## 5 Quantitative Seismic Characterization Based on the RPM-FCNN Framework  
Adaptive rock physics inversion has been executed on the well log data of shale gas reservoirs using the previously established DARPM. The results vividly portray the nonlinear quantitative relationship between reservoir physical parameters and seismic elastic parameters in the form of data and curves. We propose integrating these inversion results based on RPMs with a fully connected neural network (FCNN), thereby constructing an RPM-FCNN framework. This framework aims to establish a nonlinear quantitative relationship between reservoir physical parameters and seismic elastic parameters, facilitating regional data computation and reservoir characterization. The FCNN, also known as the multilayer perceptron (MLP), is a deep-learning model where each neuron in one layer is connected to every neuron in the previous layer [38]. This structure is one of the most common configurations in neural networks. The FCNN comprises interconnected neurons associated with three types of layers: the input layer, hidden layers, and the output layer. Figure 12 illustrates the structure of the FCNN with two hidden layers. Fundamentally, the FCNN maps inputs from the input layer to the output layer through simple arithmetic operations executed by neurons. Each neuron in the hidden and output layers takes input from the previous layer and performs the following operations:  
\[\mu_{i,k}=f(\sum_{j=1}^{n}w_{ij,k}x_{i}-b_{k}) \tag{19}\]  
In the equation, \(f\) represents the activation function, which introduces nonlinearity into the neural network's fitting process. \(\mu_{i,k}\) is the output of the \(i\)-th neuron in the \(k\)-th layer, \(b_{k}\) is the bias value for the \(k\)-th layer, \(x_{i}\) is the \(i\)-th input, and \(w_{ij,k}\) is the weight of the connection between the \(i\)-th input in the current layer and the \(j\)-th neuron in the next layer. In FCNN, the weights \(w_{\bar{y},1}\) from the input layer to the first hidden layer represent the contribution of the \(i\)-th input to the network, indicating the importance of that input in predicting the target values. If the weight associated with an input is positive, it means that the input is positively correlated with the target values, and vice versa for negative weights, indicating a negative correlation. Inputs with larger absolute weight values are typically more important in the prediction process. FCNN exhibits good flexibility and adaptability when dealing with nonlinear problems. Therefore, in this study, by integrating RPMs into FCNN, the RPM-FCNN framework is used to establish a nonlinear quantitative relationship  
Figure 12: The structure of the FCNN with two hidden layers. between reservoir physical parameters and seismic elastic parameters, enabling regional quantitative reservoir characterization. The workflow of quantitative seismic characterization of reservoir physical parameters (organic matter mixture aspect ratio \(\alpha_{k}\) and organic matter porosity \(\Phi_{k}\), anisotropy parameters \(\varepsilon\) and \(\gamma\), clay content _Vclay_, kerogen content _Vkerogen_, and total porosity \(\Phi\)) based on rock physics inversion results in the RPM-FCNN framework is as follows:  
First, we must integrate the RPMs and the inversion results into the FCNN to establish a nonlinear mapping relationship between elastic parameters and reservoir physical parameters. The FCNN is trained with elastic parameters as input elements and reservoir physical parameters as output elements. The training process aims to minimize the error, leading to the optimal weight values and thresholds for the trained FCNN. Once the FCNN is trained, it can be used for prediction. Next, the pre-stack seismic inversion elastic parameters are taken as input data, and the trained RPM-FCNN framework is used to predict the corresponding reservoir physical parameters. To avoid underdetermined problems and reduce computational ambiguity, the data are divided into three groups for computation.