Time-Scaled ETAS Advances

• Time-Scaled ETAS is a family of model extensions that transform the time coordinate to better capture clustered, heterogeneous, and multi-horizon seismicity.
• It employs diverse approaches—including coordinate transformation, clock advance, and multiscale memory crossover—to improve parameter identifiability and forecast accuracy.
• Empirical studies show that adaptive time scaling and revised productivity laws enhance declustering, optimize likelihood, and boost forecasting performance.

Time-Scaled ETAS denotes a family of extensions and reinterpretations of the Epidemic-Type Aftershock Sequence model in which earthquake triggering is analyzed on a transformed, rescaled, or explicitly multiscale temporal axis rather than on raw calendar time alone. In the cited literature, this idea appears in several non-equivalent forms: direct replacement of time by a transformed coordinate $t^Z=\phi(t,\cdot)$ for likelihood fitting and declustering; interpretation of triggering as a local clock advance of an independent background process; introduction of short- and long-time productivity regimes to reproduce observed memory crossover; fractional reformulation of temporal ETAS through a Caputo operator; and scaling relations in cluster statistics controlled by the branching ratio $n$ (Das et al., 30 May 2025, Holschneider, 2 Jan 2025, Zhang et al., 2020, Cristofaro et al., 2022). Across these formulations, the common objective is to preserve the ETAS decomposition between background seismicity and triggered cascades while making the temporal organization of seismicity more faithful to clustered, heterogeneous, and multi-horizon earthquake occurrence.

1. Standard ETAS substrate and the meaning of temporal scaling

The common substrate is the marked spatio-temporal ETAS point process. In one formulation, each event is represented by $(t_i,x_i,y_i,m_i)$, with history $\mathcal H_t=\{(t_i,x_i,y_i,m_i): t_i<t\}$, and the conditional intensity is written as

$$\lambda_\theta(t,x,y \mid \mathcal H_t) = \widetilde u(x,y) + \sum_{i:t_i<t} \kappa_{A,\alpha}(m_i)\, g_{c,p}(t-t_i)\, f_{D,\gamma,q}(x-x_i,y-y_i;m_i).$$

Here $\widetilde u(x,y)=\mu\,u(x,y)$ is the stationary background seismicity rate, $\kappa_{A,\alpha}(m_i)=A\exp[\alpha(m_i-m_0)]$ is the productivity term, $$g_{c,p}(t-t_i)=\left(\frac{p-1}{c}\right)\left(1+\frac{t-t_i}{c}\right)^{-p}$$ is the temporal Omori-type kernel, and $f_{D,\gamma,q}$ is a radially symmetric spatial triggering kernel. Magnitudes are commonly modeled by the exponential Gutenberg–Richter-type density

$$v_\beta(m)=\beta \exp[-\beta(m-m_0)], \qquad \beta>0,\; m>m_0,$$

and the paper states the stationarity condition as

$n$0

Within this baseline, background and triggered events are separated probabilistically through

$n$1

where $n$2 is the probability that event $n$3 was triggered by event $n$4 (Das et al., 30 May 2025).

In this setting, “time scaling” does not refer to a single canonical modification. Rather, it refers to distinct ways of changing how elapsed time enters ETAS inference, triggering, or interpretation.

Sense of time scaling	Mechanism	Representative source
Coordinate transformation	Replace $n$5 by $n$6 before fitting ETAS	(Das et al., 30 May 2025)
Clock advance	Triggering acts by accelerating the local time of a background jump process	(Holschneider, 2 Jan 2025)
Memory crossover	Short- and long-time regimes use different productivity or decay structure	(Zhang et al., 2020, Cristofaro et al., 2022)
Cluster scaling	Characteristic time depends on branching ratio, e.g. $n$7	(Saichev et al., 2017)

This multiplicity is important because different papers use “time-scaled ETAS” to solve different problems: identifiability in likelihood optimization, inclusion of microseismicity, representation of long memory, or improvement of forecasts across multiple horizons.

