# Algorithm for Extraction of Reflection Waves in Single-Well Imaging Based on MC-ConvTasNet
## 2 Theory and Method  
### Network Model  
Figure 1 shows a fluid-filled borehole surrounded by an elastic isotropic homogeneous formation. There is a geologic interface more than ten meters away from the borehole. The  
geologic interface hereafter is called a reflector. An array of receivers on the axis of the borehole receives both direct and reflected waves simultaneously.  
The direct waves excited by a dipole source contain a flexural wave and a longitudinal wave. Because the longitudinal wave amplitude is very weak, it is ignored in the wave separation below. The reflected waves are SH-SH, SV-SV, P-P, P-SV and SV-P waves. The received displacement signal is  
\[u=u_{1}+u_{2}, \tag{1}\]  
where \(u\) is the total displacement signal, and \(u_{1}\) and \(u_{2}\) are borehole displacements stimulated by the direct wave and reflected wave, respectively. The target of wave separation is to separate \(u_{1}\) and \(u_{2}\) in the received waveform. Traditional wave separation methods cannot effectively extract reflected waves. To improve the performance and efficiency of wave separation algorithms, we use MC-ConvTasNet for wave separation. The input data of MC-ConvTasNet are CSG-containing data from the channel currently being processed. The labels are the direct and reflected waves in the channel currently being processed. The propagation distance and velocity of the direct wave and the reflected wave are different, resulting in different arrival time differences between channels in the CSG. Therefore, a two-dimensional convolution (Conv2d) module is added to the MC-ConvTasNet to extract acoustic features between channels to help solve the problem of wave separation.  
MC-ConvTasNet is composed of four parts, an encoder, Conv2d, a separator and a decoder, as shown in Figure 2. Firstly, the encoder module is used to transform the waveform of the channel currently being processed into its encoded representation. Secondly, the Conv2d module is used to extract the inter-channel convolution difference representation in the CSG, from which spatial clues can be obtained to help to distinguish direct and reflected waves. Then, the separator estimates the multiplication function (or mask) of the direct wave and reflected wave by using the encoded representation and the inter-channel convolution difference representation. Finally, the decoder module reconstructs the encoded representation of the direct or reflected wave back into the time-domain waveform of the direct or reflected wave in the channel currently being processed. The structure of MC-ConvTasNet is described in detail in the next section.  
Figure 1: Dipole shear wave imaging schematic diagram. The logging tool is positioned at the center of the wellbore, with its dipole source responsible for exciting the signal. This signal is then captured by a receiver array consisting of eight receivers. This receiver array is capable of not only receiving direct waves propagating along the wellbore but also capturing reflected waves returned from the reflector located near the wellbore. The blue arrow indicates the propagation path of the direct wave, while the black arrow represents the propagation path of the reflected wave.