# Low-Frequency Expansion Approach for Seismic Data Based on Compressed Sensing in Low SNR
## 1 Introduction  
The propagation distance of the low-frequency components in seismic data is much longer than that of the high-frequency components, and the content it carries is more abundant. Due to the slow attenuation of low-frequency information and strong penetrating ability, it can be used to improve the imaging quality of complex structures [1; 2]. Seismic exploration on land faces challenges from complex surface conditions, such as severe lateral velocity variation and complex terrain [3]. Deep seismic exploration in this complex situation has higher requirements for low-frequency information. In the full waveform inversion, the frequency information below 10 Hz plays an important role [4]. If the seismic data contain high-fidelity low-frequency information, the accuracy requirements for the initial model will be greatly reduced. [PERSON] et al. analyzed in detail the impact of missing low-frequency information on the resolution of the profile and waveform inversion [5]. At present, low-frequency information below 10 Hz in seismic data obtained by conventional acquisition methods is severely contaminated by noise, and the effective low-frequencyinformation will be further lost during processing such as surface wave suppression. Therefore, the recovery of low-frequency information has very practical significance for deep seismic exploration. Deconvolution is an indispensable part of seismic data processing. It compresses the wavelets by eliminating the filtering effect of the earth to obtain the seismic reflection coefficient and ultimately improve the resolution of the data [6,7]. In the 1950s, [PERSON] and [PERSON] first proposed a time series convolution model [8], which regarded the seismic trace as a convolution of a white noise reflection sequence and a minimum phase seismic wavelet, and proposed a predictive deconvolution method. Then, [PERSON] proposed the minimum entropy deconvolution algorithm by adjusting the inverse wavelet to maximize the output timing variance mode, finally obtaining the reflection coefficient. The algorithm does not need to make any assumptions about the wavelet phase [9]. [PERSON], [PERSON], [PERSON], [PERSON] and [PERSON] have successively conducted in-depth research on this method [10, 11, 12, 13, 14]. [PERSON] et al. and [PERSON] et al. proposed a deconvolution algorithm based on minimizing the mutual information rate and implemented the deconvolution algorithm in the frequency domain [15,16]. [PERSON] et al. introduced the spectral simulation algorithm into the deconvolution process, which further improved the resolution of seismic data [17]. In addition, [PERSON] et al. [18] used predictive filtering methods in the frequency domain to extrapolate the frequency spectrum of seismic data and successfully expanded the low-frequency components of the seismic signal. [PERSON] et al. [19] and [PERSON] et al. [20], respectively, proposed spatial spectrum whitening and frequency division denoising processing methods, which can effectively protect the low-frequency energy in seismic data. After that, [PERSON] et al. [2] used the zero-phase matching operator to perform frequency expansion on the low-frequency part of the zero-phase wavelet to improve the amplitude spectrum of a low frequency signal and realize the expansion for the low-frequency component. [PERSON] et al. [21] and [PERSON] et al. [5] successively proposed a low-frequency expansion method based on the compressive sensing (CS) theory and achieved certain results. However, this method is greatly affected by noise. When the noise interference is weak, the reflection coefficient obtained is more accurate, but when the noise is strong, the accuracy of the obtained reflection coefficient will be greatly reduced. The collected data from well logs have a very high vertical resolution and are rich in information of different frequency bands. Well logging is a significant tool that assists geophysicists with making constraints, guidance and quantity analysis regarding low resolution seismic data [22]. [PERSON] et al.