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SUMMARY The spatio-temporal properties of seismicity give us incisive insight into the stress state evolution and fault structures of the crust. Empirical models based on self-exciting point processes continue to provide an important tool for analysing seismicity, given the epistemic uncertainty associated with physical models. In particular, the epidemic-type aftershock sequence (ETAS) model acts as a reference model for studying seismicity catalogues. The traditional ETAS model uses simple parametric definitions for the background rate of triggering-independent seismicity. This reduces the effectiveness of the basic ETAS model in modelling the temporally complex seismicity patterns seen in seismic swarms that are dominated by aseismic tectonic processes such as fluid injection rather than aftershock triggering. In order to robustly capture time-varying seismicity rates, we introduce a deep Gaussian process (GP) formulation for the background rate as an extension to ETAS. GPs are a robust non-parametric model for function spaces with covariance structure. By conditioning the length-scale structure of a GP with another GP, we have a deep-GP: a probabilistic, hierarchical model that automatically tunes its structure to match data constraints. We show how the deep-GP-ETAS model can be efficiently sampled by making use of a Metropolis-within-Gibbs scheme, taking advantage of the branching process formulation of ETAS and a stochastic partial differential equation (SPDE) approximation for Matérn GPs. We illustrate our method using synthetic examples, and show that the deep-GP-ETAS model successfully captures multiscale temporal behaviour in the background forcing rate of seismicity. We then apply the results to two real-data catalogues: the Ridgecrest, CA 2019 July 5 Mw 7.1 event catalogue, showing that deep-GP-ETAS can successfully characterize a classical aftershock sequence; and the 2016–2019 Cahuilla, CA earthquake swarm, which shows two distinct phases of aseismic forcing concordant with a fluid injection-driven initial sequence, arrest of the fluid along a physical barrier and release following the largest Mw 4.4 event of the sequence.
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This content will become publicly available on April 17, 2026
Multi-output stochastic emulation with applications to seismic response correlation estimation
Stochastic emulation techniques represent a specialized surrogate modeling branch that is appropriate for applications for which the relationship between input and output is stochastic in nature. Their objective is to address the stochastic uncertainty sources by directly predicting the output distribution for a given input. An example of such application, and the focus of this contribution, is the estimation of structural response (engineering demand parameter) distribution in seismic risk assessment. In this case, the stochastic uncertainty originates from the aleatoric variability in the seismic hazard description. Note that this is a different uncertainty-source than the potential parametric uncertainty associated with structural characteristics or explanatory variables for the seismic hazard (for example, intensity measures), that are treated as the parametric input in surrogate modeling context. The key challenge in stochastic emulation pertains to addressing heteroscedasticity in the output variability. Relevant approaches to-date for addressing this challenge have focused on scalar outputs. In contrast, this paper focuses on the multi-output stochastic emulation problem and presents a methodology for predicting the output correlation matrix, while fully addressing heteroscedastic characteristics. This is achieved by introducing a Gaussian Process (GP) regression model for approximating the components of the correlation matrix, and coupling this approximation with a correction step to guarantee positive definite properties for the resultant predictions. For obtaining the observation data to inform the GP calibration, different approaches are examined, relying-or-not on the existence of replicated samples for the response output. Such samples require that, for a portion of the training points, simulations are repeated for the same inputs and different descriptions of the stochastic uncertainty. This information can be readily used to obtain observation for the response statistics (correlation or covariance in this instance) to inform the GP development. An alternative approach is to use as observations noisy covariance samples based on the sample deviations from a primitive mean approximation. These different observation variants lead to different GP variants that are compared within a comprehensive case study. A computational framework for integrating the correlation matrix approximation within the stochastic emulation for the marginal distribution approximation of each output component is also discussed, to provide the joint response distribution approximation.
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- Award ID(s):
- 2131111
- PAR ID:
- 10586358
- Publisher / Repository:
- Elsevier
- Date Published:
- Journal Name:
- Structural safety
- Volume:
- 115
- Issue:
- 2025
- ISSN:
- 0167-4730
- Subject(s) / Keyword(s):
- Process regression Seismic risk estimation Surrogate modeling
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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