*4.1. Data Preparation*

The machine learning models are trained in this study to predict Estatic based on the RHOB, DTs, and DTc as inputs. In this study, core-derived Estatic and their corresponding well log data collected from two different sandstone wells (598 collected from Well-A and 38 from Well-B) were used. The data of Well-A was used to build and test the machine learning models, and Well-B data (unseen data) was used to validate the trained machine learning models. Both formations considered in wells A and B were sandstone formations.

Before training the machine learning models, the data were studied statistically to remove all noise, unreal values, and outliers from the training data. The standard deviation (SD) was considered for removing the outliers; based on this, all data points without the range of ± 3.0 SD were considered as outliers and removed from the input dataset. This preprocessing is very important to ensure accurate estimation of the targeted parameter by applying the machine learning techniques [53]. Out of the 598 data points collected from Well-A, 6 data points were considered as outliers, these data points were removed from the data before the start of the training process.

Since the core derived Estatic was estimated based on well log data, it was very important to perform depth matching between the well log input data and core derived Estatic. Although the gamma-ray log was not considered as input in this study, it was considered at this step to perform the depth matching.

#### *4.2. Training the Machine Learning Models*

After data preprocessing, 592 well log data points and their corresponding core derived Estatic were considered valid for machine learning models training. Four hundred and fourteen, 178, 355, and 444 well log data points (out of the 592) were considered to train ANN, M-FIS, FNN, and SVM models, respectively. The number of the training data was selected based on the optimization process, where the optimum number of the training data that optimize the predictability of the different machine learning models was selected in every case. The statistical characteristics for the training datasets for the different machine learning models are summarized in Table 1. The data of Table 1 is very important when the machine learning models are to be used for evaluating Estatic for a new dataset; the new testing data should be within the ranges in Table 1.


**Table 1.** The statistical characteristics for the training data set.

The input training well logs data were selected based on their relative importance on the actual Estatic which was determined in this study based on the correlation coefficient (R), Figure 3 compares R for the input well log data used to train the different machine learning model. As indicated in Figure 3, all well log parameters used to train the machine learning models are strongly related to Estatic with high Rs of >0.7 for the bulk density, >0.8 for the compressional transit time, and >0.95 for the shear transit time.

Figure 4 shows the inputs used to learn the machine learning models. Inserted *for* loops were designed using MATLAB software to optimize all combinations of the machine learning model's design parameters for Estatic estimation; every single *for* loop represents one design parameter. Sensitivity analysis was conducted to evaluate the effect of changing every single design parameter on the predictability of Estatic by the different machine learning models considered in this study. The sensitivity analysis is a critical step in optimizing the design parameters of the machine learning models and several previous studies considered it as a crucial step in optimizing the performance of different mathematical models [54–56]. Based on the sensitivity analysis results, the combinations of the variables in Table 2 were found to optimize Estatic estimation using the different machine learning models; these parameters predicted Estatic with the lowest average absolute percentage error (AAPE) and the highest R; the AAPE was calculated using Equation (3).

$$AAPE = \frac{1}{N} \sum\_{i=1}^{N} \left( \left| \frac{(E\_{\text{static}})\_i - (E\_{\text{static}})\_i}{(E\_{\text{static}})\_i} \right| \times 100 \right) \tag{3}$$

where *N* represents the number of the data points, *a* and *m* denote the actual and estimated Estatic, respectively.

**Figure 3.** Comparison of the relative importance of the training parameters used to develop (**a**) ANN, (**b**) the Mamdani fuzzy interference system (M-FIS), (**c**) functional neural networks (FNN), and (**d**) support vector machine (SVM) models.


**Table 2.** Optimized design parameters for the machine learning models.

**Figure 4.** The training input well log data collected from Well-A.
