⚡ electrical engineering

Quantifying Epistemic Uncertainty in Ground Motion Prediction Using Bayesian Neural Networks: Application to Peninsular India

This study develops a Bayesian neural network ground motion model for Peninsular India that effectively separates and quantifies epistemic uncertainty from aleatory variability, demonstrating that propagating this scenario-dependent epistemic uncertainty through probabilistic seismic hazard analysis yields significantly higher and more accurate spectral acceleration estimates than traditional aleatory-only approaches, particularly for long return periods in data-sparse regions.

Original authors: Yellapragada Meenakshi, STG. Raghukanth

Published 2026-10-06

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Yellapragada Meenakshi, STG. Raghukanth

Original paper licensed under CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

When the earth shakes, the violence of the ground beneath our feet is not a single, predictable number. It is a complex, chaotic event that depends on how big the earthquake is, how far away it is, what kind of soil the ground sits on, and how deep the rupture lies. For engineers designing hospitals, bridges, and skyscrapers, knowing the likely strength of this shaking is a matter of life and death. They rely on mathematical models to predict these forces, but in many parts of the world, the data needed to build these models is thin. In regions like Peninsular India, where earthquakes are less frequent than in places like Japan or California, scientists have very few recordings of actual shaking to study. This lack of data creates a specific kind of problem: not just that the models might be wrong, but that the scientists themselves do not know how wrong they might be. This is called epistemic uncertainty, a measure of the gaps in human knowledge. If a model pretends to be more certain than the data allows, it can lead to buildings that are either dangerously weak or wastefully over-engineered.

A team of researchers at the Indian Institute of Technology Madras has tackled this problem by building a new kind of prediction tool for Peninsular India. Instead of using a standard computer program that gives a single, fixed answer, they created a system based on Bayesian neural networks. Think of this as a digital brain that does not just learn a set of rules, but learns to understand the limits of its own knowledge. By feeding the system thousands of earthquake records from India and similar stable regions, the researchers taught it to predict the intensity of ground shaking, from the initial jolt to the long, rolling waves that follow. The result is a model that does not just say "the shaking will be this strong," but rather "the shaking will likely be this strong, and here is the range of possibilities we cannot rule out."

The researchers compiled a massive collection of data, pulling together nearly 40,000 recordings from over 3,000 earthquakes. Because strong-motion data in Peninsular India is scarce, they had to mix these local records with data from other stable continental regions, such as the central and eastern United States. This created a rich but uneven dataset, dominated by smaller earthquakes and lacking many recordings of the rare, massive events that occur far away. The new model learned to navigate this uneven landscape, predicting how the ground would move for different magnitudes, distances, and soil types. It successfully reproduced the physical reality of how shaking fades with distance and how it amplifies in softer soil, achieving a high level of accuracy in its predictions.

What makes this work distinct is how it separates two different types of uncertainty. The first is aleatory variability, which is the natural, random chaos of an earthquake. No two earthquakes are exactly alike, even if they are the same size and distance; the ground will always shake differently. The second is epistemic uncertainty, which is the uncertainty in the model itself. In this study, the researchers found that the natural randomness of the shaking is quite large, with standard deviations ranging from 0.76 to 0.89 in logarithmic units. In contrast, the uncertainty about the model's own predictions is much smaller, ranging from about 0.04 to 0.11. This means the model is quite confident in its structure, but the real world remains a noisy, unpredictable place. Crucially, the model showed that this uncertainty is not the same everywhere. It grows larger for long-period shaking, which affects tall buildings, and in areas where the data is sparse, such as for very large earthquakes far away from the recording stations.

The true test of this new approach came when the researchers applied it to real-world safety calculations for four major Indian cities: Mumbai, Chennai, Bangalore, and Hyderabad. They ran simulations to see how including this "uncertainty about the model" changed the final safety estimates. They found that the answer depends heavily on the specific city and the type of building. For some locations, ignoring the model's uncertainty led to underestimating the risk of rare, catastrophic shaking. For others, the difference was minimal. The study demonstrated that you cannot simply add a fixed safety margin to a prediction to account for these unknowns. Instead, the uncertainty must be woven directly into the calculation, changing the result differently for a short building in one city versus a tall tower in another.

The researchers validated their model by checking it against the few major earthquakes that have struck Peninsular India in recent history, such as the 1967 Koyna and 2001 Bhuj events. The model's predictions matched the observed shaking patterns well, falling within the expected range of uncertainty. This suggests that the tool is robust enough to be used for designing safer infrastructure in a region where data is hard to come by. By explicitly quantifying what they do not know, the researchers have provided a more honest and complete picture of seismic risk. Their work shows that in the absence of perfect data, the best way to protect a city is not to pretend the data is perfect, but to build a model that understands its own limits.

Technical Summary: Quantifying Epistemic Uncertainty in Ground Motion Prediction Using Bayesian Neural Networks: Application to Peninsular India

Problem Statement
Probabilistic Seismic Hazard Analysis (PSHA) in data-sparse regions, such as Peninsular India (PI), faces significant challenges due to limited strong-motion recordings and incomplete sampling of magnitude, distance, and site conditions. Existing Ground Motion Models (GMMs) for this region typically provide median predictions without directly quantifying posterior epistemic uncertainty (uncertainty arising from incomplete knowledge and model structure). Inadequate characterization of this uncertainty can lead to overconfident predictions and poorly constrained hazard estimates. While logic trees are commonly used to represent epistemic uncertainty, they often fail to capture the full range of uncertainty specific to the target region or the period- and scenario-dependent nature of model uncertainty.

