Magnitude-Dependent Aftershock Dynamics

• The paper demonstrates a novel nonparametric Bayesian framework with marked Hawkes processes to accurately capture the dependency of aftershock productivity and timing on mainshock magnitude.
• It employs tensor-product basis expansions and gamma process priors, allowing flexible, data-driven estimation of aftershock excitation functions without reliance on parametric assumptions.
• Results on real-world catalogs show improved immigrant/offspring classification and sharper out-of-sample forecasts, highlighting the method's practical seismic hazard implications.

Magnitude-dependent aftershock dynamics describes the phenomenon in seismology where the statistical properties of aftershocks (secondary earthquakes following a large event) exhibit explicit dependence on the magnitude of the main shock. Modeling this dependence is critical for accurate seismic hazard assessment, statistical forecasting, and understanding the physical mechanisms underlying earthquake triggering. Recent advances employ fully nonparametric Bayesian methods, particularly marked Hawkes processes with flexible, magnitude-dependent (mark-dependent) offspring and background structure, enabling the relaxation of classical parametric assumptions and offering significantly increased inferential scope (Kim et al., 27 Nov 2025).

1. Hawkes Process Framework and Mark-Dependence

Magnitude-dependent aftershock dynamics are most rigorously expressed via marked temporal Hawkes processes. In this framework, observed temporal points $\{(t_i, m_i)\}$—event times and associated magnitudes—are modeled as realizations from a self-exciting point process. The full conditional intensity function is

$$\lambda(t,m|\mathcal H_t) = \lambda_g(t|\mathcal H_t) f^*(m|t,\mathcal H_t)$$

where typically $f^*(m|t,\mathcal H_t) \equiv f(m)$ (i.e., marks are independent of history, conditioned on occurrence time), and the ground process (integrated over all marks) has intensity

$$\lambda_g(t|\mathcal H_t) = \mu(t) + \sum_{t_i < t} h(t - t_i, m_i),$$

with $\mu(t)$ a possibly time-varying background (immigrant) rate and $h(\cdot, m)$ a magnitude-dependent offspring kernel (Kim et al., 27 Nov 2025).

Magnitude-dependence is fully encoded in $h(x, \kappa)$, the expected aftershock intensity at lag $x$ from a parent event of magnitude $\kappa$. In classical ETAS models, $h(x, \kappa)$ is artificially separated into a total productivity term and a fixed lag distribution, with parametric (typically exponential or Lomax) functional forms. Nonparametric extension removes this constraint, allowing both total aftershock productivity and the temporal density of aftershocks to adapt nonparametrically as a function of mainshock magnitude (Kim et al., 27 Nov 2025).

2. Bayesian Nonparametric Modeling of Excitation Functions

The central innovation in modeling magnitude-dependent aftershock dynamics is the expansion of the excitation function $h(x, \kappa)$ as a tensor-product over basis functions for temporal lag and magnitude: $$h(x, \kappa) = \sum_{l=1}^L \sum_{m=1}^M \nu_{lm} \, \phi^{(T)}_l(x)\, \phi^{(M)}_m(\kappa),$$ where $\nu_{lm}$ are nonnegative basis weights, $\phi^{(T)}_l(\cdot)$ are temporal lag basis functions (e.g., Erlang densities), and $\phi^{(M)}_m(\cdot)$ are magnitude basis functions (e.g., monotone polynomial or power bases) (Kim et al., 27 Nov 2025).

A nonparametric gamma process prior is imposed on the set of weights $\{\nu_{lm}\}$: $$H \sim \Gamma\mathrm{P}(H_0, c_0), \quad \nu_{lm} = H(A_{lm}) \sim \mathrm{Gamma}(c_0 H_0(A_{lm}), c_0)$$ ensuring positive, sparse, and shrinkage-prior-regularized excitation surfaces. The base measure $H_0$ encodes prior beliefs as to aftershock productivity in each region of lag-magnitude space, while the concentration parameter $c_0$ controls regularization and adaptivity.

The model tractably marginalizes over the basis coefficients by choosing function families (Erlang, power basis) admitting closed-form integration, thus enabling exact evaluation of the Hawkes process likelihood without need for thinning or numerical quadrature (Kim et al., 27 Nov 2025).