2. Direct transformed-time ETAS for forecasting and declustering

The most literal use of the phrase appears in the Nepal forecasting study, which extends the standard spatio-temporal ETAS framework by transforming the time coordinate prior to fitting. The transformed time is denoted

$n$8

with five variants explicitly considered: $n$9 These are labeled, respectively, the “ideal” time scale, calibration time scale, proportional hazards time scale, log-linear time scale, and power time scale. The stated motivation is that a transformed time scale may better align the hazard with the underlying accumulation/relaxation process, improve identifiability and fit, and better separate background events from aftershocks (Das et al., 30 May 2025).

The Nepal application uses the ANSS global comprehensive catalogue, with region roughly $(t_i,x_i,y_i,m_i)$0 and $(t_i,x_i,y_i,m_i)$1, magnitude threshold $(t_i,x_i,y_i,m_i)$2, and main analysis focused on 2000–2020. Parameter estimation is based on maximized log-likelihood, with the magnitude parameter estimated by

$(t_i,x_i,y_i,m_i)$3

and the remaining ETAS parameters obtained by minimizing

$(t_i,x_i,y_i,m_i)$4

numerically. The paper compares Davidon–Fletcher–Powell optimization, an Iterative Stochastic De-clustering Method, and a simpler Nelder–Mead/simplex path. Declustering and event classification are encoded through the probabilities $(t_i,x_i,y_i,m_i)$5, $(t_i,x_i,y_i,m_i)$6, and the clustering coefficient

$(t_i,x_i,y_i,m_i)$7

Empirically, the paper reports that the ISDM-based ETAS model optimized with DFP yields the best fit, with

$(t_i,x_i,y_i,m_i)$8

Among the tested time transformations, the calibration time scale performed best; the paper notes that it makes likelihood optimization easier and yields a better fit than proportional hazards, log-linear, and power time scales. Gamma generally slightly outperformed exponential in the time-scaled ground-intensity fits. The power scale at $(t_i,x_i,y_i,m_i)$9 performed better than log-linear and proportional hazards, but worse than calibration. Using depth as the proportional-hazards usage measure was reported as not suitable. Residual diagnostics included temporal residuals, smoothed spatial residuals, transformed time residuals $\mathcal H_t=\{(t_i,x_i,y_i,m_i): t_i<t\}$0, and QQ-plots of transformed variables $\mathcal H_t=\{(t_i,x_i,y_i,m_i): t_i<t\}$1, with a Kolmogorov–Smirnov result of

$\mathcal H_t=\{(t_i,x_i,y_i,m_i): t_i<t\}$2

indicating no evidence against $\mathcal H_t=\{(t_i,x_i,y_i,m_i): t_i<t\}$3 at the 5% level (Das et al., 30 May 2025).

The same study interprets the time transformation as improving forecasting in two linked ways: sharpening the distinction between spontaneous and triggered events, and stabilizing parameter estimation in a catalogue with strong temporal clustering and heterogeneous spatial behavior. This suggests that in this formulation, time scaling is as much an inferential device as a physical modification of the triggering law.

3. Clock-advance ETAS and productivity-based time deformation

A different construction reinterprets ETAS triggering itself as a time deformation. In the “clock advance” description, the goal is to extend ETAS to micro-seismic events without imposing an artificial small magnitude cutoff $\mathcal H_t=\{(t_i,x_i,y_i,m_i): t_i<t\}$4. The central change is to replace count-based branching by productivity-based branching. Instead of asking whether the expected number of triggered events is finite, the model requires finite expected total triggering productivity,

$\mathcal H_t=\{(t_i,x_i,y_i,m_i): t_i<t\}$5

for stability, where

$\mathcal H_t=\{(t_i,x_i,y_i,m_i): t_i<t\}$6

is the productivity moment associated with magnitude $\mathcal H_t=\{(t_i,x_i,y_i,m_i): t_i<t\}$7 (Holschneider, 2 Jan 2025).

