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A Log-Gaussian Cox process framework for zone-free probabilistic seismic hazard assessment in Sumatra

This paper proposes a zone-free probabilistic seismic hazard assessment framework for Sumatra that integrates a Log-Gaussian Cox process with tectonic covariates and an ETAS aftershock model, demonstrating that this approach yields significantly higher design accelerations than current Indonesian codes by eliminating arbitrary source zone boundaries.

Original authors: Irwan Endrayanto Aluicius, Nanang Susyanto, Wiwit Suryanto, Dwi Ertiningsih, Fajar Adi-Kusumo

Published 2026-09-03

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CC BY 4.0

Original authors: Irwan Endrayanto Aluicius, Nanang Susyanto, Wiwit Suryanto, Dwi Ertiningsih, Fajar Adi-Kusumo

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

Earthquakes are among the most destructive forces on our planet, and for decades, engineers have relied on a specific method to design buildings that can survive them. This method, known as probabilistic seismic hazard analysis, attempts to predict how strong the ground shaking will be at a specific location over a long period. Traditionally, scientists have approached this by dividing a region into large, fixed zones, like drawing boxes on a map. Inside each box, they assume the earthquake risk is the same everywhere and that earthquakes happen randomly and independently of one another. However, in places where massive tectonic plates collide and grind against each other, these assumptions often fail. The stress building up underground does not stop abruptly at the edges of a drawn box; it flows smoothly across the landscape. Furthermore, when a giant earthquake strikes, it does not happen in isolation; it triggers a long chain of smaller, clustered aftershocks that follow a different pattern than simple randomness.

In a new study focused on the island of Sumatra in Indonesia, researchers have proposed a way to move beyond these rigid boxes. Instead of forcing the earthquake risk into pre-defined zones, they used a flexible mathematical framework that lets the risk map itself emerge from the physical reality of the ground. By analyzing decades of earthquake records and combining them with measurements of how tightly the tectonic plates are locked together, the team created a continuous, flowing map of seismic danger. This approach revealed that the risk is far more concentrated in certain areas than previously thought, particularly in regions that have been quiet for a long time but are actually storing up immense energy. The findings suggest that current building codes in the region may significantly underestimate the shaking that could occur, especially in areas where the ground is most tightly locked and waiting to slip.

The researchers applied this new method to Sumatra, a long, narrow island sitting on one of the most active and dangerous plate boundaries on Earth. Here, the massive Indo-Australian plate dives beneath the Sunda plate, creating a zone of intense pressure. The team started with a catalog of over 5,000 earthquakes recorded between 1998 and 2024. To understand the true background risk, they first had to separate the main earthquakes from the aftershocks that follow them, much like distinguishing the main event from the echoes. Once they isolated the main shocks, they looked for the physical factors that controlled where these earthquakes happened. They found two key drivers: the distance to the Great Sumatran Fault, a major crack running down the island, and the "inter-seismic coupling," a measure of how tightly the two tectonic plates are stuck together at their boundary.

Using these physical measurements, the team built a model that treats the earthquake risk as a smooth, invisible surface rather than a collection of separate zones. In this model, the risk is highest where the plates are most tightly locked and where the fault is closest. This approach allowed them to see patterns that traditional methods missed. For instance, the model clearly identified a "seismic gap" off the coast of the Mentawai Islands. This area has not experienced a massive rupture in over two centuries, yet the plates there are locked so tightly that the model predicts a very high risk of a future giant earthquake. Because the model does not rely on past earthquake locations to define the zones, it can predict high risk in areas that have been quiet for a long time, provided the physical conditions—like the tight locking of the plates—are right.

When the researchers used this new, continuous risk map to calculate the expected ground shaking for a 475-year return period—a standard benchmark for building design—they found results that were startlingly different from the current national building code. In the city of Padang, located directly above the locked Mentawai gap, the new model predicted ground accelerations nearly three times higher than what the current code requires. In other cities like Banda Aceh, the difference was smaller, but still significant. The researchers traced this gap to two specific choices in their modeling: they accounted for the full range of possible earthquake magnitudes without cutting off the largest ones, and they included the full statistical variability of how ground shaking behaves, which tends to produce higher peaks than average estimates.

