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A Novel Denoising Model of Underwater Drilling and Blasting Vibration Signal Based on CEEMDAN
基于 CEEMDAN 的新型水下钻孔和爆破振动信号去噪模型

Yaxiong Peng 1 ( 0 ) ⋅ 1 ( 0 ) ⋅ ^(1)^((0))*{ }^{1}{ }^{(0)} \cdot Yunsi Liu 1 ⋅ 1 ⋅ ^(1)*^{1} \cdot Chao Zhang 1 ⋅ Li Wu 2 1 ⋅ Li Wu 2 ^(1)*LiWu^(2)^{1} \cdot \mathrm{Li} \mathrm{Wu}^{\mathbf{2}}
彭亚雄 1 ( 0 ) ⋅ 1 ( 0 ) ⋅ ^(1)^((0))*{ }^{1}{ }^{(0)} \cdot 刘云思 1 ⋅ 1 ⋅ ^(1)*^{1} \cdot 张超 1 ⋅ Li Wu 2 1 ⋅ Li Wu 2 ^(1)*LiWu^(2)^{1} \cdot \mathrm{Li} \mathrm{Wu}^{\mathbf{2}}

Received: 5 April 2020 / Accepted: 21 December 2020 / Published online: 5 January 2021
收到：接收：2020 年 4 月 5 日 / 接受：2020 年 12 月 21 日 / 在线发表：2021 年 1 月 5 日
© King Fahd University of Petroleum & Minerals 2021
© 法赫德国王石油与矿业大学 2021

Abstract  摘要

In underwater drilling and blasting engineering, the blasting vibration signal is mixed with a mass of noises due to the complexity of monitoring environment, the error of monitoring sensors and the reflection of propagation medium. In order to accurately obtain the characteristics of vibration signal, a novel denoising model is established. The complete ensemble empirical mode decomposition with adaptive noise is used to decompose the original signal, and the objective function of the filtering algorithm is used to obtain the optimal denoising signal. The results indicate that the model can not only successfully remove the high-frequency noise but also has no effect on the low-frequency signal components, which verifies the reliability and validity of the denoising model.
在水下钻孔爆破工程中，由于监测环境的复杂性、监测传感器的误差和传播介质的反射，爆破振动信号中混杂着大量的噪声。为了准确获取振动信号的特征，本文建立了一个新颖的去噪模型。利用带有自适应噪声的完全集合经验模态分解法对原始信号进行分解，并利用滤波算法的目标函数获得最优去噪信号。结果表明，该模型不仅能成功去除高频噪声，而且对低频信号成分没有影响，验证了去噪模型的可靠性和有效性。

Keywords Underwater drilling and blasting ⋅ ⋅ *\cdot Vibration signal ⋅ ⋅ *\cdot Denoising model ⋅ ⋅ *\cdot CEEMDAN
关键词 水下钻孔和爆破 ⋅ ⋅ *\cdot 振动信号 ⋅ ⋅ *\cdot 去噪模型 ⋅ ⋅ *\cdot CEEMDAN

List of Symbols  符号列表

IMF Intrinsic mode function
IMF 固有模式功能
x ( t ) x ( t ) x(t)quadx(t) \quad Original signal
x ( t ) x ( t ) x(t)quadx(t) \quad 原始信号
ε 0 ε 0 epsi_(0)quad\varepsilon_{0} \quad Noise coefficient
ε 0 ε 0 epsi_(0)quad\varepsilon_{0} \quad 噪声系数
r ( t ) r ( t ) r(t)quadr(t) \quad Residual component
r ( t ) r ( t ) r(t)quadr(t) \quad 剩余部分
n j ( t ) n j ( t ) n_(j)(t)quadn_{j}(t) \quad White noises   n j ( t ) n j ( t ) n_(j)(t)quadn_{j}(t) \quad 白噪声
x m m x m m x_(m)quad mx_{m} \quad m th sample point of original signal
x m m x m m x_(m)quad mx_{m} \quad m 原始信号的第 1 个采样点
x ~ m M x ~ m M tilde(x)_(m)quad M\tilde{x}_{m} \quad M th sample point of denoised signal
去噪信号的 x ~ m M x ~ m M tilde(x)_(m)quad M\tilde{x}_{m} \quad M 个采样点
MSE f MSE f MSE_(f)quad\operatorname{MSE}_{f} \quad Mean square error
MSE f MSE f MSE_(f)quad\operatorname{MSE}_{f} \quad 均方误差
u ( x ) , v ( x ) , f ( x ) u ( x ) , v ( x ) , f ( x ) u(x),v(x),f(x)quadu(x), v(x), f(x) \quad Smooth curves   u ( x ) , v ( x ) , f ( x ) u ( x ) , v ( x ) , f ( x ) u(x),v(x),f(x)quadu(x), v(x), f(x) \quad 平滑曲线
h h h quadh \quad Sampling interval
h h h quadh \quad 采样间隔
SMSE f f _(f)quad_{f} \quad Mean square error of smoothness
SMSE f f _(f)quad_{f} \quad 平滑度的均方误差
F F F quadF \quad Objective function
F F F quadF \quad 目标函数
μ μ muquad\mu \quad Weight coefficient
μ μ muquad\mu \quad 重量系数
ξ ξ xiquad\xi \quad Signal-to-noise ratio
ξ ξ xiquad\xi \quad 信噪比
ε ε epsiquad\varepsilon \quad Mean absolute error
ε ε epsiquad\varepsilon \quad 平均绝对误差