In productivity scale, the background process becomes a pure-jump process with intensity

$\mathcal H_t=\{(t_i,x_i,y_i,m_i): t_i<t\}$8

under the condition $\mathcal H_t=\{(t_i,x_i,y_i,m_i): t_i<t\}$9, so that the first moment exists. For a triggering event at time $$\lambda_\theta(t,x,y \mid \mathcal H_t) = \widetilde u(x,y) + \sum_{i:t_i<t} \kappa_{A,\alpha}(m_i)\, g_{c,p}(t-t_i)\, f_{D,\gamma,q}(x-x_i,y-y_i;m_i).$$0 with productivity $$\lambda_\theta(t,x,y \mid \mathcal H_t) = \widetilde u(x,y) + \sum_{i:t_i<t} \kappa_{A,\alpha}(m_i)\, g_{c,p}(t-t_i)\, f_{D,\gamma,q}(x-x_i,y-y_i;m_i).$$1, the offspring intensity is

$$\lambda_\theta(t,x,y \mid \mathcal H_t) = \widetilde u(x,y) + \sum_{i:t_i<t} \kappa_{A,\alpha}(m_i)\, g_{c,p}(t-t_i)\, f_{D,\gamma,q}(x-x_i,y-y_i;m_i).$$2

with cumulative Omori kernel

$$\lambda_\theta(t,x,y \mid \mathcal H_t) = \widetilde u(x,y) + \sum_{i:t_i<t} \kappa_{A,\alpha}(m_i)\, g_{c,p}(t-t_i)\, f_{D,\gamma,q}(x-x_i,y-y_i;m_i).$$3

The paper then represents the triggered sequence as a time-deformed independent background process: $$\lambda_\theta(t,x,y \mid \mathcal H_t) = \widetilde u(x,y) + \sum_{i:t_i<t} \kappa_{A,\alpha}(m_i)\, g_{c,p}(t-t_i)\, f_{D,\gamma,q}(x-x_i,y-y_i;m_i).$$4

The same mechanism is iterated generation by generation: $$\lambda_\theta(t,x,y \mid \mathcal H_t) = \widetilde u(x,y) + \sum_{i:t_i<t} \kappa_{A,\alpha}(m_i)\, g_{c,p}(t-t_i)\, f_{D,\gamma,q}(x-x_i,y-y_i;m_i).$$5 so that the full ETAS catalogue is

$$\lambda_\theta(t,x,y \mid \mathcal H_t) = \widetilde u(x,y) + \sum_{i:t_i<t} \kappa_{A,\alpha}(m_i)\, g_{c,p}(t-t_i)\, f_{D,\gamma,q}(x-x_i,y-y_i;m_i).$$6

The process is described as an infinite Markov chain of independent background processes, each time-rescaled by the triggering accumulated from the previous generation. In this formulation, triggered seismicity is not a separate offspring process in the usual literal branching sense; it is a local acceleration of an independent background process. The paper also states that the logic extends to space-time ETAS through spatial kernels $$\lambda_\theta(t,x,y \mid \mathcal H_t) = \widetilde u(x,y) + \sum_{i:t_i<t} \kappa_{A,\alpha}(m_i)\, g_{c,p}(t-t_i)\, f_{D,\gamma,q}(x-x_i,y-y_i;m_i).$$7, with normalization $$\lambda_\theta(t,x,y \mid \mathcal H_t) = \widetilde u(x,y) + \sum_{i:t_i<t} \kappa_{A,\alpha}(m_i)\, g_{c,p}(t-t_i)\, f_{D,\gamma,q}(x-x_i,y-y_i;m_i).$$8 (Holschneider, 2 Jan 2025).

This construction is significant because it makes time scaling literal: triggering is a clock deformation. It also permits inclusion of micro-events down to $$\lambda_\theta(t,x,y \mid \mathcal H_t) = \widetilde u(x,y) + \sum_{i:t_i<t} \kappa_{A,\alpha}(m_i)\, g_{c,p}(t-t_i)\, f_{D,\gamma,q}(x-x_i,y-y_i;m_i).$$9, provided mean productivity remains finite, and shifts the meaning of ETAS stability from expected offspring count to expected productivity moment.

