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Development and Application of a Non-ergodic Ground-Motion Model for the Czech Republic

This paper presents a non-ergodic ground-motion model for the Czech Republic that utilizes regional data to derive site- and path-specific terms, significantly reducing aleatory variability and altering seismic hazard estimates compared to traditional ergodic models.

Original authors: Chih-Hsuan Sung, Norman Abrahamson, Ivan prachar, Jiri Vackar, Jiri Malek

Published 2026-09-22

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Chih-Hsuan Sung, Norman Abrahamson, Ivan prachar, Jiri Vackar, Jiri Malek

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

In the quiet, stable heart of Europe, the Czech Republic sits on a geological foundation that rarely shakes. Unlike the volatile edges of tectonic plates where massive earthquakes are common, this region experiences only small to moderate tremors. For decades, scientists have tried to predict how strong the ground might shake during these rare events, a task essential for designing safe buildings and protecting critical infrastructure like nuclear power plants. To do this, they rely on mathematical models that estimate ground shaking based on three main factors: how big the earthquake is, how far away it is, and what kind of soil the shaking hits. Traditionally, these models have treated the Earth as a uniform, average place. They assume that if two earthquakes of the same size happen at the same distance, they will shake the ground with the same intensity, regardless of the specific path the energy takes through the rock or the exact location of the building. This "average" approach works well for broad, global predictions, but it misses the unique, local quirks of the Earth's crust that can make a specific spot shake much harder or softer than expected.

A team of researchers has now taken a closer look at the Czech landscape to see if these local quirks matter. They gathered data from a temporary network of sensors deployed across the country between 2016 and 2019. During this time, the network recorded 14 small earthquakes, ranging from magnitude 2.0 to 4.8, capturing how the ground moved at dozens of different locations. Instead of treating all these movements as random noise, the scientists looked for patterns. They found that the Earth in this region is not uniform; the rock layers and the paths the seismic waves travel create specific, repeatable effects. Some paths through the crust carry energy more efficiently, while others dampen it. By analyzing these specific paths and the unique characteristics of the ground at each recording station, the team built a new kind of model. This new model, called a non-ergodic ground-motion model, stops treating the Earth as a generic average and starts accounting for the specific, systematic differences between any given earthquake source and a specific site.

The results of this new approach were striking. When the researchers compared their new model to the old, traditional one, they found that the old model was hiding a lot of uncertainty. The traditional model assumed that the variations in shaking were random and unpredictable, leading to a wide range of possible outcomes. The new model, however, explained much of that variation by identifying the specific physical reasons behind it. For instance, they discovered that the path from an earthquake in the Eastern Alps to the Czech region behaves differently than other paths, carrying energy with less loss than previously thought. This specific path effect can change the expected shaking intensity by a factor of two. By accounting for these specific path and site effects, the new model reduced the "random" uncertainty significantly. The standard deviation, a measure of how much the shaking might vary, dropped from 0.76 to 0.33 for peak ground acceleration, and from 0.65 to 0.25 for peak ground velocity. In simpler terms, the new model is much more precise because it stops guessing and starts measuring the actual behavior of the local geology.

This increased precision has profound implications for how we assess risk. When scientists calculate the probability of a dangerous earthquake, they use these models to draw hazard curves. The new, more precise model produces steeper curves, meaning that for very rare, long-term events, the predicted shaking is either significantly higher or significantly lower than the old model suggested, depending on the specific location. For example, at a site in Prague, the new model predicts much less shaking than the old one because the specific ground conditions there tend to dampen the waves. Conversely, at a site near a nuclear power plant in the south, the model predicts slightly more shaking due to the efficient path the waves take from certain directions. The trade-off is that while the random uncertainty is lower, the uncertainty about the model itself—what scientists call epistemic uncertainty—has increased. This is because the new model relies on estimating many specific local parameters from a relatively small dataset. However, as more data is collected over time, this uncertainty will shrink, and the model will become even more reliable.

The study demonstrates that even in regions with low seismic activity, where large earthquakes are rare, it is possible to build highly accurate, location-specific models using data from smaller tremors. The researchers showed that temporary sensor networks, which can be deployed quickly and inexpensively, are powerful tools for mapping the hidden characteristics of the Earth's crust. By moving away from the idea of a generic average and embracing the specific details of local geology, scientists can provide much clearer answers about what to expect when the ground shakes. This shift from a one-size-fits-all approach to a tailored, non-ergodic understanding represents a significant step forward in seismic safety, offering a clearer picture of risk for communities and infrastructure in stable continental regions.

Technical Summary: Development and Application of a Non-ergodic Ground-Motion Model for the Czech Republic

Problem Statement
The Czech Republic is a stable continental region characterized by low seismicity. Traditional probabilistic seismic hazard analyses (PSHA) in this region have relied on ergodic ground-motion models (GMMs). These models assume that ground-motion scaling is uniform for all source-site pairs within a tectonic class, treating unmodeled physical effects as random "aleatory" variability. However, instrumental data and macroseismic observations in the Czech region reveal strong systematic effects on ground motion that depend on specific ray paths. For instance, attenuation from Eastern Alpine earthquakes into the Bohemian Massif is significantly weaker than for other paths, resulting in pear-shaped macroseismic fields. The ergodic approach fails to capture these systematic path and site variations, leading to inflated apparent aleatory variability and reduced accuracy in hazard estimation.

