CL-ETAS Model Extensions

• CL-ETAS Model is a generalized extension of the classical ETAS framework that incorporates innovative triggering mechanisms like clock-advance and regime-switching to enhance seismic forecasting.
• It replaces traditional magnitude cutoffs with productivity moments and Lévy jump processes, enabling refined analysis of microseismicity and infinite-generation aftershock sequences.
• Hybrid models that integrate deep learning with physical ETAS structures significantly improve forecast accuracy by capturing non-Poisson clustering and complex memory effects.

The term CL-ETAS refers to several distinct, advanced formulations of the Epidemic-Type Aftershock Sequence (ETAS) model for earthquake clustering, each extending the classical ETAS framework in fundamental ways to enhance the modeling of seismicity. These extensions address issues such as the incorporation of microseismicity without magnitude cutoffs, the empirical two-regime memory in aftershock sequences, catalog-level non-Poissonian clustering, and hybridization with deep learning architectures. The principal variations, as detailed in recent literature, include (1) the Clock-Advance ETAS (CL-ETAS) model, (2) the two-regime memory CL-ETAS/ETAS2, (3) the Catalog-Level ETAS (CL-ETAS) with non-Poisson clustering, and (4) the ConvLSTM–ETAS hybrid (CL-ETAS). Each targets distinct theoretical or operational shortcomings in the standard ETAS approach.

1. Core Principles and Theoretical Modifications

The CL-ETAS family generalizes the ETAS paradigm by reinterpreting event triggering and branching mechanisms. A central innovation is the shift from direct additive rate modification to either time-changed, infinite-generation Markovian superpositions (clock-advance) or to catalog-level, non-Poissonian resampling procedures (catalog-level clustering). In certain formulations, the model replaces the classical magnitude process with productivity moments, introducing Lévy jump processes to address infinite-branching rates at low magnitudes, with convergence controlled by the mean productivity rather than ad-hoc cutoffs (Holschneider, 2 Jan 2025).

Alternate formulations introduce regime-switching or random aftershock compression factors to more faithfully reproduce observed short- and long-term memory exponents and the high-variance clustering signatures that characterize real earthquake sequences (Zhang et al., 2020, Baughman et al., 2023). Hybridizations with deep spatiotemporal neural networks further integrate data-driven pattern extraction with physical rate models (Zhang et al., 2023).

2. Mathematical Structure and Conditional Intensity

a) Clock-Advance ETAS (Holschneider, 2 Jan 2025)

The conditional rate function is reparameterized to describe a process on productivity moment $Y = 10^{\kappa + \alpha m}$, with Lévy jump density: $$\psi(Y) = \gamma Y^{-1-b/\alpha}, \;\; 0 < Y \leq Y_{\max}, \quad \gamma = \frac{10^{a+\kappa b/\alpha}}{\ln 10 \,\alpha},$$ assuming $\alpha > b$ for finite mean.

The conditional intensity (hazard) becomes: $$\psi(t,Y|\mathcal H_t) = \psi(Y)\left(1 + \sum_{\tau \leq t} Y(\tau) h(t-\tau)\right)$$ where $h$ is the Omori–Utsu kernel.

Triggered events correspond to copies of the background Lévy process, but run on accelerated "clocks" determined by the moment and Omori integral of each ancestor event: $$C^{\rm triggered}_Y(t) \overset{d}{=} C^{B}_{Y}(Y(\tau) H(t-\tau)), \quad H(t) = \int_0^t h(u)\, du.$$

Infinite generations are constructed by Markovian recursion and summed for the total process. No hard lower magnitude cutoff is applied; the convergence of the infinite sum is controlled by the mean productivity $D = \int_0^{Y_{\max}} Y\,\psi(Y) dY < 1$.

b) Two-Regime Memory CL-ETAS/ETAS2 (Zhang et al., 2020)

Here, the conditional intensity includes a dual-regime aftershock productivity: $$\lambda(x,y,t|H_t) = \mu(x,y) + \sum_{i:t_i < t} k(M_i)g(t-t_i) f(x-x_i, y-y_i|M_i)$$ with

$$k(M_i) = \begin{cases} A \exp\left[\alpha_1(M_i-M_0)\right] & \text{for } i > n - N_c \ A \exp\left[\alpha_2(M_i-M_0)\right] & \text{otherwise} \end{cases}$$

where $N_c = h\,10^{-bM_0}$ is the crossover count, and spatial and temporal kernels follow established ETAS forms.

c) Catalog-level Non-Poisson Clustering (Baughman et al., 2023)

The triggering kernel is modulated by a random cluster compression factor $f_i = 1-\epsilon_i$, where $\epsilon_i \sim Uniform(0,\epsilon_{max})$. Aftershock clusters are compressed in time post hoc, producing observed clusters that deviate from the standard Poisson hypothesis while preserving productivity and magnitude–frequency scaling: $$\lambda(t,x,y,m) = \mu(x,y)s(m) + \sum_{i:t_i < t} K10^{\alpha(m_i-m_0)}f_i (t-t_i + c)^{-p}(r_{ij}+d)^{-q} s(m)$$ with $s(m) = \beta e^{-\beta(m-m_0)}$, $\beta = b\ln10$.

d) ConvLSTM–ETAS Hybridization (Zhang et al., 2023)

The architecture fuses ETAS one-step-ahead forecasts with observed counts as separate input channels to a 4-layer ConvLSTM with Conv3D readout, mapping sequences of $5\times20$ magnitude-space binned tensors to next-day forecasts. The overall prediction can be characterized as: $$\lambda_{CL-ETAS}(t+1,x,y) \approx F_{CL}(X_{t-L+1:t},\,\widehat X^{ETAS}_{t+1}),$$ where $F_{CL}$ is the trained network, implicitly discovering optimal combinations of physical ETAS structure and data-driven spatiotemporal corrections.

