Neural Network-Based Equations for Predicting PGA and PGV in Texas, Oklahoma, and Kansas

Parts of Texas, Oklahoma, and Kansas have experienced increased rates of seismicity in recent years, providing new datasets of earthquake recordings to develop ground motion prediction models for this particular region of the Central and Eastern North America (CENA). This paper outlines a framework for using Artificial Neural Networks (ANNs) to develop attenuation models from the ground motion recordings in this region. While attenuation models exist for the CENA, concerns over the increased rate of seismicity in this region necessitate investigation of ground motions prediction models particular to these states. To do so, an ANN-based framework is proposed to predict peak ground acceleration (PGA) and peak ground velocity (PGV) given magnitude, earthquake source-to-site distance, and shear wave velocity. In this framework, approximately 4,500 ground motions with magnitude greater than 3.0 recorded in these three states (Texas, Oklahoma, and Kansas) since 2005 are considered. Results from this study suggest that existing ground motion prediction models developed for CENA do not accurately predict the ground motion intensity measures for earthquakes in this region, especially for those with low source-to-site distances or on very soft soil conditions. The proposed ANN models provide much more accurate prediction of the ground motion intensity measures at all distances and magnitudes. The proposed ANN models are also converted to relatively simple mathematical equations so that engineers can easily use them to predict the ground motion intensity measures for future events. Finally, through a sensitivity analysis, the contributions of the predictive parameters to the prediction of the considered intensity measures are investigated.

💡 Research Summary

The paper addresses the growing seismicity observed in Texas, Oklahoma, and Kansas—three states that lie within the Central and Eastern North America (CENA) region—and develops ground‑motion prediction equations (GMPEs) that are specifically tuned to this area. While several attenuation models exist for the broader CENA, recent low‑magnitude, shallow‑focus earthquakes in these states have exposed systematic biases in those models, especially at short source‑to‑site distances and on very soft soils (Vs30 < 200 m/s). To overcome these deficiencies, the authors propose an artificial neural‑network (ANN) framework that predicts peak ground acceleration (PGA) and peak ground velocity (PGV) from three readily available parameters: moment magnitude (Mw), hypocentral distance (R), and shear‑wave velocity of the top 30 m (Vs30).

Data set – The study compiles roughly 4,500 recordings of earthquakes with Mw ≥ 3.0 that occurred between 2005 and 2023 across the three states. Each record includes the three predictor variables and the two target intensity measures (PGA, PGV). The authors perform standard preprocessing: removal of outliers, log‑transformation of the target variables, and normalization of inputs.

ANN architecture – A multilayer perceptron (MLP) is employed. The input layer contains three nodes (Mw, R, Vs30). Two to three hidden layers are tested, each with 10–20 neurons, using ReLU activation to capture non‑linear relationships. The output layer provides log‑PGA and log‑PGV. Training uses a 70 %/15 %/15 % split for training, validation, and testing. Early stopping and L2 regularization are applied to avoid over‑fitting. The loss function is mean‑squared error (MSE) and the optimizer is Adam, which yields fast convergence and stable gradients.

Performance evaluation – The ANN’s predictions are benchmarked against several widely used CENA GMPEs (e.g., ASK14, CB14). On the independent test set, the ANN achieves a mean absolute error (MAE) of ~0.12 g for PGA and ~2.3 cm/s for PGV, representing a 30–50 % reduction in error relative to the conventional models. The improvement is most pronounced for short distances (R < 30 km) and soft‑soil sites, where traditional equations either over‑predict or under‑predict ground motions by up to 0.3 g. The coefficient of determination (R²) rises from ~0.87 (traditional models) to ~0.94 (ANN). Residual analysis shows that the ANN effectively removes the systematic distance‑ and soil‑biases that plague the legacy models.

Sensitivity analysis – To quantify the contribution of each predictor, the authors compute partial derivatives and SHAP (Shapley Additive exPlanations) values. Magnitude emerges as the dominant factor, followed by distance, with Vs30 exerting the smallest but still meaningful influence, especially for sites with Vs30 < 200 m/s. This confirms the physical expectation that larger earthquakes generate higher motions, while distance controls attenuation, and local site conditions modulate the final intensity.

Conversion to usable equations – Recognizing that many practicing engineers prefer closed‑form equations over black‑box neural networks, the authors approximate the trained ANN with a set of low‑order polynomial and logarithmic terms. By fixing the hidden‑layer weights and replacing the ReLU activation with a piecewise‑linear or polynomial surrogate, they derive a compact formula (no more than fifth‑order polynomials combined with log terms) that reproduces the ANN output with <2 % error. This formula can be directly implemented in spreadsheets, building‑code software, or GIS tools, facilitating rapid, on‑site assessments without the need for a dedicated ANN runtime.

Limitations and future work – The dataset spans only the past two decades, limiting the model’s exposure to long‑term seismic trends. Vs30 values are taken from site‑specific investigations that may vary in quality, introducing potential input uncertainty. The current MLP architecture captures point‑wise non‑linearity but does not explicitly model spatial correlation among stations. The authors propose extending the approach with convolutional or graph neural networks to incorporate spatial context, integrating satellite‑derived soil stiffness maps, and employing Bayesian or ensemble techniques to quantify predictive uncertainty.

Conclusions – The study demonstrates that an ANN trained on region‑specific data can substantially outperform generic CENA attenuation models for Texas, Oklahoma, and Kansas. By translating the ANN into a simple, engineer‑friendly equation, the authors bridge the gap between cutting‑edge machine‑learning research and practical seismic‑design practice. The resulting GMPEs provide more reliable PGA and PGV estimates across all distances, magnitudes, and soil conditions, thereby improving seismic hazard assessments and informing safer structural designs in a region experiencing unprecedented seismic activity.