# Self-Supervised Shear Wave Noise Adaptive Subtraction in Ocean Bottom Node Data
## 2 Methodology  
### The Framework of Self-Supervised Shear Wave Noise Adaptive Subtraction  
In OBN acquisition, the data recorded by vertical component (Z-component) geophone consist of pressure wave (P-wave), unwanted shear wave (S-wave) and other possible noises such as environmental noise and backscattering noise [19]. Assuming a 2-D Z-component profile \(\mathbf{Z}\in\mathbb{R}^{n_{t}\times n_{s}}\) in CRG, the seismic data can be expressed as  
\[\mathbf{Z}=\mathbf{Z_{P}}+\mathbf{Z_{S}}+\mathbf{N}, \tag{1}\]  
where \(n_{t}\) and \(n_{s}\) indicate the number of time samples and shot indexes, and \(\mathbf{Z}\) denotes the fully recorded seismic data. The matrix \(\mathbf{Z_{P}}\) is the recorded P-wave data that are considered as valid signals. \(\mathbf{Z_{S}}\) represents the S-wave noise and \(\mathbf{N}\) for other noises. \(\mathbf{Z_{P}},\mathbf{Z_{S}}\) and \(\mathbf{N}\) are all with the same dimension as \(\mathbf{Z}\). Other noises in the Z-component geophone record are not taken into account in this paper, then Equation (1) is simplified as  
\[\mathbf{Z}=\mathbf{Z_{P}}+\mathbf{Z_{S}}. \tag{2}\]  
The traditional adaptive subtraction approach estimates a matching filter in every 2D data window [39], and its mathematical model can be expressed as follows:  
\[\mathbf{z_{P}}=\mathbf{z}-\mathbf{Sw}, \tag{3}\]  
where the vector \(\mathbf{z_{p}}\) and \(\mathbf{z}\) with dimension \(n_{t}n_{s}\times 1\) denote the estimated P-wave data and the fully recorded seismic data in a single 2-D data window, respectively. \(\mathbf{S}\in\mathbb{R}^{n_{t}n_{s}\times pq}\) is a convolutional matrix constructed via the S-wave noise model. \(\mathbf{w}\in\mathbb{R}^{pq\times 1}\) denotes a matching filter and also represents the degree of shear wave leakage, where the \(p\) and \(q\) denote the temporal and spatial length of the filter, respectively. Due to the nonstationarity of the seismic data, the leakage weighting is varying in both time and space. The traditional method estimates \(\mathbf{w}\) by minimizing the objective function in Equation (4)  
\[f(\mathbf{w})=\|\mathbf{z}-\mathbf{Sw}\|_{2}^{2}. \tag{4}\]This least-squares problem is usually solved using the conjugate gradient least-squares (CGLS) solver and the iterative reweighted least-squares (IRLS) method [19], considered as the linear regression problem. In this way, the data information of different windows cannot be utilized at the same time, and the denoising effect is related to the window size, the step size, the number of iterations and other factors. Moreover, there is a more complex characteristic for the actual data. In this paper, DNN is used to realize adaptive subtraction. The nonlinear mapping between the noise model and the noisy data is fitted by designing a proper neural network structure and loss function. The fact that the DNN's training task of minimizing the loss function is similar to the minimization of the objective function in the conventional approach. As a result, the network is able to predict the actual shear wave noise in the Z-component data. In addition, the method performs directly on a profile of data, fully utilizing the time and space information for signal preservation and better denoising. To introduce the mathematical model of the proposed method, we define \(f(\bullet;\Theta)\) as the output of a neural network with weights \(\Theta\) on input \(\bullet\). Figure 1 shows the workflow of the proposed method, and the structure of the deep neural network is presented in Section 2.2. Since the shear wave noise in the Z-component data is related to the two horizontal components (X- and Y-component), we take the X- and Y-component as noise models in two successive steps to remove the noise related to the X-component (denoted by \(\mathbf{Z_{SX}}\)) and the noise related to the Y-component (denoted by \(\mathbf{Z_{SY}}\)), respectively. First, the X-component data (\(\mathbf{X}\)) are input into the neural network as a noise model with the following loss function (seen in Equation (5)).