Methodology
The authors developed a non-ergodic GMM for Peak Ground Acceleration (PGA) and Peak Ground Velocity (PGV) using ground-motion data from 14 crustal earthquakes (magnitude M2.0–M4.8) recorded by the temporary AlpArray seismic network between 2016 and 2019. The dataset comprises 792 recordings from stations within 400 km of the hypocenters, subject to strict quality control regarding microseism power, teleseismic wave amplitudes, and signal-to-noise ratios.

The development process followed a hierarchical approach:

Regionalization of an Ergodic Baseline: The study began with the Chiou and Youngs (2014) (CY14) global ergodic GMM. Distance scaling was adjusted to fit the Czech data, while magnitude scaling was preserved from the global dataset. A separate constant term was introduced to account for four specific events (3, 7, 10, and 11) that exhibited large negative residuals.
Decomposition of Residuals: The within-event residuals were decomposed into non-ergodic site terms and path terms.
- Site Terms: Modeled as a sum of a spatially correlated component ( $\delta S2S_{reg}$ ) and an uncorrelated site-specific component ( $\delta S2S_{uncor}$ ).
- Path Terms: Modeled as a sum of effects from 3-D anelastic attenuation ( $Q$ structure, $\delta P2P_Q$ ) and 3-D velocity structure ( $\delta P2P_V$ ).
Estimation Techniques:
- Variable Coefficient Model (VCM): Used Gaussian Process regression with a Matern covariance function to estimate the spatially correlated site terms and the velocity-structure path terms ( $\delta P2P_V$ ). This allows for interpolation and extrapolation to unsampled locations, providing both median estimates and epistemic uncertainty.
- Cell-Specific Approach: Used to estimate the anelastic attenuation terms ( $\delta P2P_Q$ ) by dividing the region into 20 km $\times$ 20 km cells and calculating ray path lengths within each cell.
Hazard Application: The non-ergodic GMM was applied to PSHA calculations for three sites: the Dukovany and Temelín nuclear power plants and the city of Prague. The analysis utilized the A3 LASZ v25 areal source model (combined with fault model F3 parameters where applicable) to compute hazard curves, comparing the ergodic and non-ergodic approaches.

Key Results

Reduction in Aleatory Variability: The primary result is a significant reduction in the apparent aleatory standard deviation. For the ergodic GMM, the within-event standard deviations were 0.76 (PGA) and 0.65 (PGV) natural log units. For the non-ergodic GMM, the within-site within-path residuals were reduced to 0.33 (PGA) and 0.25 (PGV).
Systematic Path Effects: The non-ergodic path terms capture strong source-site-specific variations. The study notes that path terms can change the median ground motion by up to a factor of 2 compared to the ergodic GMM. Specifically, the 3-D velocity structure ( $\delta P2P_V$ ) significantly influences short-period ground motions at shorter distances, while the 3-D $Q$ structure ( $\delta P2P_Q$ ) affects long-distance attenuation.
Hazard Curve Implications: The reduction in aleatory variability leads to steeper hazard curves for the non-ergodic model. The net effect on hazard depends on the specific site and path terms:
- Temelín (Site 1): A moderate positive site term was offset by reduced variability at the 1E-3 annual hazard level, but a 15–20% reduction in PGA/PGV was observed at the 1E-5 level.
- Dukovany (Site 2): A small negative site term resulted in a larger reduction in hazard compared to Temelín.
- Prague (Site 3): A large negative site term (-0.32 ln units), combined with reduced variability, resulted in hazard curves significantly lower than the ergodic case (reduction factors of 1.5 and 2.0 for PGA at 1E-3 and 1E-5 levels, respectively).
Epistemic Uncertainty: The study highlights that the reduction in aleatory variability is accompanied by an increase in epistemic uncertainty due to the estimation of additional parameters. For locations with sparse data, the median non-ergodic terms approach zero, but the epistemic uncertainty remains high.

Significance and Claims
The paper claims that the successful application of the non-ergodic GMM approach to the Czech Republic demonstrates that ground-motion data from temporary deployments of small-to-moderate magnitude earthquakes are valuable for constraining non-ergodic terms in low-seismicity regions. The study asserts that by modeling systematic source, path, and site effects, non-ergodic GMMs offer improved accuracy over traditional ergodic models, evidenced by the substantial reduction in standard deviation.

The authors emphasize that while the non-ergodic approach reduces aleatory uncertainty, it increases epistemic uncertainty. They conclude that as more local data are collected over time, the epistemic uncertainty will decrease, allowing for further refinement of the aleatory variability. The paper does not claim that the non-ergodic model universally lowers hazard; rather, the impact is site-specific, driven by the interplay between the non-ergodic site/path adjustments and the reduced aleatory variability. The largest effects on seismic hazard are observed for long return periods.

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