1 Introduction  1 引言

In the constructions of wading engineering, such as harbors, wharfs and channels, underwater drilling and blasting plays a critical role in the excavation of rock mass. In such engineering, merely 20 − 30 % 20 − 30 % 20-30%20-30 \% energy released by the explosion of explosive is used to break rock mass, while the main part adversely affects the surrounding environment [1-3]. Blasting vibration is one of the main harmful effects, which will adversely affect the buildings, bridges and slopes near the blasting area.
在港口、码头、航道等涉水工程建设中，水下钻孔爆破对岩体开挖起着至关重要的作用。在这类工程中，仅仅利用炸药爆炸释放的 20 − 30 % 20 − 30 % 20-30%20-30 \% 能量来破碎岩体，而主要部分则对周围环境造成不利影响[1-3]。爆破振动是主要的有害影响之一，会对爆破区域附近的建筑物、桥梁和斜坡造成不利影响。
Blasting vibration signal is the basis for analyzing blasting vibration effects, which reflects the dynamic response characteristics of structures. Noise is an inevitable problem for signals in the process of vibration monitoring due to the complexity of monitoring environment, the error of monitoring sensors, the reflection of propagation medium and the interference of magnetic field [4]. The high-frequency noise will distort blasting vibration signals and conceal the real information of signal, which result in the inaccurate description of vibration characteristics. Therefore, blasting vibration signals must be denoised.
爆破振动信号是分析爆破振动效应的基础，它反映了结构的动态响应特性。在振动监测过程中，由于监测环境的复杂性、监测传感器的误差、传播介质的反射以及磁场的干扰，噪声是信号不可避免的问题[4]。高频噪声会使爆破振动信号失真，掩盖信号的真实信息，导致振动特征描述不准确。因此，必须对爆破振动信号进行去噪处理。
Blasting vibration signal is a sort of non-stationary random signal, and the denoising methods for the signal mainly contain the wavelet decomposition technology [5-7] and the empirical mode decomposition (EMD) technology [8-10]. Wavelet decomposition technology is characterized by multi-resolution analysis and excellent time-frequency
爆破振动信号是一种非稳态随机信号，其去噪方法主要包括小波分解技术[5-7]和经验模态分解（EMD）技术[8-10]。小波分解技术的特点是多分辨率分析和出色的时频分析能力。
Springer  施普林格
localization capability. The real signal and noises can be separated according to different characteristics of wavelet coefficients. Zhang et al. [11] applied the wavelet threshold denoising method to process the blast vibration signal, and results demonstrated that noise could be removed by Rigrsure threshold based on Stein unbiased likelihood estimation. Xie et al. [12] and Xia et al. [13], respectively, proposed different types of wavelet threshold denoising algorithms and applied the algorithms to process the measured vibration signals. However, wavelet denoising methods have to choose basis functions, which inevitably generates defects as Fourier transform.
定位能力根据小波系数的不同特性，可以分离真实信号和噪声。Zhang 等[11]应用小波阈值去噪方法处理爆炸振动信号，结果表明基于 Stein 无偏似然估计的 Rigrsure 阈值可以去除噪声。Xie 等人[12]和 Xia 等人[13]分别提出了不同类型的小波阈值去噪算法，并将算法应用于处理实测振动信号。然而，小波去噪方法必须选择基函数，这不可避免地会产生傅里叶变换的缺陷。
EMD is characterized by multi-resolution and excellent adaptability to non-stationary signals, which is widely applied to process vibration signals. Zhao et al. [14] employed the ensemble empirical mode decomposition (EEMD) to denoise blasting vibration signals and compared it with wavelet denoising method. Yuan et al. [15] applied EMD denoising method and Hilbert transform to analyze the blasting vibration signals in lead-zinc mine. EMD is adaptive and need not to select basis functions, but the method will result in mode mixing when processing abnormal noise signals [16].
EMD具有多分辨率、对非平稳信号适应性强等特点，被广泛应用于振动信号的处理。Zhao 等[14]采用集合经验模态分解（EEMD）对爆破振动信号进行去噪，并与小波去噪方法进行了比较。Yuan等人[15]应用EMD去噪方法和希尔伯特变换分析了铅锌矿的爆破振动信号。EMD 是自适应的，无需选择基函数，但该方法在处理异常噪声信号时会导致模态混合[16]。
In order to accurately analyze the vibration characteristics of underwater drilling and blasting, it is necessary to reduce the noise of the measured signals. The existing denoising methods still have defects and are not effective in more complex underwater environment. Therefore, it is necessary to establish a denoising method suitable for vibration signal induced by underwater drilling and blasting.
为了准确分析水下钻孔和爆破的振动特性，有必要降低测量信号的噪声。现有的去噪方法仍存在缺陷，在较为复杂的水下环境中效果不佳。因此，有必要建立一种适合水下钻孔和爆破引起的振动信号的去噪方法。
The complete ensemble empirical mode decomposition with adaptive noise (CEEMDAN) can not only reduce the phenomenon of mode mixing but also accurately reconstruct the original signal [ 17 , 18 ] [ 17 , 18 ] [17,18][17,18]. The optimal smooth denoising algorithm can get better filtering signals. This paper proposes a novel denoising model to process vibration signals induced by underwater drilling and blasting, and the model is consisted of above two parts. CEEMDAN is used to decompose the original signal, and the objective function of the filtering algorithm is used to obtain the optimal denoised signal. The novel denoising model can effectively improve the denoising effect and preserve the authenticity and integrity of waveform better.
具有自适应噪声的完全集合经验模态分解（CEEMDAN）不仅能减少模态混合现象，还能准确地重建原始信号 [ 17 , 18 ] [ 17 , 18 ] [17,18][17,18] 。最优平滑去噪算法可以获得更好的滤波信号。本文提出了一种处理水下钻孔爆破引起的振动信号的新型去噪模型，该模型由以上两部分组成。利用 CEEMDAN 对原始信号进行分解，并利用滤波算法的目标函数获得最优去噪信号。该新型去噪模型能有效提高去噪效果，更好地保持波形的真实性和完整性。

2 CEEMDAN Theory  2 CEEMDAN 理论

2.1 Empirical Mode Decomposition
2.1 经验模式分解

Empirical mode decomposition (EMD) is an adaptive signal decomposition method for processing the nonlinear and non-stationary signals proposed by Huang et al. [19]. Based on the time-scale characteristics of signals, the
经验模态分解（EMD）是 Huang 等人[19]提出的一种处理非线性和非平稳信号的自适应信号分解方法。根据信号的时间尺度特征，EMD
multi-component signals can be decomposed into a series of intrinsic mode function (IMF) components and residual components, and the IMF components are arranged in the order of instantaneous frequencies from high to low. The method possesses favorable adaptability, completeness and orthogonality. However, there is mode mixing problem in processing signals containing discontinuities, impulses and noises [20].
多分量信号可被分解为一系列本征模态函数（IMF）分量和残差分量，IMF分量按瞬时频率从高到低的顺序排列。该方法具有良好的适应性、完整性和正交性。然而，在处理含有不连续、脉冲和噪声的信号时，存在模式混合问题[20]。
To solve the above problems, Huang et al. [21] added the even-distributed white noises to the original signal and decomposed the aggregate signals into IMFs by EMD method for several times, and the final IMF was obtained by the averaged IMFs. This method is called the ensemble empirical mode decomposition (EEMD), which can weaken the mode mixing phenomenon of EMD algorithm to a certain extent. The algorithm tries to eliminate the influence of white noise on decomposition results through integration averaging, but it cannot be entirely suppressed. The reconstruction error is highly dependent on the number of integration times.
为了解决上述问题，Huang 等人[21]在原始信号中加入偶数分布的白噪声，用 EMD 方法将集合信号多次分解为 IMF，通过平均 IMF 得到最终的 IMF。这种方法被称为集合经验模态分解（EEMD），可以在一定程度上削弱 EMD 算法的模态混合现象。该算法试图通过积分平均消除白噪声对分解结果的影响，但并不能完全抑制白噪声。重构误差与积分次数有很大关系。