Methodology
This study develops a Bayesian Neural Network-based Ground Motion Model (BNN-GMM) specifically for Peninsular India to address these limitations.

Database: The model utilizes a composite database of 39,835 records from 3,010 events, compiled from the PEER NGA-East database, the Gupta et al. (2017) dataset, the NEar-Source Strong-motion (NESS) flat file, and region-specific recordings from the Bhuj, Jabalpur, and Koyna earthquakes. The dataset is dominated by small-to-moderate magnitude events (Mw 3.0–4.0) and is heavily reliant on proxy-based $V_{S30}$ estimates.
Model Architecture: The model predicts Peak Ground Acceleration (PGA), Peak Ground Velocity (PGV), and 5%-damped pseudo-spectral acceleration (PSA) at 20 periods (0.01–3.0 s) as functions of Moment Magnitude ( $M_w$ $M_{w}$ ), Joyner–Boore distance ( $R_{jb}$ $R_{j b}$ ), $V_{S30}$ $V_{S 30}$ , and focal depth.
- The network employs a partially Bayesian formulation with two hidden layers where weights and biases are treated as random variables with mean-field Gaussian distributions.
- The output layer remains deterministic, predicting the conditional mean and log-variance for each intensity measure.
- Training is performed using variational inference to approximate the posterior distribution of network parameters, minimizing the negative evidence lower bound (ELBO).
- Hyperparameters (e.g., learning rate, hidden layer width) were optimized using Bayesian optimization (Optuna framework).
Uncertainty Quantification:
- Epistemic Uncertainty: Quantified via Monte Carlo sampling of the posterior network parameters (100 samples per prediction). The variability of these samples represents the uncertainty in the model's median prediction.
- Aleatory Variability: Estimated independently by decomposing logarithmic residuals using a mixed-effects model (REML) to separate inter-event, site-to-site, and within-event components.
Validation and PSHA: The model was validated against statistical metrics, residual trends, and recordings from significant PI earthquakes (Bhuj and Koyna). A PSHA sensitivity analysis was conducted for Mumbai, Chennai, Bangalore, and Hyderabad, comparing three scenarios: (i) total variability (aleatory + propagated epistemic), (ii) aleatory-only, and (iii) aleatory with a three-branch logic tree approximation of the epistemic distribution.

Key Results

Predictive Performance: The BNN-GMM achieved a test $R^2$ of 0.817 and a Mean Squared Error (MSE) of 0.824. The model successfully reproduced physically consistent scaling for magnitude, distance, site conditions, and focal depth, with no significant systematic bias in residuals.
Uncertainty Characterization:
- Aleatory Variability: Total aleatory standard deviations ( $\sigma_{total}$ ) ranged from 0.76 to 0.89 natural-log units. Notably, the site-to-site component ( $\phi_{S2S}$ ) exceeded the single-station component ( $\phi_{SS}$ ) across all periods, attributed to the reliance on proxy-based $V_{S30}$ estimates in the database.
- Epistemic Uncertainty: Epistemic standard deviations ranged from approximately 0.04 to 0.11 natural-log units. This uncertainty was period-dependent and scenario-dependent, increasing in sparsely sampled regions of the predictor space (e.g., large distances or specific magnitude ranges) and generally increasing with oscillator period.
PSHA Sensitivity: Propagating BNN-derived epistemic uncertainty through the hazard integral resulted in systematic increases in spectral acceleration estimates compared to aleatory-only treatments, particularly at long return periods (2,475 years) for Mumbai and Hyderabad. The study found that epistemic uncertainty cannot be adequately represented by a uniform adjustment to the median ground motion; its impact varies significantly with site, spectral period, and annual exceedance rate.
Logic Tree Approximation: The study demonstrated that a three-branch logic tree (using the 5th, 50th, and 95th percentiles of the epistemic distribution) effectively approximates the continuous posterior propagation for the cases considered.

Significance and Claims
The paper claims that the developed BNN-GMM provides a robust framework for ground-motion prediction in data-sparse regions by explicitly separating and quantifying epistemic uncertainty from aleatory variability. Key contributions include:

Methodological Advancement: Demonstrating the application of variational Bayesian inference to ground-motion modeling for stable continental regions, moving beyond deterministic point estimates to full predictive distributions.
Hazard Characterization: Showing that epistemic uncertainty is not a constant factor but is highly dependent on the specific scenario (magnitude, distance, period) and site conditions.
Practical Implication: Arguing that for regions with sparse data like Peninsular India, hazard calculations should propagate period- and scenario-dependent epistemic uncertainty (either via continuous sampling or discrete logic tree branches) rather than relying solely on aleatory variability or uniform adjustments. This leads to a more complete characterization of ground-motion exceedance probabilities, particularly for low-probability, high-consequence events.

The authors acknowledge limitations, including the scarcity of large-magnitude near-source recordings and the extensive use of proxy-based site parameters, which likely contribute to the observed site-to-site variability. They conclude that while the model provides reliable estimates within the interpolation range of the available data, future work should incorporate measured site parameters and additional regional recordings to further reduce uncertainty.

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