## 1 Introduction  
Microseismic (MS) monitoring technology, as an advanced dynamic disaster monitoring method, has been widely used in deep-buried tunnels [1], coal mines [2], underground chambers [3], rock slopes [4] and other engineering fields. Sensors are used to receive vibration signals induced by mass rock rupture. However, monitoring systems often work in high-noise environments. The signals picked up by each channel contain a large amount of external noise. These external noises are used for P and S wave pick-up, source location, source mechanism interpretation, and so on, which has caused significant interference for subsequent research work. Therefore, the signal recorded by the vibrator must be filtered to separate the MS effective signal from the noise. MS signal has the characteristics of randomness, non-stationarity, low signal-to-noise ratio, high complexity, and so on. Scholars have done a lot of research on denoising of MS signals. At present, there are many methods for denoising MS signals widely used. The characteristics and application scope of the widely used MS signals can be roughly divided into three categories: estimation filtering method [5; 6], wavelet threshold filtering method [7; 8], and adaptive time-frequency analysis method [9; 10; 11]. These methods have certain limitations in practical engineering applications. In recent years, there has been wide application of compressed sensing technology in information [12], image optimization [13], health care [14; 15], inverse scattering problems [16; 17] and other fields. Due to the limitation of acquisition conditions, the phenomenon of traces missing in prestack seismic data is severe. Some scholars apply compressed sensing theory to the reconstruction of seismic data. [PERSON] et al. [18] proposed a method of seismic data reconstruction based on the theory of compressed perception, which can better restore some essential traces lost in data acquisition. [PERSON] et al. [19] proposed the seismic data based on the optimized Poisson disk sampling and compression sensing reconstruction of the missing track, which much restored the integrity and regularity of the data. [PERSON] et al. [20] proposed a weighted MCA seismic data reconstruction method based on compression perception, which can restore the internal structural characteristics of the data more realistically. According to the CS theory, if the signal is sparse in a certain transformation domain, then the high-dimensional signal can be projected into the low-dimensional space through the measurement matrix irrelevant to the transformation domain. Then high-dimensional signals are reconstructed with high probability through low-dimensional signal. It is an optimization problem [21]. Because noise is not sparse in sparse domains, it cannot be recovered in sparse domains [22]. In most cases, the signal is contaminated by noise. In the face of high signal-to-noise ratio signals, compressed sensing can restore the signal and remove noises with high probability. However, in low signal-to-noise ratio signal reconstruction, since the signal does not have enough sparsity in the sparse domain, part of the noise is also reconstructed, and it is difficult to achieve better denoising effect. Recent studies have shown that the noise signal is amplified by a random measurement process, which causes a decrease in the reconstructed signal-to-noise ratio. This phenomenon is called the Noise Folding (NF) phenomenon in CS [23]. In response to this phenomenon, some scholars have proposed an enhanced minimum L\({}_{1}\) norm algorithm [24] to suppress noise signals and achieve better reconstruction effects. At the same time, some scholars believe that the random signal of the random Gaussian measurement matrix brings poor reconstruction signals, which weakens the NF phenomenon by optimizing the measurement matrix algorithm [25]. Similarly, in the face of low signal-to-noise MS signals, CS denoising will present the NF phenomenon.