## 3 Methodology
### Rock Physics Workflow
#### 3.1.4 Petrophysical Seismic Inversion
The low-frequency earth model is derived from the interpolation of the well-log curves (density, compressional, and shear wave velocities) and constrained by seismic interpretations or horizons influenced by the petrophysical interpretation of formation well-tops and detailed synthetic seismic data to well-tie. This phase is crucial, as it assumes that the properties of geological formations/facies are interpolated within an interval, as guided by horizons. Finally, a moving-average filter is applied to produce the low-frequency component at 10 Hz and below. The forward model for the simultaneous seismic inversion discussed above and the Bayesian inversion uses [PERSON]'s three-term equation. In contrast, simultaneous inversion utilizes regression parameters (gradient m and intercept b) estimated from the regional rock. Bayesian inversion constrains statistical properties such as variance and the correlation between the elastic logs. Finally, due to the limited range of angles, the workflow is restricted to applying only the acoustic and shear impedance for prediction. The seismically derived density volume from AVO inversion is unreliable and excluded from the training data. Using the inverted attributes obtained from the elastic inversion and using the well logs as training dataset, the second phase entails predicting the petrophysical properties using a supervised machine-learning (SML) approach. SML involves the process of finding relationships between provided inputs and outputs and using the relationship to predict an event or outcome. Machine Learning: Neural Network  
This two-step process operates within a probabilistic framework, due to the inherent complexity of the relationship between seismic attributes and petrophysical parameters. In the first stage, it employs the Bayesian time-lapse algorithm shown in [37] to invert changes in elastic properties and their associated uncertainties. Unlike deterministic regularization methods like damped least-squares or maximum likelihood, Bayesian treatment mitigates the impact of noisy data. The a priori model guides the inversion, making priors more influential in low signal-to-noise ratio (SNR) scenarios, while seismic signals dominate in high SNR situations. Complex patterns in datasets can be modeled with neural networks, which are widely used in machine-learning algorithms [19; 42]. A neural network takes inputs, processes them through multiple layers of hidden neurons, and establishes complex relationships with outputs (predictions based on the combined input of all of the neurons). In preparation for the neural network training, input data (acoustic and shear impedance) and output data (petrophysical properties: volume of clay, porosity, and water-saturation) were initially normalized (within the range of 0 to 1). The neural network architecture includes 20 hidden layers, the sigmoid function (suitable for predicting continuous parameters), which serves as the kernel, and an Adams optimizer for finding the optimal weights [43]. Due to multiple tests, the optimal number of iterations of 5000 is used to train the model. To limit bias in the training process, 70% of the data picked at random is used for training and the remaining 30% for validation purposes. See Appendix A for further information on our machine-learning algorithm. For the prediction, we apply the neural network to simultaneously predict the shale volume, porosity, and water saturation (simultaneous workflow). After this, we adopt a sequential workflow, where the shale volume is predicted initially, and subsequently include the shale volume in the training input and predict porosity. Finally, both shale volume and porosity are used to predict water saturation. The workflow adopted for this research is summarized in Figure 3. Through the seismic inversion process, we derive impedance volumes, specifically the primary wave impedance (AI) and shear wave impedance (SI) volumes. Utilizing these AI and SI volumes, our algorithm sequentially predicts the lithology volume.