# A Strong Noise Reduction Network for Seismic Records
## 2 Methodology  
Time-frequency domain representation is a way that concurrently analyzes signals in terms of both time and frequency, offering insights into signal variations across these dimensions. Consequently, converting seismic signals from the time domain to the time-frequency domain is advantageous for models to harness frequency information in learning the distinct features of seismic signals and noise. In this study, we employ the short-time Fourier transform (STFT) [46] for this conversion to the time-frequency domain, which is particularly suitable for analyzing non-stationary signals. The STFT divides the signal into shorter time periods and applies the Fourier transform (FT) to these time periods to obtain the time-frequency representation of the signal. This allows us to observe the spectral characteristics of the signal at different time periods. The mathematical expression for the STFT is as follows:  
\[\mathrm{X(t,f)}=\int_{-\infty}^{\infty}\mathrm{x(\tau)w(t-\tau)e^{-j2\pi ft}d\tau} \tag{1}\]  
where \(\mathrm{X(t,f)}\) is the result of the STFT, \(\mathrm{x(\tau)}\) is the input discrete signal, \(\mathrm{w(t-\tau)}\) is the window function, and f is the frequency. In the time-frequency domain, the noisy seismic signal \(\mathrm{Y(t,f)}\) consists of the summation of the clean seismic signal \(\mathrm{S(t,f)}\) and the noise \(\mathrm{N(t,f)}\), expressed by the subsequent equation. This representation facilitates the separation and analysis of the signal and noise components within the seismic data. \[\mathrm{Y(t,f)}=\mathrm{S(t,f)}+\mathrm{N(t,f)} \tag{2}\]  
The objective of denoising is to maximize the removal of noise \(\mathrm{N(t,f)}\) so that the predicted clean seismic signal \(\mathrm{\widehat{S}(t,f)}\) closely approximates \(\mathrm{S(t,f)}\). The equation to obtain \(\mathrm{\widehat{S}(t,f)}\) is as follows:  
\[\mathrm{\widehat{S}(t,f)}=\mathrm{\widehat{M_{S}}(t,f)Y(t,f)} \tag{3}\]  
where \(\mathrm{\widehat{M_{S}}(t,f)}\) represents the time-frequency domain mask of the predicted signal. The network takes the real and imaginary components of the time-frequency coefficients of the noise-containing seismic signals as inputs and outputs the masks of the signal and noise in the time-frequency domain. These predicted time-frequency masks can more accurately estimate the distribution of seismic signals and noise across the frequency spectrum, thereby enhancing the network's ability to discern between signal and noise, resulting in more effective denoising. This study adopts a magnitude-time-frequency mask, influenced by those commonly employed in speech noise reduction algorithms [47, 48]. The formulas for the signal's mask \(\mathrm{M_{S}(t,f)}\) and the noise's mask \(\mathrm{M_{N}(t,f)}\) are as follows:  
\[\mathrm{M_{S}(t,f)}=\frac{1}{1+\frac{|\mathrm{N(t,f)}|}{|\mathrm{S(t,f)}|}} \tag{4}\]  
\[\mathrm{M_{N}(t,f)}=\frac{\frac{|\mathrm{N(t,f)}|}{|\mathrm{S(t,f)}|}}{1+\frac {|\mathrm{N(t,f)}|}{|\mathrm{S(t,f)}|}} \tag{5}\]  
The size of each mask is the same as the input time-frequency domain seismic signal \(\mathrm{Y(t,f)}\), where \(\mathrm{M_{S}(t,f)}\) represents the proportion of the clean seismic signal \(\mathrm{S(t,f)}\) in the noise-containing seismic signal \(\mathrm{Y(t,f)}\) in the range of 0-1 [49]. After obtaining the predicted clean signal \(\mathrm{\widehat{S}(t,f)}\) in the time-frequency domain, the predicted seismic signal \(\mathrm{\widehat{S}(t)}\) in the time domain can be obtained by performing a short-time Fourier inversion [46] on it with the following equation:  
\[\mathrm{\widehat{S}(t)}=\mathrm{STFT}^{-1}\Big{\{}\mathrm{\widehat{S}(t,f)} \Big{\}} \tag{6}\]  
In this paper, we integrate an inception module and CA module into the U-Net, with the network structure depicted in Figure 1a. We use a convolutional layer with a 3 \(\times\) 3 sizeconvolutional kernel with a step size of 2 and padding of 1 to supplant the maximum pooling layer for down-sampling, thereby mitigating the loss of high-frequency seismic information typically incurred during pooling.