2.2 CEEMDAN

Torres et al. [17] proposed the complete ensemble empirical mode decomposition with adaptive noise (CEEMDAN) based on EEMD. Several adaptive white noises are added to the original signal during each EMD decomposition. Even if the integration times are limited, the reconstruction error is almost 0 ; that is, the reconstructed signal is almost identical to the original signal. The main steps of CEEMDAN are as follows.
Torres 等人[17] 在 EEMD 的基础上提出了带有自适应噪声的完整集合经验模式分解（CEEMDAN）。在每次 EMD 分解过程中，都会向原始信号添加若干自适应白噪声。即使积分时间有限，重建误差也几乎为 0；也就是说，重建后的信号与原始信号几乎相同。CEEMDAN 的主要步骤如下。
Add the white noises n j ( t ) n j ( t ) n_(j)(t)n_{j}(t) with different amplitudes to the original signal x ( t ) x ( t ) x(t)x(t). Then, the signal can be expressed as x ( t ) + ε 0 n j ( t ) x ( t ) + ε 0 n j ( t ) x(t)+epsi_(0)n_(j)(t)x(t)+\varepsilon_{0} n_{j}(t), and ε 0 ε 0 epsi_(0)\varepsilon_{0} is the noise coefficient. EMD is used to carry out I I II times of decomposition, and the first IMF can be obtained by the integrated averaging. The first IMF and the first residual component are shown as follows.
在原始信号 x ( t ) x ( t ) x(t)x(t) 中加入不同振幅的白噪声 n j ( t ) n j ( t ) n_(j)(t)n_{j}(t) 。然后，信号可以表示为 x ( t ) + ε 0 n j ( t ) x ( t ) + ε 0 n j ( t ) x(t)+epsi_(0)n_(j)(t)x(t)+\varepsilon_{0} n_{j}(t) ， ε 0 ε 0 epsi_(0)\varepsilon_{0} 是噪声系数。利用 EMD 进行 I I II 次分解，通过积分平均可以得到第一个 IMF。第一个 IMF 和第一个残差分量如下所示。
IMF 1 ( t ) = 1 I ∑ i = 1 I IMF i 1 ( t ) IMF 1 ( t ) = 1 I ∑ i = 1 I   IMF i 1 ( t ) IMF_(1)(t)=(1)/(I)sum_(i=1)^(I)IMF_(i1)(t)\operatorname{IMF}_{1}(t)=\frac{1}{I} \sum_{i=1}^{I} \operatorname{IMF}_{i 1}(t)
r 1 ( t ) = x ( t ) − IMF 1 ( t ) r 1 ( t ) = x ( t ) − IMF 1 ( t ) r_(1)(t)=x(t)-IMF_(1)(t)r_{1}(t)=x(t)-\mathrm{IMF}_{1}(t).
EMD j ( ⋅ ) EMD j ( ⋅ ) EMD_(j)(*)\mathrm{EMD}_{j}(\cdot) is the j j jj th modal component generated by the EMD algorithm. After the signal r 1 ( t ) + ε 1 ⋅ EMD 1 ( n j ( t ) ) r 1 ( t ) + ε 1 ⋅ EMD 1 n j ( t ) r_(1)(t)+epsi_(1)*EMD_(1)(n_(j)(t))r_{1}(t)+\varepsilon_{1} \cdot \operatorname{EMD}_{1}\left(n_{j}(t)\right) is further decomposed for I I II times, the second IMF can be obtained as follows.
EMD j ( ⋅ ) EMD j ( ⋅ ) EMD_(j)(*)\mathrm{EMD}_{j}(\cdot) 是 EMD 算法生成的 j j jj 第三模态分量。在对信号 r 1 ( t ) + ε 1 ⋅ EMD 1 ( n j ( t ) ) r 1 ( t ) + ε 1 ⋅ EMD 1 n j ( t ) r_(1)(t)+epsi_(1)*EMD_(1)(n_(j)(t))r_{1}(t)+\varepsilon_{1} \cdot \operatorname{EMD}_{1}\left(n_{j}(t)\right) 进一步分解 I I II 次后，可得到第二个 IMF 如下。
IMF 2 ( t ) = 1 I ∑ i = 1 I EMD 1 [ r 1 ( t ) + ε 1 EMD 1 ( n i ( t ) ) ] IMF 2 ( t ) = 1 I ∑ i = 1 I   EMD 1 r 1 ( t ) + ε 1 EMD 1 n i ( t ) IMF_(2)(t)=(1)/(I)sum_(i=1)^(I)EMD_(1)[r_(1)(t)+epsi_(1)EMD_(1)(n_(i)(t))]\operatorname{IMF}_{2}(t)=\frac{1}{I} \sum_{i=1}^{I} \operatorname{EMD}_{1}\left[r_{1}(t)+\varepsilon_{1} \operatorname{EMD}_{1}\left(n_{i}(t)\right)\right]
Calculate the residual components in the sequence of 2, 3 , … , k 3 , … , k 3,dots,k3, \ldots, k, which is shown as follows.
计算 2、 3 , … , k 3 , … , k 3,dots,k3, \ldots, k 序列中的残差分量，如下所示。
r k ( t ) = r k − 1 ( t ) − IMF k ( t ) r k ( t ) = r k − 1 ( t ) − IMF k ( t ) r_(k)(t)=r_(k-1)(t)-IMF_(k)(t)r_{k}(t)=r_{k-1}(t)-\operatorname{IMF}_{k}(t)
Extract the first IMF from the signal r 1 ( t ) + ε 1 ⋅ EMD 1 ( n j ( t ) ) r 1 ( t ) + ε 1 ⋅ EMD 1 n j ( t ) r_(1)(t)+epsi_(1)*EMD_(1)(n_(j)(t))r_{1}(t)+\varepsilon_{1} \cdot \operatorname{EMD}_{1}\left(n_{j}(t)\right), and the IMF k + 1 IMF k + 1 IMF_(k+1)\mathrm{IMF}_{k+1} is obtained.
从信号 r 1 ( t ) + ε 1 ⋅ EMD 1 ( n j ( t ) ) r 1 ( t ) + ε 1 ⋅ EMD 1 n j ( t ) r_(1)(t)+epsi_(1)*EMD_(1)(n_(j)(t))r_{1}(t)+\varepsilon_{1} \cdot \operatorname{EMD}_{1}\left(n_{j}(t)\right) 中提取第一个 IMF，得到 IMF k + 1 IMF k + 1 IMF_(k+1)\mathrm{IMF}_{k+1} 。
IMF k + 1 ( t ) = 1 I ∑ i = 1 I EMD k [ r k ( t ) + ε k EMD k ( n k ( t ) ) ] IMF k + 1 ( t ) = 1 I ∑ i = 1 I   EMD k r k ( t ) + ε k EMD k n k ( t ) IMF_(k+1)(t)=(1)/(I)sum_(i=1)^(I)EMD_(k)[r_(k)(t)+epsi_(k)EMD_(k)(n_(k)(t))]\operatorname{IMF}_{k+1}(t)=\frac{1}{I} \sum_{i=1}^{I} \operatorname{EMD}_{k}\left[r_{k}(t)+\varepsilon_{k} \operatorname{EMD}_{k}\left(n_{k}(t)\right)\right]
Until the residual signal can no longer be decomposed, all the IMFs can be obtained, and the final residual component is shown as follows.
直到残差信号无法再被分解为止，所有的 IMF 都可以得到，最终的残差分量如下所示。
r ( t ) = x ( t ) − ∑ k = 1 K IMF k ( t ) r ( t ) = x ( t ) − ∑ k = 1 K   IMF k ( t ) r(t)=x(t)-sum_(k=1)^(K)IMF_(k)(t)r(t)=x(t)-\sum_{k=1}^{K} \mathrm{IMF}_{k}(t)
The original signal x ( t ) x ( t ) x(t)x(t) can be expressed as follows.
原始信号 x ( t ) x ( t ) x(t)x(t) 可以表示如下。
x ( t ) = r ( t ) + ∑ k = 1 K IMF k ( t ) x ( t ) = r ( t ) + ∑ k = 1 K   IMF k ( t ) x(t)=r(t)+sum_(k=1)^(K)IMF_(k)(t)x(t)=r(t)+\sum_{k=1}^{K} \operatorname{IMF}_{k}(t)
CEEMDAN makes full use of the noise-assisted analysis and can reconstruct the original signal accurately and completely. And adjusting the noise coefficient ε k ε k epsi_(k)\varepsilon_{k}, the noises with different ratios are selected during each decomposition to calculate IMFs.
CEEMDAN 充分利用噪声辅助分析技术，可以准确、完整地重建原始信号。通过调整噪声系数 ε k ε k epsi_(k)\varepsilon_{k} ，在每次分解时选择不同比例的噪声来计算 IMF。