The study also tested whether this new map could predict future earthquakes. They trained the model on data from 1998 to 2015 and then asked it to forecast the pattern of earthquakes from 2016 to 2024. The model successfully predicted the general locations where earthquakes would occur, confirming that the physical factors they used—plate locking and fault distance—are reliable guides to where the ground is likely to shake. However, the model did predict a higher total number of earthquakes than actually happened in the test period. The researchers explained this by noting that the training period included a time of unusually high activity following a massive 2004 earthquake, which temporarily inflated the background rate. As the region settles back to a normal pace, the predicted rates are expected to decrease, but the spatial pattern of where the risk is highest remains stable.

The implications of these findings are profound for how we think about safety in earthquake-prone regions. The traditional method of drawing boxes on a map can create false security in areas just outside a high-risk zone, or miss danger in quiet areas that are physically primed for a rupture. By using a framework that flows with the physics of the earth, the researchers have shown that the risk in Sumatra is not uniform and not static. The highest danger lies in the Mentawai gap, a place that has been quiet for centuries but is currently the most tightly locked part of the plate boundary. The study suggests that building codes need to be re-evaluated to reflect this continuous, physics-based reality, ensuring that structures are designed to withstand the true potential of the ground beneath them. This work does not just offer a new map; it offers a new way of seeing the earth, one that recognizes that the risk of a disaster is a smooth, shifting landscape rather than a set of fixed boundaries.

Technical Summary: A Log-Gaussian Cox Process Framework for Zone-Free Probabilistic Seismic Hazard Assessment in Sumatra

Problem Statement
Conventional Probabilistic Seismic Hazard Analysis (PSHA), following the Cornell (1968) framework, relies on two primary assumptions that are difficult to justify in subduction settings like Sumatra: (1) the seismogenic region is partitioned into area source zones with spatially uniform background rates, and (2) earthquake events occur independently in time. In reality, tectonic loading varies smoothly across space rather than in discrete steps, and large mainshocks generate strongly clustered aftershock sequences in both space and time. The authors argue that these assumptions obscure the physical drivers of seismicity and necessitate a "zone-free" alternative where the spatial source term is a fitted stochastic intensity derived from physical covariates.

Methodology
The study proposes a framework that replaces predefined source zones with a Log-Gaussian Cox Process (LGCP) for the background seismicity, coupled with a self-exciting Epidemic-Type Aftershock Sequence (ETAS) model for triggered clustering.

Data and Preprocessing:
- Catalogue: A working dataset of 5,356 events ( $M \ge 4.5$ ) from 1998–2024 in Sumatra ( $95^\circ\text{E}–108^\circ\text{E}$ , $6^\circ\text{S}–6^\circ\text{N}$ ) was used.
- Declustering: To isolate background seismicity, the Gardner–Knopoff (1974) space-time window method was applied, reducing the catalogue to 693 mainshocks. This step was necessary because the 2004–2005 megathrust aftershock sequences would otherwise inflate the apparent background rate by an order of magnitude.
Background Intensity Modeling (LGCP):
- The background intensity $\lambda(s)$ is modeled as a doubly stochastic process where $\log \lambda(s) = \mu(s) + Z(s)$ .
- Covariates: The mean trend $\mu(s)$ $μ (s)$ is a log-linear function of two physically interpretable tectonic covariates:
  - Inter-seismic coupling ( $\phi$ ): The ratio of coupling on the Sunda megathrust interface (derived from geodetic inversion), which peaks in the still-locked Mentawai seismic gap.
  - Fault distance ( $D_{fault}$ ): The perpendicular distance to the Great Sumatran Fault (GSF).
- Latent Field: $Z(s)$ is a Gaussian Random Field (GRF) with an exponential covariance structure to capture residual meso-scale variability not explained by the covariates.
- Estimation: Parameters were estimated via Poisson log-linear maximum likelihood estimation (MLE) on a $60 \times 60$ grid.
Triggered Component:
- The triggered, clustered component is described separately by a self-exciting ETAS model with an Omori–Utsu temporal kernel. This model accounts for event-to-event triggering that the covariate-driven background leaves unexplained.
Hazard Integration:
- The fitted LGCP intensity surface serves directly as the spatial source term in the Cornell hazard integral, eliminating the need for zone delineation.
- The annual exceedance rate of Peak Ground Acceleration (PGA) is calculated using a Gutenberg–Richter magnitude distribution and the Youngs et al. (1997) subduction-interface Ground Motion Prediction Equation (GMPE).
Validation and Comparison:
- Model Selection: The LGCP was compared against a Neyman–Scott Thomas cluster process (a statistical baseline without physical covariates) using the Akaike Information Criterion (AIC) and the inhomogeneous L-function.
- Forecast Validation: A temporal hold-out test (training on 1998–2015, testing on 2016–2024) and the Regional Earthquake Likelihood Model (RELM) testing suite were used to assess spatial forecast performance.

