Synthesizing Missing Travel Time of P-Wave and S-Wave: A Two-Stage Evolutionary Modeling Approach

Acquiring sonic waves is an essential part of oil and gas exploration as they give critical information about the well’s data and lithography at each well depth progression. However, these measurements are not always accessible, making analysis challenging. As computational power has improved, machine learning methods may now be used to predict these values from other data. Nonetheless, one shortcoming of existing models is that most of them are not transparent (i.e., black-box models). As a result, although promising great performance, they do not offer much insight to petrophysicists and geologists. This research aims to generate mathematical models for predicting compressional wave (P-wave) and shear wave (S-wave) readings using a multistage evolutionary modeling approach. In particular, a multistage equation modeling approach using tree-based genetic programming (GP) and adaptive differential evolution (ADE) is proposed. The obtained best mathematical models yield ${R}^{{2}}$ of 0.745 and 0.9066 for P-wave and S-wave regression on normalized data, respectively. The average performance of models is ${R}^{{2}}={0}.{90}$ (P-Wave) and ${R}^{{2}}={0}.{75}$ (S-Wave). The performance of these mathematical models is comparable with other “black-box” models but with more compact mathematical approach in regression, thereby opening opportunities for interpretability and analysis. Finally, the “white-box” models presented in this article can be fine-tuned further as needed.