3 Optimal Smooth Denoising Model
3 最佳平滑去噪模型

A low-pass filtering algorithm can be obtained by IMFs with CEEMDAN decomposition. The optimal smooth denoising model is established by combining the smoothness and deviation of the filtering algorithm [22].
低通滤波算法可以通过 IMF 与 CEEMDAN 分解得到。结合滤波算法的平滑度和偏差，可以建立最佳平滑去噪模型[22]。

3.1 Deviation of the Filtering Algorithm
3.1 过滤算法的偏差

After the original signal x ( t ) x ( t ) x(t)x(t) is decomposed by CEEMDAN, the low-pass filtering algorithm can be established, as shown in Eq. (8).
原始信号 x ( t ) x ( t ) x(t)x(t) 经 CEEMDAN 分解后，可建立低通滤波算法，如式（8）所示。
x ~ k ( t ) = x ( t ) − ∑ 1 k IMF k 1 ≤ k ≤ K x ~ k ( t ) = x ( t ) − ∑ 1 k   IMF k 1 ≤ k ≤ K tilde(x)_(k)(t)=x(t)-sum_(1)^(k)IMF_(k)1 <= k <= K\tilde{x}_{k}(t)=x(t)-\sum_{1}^{k} \operatorname{IMF}_{k} 1 \leq k \leq K
Then, the mean square error between the denoised signal and the original signal is shown as follows.
然后，去噪信号与原始信号之间的均方误差如下所示。
MSE f = ∑ m = 1 M ( x ~ m − x m ) 2 M MSE f = ∑ m = 1 M   x ~ m − x m 2 M MSE_(f)=sqrt((sum_(m=1)^(M)( tilde(x)_(m)-x_(m))^(2))/(M))\mathrm{MSE}_{f}=\sqrt{\frac{\sum_{m=1}^{M}\left(\tilde{x}_{m}-x_{m}\right)^{2}}{M}}
where M M MM is the number of sample points; x m x m x_(m)x_{m} is the value of the m m mm th sample point of original signal; and x ~ m x ~ m tilde(x)_(m)\tilde{x}_{m} is the value of the m m mm th sample point of denoised signal. MSE f MSE f MSE_(f)\mathrm{MSE}_{f} reflects the closeness degree between filtered data and original
其中， M M MM 为采样点的个数； x m x m x_(m)x_{m} 为原始信号的第 m m mm 个采样点的值； x ~ m x ~ m tilde(x)_(m)\tilde{x}_{m} 为去噪信号的第 m m mm 个采样点的值。 MSE f MSE f MSE_(f)\mathrm{MSE}_{f} 反映了滤波数据与原始数据的接近程度。
data, which is used to evaluate the quality of the filtering algorithm.
数据，用于评估过滤算法的质量。