4. Multiscale memory, productivity crossover, and fractional temporal dynamics

Another line of work introduces time scaling through explicit temporal crossover in the memory structure of ETAS. One generalized model, called ETAS2, keeps the standard background, Omori, and spatial kernels but replaces the single productivity exponent $\widetilde u(x,y)=\mu\,u(x,y)$0 with two productivity exponents: $\widetilde u(x,y)=\mu\,u(x,y)$1 Here $\widetilde u(x,y)=\mu\,u(x,y)$2 governs short-time-scale triggering and $\widetilde u(x,y)=\mu\,u(x,y)$3 long-time-scale triggering. The empirical motivation is a double power law in the rescaled memory statistic

$\widetilde u(x,y)=\mu\,u(x,y)$4

with

$\widetilde u(x,y)=\mu\,u(x,y)$5

and crossover around

$\widetilde u(x,y)=\mu\,u(x,y)$6

corresponding to an average time of about 100 days. Using Italian and Southern California catalogues, the paper reports that ETAS2 reproduces the observed short-time and long-time memory scaling, whereas standard ETAS1 gives only a single power law and no crossover. It also reports improved forecasting after the 2009 L’Aquila mainshock and after the six largest Italian events ($\widetilde u(x,y)=\mu\,u(x,y)$7) in 1981–2017 (Zhang et al., 2020).

A related but mathematically distinct approach rewrites pure-temporal ETAS as a fractional differential equation. Starting from the expected intensity self-consistency equation and assuming a modified Omori-Utsu law

$\widetilde u(x,y)=\mu\,u(x,y)$8

the mean rate is transformed into a non-homogeneous Caputo fractional differential equation: $\widetilde u(x,y)=\mu\,u(x,y)$9 Its solution is expressed through the two-parameter Mittag-Leffler function, and the paper identifies a crossover time

$\kappa_{A,\alpha}(m_i)=A\exp[\alpha(m_i-m_0)]$0

The resulting decay has two regimes: $\kappa_{A,\alpha}(m_i)=A\exp[\alpha(m_i-m_0)]$1 and

$\kappa_{A,\alpha}(m_i)=A\exp[\alpha(m_i-m_0)]$2

Applied to the Japan Meteorological Agency catalogue from 1965/01/01 to 2003/09/23, with parameters $\kappa_{A,\alpha}(m_i)=A\exp[\alpha(m_i-m_0)]$3 and completeness magnitude $\kappa_{A,\alpha}(m_i)=A\exp[\alpha(m_i-m_0)]$4, the paper concludes that larger representative aftershock magnitude $\kappa_{A,\alpha}(m_i)=A\exp[\alpha(m_i-m_0)]$5 leads to a longer crossover time $\kappa_{A,\alpha}(m_i)=A\exp[\alpha(m_i-m_0)]$6, while small $\kappa_{A,\alpha}(m_i)=A\exp[\alpha(m_i-m_0)]$7 leaves the global rate close to the original Omori decay (Cristofaro et al., 2022).

Taken together, these models show that time scaling in ETAS need not be a coordinate transform. It can instead be encoded in the productivity law or in the evolution operator governing the expected rate, thereby creating explicit short/long temporal regimes.

5. Branching-ratio scaling, cluster-duration statistics, and forecasting of all generations

Time scaling also emerges from the statistics of triggered clusters. In the analysis of temporal seismic clusters, a main shock of magnitude $\kappa_{A,\alpha}(m_i)=A\exp[\alpha(m_i-m_0)]$8 at $\kappa_{A,\alpha}(m_i)=A\exp[\alpha(m_i-m_0)]$9 generates a branching cascade governed by Gutenberg–Richter magnitudes,

$$g_{c,p}(t-t_i)=\left(\frac{p-1}{c}\right)\left(1+\frac{t-t_i}{c}\right)^{-p}$$0

fertility law

$$g_{c,p}(t-t_i)=\left(\frac{p-1}{c}\right)\left(1+\frac{t-t_i}{c}\right)^{-p}$$1

and Omori–Utsu waiting-time density

$$g_{c,p}(t-t_i)=\left(\frac{p-1}{c}\right)\left(1+\frac{t-t_i}{c}\right)^{-p}$$2