3. Simulation Algorithms and Computational Methods

a) Markov Chain Superposition (Holschneider, 2 Jan 2025)

The core simulation proceeds generation-wise, each iteration sampling an independent realization of the background jump process, which is then time-warped according to the cumulative moment and Omori kernel of prior generations: $$Y^{T,0}(t) = Y^B(t), \quad H^{*,k}(t) = h * C^{T,k}(t),$$

$Y^{T,{k+1}}(t) = Y^{B,{k+1}}(H^{*,k}(t)),$

summed to convergence, controlling for subcriticality ($D<1$). Each realization does not require partitioning by magnitude, but rather aggregates over the productivity moment variable directly.

b) Catalog Resampling Algorithm (Baughman et al., 2023)

Simulation proceeds in two phases:

1. Bare ETAS Catalog: Generate using standard maximum-likelihood–estimated branching and background parameters, in time, space, and magnitude.
2. Cluster Compression: For each mainshock "cycle," sample a cluster compression factor $f_c$ (from $1 - \epsilon_c$), then compress all descendant event times within the cycle: $$t_j^{\rm new} = t_{\rm ms} + f_c (t_j^{\rm old} - t_{\rm ms})$$.

A high-level pseudocode appears in the literature, providing explicit workflow for catalog generation and cluster modification. Calibration of the compression parameter $\epsilon_{max}$ is performed such that Omori-law decay in the simulated catalog matches real observed $p$-values.

c) Deep Learning Fusion (Zhang et al., 2023)

The ConvLSTM–ETAS hybrid is trained to minimize mean squared error over daily binned event tensors, with the network ingesting both raw event count tensors and one-day–ahead ETAS forecast maps. The network comprises four ConvLSTM2D layers (each with 40 $3 \times 3$ filters), with a Conv3D output layer, optimized using Adam on batches of index-windowed time sequences.

Roll-forward simulation for operational forecasting involves iterative ETAS reparameterization, generation of forecast tensors, and ConvLSTM prediction updates in a moving time window.

4. Empirical Validation and Performance Characteristics

a) Clock-Advance ETAS

The convergence of the Markov sum—and therefore the model's well-posedness—is governed solely by the mean productivity $D$. Arbitrary inclusion of microseismicity is permitted, with the total process never exploding provided $D<1$. Simulations admit analytical tractability for higher moments and facilitate modular, generation-wise parallelization (Holschneider, 2 Jan 2025).

b) Memory and Forecasting Improvements

The two-regime CL-ETAS reproduces empirical scaling crossovers in conditional CDF memory measures, notably: $$F(x) \sim x^{-\gamma_1},\, x<x_c;\quad x^{-\gamma_2},\, x>x_c,$$ with observed exponents $a\approx0.07$, $\gamma_1\approx0.25$, $\gamma_2\approx1.11$ for Italy, closely matching real data and outperforming the standard ETAS's single-regime scaling (Zhang et al., 2020). In forecasting applications (e.g., after the L’Aquila Mw6.3 mainshock), the model consistently contains observed event counts within $90\%$ model confidence bands, an outcome sporadic for the classical ETAS.

c) Information-Theoretic Nowcasting (Baughman et al., 2023)

Catalog-level CL-ETAS yields enhanced Predictive Positive Value (PPV) and Shannon information content (up to $\sim$0.88 bits for $\epsilon_{max}=0.005$ in synthetic catalogs), with gains aligning with observed California catalogs (MaxPG $\approx236\%$, MaxIG $\approx1.75$ bits). Declustering, which removes aftershock clusters, erases the surplus information carried by non-Poissonian temporal clustering, indicating that maximal nowcasting skill requires retaining full clustered structure.

d) Hybrid Deep Learning (Zhang et al., 2023)

The hybrid CL-ETAS produces significant quantitative improvements—$25$–$35\%$ gain in pseudo-log-likelihood and $~30\%$ reduction in event count RMSE relative to stand-alone ETAS or ConvLSTM—across three mainshock aftershock sequences and multiple performance metrics (N/M/S/PL tests). CL-ETAS forecasts are both more stable (narrower CI) and more physically interpretable due to the explicit physical/dynamical-ETAS input.

5. Implementation, Stability, and Inference

The modular construction—especially in the clock-advance and catalog-level models—separates independent increments (background process) from deterministic or random temporal/spatial modifications, supporting statistical inference techniques such as Poisson/Lévy-based likelihood maximization and moment-matching estimation for kernel parameters. The absence of a magnitude cutoff and applicability to microseismicity enhances analytical tractability for stability and long-time rate calculations.

Deep learning hybridizations require local retraining and regional ETAS re-parameterization, with operational usage favoring periodic re-calibration of ETAS parameters and rolling window network updates. Simulation for both "pure" and hybrid models is highly parallelizable due to independence across background or generation layers, or across unconnected spatial cells in the ConvLSTM case.

6. Operational and Research Implications

The CL-ETAS framework, in its various instantiations, enables more faithful modeling of high-density microseismic catalogs, more accurate and stable operational aftershock forecasts, enhanced interpretability (especially in fused deep learning systems), and the rigorous retention of information-theoretic content essential for nowcasting. Removal of aftershock clusters (declustering) is contraindicated when maximizing predictive content (Baughman et al., 2023).

A plausible implication is that future ETAS-based systems will increasingly adopt either time-change/Lévy-process structures or hybrid data-driven/physical models to capture non-Poissonian clustering, scale breaks in memory exponents, and microseismicity, leveraging these innovations for enhanced hazard quantification and ensemble-based operational forecasting.