3.2 Smoothness of the Filtering Algorithm
3.2 滤波算法的平滑性

Smooth curve is that the function x ( t ) x ( t ) x(t)x(t) has a first continuous derivative in the defined interval. For all the points in a certain curve, the left and right derivatives exist and are equal. For a smooth curve formed by more than two curves, the derivatives at the junction of the curves are equal.
平滑曲线是指函数 x ( t ) x ( t ) x(t)x(t) 在定义的区间内具有一阶连续导数。对于某条曲线上的所有点，左导数和右导数都存在且相等。对于由两条以上曲线构成的光滑曲线，曲线交点处的导数相等。
Suppose that the smooth curve formed by two curves u ( x ) u ( x ) u(x)u(x) and v ( x ) v ( x ) v(x)v(x) joins at the point x 0 x 0 x_(0)x_{0} and the curves are continuous and derivable in the point x 0 x 0 x_(0)x_{0}, the curves must satisfy Eq. (10).
假设由两条曲线 u ( x ) u ( x ) u(x)u(x) 和 v ( x ) v ( x ) v(x)v(x) 形成的光滑曲线在点 x 0 x 0 x_(0)x_{0} 相接，且曲线连续并可在点 x 0 x 0 x_(0)x_{0} 中导出，则曲线必须满足公式 (10)。
u ( x 0 ) = v ( x 0 ) u ′ ( x 0 ) = v ′ ( x 0 ) } u x 0 = v x 0 u ′ x 0 = v ′ x 0 {:[u(x_(0)),=v(x_(0))],[u^(')(x_(0)),=v^(')(x_(0))]}\left.\begin{array}{rl}u\left(x_{0}\right) & =v\left(x_{0}\right) \\ u^{\prime}\left(x_{0}\right) & =v^{\prime}\left(x_{0}\right)\end{array}\right\}
The smooth curve should satisfy that the curvatures of u ( x ) u ( x ) u(x)u(x) and v ( x ) v ( x ) v(x)v(x) at the point x 0 x 0 x_(0)x_{0} are equal.
平滑曲线应满足 u ( x ) u ( x ) u(x)u(x) 和 v ( x ) v ( x ) v(x)v(x) 在 x 0 x 0 x_(0)x_{0} 点的曲率相等。
K u | x = x 0 = K v | x = x 0 | u ′ ′ ( x 0 ) | { 1 + [ u ′ ( x 0 ) ] 2 } 3 / 2 = | v ′ ′ ( x 0 ) | { 1 + [ v ′ ( x 0 ) ] 2 } 3 / 2 K u x = x 0 = K v x = x 0 u ′ ′ x 0 1 + u ′ x 0 2 3 / 2 = v ′ ′ x 0 1 + v ′ x 0 2 3 / 2 {:[K_(u)|_(x=x_(0))=K_(v)|_(x=x_(0))],[(|u^('')(x_(0))|)/({1+[u^(')(x_(0))]^(2)}^(3//2))=(|v^('')(x_(0))|)/({1+[v^(')(x_(0))]^(2)}^(3//2))]:}\begin{aligned} & \left.K_{u}\right|_{x=x_{0}}=\left.K_{v}\right|_{x=x_{0}} \\ \frac{\left|u^{\prime \prime}\left(x_{0}\right)\right|}{\left\{1+\left[u^{\prime}\left(x_{0}\right)\right]^{2}\right\}^{3 / 2}}= & \frac{\left|v^{\prime \prime}\left(x_{0}\right)\right|}{\left\{1+\left[v^{\prime}\left(x_{0}\right)\right]^{2}\right\}^{3 / 2}}\end{aligned}
Substitute Eq. (10) into Eq. (11).
将公式 (10) 代入公式 (11)。
u ′ ′ ( x 0 ) = v ′ ′ ( x 0 ) u ′ ′ x 0 = v ′ ′ x 0 u^('')(x_(0))=v^('')(x_(0))u^{\prime \prime}\left(x_{0}\right)=v^{\prime \prime}\left(x_{0}\right)
By the definition of derivative, u u uu " ( x ) ( x ) (x)(x) and v " ( x ) v " ( x ) v"(x)v "(x) can be represented as follows.
根据导数的定义， u u uu " ( x ) ( x ) (x)(x) 和 v " ( x ) v " ( x ) v"(x)v "(x) 可以表示如下。
u ′ ′ ( x 0 ) = lim h → 0 u ( x 0 − 2 h ) − 2 u ( x 0 − h ) + u ( x 0 ) h 2 u ′ ′ x 0 = lim h → 0   u x 0 − 2 h − 2 u x 0 − h + u x 0 h 2 u^('')(x_(0))=lim_(h rarr0)(u(x_(0)-2h)-2u(x_(0)-h)+u(x_(0)))/(h^(2))u^{\prime \prime}\left(x_{0}\right)=\lim _{h \rightarrow 0} \frac{u\left(x_{0}-2 h\right)-2 u\left(x_{0}-h\right)+u\left(x_{0}\right)}{h^{2}}
v ′ ′ ( x 0 ) = lim h → 0 v ( x 0 + 2 h ) − 2 v ( x 0 − h ) + v ( x 0 ) h 2 v ′ ′ x 0 = lim h → 0   v x 0 + 2 h − 2 v x 0 − h + v x 0 h 2 v^('')(x_(0))=lim_(h rarr0)(v(x_(0)+2h)-2v(x_(0)-h)+v(x_(0)))/(h^(2))v^{\prime \prime}\left(x_{0}\right)=\lim _{h \rightarrow 0} \frac{v\left(x_{0}+2 h\right)-2 v\left(x_{0}-h\right)+v\left(x_{0}\right)}{h^{2}}
Substitute Eq. (10) and Eq. (13-14) into Eq. (12).
将式 (10) 和式 (13-14) 代入式 (12)。
v ( x 0 + 2 h ) − u ( x 0 − 2 h ) − 2 [ u ( x 0 − h ) − u ( x 0 − h ) ] = 0 v x 0 + 2 h − u x 0 − 2 h − 2 u x 0 − h − u x 0 − h = 0 v(x_(0)+2h)-u(x_(0)-2h)-2[u(x_(0)-h)-u(x_(0)-h)]=0v\left(x_{0}+2 h\right)-u\left(x_{0}-2 h\right)-2\left[u\left(x_{0}-h\right)-u\left(x_{0}-h\right)\right]=0
Therefore, the smoothness value of the curve f ( x ) f ( x ) f(x)f(x) at the point x 0 x 0 x_(0)x_{0} can be represented as follows.
因此，曲线 f ( x ) f ( x ) f(x)f(x) 在点 x 0 x 0 x_(0)x_{0} 上的平滑度值可以表示如下。
S N | x = x 0 = f ( x 0 + 2 h ) − f ( x 0 − 2 h ) − 2 [ f ( x 0 − h ) − f ( x 0 − h ) ] S N x = x 0 = f x 0 + 2 h − f x 0 − 2 h − 2 f x 0 − h − f x 0 − h {:[SN|_(x=x_(0))=f(x_(0)+2h)-f(x_(0)-2h)],[-2[f(x_(0)-h)-f(x_(0)-h)]]:}\begin{aligned} \left.S N\right|_{x=x_{0}} & =f\left(x_{0}+2 h\right)-f\left(x_{0}-2 h\right) \\ & -2\left[f\left(x_{0}-h\right)-f\left(x_{0}-h\right)\right] \end{aligned}
where h h hh is the sampling interval. The smoothness value is smaller, and the curve is smoother.
其中 h h hh 为采样间隔。平滑度值越小，曲线越平滑。
The mean square error of all the smoothness values of the filtering curve (without 2 endpoints) is called the flatness of the filtering algorithm, which can be denoted as SMSE f SMSE f SMSE_(f)\mathrm{SMSE}_{f}.
滤波曲线（不含 2 个端点）所有平滑度值的均方误差称为滤波算法的平滑度，可表示为 SMSE f SMSE f SMSE_(f)\mathrm{SMSE}_{f} 。
Springer  施普林格

3.3 Objective Function  3.3 目标函数

In order to obtain the optimal smooth denoising model, a weight coefficient μ μ mu\mu is set to embody the smoothness of the filtering algorithm and the error between the reconstruction and original signal. Then, the objective function is shown as follows.
为了获得最佳的平滑去噪模型，需要设置一个权重系数 μ μ mu\mu 来体现滤波算法的平滑度以及重建信号与原始信号之间的误差。然后，目标函数如下所示。
F = μ ⋅ MSE f + ( 1 − μ ) ⋅ SMSE f F = μ ⋅ MSE f + ( 1 − μ ) ⋅ SMSE f F=mu*MSE_(f)+(1-mu)*SMSE_(f)F=\mu \cdot \mathrm{MSE}_{f}+(1-\mu) \cdot \mathrm{SMSE}_{f}
The smaller the weight coefficient μ μ mu\mu, the greater the smoothness, and the signal has higher denoising degree. According to the literature [22], the weight coefficient μ μ mu\mu normally is set as 0.3 , and F 0 F 0 F_(0)F_{0} is the minimum value, i.e., the objective function can be used to calculate the optimal solution, which possesses the greatest noise reduction effect in smooth denoising model.
权重系数 μ μ mu\mu 越小，平滑度越高，信号的去噪程度越高。根据文献[22]，权重系数 μ μ mu\mu 一般设置为 0.3， F 0 F 0 F_(0)F_{0} 为最小值，即可以利用目标函数计算出最优解，在平滑去噪模型中具有最大的降噪效果。

4 Blasting Vibration Signal Denoising
4 爆破振动信号去噪

4.1 Project Profile  4.1 项目简介

The Jiulongpo-Chaotianmen channel regulation project is located in the upper Yangtze River, Chongqing, China. Five reefs in the channel are to be exploded, and the total quantity of rock mass is about 1 , 54 , 000 m 3 1 , 54 , 000 m 3 1,54,000m^(3)1,54,000 \mathrm{~m}^{3}. Zuanzhaozi Reef, the largest reef to be exploded, is in the mileage section from 675 to 678 km in Yangtze River. The blasting vibration signals are collected from the above reef. The location and the site conditions are demonstrated in Fig. 1.
九龙坡-朝天门航道整治工程位于中国重庆长江上游。该航道共需爆破 5 座礁石，爆破岩体总量约 1 , 54 , 000 m 3 1 , 54 , 000 m 3 1,54,000m^(3)1,54,000 \mathrm{~m}^{3} 。其中最大的爆破礁盘--斑掌子礁位于长江675至678公里里程段。爆破振动信号采集自上述礁盘。位置和现场情况如图 1 所示。
The signals are collected by TC-4850 vibration monitors. The vibration sensors are placed on the shore. The monitor has three channels and can collect seismic signals from the horizontal radial, the horizontal tangential and the vertical
信号由 TC-4850 振动监测器收集。振动传感器放置在岸边。监测器有三个通道，可收集水平径向、水平切向和垂直方向的地震信号。
direction. The received frequency is set at 4000 sps , and the sampling interval is 1 s .
方向。接收频率设定为 4000 sps，采样间隔为 1 秒。

