# Algorithm for Extraction of Reflection Waves in Single-Well Imaging Based on MC-ConvTasNet
## 2 Theory and Method
### Dataset  
Field logging experiments cannot provide data for training because the interface outside the borehole is always unknown. We need to simulate theoretical waveforms as a dataset. In single-well imaging, 3-D finite-difference simulation is often used for forward simulation. However, it takes several hours to simulate the wave field for single-well imaging [30]. It is difficult to use for synthesizing a large number of waveforms in a dataset. [PERSON] and [PERSON] [28] propose an analytical three-step algorithm for quickly calculating theoretical reflection waves. Firstly, the far-field expression of radiated waves is obtained through the steepest-descent integration method [31]. Secondly, the reflected wave is equivalent to the radiation wave of a virtual source. The location and intensity of the virtual source are determined. Thirdly, an analytical expression for the borehole displacement response is obtained through the reciprocity relationship between point sources in a fluid-solid configuration [32]. Therefore, the theoretical full waveform can be synthesized by calculating the direct wave through the real axis integration method [27] and the reflected wave through the fast forward algorithm of single-well imaging [28; 33]. Detailed calculation formulas for direct and reflected waves are shown in Appendix A. When calculating the dataset, we only consider faults. Although cavities are also a common geological feature in single-well imaging, in certain situations, they can be simplified and considered as faults. In the forward calculation model of dipole S-wave imaging, the borehole radius is 0.1 m. The vibration of the acoustic source over time is cosine enveloped, with a center frequency of 3 kHz and a half-bandwidth of 2 kHz. The distance between the acoustic source and the first receiver is 112 inches. There are a total of eight receivers, with a spacing of 6 inches between them. The recording length in time of the full wave is 14.4 ms, with a time sampling interval of 10 \(\mu\)s.  
The definitions of the distance, dip angle and azimuth angle for the reflector are illustrated in Figure 5. By changing the distance, dip angle and azimuth angle (as shown in Table 1), a total of 1092 spatial positions (=13 distances \(\times\) 14 dip angles \(\times\) 3 azimuth angles) of the reflectors are obtained. We generate 245,700 waveforms (=1092 spatial locations \(\times\) 225 formation combinations \(\times\) 8 receivers) as the dataset. The formations on both sides of the reflector are shown in Table 2. In the process of selecting formation parameters, we first determine the S-wave velocity. On the basis of [PERSON]'s mudrock line formula [34] (\(\alpha=1.360+1.16\beta\)) and [PERSON]'s empirical formula [35] (\(\alpha=0.0874\beta^{2}+0.994\beta+1.250\)) for P-wave and S-wave velocities, we determine the corresponding P-wave velocity by incorporating computer-generated random values. The density is not completely independent and is related to the P-wave and S-wave velocities. [PERSON] et al. [36] propose an empirical formula that fits all data without distinguishing rock types, revealing the relationship between density, P-wave velocity and S-wave velocity. The formula is expressed as \(\rho=1.6289a^{0.2254}\beta^{0.0924}\), where \(\rho\) is the formation density. Based on this empirical formula, we generate the density values in the dataset by adding some computer-generated random values. Sedimentary rocks (such as shale, sandstone, limestone, mudstone and dolomite) are the most common geological conditions encountered in oil exploration. The ranges of P-wave velocity, S-wave velocity and density provided in Table 2 encompass these several common types of rocks. Although the formation medium parameters of the dataset are only a subset of the complex geological conditions, the trained neural network shows good generalization ability and can be applied to a variety of geological conditions. The range of the Poisson's ratio of the formation in Table 2 is 0.095-0.462.