# Microseismic Signal Denoising and Separation Based on Fully Convolutional Encoder-Decoder Network
## 2 Methods and Network Training  
### Methods  
The noisy signal, defined as \(NS(t,f)\), represents the superposition of real microseismic signals \(MS(t,f)\), and noise \(N(t,f)\) that includes some instrumental noise or unknown noise in the time-frequency domain as follows:  
\[NS(t,f)=MS(t,f)+N(t,f) \tag{1}\]  
where \(t\) and \(f\) represents the sampling point and frequency bins, respectively. The purpose of denoising is to minimize the expected error between the actual microseismic signal \(MS(t,f)\) and the estimated microseismic signal \(\overline{MS(t,f)}\):  
\[Error=\frac{1}{n}\sum_{i=1}^{n}\left|MS(t,f)_{i}-\overline{MS(t,f)}_{i} \right|^{2} \tag{2}\]  
where \(\overline{MS(t,f)}=f(t,f)\cdot NS(t,f)\), \(f(t,f)\) is a function that maps \(NS(t,f)\) to the \(\overline{MS(t,f)}\), and \(n\) is the number of samples. In this paper, the error minimization problem is solved in a supervised learning manner in which a deep neural network learns to extract a sparse representation of noisy input waveform \(NS(t,f)\) and to map it to a clean microseismic signal. The above theory is also used for estimation of noise, i.e., \(\overline{N(t,f)}=f(t,f)\cdot NS(t,f)\). Then, the mapping functions \(f(t,f)\) are defined by two individual masks, modified from [PERSON] et al. [25], \(M_{MS}(t,f)\) and \(M_{N}(t,f)\) for estimating microseismic signal and noise, respectively. \[f(t,f)= \begin{cases}M_{\text{MS}}(t,f)=\frac{\frac{\left|\text{ MS}(t,f)\right|}{\left|N(t,f)\right|}}{1+\frac{\left|\text{MS}(t,f)\right|}{ \left|N(t,f)\right|}}\\ M_{\text{N}}(t,f)=\frac{1}{1+\frac{\left|\text{MS}(t,f)\right|}{\left|N(t,f) \right|}}\end{cases}  \tag{3}\]  
Both the two individual masks have the same sizes as the noisy signal \(NS(t,f)\), and they are merged as the targets for optimizing the performance of the neural network during training. Two masks contain a series of values between 0 and 1 to attenuate the noisy signal. The real and imaginary parts of time-frequency coefficients of noisy signals (see Figure 1), as input vector, are firstly reshaped (from 129, 236, 2 into 256, 256, 2) by a zero-padding for the further encodingprocess. This is adapted from the existing method [25] to avoid a lack of noisy signal information in the encoding and decoding process. Otherwise, the feature space without reshaping will be shrunk by multiple downsampling operations (e.g., the shrinking process of the feature space as follows: 129, 236 firstly to 64, 118, then gradually to 32 and 59, 16 and 29, 8 and 14, 4 and 7, 2 and 3, and finally to 1, 1, which causes the loss of signal information, and the feature cannot be reconstructed entirely in the decoding process. The reshaped vector has then been transformed into new layers through a series of encoding operators that consist of a convolution, a ReLU (rectified linear unit) activation, and a batch normalization layer [26]. A stride of 2 is applied after every two successive layers to shrink the feature space gradually and improve the computational efficiency during the encoding process. For the convolution calculation, a larger kernel has a wider receptive field, which obtains more features of neighboring signals. However, the large convolution kernel leads to a dramatic increase in computational time limiting the depth of the neural network. Therefore, the kernel size of convolution/deconvolution layers is set to be 3 through the entire work. The decoding operators generate the masks \(M_{\text{MS}}(t,f)\) and \(M_{\text{N}}(t,f)\) in the decoding process. The corresponding feature maps in the encoding and decoding process are concatenated with skip connections, which improve the convergence of training and the reconstruction information of signal [27]. In the penultimate layer of the denoising network, a softmax activation function is used to produce masks. In the last layer, the masks are reshaped into 129, 236, 2, where each channel represents the time-frequency coefficients of microseismic signal and noise, respectively. Figure 2 shows the flow diagram of the proposed method.