Key Results

Model Performance: The covariate-driven LGCP decisively outperformed the Thomas cluster baseline, with a likelihood-ratio improvement of $2\Delta\ell = 645.7$ ( $\chi^2_2, p < 10^{-4}$ ) and a $\Delta$ AIC of 643.7. The LGCP successfully reproduced the observed hazard gradient, including the high-risk Mentawai gap, without predefined boundaries.
Covariate Significance: Both covariates were highly significant (Cox–Snell pseudo- $R^2 \approx 0.61$ ). The coupling coefficient was positive (higher coupling $\rightarrow$ higher rate), and the fault distance coefficient was negative (closer to GSF $\rightarrow$ higher rate).
Hazard Estimates: The framework yielded 475-year design accelerations significantly higher than the Indonesian national code (SNI 1726:2019). The ratios of LGCP-derived PGA to code PGA were approximately:
- Padang: 2.7x (1.79 g)
- Pekanbaru: 2.5x
- Medan: 2.1x
- Banda Aceh: 1.5x
Spatial Pattern: The highest hazard was concentrated in the northwest and along the Padang corridor, directly corresponding to the high-coupling Mentawai gap. In contrast, Banda Aceh showed lower hazard due to the low residual coupling following the 2004 rupture.
Validation: The temporal hold-out confirmed that the spatial pattern of the LGCP is preserved on unseen data (L-test quantile $\gamma = 0.21$ ). However, the absolute rate was over-predicted by a factor of 3.3 in the test period, attributed to the training window including elevated post-2004 aftershock activity.

Significance and Claims
The paper claims to demonstrate a direct mathematical coupling between a fitted, covariate-driven Cox intensity and the Cornell hazard integral, a combination not previously demonstrated in a subduction setting. The primary significance lies in the ability to derive design ground motions from a continuous stochastic source model rather than predefined zones.

The authors emphasize that the framework:

Reproduces physical gradients: It captures the Mentawai seismic gap and the hazard gradient across Sumatra using only physical covariates (coupling and fault distance), without inputting rupture history or zone boundaries.
Offers a conservative guide: The higher hazard values compared to the national code are traced to two specific, defensible modeling choices: (a) using a magnitude completeness floor ( $M_c=4.5$ ) consistent with the catalogue, which weights larger events more heavily, and (b) applying an untruncated aleatory variability term in the GMPE.
Distinguishes time scales: The authors clarify that while the LGCP is appropriate for long-term engineering design (decades/centuries) where tectonic loading governs the rate, the self-exciting ETAS model is better suited for short-term operational forecasting (days/months) where triggered clustering dominates.

The paper concludes that the zone-free LGCP framework provides a geologically sound, reproducible alternative to traditional area-source PSHA, particularly in regions where tectonic loading varies smoothly and historical rupture data is sparse (e.g., the locked Mentawai gap).

Technical Summary: A Log-Gaussian Cox Process Framework for Zone-Free Probabilistic Seismic Hazard Assessment in Sumatra

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