4.2 Denoising Process and Analysis
4.2 去噪过程与分析

A typical measured signal with a sampling frequency of 4 kHz is selected for study. According to Nyquist sampling theorem [23], the Nyquist frequency of the measured signal is 2 kHz . The sampling interval is 1 s , and a total of 4000 sampling points are collected. The typical measured signal is shown in Fig. 2. In the decomposition process of measured signals by CEEMDAN, 50 groups of Gaussian white noises with a standard deviation of 0.2 are added. Therefore, the Nstd (noise standard deviation) and Nr (number of realizations) control parameters of CEEMDAN, respectively, are 50 and 0.2 . The maximum number of sifting iterations allowed (MaxIter) is set to 500 .
研究选择了采样频率为 4 kHz 的典型测量信号。根据奈奎斯特采样定理[23]，测量信号的奈奎斯特频率为 2 kHz。采样间隔为 1 秒，共采集 4000 个采样点。典型的测量信号如图 2 所示。在 CEEMDAN 对测量信号进行分解的过程中，会加入 50 组标准偏差为 0.2 的高斯白噪声。因此，CEEMDAN 的控制参数 Nstd（噪声标准偏差）和 Nr（实现次数）分别为 50 和 0.2。允许的最大筛选迭代次数（MaxIter）设为 500。
The above signal in Fig. 2 is decomposed by CEEMDAN. IMF1-IMF9 and the residue, totally 10 components, can be obtained, which are shown as follows (Fig. 3).
图 2 中的上述信号由 CEEMDAN 进行分解。可以得到 IMF1-IMF9 和残差，共 10 个分量，如下图所示（图 3）。
Fig. 2 Typical measured signal
图 2 典型测量信号
Fig. 1 Project location and site circumstance
图 1 项目位置和现场环境
According to the above IMFs decomposed by CEEMDAN, the denoising models can be established, as shown in Eq. (18).
根据 CEEMDAN 分解的上述 IMF，可以建立去噪模型，如公式（18）所示。
DMk = x ~ k ( t ) = x ( t ) − ∑ 1 k IMF k 1 ≤ k ≤ 10 DMk = x ~ k ( t ) = x ( t ) − ∑ 1 k   IMF k 1 ≤ k ≤ 10 DMk= tilde(x)_(k)(t)=x(t)-sum_(1)^(k)IMF_(k)1 <= k <= 10\mathrm{DMk}=\tilde{x}_{k}(t)=x(t)-\sum_{1}^{k} \operatorname{IMF}_{k} 1 \leq k \leq 10
The smoothness index SMSE f SMSE f SMSE_(f)\mathrm{SMSE}_{f} and reconstruction error index MSE f MSE f MSE_(f)\mathrm{MSE}_{f} of each filtering model can be calculated. Then, the objective function F F FF can be obtained by Eq. (11). The results of each index are shown as follows.
可以计算出每个滤波模型的平滑指数 SMSE f SMSE f SMSE_(f)\mathrm{SMSE}_{f} 和重建误差指数 MSE f MSE f MSE_(f)\mathrm{MSE}_{f} 。然后，由式（11）可求得目标函数 F F FF 。各指标的计算结果如下。
As seen from Fig. 4, from DM1 to DM10, SMSE f f _(f)_{f} gradually increases, while MSE f MSE f MSE_(f)\mathrm{MSE}_{f} gradually decreases. It indicates that the similarity of the denoised signal to the original signal decreases, while the smoothness of the denoised signal improves. The objective function F 2 F 2 F_(2)F_{2} of DM2 is minimum, and the value is 0.114 . Therefore, the optimal denoising model of the typical measured signal is DM2. Comparing the results of the objective function, the F F FF values of DM2～DM5 are not very different. The comparison diagrams between the original signal and the denoised signal are shown as follows.
从图 4 可以看出，从 DM1 到 DM10，SMSE f f _(f)_{f} 逐渐增大，而 MSE f MSE f MSE_(f)\mathrm{MSE}_{f} 逐渐减小。这表明去噪信号与原始信号的相似度降低，而去噪信号的平滑度提高。DM2 的目标函数 F 2 F 2 F_(2)F_{2} 最小，其值为 0.114 。因此，典型测量信号的最优去噪模型是 DM2。比较目标函数的结果，DM2～DM5 的 F F FF 值相差不大。原始信号与去噪信号的对比图如下所示。
As seen from Fig. 5, the noise reduction effect of DM2 is the best and the similarity compared with the original signal is the highest. The similarity of DM3 to the original signal is relatively high, but the signal is excessively denoised in the peak point, which cannot reflect the characteristics of the
从图 5 可以看出，DM2 的降噪效果最好，与原始信号的相似度最高。DM3 与原始信号的相似度相对较高，但信号在峰值点处去噪过度，不能反映信号的特征。
Fig. 3 IMF1-IMF9 and the residue components
图 3 IMF1-IMF9 和残留成分
DM1 DM2 DM3 DM4 DM5 DM6 DM7 DM8 DM9 DM10
Fig. 4 Relationship among MSE f , SMSE f MSE f , SMSE f MSE_(f),SMSE_(f)\mathrm{MSE}_{f}, \mathrm{SMSE}_{f} and F F FF
图 4 MSE f , SMSE f MSE f , SMSE f MSE_(f),SMSE_(f)\mathrm{MSE}_{f}, \mathrm{SMSE}_{f} 和 F F FF 之间的关系
real signal. The smooth effects of DM4 and DM5 are great, but the denoised signals have been obviously distorted. Therefore, DM2 has the best noise reduction effect according to the objective function and the comparison between the original signal and the denoised signal.
真实信号。DM4 和 DM5 的平滑效果很好，但去噪信号明显失真。因此，根据目标函数和原始信号与去噪信号的比较，DM2 的降噪效果最好。

