## 2 Theory
### Generation of the Training Set  
For supervised DL methods, sufficiently large training sets that fit the characteristics of actual data are required for complex data applications. In this study, the reflectivity distribution is initially extracted based on well log data to randomly generate multiple reflectivity models. Subsequently, the non-stationary characteristics of the wavelet over time are taken into account due to the change in frequency content for generating the wavelet library. Finally, a large number of training sets are generated using the generated reflection model and wavelets with different frequency bands. This can be achieved through the following two steps:  
1. [label=(0)]
2. Obtain the statistical distribution of the reflection coefficient based on logging data and generate multiple reflectivity models. First, we equally divide the reflection coefficient into several segments based on its values \[\left\{ \begin{array}{l}s_{l}^{lower}=r_{\text{min}}+(l-1)\times\frac{r_{ \text{max}}-r_{\text{min}}}{L}\\ s_{l}^{upper}=r_{\text{min}}+l\times\frac{r_{\text{max}}-r_{\text{min}}}{L} \end{array} \right.,l=1,2,\ldots,L,\] (5)  
where \(r_{\text{min}}\) and \(r_{\text{max}}\) are the maximum value and minimum value of the reflection coefficient, respectively. \(s_{l}^{lower}\) and \(s_{l}^{upper}\) are the lower and upper bounds of segment \(s_{l}\). \(L\) is the total number of segments. Then, according to the number of data points in each part, the probability is defined as  
\[P(s_{l})=\frac{M_{s_{l}}}{M_{T}}, \tag{6}\]  
in which \(M_{s_{l}}\) is the number of data points in the segment \(s_{l}\), and \(M_{T}\) is the total data points of the entire reflectivity. As shown in the histogram in Figure 2a, the probability is calculated with the reflection coefficient which is based on logging data (the first column in Figure 2b), where \(r_{\text{min}}\)= -0.2, \(r_{\text{max}}\)= 0.2, and \(L\) = 81. Finally, the Gaussian fit function is used to match the probability, and Acceptance-Rejection Sampling is used to generate a large number of reflection coefficient models that honor the extracted statistical distribution. The Gaussian fit function is expressed as \[f(a_{i},b_{i},c_{i})=\sum_{i=1}^{n}a_{i}e^{-((x-b_{i})/c_{i})^{2}}, \tag{7}\]  
in which \(a_{i}\) denotes the amplitude of the Gaussian function, \(b_{i}\) signifies the central position, \(c_{i}\) represents the standard deviation, and \(n\) denotes the order of the Gaussian curve. Within the histogram depicted in Figure 2a, the parameters configuring the curve are established with \(n\) assigned a value of six as the Gaussian curve fits best, [\(a_{1}\), \(b_{1}\), \(c_{1}\), \(a_{2}\), \(b_{2}\), \(c_{2}\), \(a_{3}\), \(b_{3}\), \(c_{3}\), \(a_{4}\), \(b_{4}\), \(c_{4}\), \(a_{5}\), \(b_{5}\), \(c_{5}\), \(a_{6}\), \(b_{6}\), \(c_{6}\)] = [8.455, 0.00308, 0.02116, 4.751, -0.006751, 0.005758, -0.3839, -0.04897, 0.000609, 4.58, -0.01605, 0.09542, -1.403, -0.06501, 0.02489, -0.9482, -0.1161, 0.04361]. The second to fifth columns in Figure 2b show the generated reflectivity models, which are similar to the reflection coefficient (the first column) generated from actual logging data. (2) Build a wavelet library with a variable frequency band and convolve it with generated reflection coefficient models. In order to alleviate the change in the frequency content of the non-stationary wavelet, the statistical wavelet is compressed and stretched through the stretch factor \(a\) by us to generate a series of wavelets with varying bandwidths. This process can be formulated as follows:  
\[\bar{w}(t)=w(at),\bar{W}=\frac{1}{a}W\bigg{(}\frac{f}{a}\bigg{)}, \tag{8}\]  
in which \(w\) and \(\bar{w}\) are the original wavelet and the wavelet changed by a stretch factor \(a\) in the time domain, respectively, and \(W\) and \(\bar{W}\) are their Fourier transforms.