## 1 Introduction  
Ideal broadband seismic data can reveal intricate geological details and provide high-resolution images of subsurface structures. However, due to the excitation characteristics of the explosive source and the absorption of high frequencies by the earth's layers, as well as the suppression of low-frequency noise in data processing, the actual seismic data often partially lack high-frequency and low-frequency signals, resulting in insufficient imaging resolution and limited detection capabilities of deep geological structures [1]. To mitigate these challenges, frequency augmentation processing becomes essential. Frequency extension processing of seismic data plays an important role in data interpretation and inversion. Extending the high-frequency information can help to shorten the wavelet width and improve the spatial resolution in imaging results. At the same time, enhancing the low-frequency information can weaken the energy of the side lobe, dividing the layers more explicitly and precisely [2]. In full waveform inversion, the low-frequency information can improve the inversion's accuracy and estimate the parameters of the subsurface strata closer to the actual situation [3]. Therefore, the research and application of frequency extension methods for seismic data have become one of the key means to improve data resolution. Traditional methods for extending the frequency range of seismic data typically involve deconvolution, inverse Q filtering, and time-frequency transformation [4]. However, these techniques come with their own set of challenges. Deconvolution may introduce some unstable factors, leading to inaccurate processing results. Inverse Q filtering is effective at enhancing high frequencies but falls short in amplifying low frequencies. The time-frequency transformation method may be affected by noise and other interference, resulting in incomplete and inaccurate frequency recovery. In recent years, some new seismic data frequency extension methods have begun to emerge with the continuous development of technology. [PERSON] et al. [5] proposed a spectrum extension technique that combined broadband processing and a tuned energy enhancement method, which effectively improved the seismic identification and prediction capabilities of thin-crossing reservoirs. [PERSON] et al. [6] solved the sparse inversion problem based on compressed sensing theory and achieved low-frequency compensation of seismic data. [PERSON] et al. [7] proposed a deconvolution low-frequency recovery method based on high- and low-frequency joint measurement, solving the amplitude and phase distortion generated during the signal acquisition process. [PERSON] et al. [8] used compressed sensing technology and a broadband Yu-type low-pass shaping filter to analyze and compensate for seismic low-frequency information. [PERSON] et al. [9] realized the expansion of lacking low-frequency signals based on compressed sensing theory. [PERSON] et al. [10] proposed a time-domain seismic data compression method and frequency band extension using nth power operations, which effectively enhanced the low-frequency components of seismic data. In recent years, the advancement of computer hardware resources has significantly facilitated the application of deep learning in geophysical research [11; 12; 13]. [PERSON] et al. [14] introduced the GAN and combined it with an adaptive bandwidth extension method to use deep learning to improve the resolution of seismic data. [PERSON] et al. [15; 16; 17] studied low-frequency bandwidth extension methods through convolutional neural networks (CNNs). [PERSON] et al. [18] proposed a data-driven low-frequency recovery method based on high-frequency signals to improve the reliability of full waveform inversion. [PERSON] et al. [19] enhanced the spectrum by training the U-Net network by considering data characteristics and prior information. [PERSON] et al. [20] achieved low-frequency extrapolation through supervised learning. [PERSON] and [PERSON] et al.