation of the power spectral density estimation and noise reduction effect indexes, which is critical for noise reduction and analysis of demolition blast vibration signals.

Data availability statement

The data generated and analyzed for this study are available from the corresponding author upon reasonable request. Due to the contractual agreement with the participants, if there is a need for these data, they can be obtained by contacting the corresponding author.

CRediT authorship contribution statement

Yi-ze Kang: Writing – review & editing, Writing – original draft, Visualization, Methodology, Formal analysis, Data curation, Conceptualization. Ying-kang Yao: Supervision, Funding acquisition, Conceptualization. Run-long Dong: Data curation. Yong-sheng Jia: Supervision, Resources. Quan-min Xie: Conceptualization. Jian-ning Wang: Software, Funding acquisition.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

This research was funded by The National 10.13039/501100001809 Natural Science Foundation of China (Grant No. 52478525 ), State Key Laboratory of Precision Blasting, 10.13039/501100011319 Jianghan University (No. PBSKL-2022-QD-02 ), the State Key Laboratory of Precision Blasting and Hubei Key Laboratory of Blasting 10.13039/100000084 Engineering , 10.13039/501100011319 Jianghan University (No. PBSKL2023A9 ) and the SINOMACH Youth Science and Technology Fund (QNJJ-PY-2022-02 ). The supports are gratefully acknowledged.
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