4.3 Time-Frequency Energy Analysis
4.3 时频能量分析

AOK time-frequency analysis technique can well reflect the time-frequency characteristics of vibrations induced by underwater drilling and blasting [24]. To further manifest the effect of the optimal smooth denoising model, the time-frequency energy of original signal and denoised signal is compared by AOK time-frequency analysis. The time-frequency energy spectrums of the signals are shown in Fig. 6. X is the peak value of time-frequency energy, and Y is the dominant frequency.
AOK 时频分析技术能很好地反映水下钻孔爆破引起的振动的时频特征[24]。为了进一步体现最优平滑去噪模型的效果，利用 AOK 时频分析技术比较了原始信号和去噪信号的时频能量。信号的时频能量谱如图 6 所示。X 为时频能量的峰值，Y 为主频。
As seen from Fig. 6, the time-frequency energies of signals are mainly distributed in the frequency range of 0 ∼ 250 Hz 0 ∼ 250 Hz 0∼250Hz0 \sim 250 \mathrm{~Hz} and time range of 0 ∼ 0.5 s 0 ∼ 0.5 s 0∼0.5s0 \sim 0.5 \mathrm{~s}. Noise components in the frequency range of 250 ∼ 500 Hz 250 ∼ 500 Hz 250∼500Hz250 \sim 500 \mathrm{~Hz} and near 1000 Hz are the main cause for the original signal distortion. These noise components are successfully removed by the optimal smooth denoising model of CEEMDAN. The low-frequency energy components, in the range of 0 ∼ 250 Hz 0 ∼ 250 Hz 0∼250Hz0 \sim 250 \mathrm{~Hz}, does not experience significant reduction. The dominant frequencies of the signals are both 25.41 Hz , and the peak value of time-frequency energies, respectively, is 386.1 cm 2 / s 2 386.1 cm 2 / s 2 386.1cm^(2)//s^(2)386.1 \mathrm{~cm}^{2} / \mathrm{s}^{2} and 380.6 cm 2 / s 2 380.6 cm 2 / s 2 380.6cm^(2)//s^(2)380.6 \mathrm{~cm}^{2} / \mathrm{s}^{2}. This indicates that the optimal smooth denoising model of CEEMDAN can not only successfully remove the high-frequency noise but also has no effect on the lowfrequency signal components.
从图 6 可以看出，信号的时频能量主要分布在 0 ∼ 250 Hz 0 ∼ 250 Hz 0∼250Hz0 \sim 250 \mathrm{~Hz} 的频率范围和 0 ∼ 0.5 s 0 ∼ 0.5 s 0∼0.5s0 \sim 0.5 \mathrm{~s} 的时间范围内。频率范围 250 ∼ 500 Hz 250 ∼ 500 Hz 250∼500Hz250 \sim 500 \mathrm{~Hz} 和 1000 Hz 附近的噪声成分是原始信号失真的主要原因。CEEMDAN 的最优平滑去噪模型成功去除了这些噪声成分。在 0 ∼ 250 Hz 0 ∼ 250 Hz 0∼250Hz0 \sim 250 \mathrm{~Hz} 范围内的低频能量成分并没有明显减少。信号的主频均为 25.41 Hz，时频能量的峰值分别为 386.1 cm 2 / s 2 386.1 cm 2 / s 2 386.1cm^(2)//s^(2)386.1 \mathrm{~cm}^{2} / \mathrm{s}^{2} 和 380.6 cm 2 / s 2 380.6 cm 2 / s 2 380.6cm^(2)//s^(2)380.6 \mathrm{~cm}^{2} / \mathrm{s}^{2} 。这表明 CEEMDAN 的最优平滑去噪模型不仅能成功去除高频噪声，而且对低频信号成分没有影响。
Springer  施普林格
Fig. 5 Comparison between the original signal and the denoised signal (DM2～DM5)
图 5 原始信号与去噪信号（DM2～DM5）的比较

Fig. 6 Time-frequency spectrum for Original signal and Denoised signal
图 6 原始信号和去噪信号的时频谱

5 Discussion  5 讨论

The optimal smooth denoising model can be established by EMD and EEMD method [22, 25]. The noise reduction effects of these models are compared in this section. In order to compare the advantages and disadvantages of these models, the signal-to-noise ratio ξ ξ xi\xi and mean absolute error ε ε epsi\varepsilon are used as the evaluation criterion [26], which are defined as follows.
最佳平滑去噪模型可通过 EMD 和 EEMD 方法 [22, 25] 建立。本节将比较这些模型的降噪效果。为了比较这些模型的优劣，采用信噪比 ξ ξ xi\xi 和平均绝对误差 ε ε epsi\varepsilon 作为评价标准 [26]，其定义如下。
ξ = 10 lg ⁡ ( ∑ m = 1 M ( x m ) 2 ∑ m = 1 M ( x m − x ~ m ) 2 ) ξ = 10 lg ⁡ ∑ m = 1 M   x m 2 ∑ m = 1 M   x m − x ~ m 2 xi=10 lg((sum_(m=1)^(M)(x_(m))^(2))/(sum_(m=1)^(M)(x_(m)- tilde(x)_(m))^(2)))\xi=10 \lg \left(\frac{\sum_{m=1}^{M}\left(x_{m}\right)^{2}}{\sum_{m=1}^{M}\left(x_{m}-\tilde{x}_{m}\right)^{2}}\right)
ε = 1 M ∑ m = 1 M ( x m − x ~ m ) 2 ε = 1 M ∑ m = 1 M   x m − x ~ m 2 epsi=(1)/(M)sum_(m=1)^(M)(x_(m)- tilde(x)_(m))^(2)\varepsilon=\frac{1}{M} \sum_{m=1}^{M}\left(x_{m}-\tilde{x}_{m}\right)^{2}
where M M MM is the number of sample points; x m x m x_(m)x_{\mathrm{m}} is the value of the m m mm th sample point of original signal; and x ~ m x ~ m tilde(x)_(m)\tilde{x}_{m} is the value of the m m mm th sample point of denoised signal. The signal-to-noise ratio ξ ξ xi\xi reflects the relationship between signal energy and noise energy, and the mean absolute error ε ε epsi\varepsilon reflects the average energy of noise. It is generally acknowledged that the method with larger signal noise and smaller absolute mean error has better noise reduction effect. The noise reduction effect should not only evaluate the objective measurement index, but also ensure that before and after noise reduction, the peak signal does not change, local waveform does not generate excessive deviation, and signal noise is basically removed.
其中， M M MM 为采样点数； x m x m x_(m)x_{\mathrm{m}} 为原始信号中第 m m mm 个采样点的值； x ~ m x ~ m tilde(x)_(m)\tilde{x}_{m} 为去噪信号中第 m m mm 个采样点的值。信噪比 ξ ξ xi\xi 反映了信号能量与噪声能量之间的关系，平均绝对误差 ε ε epsi\varepsilon 反映了噪声的平均能量。一般认为，信号噪声越大、绝对平均误差越小的方法降噪效果越好。降噪效果不仅要评价客观测量指标，还要保证降噪前后峰值信号不发生变化，局部波形不产生过大偏差，信号噪声基本消除。
The optimal smooth denoising models of EMD, EEMD and CEEMDAN are used to denoise the typical measured
使用 EMD、EEMD 和 CEEMDAN 等最优平滑去噪模型对典型测量值进行去噪。
signal. The denoising indexes obtained by the 3 methods are shown in Table 1. The original signal and the denoised signals are shown in Fig. 7.
信号。三种方法得到的去噪指标如表 1 所示。原始信号和去噪后的信号如图 7 所示。
As seen from Table 1, the signal-to-noise ratio ξ ξ xi\xi of CEEMDAN is the largest, and the value is 19.03 db . The signal-to-noise ratio ξ ξ xi\xi of EMD is 10.33 db , which is the minimum, while the sort for mean absolute error ε ε epsi\varepsilon is just the other way around (EMD > EEMD > CEEMDAN). It indicates that the noise reduction effect of CEEMDAN is the optimal one. As seen from Fig. 7, the noise reduction effects of CEEMDAN and EMD are more significant, but the denoised signal by EEMD still has apparent noises after 0.5 s . The reason is elaborated as follows. The EEMD method adds white noises to the original signal during the decomposition process, which reduces the phenomenon of mode aliasing, but the noise reduction effect is unsatisfactory. And CEEMDAN method adds white noises in pairs, which eliminate the impact of white noises. Comparing the signals before and after noise reduction, the
从表 1 中可以看出，CEEMDAN 的信噪比 ξ ξ xi\xi 最大，其值为 19.03 db。EMD 的信噪比 ξ ξ xi\xi 为 10.33 db，最小，而平均绝对误差 ε ε epsi\varepsilon 的排序正好相反（EMD > EEMD > CEEMDAN）。这表明 CEEMDAN 的降噪效果是最佳的。从图 7 可以看出，CEEMDAN 和 EMD 的降噪效果更为显著，但 EEMD 去噪后的信号在 0.5 s 后仍有明显的噪声。EEMD 方法在分解过程中向原始信号添加了白噪声，虽然减少了模态混叠现象，但降噪效果并不理想。而 CEEMDAN 方法成对添加白噪声，消除了白噪声的影响。对比降噪前后的信号，可以发现
Table 1 Signal denoising index
表 1 信号去噪指数

