# Iterative Interferometric Denoising Filter for Traveltime Picking
## 2 Methods  
The 1D pseudo-Wigner distribution function is used to apply auto-correlation to a window centered on each sample of the given signal. Here, the auto-correlation operation can capture the repeated pattern implicated in the signal. The corresponding formula is displayed as Equation (1):  
\[d^{int}(x)=\int_{x_{1}\leq|X|}d(x-x_{1})d(x+x_{1})dx_{1} \tag{1}\]  
where \(x\) is the independent variable, \(d\) is the original signal regarded as input, and \(d^{int}\) is the output signal. The variable \(X\) represents a window, and \(x_{1}\) indicates each variable in this window. For a seismic dataset consisting of useful signals and incoherent noises that can be viewed as Gaussian random noise, the seismic data equal the sum of the signal and the noise as shown in Equation (2):  
\[d(x,t)=d_{s}(x,t)+d_{n}(x,t) \tag{2}\]  
where \(d\) is the seismic dataset, \(d_{s}\) is the useful signal, and \(d_{n}\) represents the incoherent noise. In this equation, \(x\) and \(t\) represent offset and time variables, respectively. In addition, as seen in Equation (3), the useful signal \(d_{s}\) is equal to the convolution of the source wavelet with regular characters and Green's function (the response of an impulse):  
\[d_{s}(x,t)=R(x)*s(t) \tag{3}\]  
Here, \(R(x)\) is related to [PERSON]'s function for the subsurface medium, and \(s(t)\) is the source wavelet. Seismic data are recorded in the time domain, where the horizontal axis represents the offset, and the vertical axis represents the time. In the time dimension, the source wavelet convolved with the reflectivity series has a symmetric and regular pattern, which can be captured by Equation (1). In the offset dimension, refraction wave mode has its characteristic pattern and can be highlighted using pseudo-WDF. In this case, we apply pseudo-WDF to both offset and time directions of the data domain (Equation (4)):  
\[d^{int}(x,t)=\int_{x_{1}\leq|X|}\int_{t_{1}\leq|T|}d(x-x_{1},t-t_{1})d(x+x_{1},t+t_{1})dt_{1}dx_{1} \tag{4}\]  
Here, \(d^{int}(x,t)\) is the data processed using our IDF method. \(|X|\) and \(|T|\) are the local offset and time windows used in our proposed IDF procedure. The independent parameters \(x_{1}\) and \(t_{1}\) are space and time variables in a local window scanning in the \(x-t\) domain. This function essentially imposes autocorrelation operations in each direction in one offset-time local window. For example, Figure 1a shows a seismic dataset without noise, with a regular pattern in both the time (Figure 1b) and offset (Figure 1c) dimensions. In contrast, Figure 1d-f shows an example with only noise. It is apparent that the useful signal convolved with the source wavelet exhibits a regular and symmetric pattern in both time and space domains (Figure 1b,c). Still, incoherent data noises do not (Figure 1e,f). So, the convolution of the source wavelet and Green's function component, which has a symmetric kinematic character in both the time and offset domains, can be highlighted using the pseudo-WDF formula shown in Equation (4). In the meantime, the incoherent data noises, which do not have this type of regular character in either the time or spatial domain, are suppressed relative to useful signals. It is noticeable that this function increases the difference between each wave mode in a trace. Hence, the size of the local window used to apply the IDF should be as small as possible to avoid including other wave modes in the IDF window. However, the window size should also be more significant than half the wavelength (in offset dimension) and half the period (in time dimension). These criteria guarantee that the regular pattern around the wavelet center isincluded in the selected window [36]. However, in real applications, the wavelength and period are not only determined by the source wavelet but also by other factors, such as velocity variation, attenuation, and subsurface structure, so it is impossible to define the constant time and spatial value of the interferometric window used in this application.