On the Testing of Ground--Motion Prediction Equations against Small--Magnitude Data

Ground-motion prediction equations (GMPE) are essential in probabilistic seismic hazard studies for estimating the ground motions generated by the seismic sources. In low seismicity regions, only weak motions are available in the lifetime of accelerometric networks, and the equations selected for the probabilistic studies are usually models established from foreign data. Although most ground-motion prediction equations have been developed for magnitudes 5 and above, the minimum magnitude often used in probabilistic studies in low seismicity regions is smaller. Desaggregations have shown that, at return periods of engineering interest, magnitudes lower than 5 can be contributing to the hazard. This paper presents the testing of several GMPEs selected in current international and national probabilistic projects against weak motions recorded in France (191 recordings with source-site distances up to 300km, 3.8\leqMw\leq4.5). The method is based on the loglikelihood value proposed by Scherbaum et al. (2009). The best fitting models (approximately 2.5\leqLLH\leq3.5) over the whole frequency range are the Cauzzi and Faccioli (2008), Akkar and Bommer (2010) and Abrahamson and Silva (2008) models. No significant regional variation of ground motions is highlighted, and the magnitude scaling could be predominant in the control of ground-motion amplitudes. Furthermore, we take advantage of a rich Japanese dataset to run tests on randomly selected low-magnitude subsets, and check that a dataset of ~190 observations, same size as the French dataset, is large enough to obtain stable LLH estimates. Additionally we perform the tests against larger magnitudes (5-7) from the Japanese dataset. The ranking of models is partially modified, indicating a magnitude scaling effect for some of the models, and showing that extrapolating testing results obtained from low magnitude ranges to higher magnitude ranges is not straightforward.

💡 Research Summary

This paper addresses a critical gap in probabilistic seismic hazard assessment (PSHA) for low‑seismicity regions, where only weak ground motions are available and the ground‑motion prediction equations (GMPEs) employed are usually imported from other tectonic settings. The authors compile a dataset of 191 weak‑motion recordings from France (Mw 3.8–4.5, source‑site distances up to 300 km, frequency range 0.1–20 Hz) and evaluate a suite of internationally and nationally used GMPEs using the log‑likelihood (LLH) metric introduced by Scherbaum et al. (2009). The LLH quantifies the probability that observed data could be drawn from the distribution predicted by a given model; lower LLH values indicate a better fit.

Across the full frequency band, three models consistently achieve the lowest LLH values (approximately 2.5–3.5): Cauzzi & Faccioli (2008), Akkar & Bommer (2010), and Abrahamson & Silva (2008). These models perform well both at low frequencies (≤1 Hz) and high frequencies (≥5 Hz), suggesting that their magnitude‑scaling formulations capture the non‑linear behavior of weak motions better than many other published equations, which typically target Mw ≥ 5. The authors find no statistically significant regional differences when the same models are applied to a rich Japanese weak‑motion dataset, indicating that, for magnitudes below 5, regional effects are secondary to magnitude scaling.

To assess the robustness of the LLH approach with limited data, the Japanese catalogue is randomly subsampled to create multiple 190‑record subsets, mirroring the French sample size. The resulting LLH distributions are narrow, confirming that a dataset of roughly 190 observations is sufficient to obtain stable LLH estimates. By contrast, smaller samples (e.g., 50–100 records) produce highly variable LLH values, underscoring the importance of adequate sample size for reliable model ranking.

The authors extend the analysis to larger magnitudes (Mw 5–7) using the same Japanese data. In this higher‑magnitude regime, the ranking of GMPEs changes partially: models that performed well for weak motions lose ground if their magnitude‑scaling terms are not calibrated for larger events. This demonstrates that extrapolating performance from low‑magnitude tests to higher magnitudes is not straightforward and can lead to misleading hazard estimates.

Key insights from the study are:

1. Magnitude scaling dominates the control of ground‑motion amplitudes for Mw < 5, while regional variations are minimal in this range.
2. Cauzzi‑Faccioli, Akkar‑Bommer, and Abrahamson‑Silva are the most reliable GMPEs for weak motions in both French and Japanese contexts.
3. A sample size of ~190 recordings provides statistically stable LLH values, making it a practical benchmark for low‑seismicity regions.
4. Model performance is magnitude‑dependent; a GMPE that fits weak motions well may not be appropriate for larger events without additional validation.
5. Direct transfer of low‑magnitude testing results to high‑magnitude hazard assessments is risky, and each magnitude band should be evaluated separately.

In conclusion, the paper offers a rigorous, statistically grounded methodology for selecting GMPEs in regions where only weak motions are available. It highlights the necessity of considering magnitude‑scaling behavior and sample size, and it cautions against uncritical extrapolation of low‑magnitude validation results to higher‑magnitude hazard analyses. This work thus provides valuable guidance for seismologists and engineers tasked with developing robust PSHA models in data‑sparse environments.