最佳平滑去噪模型
Optimal smooth denoising
model
Optimal smooth denoising model| Optimal smooth denoising | | :--- | | model |

信噪比 ξ / dB ξ / dB xi//dB\xi / \mathrm{dB}
Signal-to-noise ratio
ξ / dB ξ / dB xi//dB\xi / \mathrm{dB}
Signal-to-noise ratio xi//dB| Signal-to-noise ratio | | :--- | | $\xi / \mathrm{dB}$ |

平均绝对误差 ε / 10 − 3 ε / 10 − 3 epsi//10^(-3)\varepsilon / 10^{-3}
Mean abso-
lute error
ε / 10 − 3 ε / 10 − 3 epsi//10^(-3)\varepsilon / 10^{-3}
Mean abso- lute error epsi//10^(-3)| Mean abso- | | :--- | | lute error | | $\varepsilon / 10^{-3}$ |
EMD 10.33 6.93
EEMD 18.13 1.20
CEEMDAN 19.03 0.94
"Optimal smooth denoising model" "Signal-to-noise ratio xi//dB" "Mean abso- lute error epsi//10^(-3)" EMD 10.33 6.93 EEMD 18.13 1.20 CEEMDAN 19.03 0.94| Optimal smooth denoising <br> model | Signal-to-noise ratio <br> $\xi / \mathrm{dB}$ | Mean abso- <br> lute error <br> $\varepsilon / 10^{-3}$ | | :--- | :--- | :--- | | EMD | 10.33 | 6.93 | | EEMD | 18.13 | 1.20 | | CEEMDAN | 19.03 | 0.94 |
noise reduction curves of CEEMDAN and EEMD are more similar. But the denoised signal by EMD is quite different from the original signal. The local waveform has changed near 0.2 s . Therefore, for signal-to-noise ratio, mean absolute error and denoising signal comparisons, the noise reduction effect of the optimal smooth denoising model of CEEMDAN is superior than the others.
CEEMDAN 和 EEMD 的降噪曲线较为相似。但 EMD 的去噪信号与原始信号有很大不同。因此，在信噪比、平均绝对误差和去噪信号的比较中，CEEMDAN 的最优平滑去噪模型的降噪效果优于其他模型。
In addition, some scholars proposed a variety of denoising methods to deal with the vibration signals induced by tunnel or mine blasting, such as translation invariant wavelet method [27], CEEMD and correlation function [28] and variational mode decomposition [29]. Blasting vibration characteristics are obviously different in diverse engineering environments. The optimal smooth denoising model of CEEMDAN has been effectively applied in underwater drilling and blasting engineering, but its application in other blasting engineering needs to be further studied.
此外，一些学者针对隧道或矿山爆破引起的振动信号提出了多种去噪方法，如平移不变小波法[27]、CEEMD 和相关函数[28]以及变模分解[29]等。在不同的工程环境中，爆破振动特性明显不同。CEEMDAN 的最优平滑去噪模型在水下钻孔爆破工程中得到了有效应用，但在其他爆破工程中的应用还有待进一步研究。

6 Conclusion  6 结论

In order to solve the noise problems of vibration signal induced by underwater drilling and blasting, the optimal smooth denoising model of CEEMDAN is proposed to decompose the original signals. The different denoised signals are obtained by low-pass filtering algorithm. According to the objective function of the filtering algorithm based on the deviation and smoothness of the denoised signal, the
为了解决水下钻孔爆破引起的振动信号噪声问题，提出了 CEEMDAN 的最优平滑去噪模型来分解原始信号。通过低通滤波算法得到不同的去噪信号。根据滤波算法的目标函数（基于去噪信号的偏差和平滑度），可以得出以下结果
Fig. 7 Original signal and denoised signals (CEEMDAN, EEMD, CEEMDAN)
图 7 原始信号和去噪信号（CEEMDAN、EEMD、CEEMDAN）
Springer  施普林格
optimal denoised signal can be obtained to achieve the purpose of noise reduction.
这样就能获得最佳的去噪信号，达到降噪的目的。
The method is used to denoise the measured signals. The results indicate that the model can not only successfully remove the high-frequency noise but also has no effect on the low-frequency signal components. Compared the noise reduction effects of the optimal smooth denoising models based on EMD, EEMD and CEEMDAN, the model of CEEMDAN can denoise the vibration signal more effectively. The denoised signal of CEEMDAN is characterized by better smoothness, less noise and higher similarity to the original signal. The reliability and validity of the optimal smooth denoising model of CEEMDAN are verified, which is of great guiding significance to practical engineering.
该方法用于对测量信号进行去噪。结果表明，该模型不仅能成功去除高频噪声，而且对低频信号成分没有影响。与基于 EMD、EEMD 和 CEEMDAN 的最优平滑去噪模型的降噪效果相比，CEEMDAN 模型能更有效地对振动信号进行去噪。CEEMDAN 的去噪信号具有更好的平滑性、更少的噪声以及与原始信号更高的相似性。验证了 CEEMDAN 最佳平滑去噪模型的可靠性和有效性，对实际工程具有重要的指导意义。
In this paper, the vibration sensors are placed on the shore and the monitoring results are collected on the shore. The novel denoising model is appropriate for the vibration signals on land. However, the vibration signals on water bottom are more complex due to water environment. In addition, the blasting vibration signals in tunnel and mine are also quite different from the object of this paper. The denoising effect of this method for these signals needs further study.
本文将振动传感器放置在岸上，并在岸上收集监测结果。新颖的去噪模型适用于陆地上的振动信号。然而，由于水环境的原因，水底的振动信号更为复杂。此外，隧道和矿井中的爆破振动信号也与本文的研究对象有很大不同。该方法对这些信号的去噪效果有待进一步研究。
Acknowledgement This work was supported by the National Natural Science Foundation of China (Grant No. 41672260, 51704109), Natural Science Foundation of Hunan (Grant No. 2020JJ5163; 2020JJ4300) and Science Foundation of Hunan University of Science and Technology (Grant No. E51884).
致谢 本研究得到了国家自然科学基金（批准号：41672260；51704109）、湖南省自然科学基金（批准号：2020JJ5163；2020JJ4300）和湖南科技大学科学基金（批准号：E51884）的资助。

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  1. Yaxiong Peng  彭亚雄
    1020172@hnust.edu.cn
    1 Hunan Provincial Key Laboratory of Geotechnical Engineering for Stability Control and Health Monitoring, School of Civil Engineering, Hunan University of Science and Technology, Xiangtan, China
    1 湖南科技大学土木工程学院岩土工程稳定性控制与健康监测湖南省重点实验室，湘潭，中国
    2 Faculty of Engineering, China University of Geosciences, Wuhan, China
    2 中国地质大学工程学院，中国武汉