3. Background Intensity and Extensions

Nonparametric modeling is also extended to the background/immigrant rate $\mu(t)$, which may be constant, $\mu \sim \mathrm{Gamma}(\alpha_\mu, \beta_\mu)$, or represented as an Erlang mixture: $$\mu(t) = \sum_{j=1}^J \omega_j\, \mathrm{ga}(t \mid j, \phi^{-1}),$$ with Dirichlet/gamma-process mixture weights $(\omega_j)$, further enhancing model flexibility (especially for highly nonstationary or nonhomogeneous sequences) (Kim et al., 27 Nov 2025).

This generalized construction subsumes classical ETAS as a special case (when the mark-dependence is parametric and separable) and allows estimation of both the total aftershock productivity $\alpha(\kappa)$ and the normalized aftershock delay density $g_\kappa(x)$ as arbitrary, smoothly varying functions of parent magnitude: $$\alpha(\kappa) = \int_0^\infty h(x, \kappa)\,dx, \qquad g_\kappa(x) = h(x, \kappa)/\alpha(\kappa).$$ Such flexibility is crucial for capturing empirically observed features—e.g., that large mainshocks trigger both more and relatively later aftershocks, violating separability (Kim et al., 27 Nov 2025).

4. Posterior Inference and Computational Aspects

Inference proceeds by Markov chain Monte Carlo combining efficient data augmentation for branching structure (latent parent indices $y_i$ indicating whether event $i$ is an immigrant or offspring, and the parent for the latter) and basis assignments ($\xi_i$ indicating which temporal and magnitude basis functions generated each event). Updates for basis weights and background rates are direct gamma draws (due to conjugacy of the gamma process), while hyperparameters employ Metropolis–Hastings. All normalizing integrals reduce to closed forms depending on the chosen basis, avoiding any need for simulation-based or thinning approximations to the Hawkes likelihood (Kim et al., 27 Nov 2025).

The model is fully tractable despite its nonparametric character, supporting exact inference on mark-dependent productivity, lag structure, and time-varying background rates.

5. Empirical Performance and Significance

Empirical investigations on both synthetic and real-world earthquake catalogs (notably the 1885–1980 Japan catalogue) demonstrate several key benefits relative to classical and semi-parametric Hawkes/ETAS variants:

• Accurate recovery of smoothly mark-varying aftershock productivity and shape functions.
• Improved immigrant/offspring classification, as assignment probabilities exploit all available structure in the observed lag-magnitude distribution.
• Sharper out-of-sample forecasts for subsequent seismicity.
• Superior flexibility, capturing observed magnitude-induced modulation in both the number and timing of aftershocks, free from parametric separability or monotonicity constraints (Kim et al., 27 Nov 2025).

Comparisons show that the nonparametric approach substantially outperforms classical ETAS and uniform-mixture semiparametrics both in out-of-sample predictive log-likelihood and in clustering accuracy for the latent branching structure.

6. Dynamics, Interpretation, and Connection to Broader Literature

In this framework, magnitude-dependent aftershock dynamics emerge mechanistically as a consequence of the marked self-exciting process with nonparametric offspring kernels. Each mainshock's magnitude modulates:

• The expected number of aftershocks
• The temporal profile (immediate vs delayed aftershocks)
• The mark-distribution of offspring events (when generalized to multivariate marks)

This generalizes point process self-excitation paradigms common in finance, social contagion, and epidemiology, but is specifically adapted to the marked, magnitude-sensitive setting of seismological applications.

Mathematically, the construction supports rigorous model assessment—posterior sampling of all functionals, credible intervals for productivity curves, and probabilistic branching analysis—without reliance on marginal approximations.

7. Outlook: Future Directions and Open Challenges

Magnitude-dependent aftershock dynamics, as encoded via fully nonparametric marked Hawkes processes, open several further research avenues:

• Extension to spatio-temporal marked Hawkes processes with spatial mark-dependence
• High-dimensional mark structures (joint modeling of magnitude and other event covariates, e.g., fault type)
• Nonparametric causal discovery in interacting self-exciting systems
• Scalable inference for large-scale catalogs with rapidly increasing event and mark complexity

Recent work (Kim et al., 27 Nov 2025) demonstrates that tractable, gamma-process-prior-equipped basis expansions form an effective foundation for these generalizations while retaining analytical and computational tractability.