The branching ratio is

$$g_{c,p}(t-t_i)=\left(\frac{p-1}{c}\right)\left(1+\frac{t-t_i}{c}\right)^{-p}$$3

Under the one-daughter approximation, the paper derives an explicit time-scaling function

$$g_{c,p}(t-t_i)=\left(\frac{p-1}{c}\right)\left(1+\frac{t-t_i}{c}\right)^{-p}$$4

so that the effective temporal variable is rescaled by $$g_{c,p}(t-t_i)=\left(\frac{p-1}{c}\right)\left(1+\frac{t-t_i}{c}\right)^{-p}$$5. As $$g_{c,p}(t-t_i)=\left(\frac{p-1}{c}\right)\left(1+\frac{t-t_i}{c}\right)^{-p}$$6, the characteristic time diverges. For cluster-duration densities, the large-time asymptotics are

$$g_{c,p}(t-t_i)=\left(\frac{p-1}{c}\right)\left(1+\frac{t-t_i}{c}\right)^{-p}$$7

and

$$g_{c,p}(t-t_i)=\left(\frac{p-1}{c}\right)\left(1+\frac{t-t_i}{c}\right)^{-p}$$8

The paper emphasizes the paradoxical result that the duration tail is fatter in the subcritical regime $$g_{c,p}(t-t_i)=\left(\frac{p-1}{c}\right)\left(1+\frac{t-t_i}{c}\right)^{-p}$$9 than in the critical case $f_{D,\gamma,q}$0, because generations of all orders cascade very fast in the critical regime and accelerate the temporal decay of the cluster dynamics (Saichev et al., 2017).

Operational forecasting studies reinforce the importance of this all-generation temporal structure. In California pseudo-prospective experiments, one ETAS-based approach forecasts background seismicity plus the rates of future aftershocks of all generations over 1-year and 5-year windows, with target magnitude thresholds $f_{D,\gamma,q}$1, $f_{D,\gamma,q}$2, and $f_{D,\gamma,q}$3. The forecast explicitly includes Type I aftershocks, defined as direct aftershocks of training-catalog earthquakes plus the cascades they trigger, and Type II aftershocks, defined as descendants of background seed events occurring during the testing period. Six forecasting experiments compare GETAS, SVETAS, three declustering-based smoothed seismicity models, a simple undeclustered model, and a strain-rate model. The paper reports that SVETAS consistently yields the highest mean information gain per earthquake across all six experiments, and that pairwise comparisons give p-values far below $f_{D,\gamma,q}$4. It interprets this as evidence that forecasting the future aftershocks of all generations is essential, and that accounting for spatially variable ETAS parameters leads to strong and statistically significant improvements in forecasting performance (Nandan et al., 2019).

The connection to time-scaled ETAS is direct: once the forecast horizon is extended beyond the immediate aftershock window, the time axis is no longer a passive index. It becomes the medium through which branching cascades, relaxation times, and horizon-dependent triggered contributions accumulate.

6. Information content, declustering, hybrid corrections, and adaptive background rates

A further strand of work links temporal scaling to the information content of seismic catalogues. In ETAS simulations designed to preserve magnitude-frequency scaling, aftershock scaling, Bath’s law, and the productivity relation while adding recursive non-Poisson clustering, the hypothesis tested is that the information in earthquake catalogues comes from non-Poisson clustering, especially aftershock clustering, and from the quiescent intervals that follow clustered activity. The modified model introduces a geometric clustering factor

$f_{D,\gamma,q}$5

which recursively compresses inter-event times. Catalogues with $f_{D,\gamma,q}$6 and $f_{D,\gamma,q}$7 are compared to the California catalogue. Information content is quantified through

$f_{D,\gamma,q}$8

The reported values are $f_{D,\gamma,q}$9 and $$v_\beta(m)=\beta \exp[-\beta(m-m_0)], \qquad \beta>0,\; m>m_0,$$0 bits for California, $$v_\beta(m)=\beta \exp[-\beta(m-m_0)], \qquad \beta>0,\; m>m_0,$$1 and $$v_\beta(m)=\beta \exp[-\beta(m-m_0)], \qquad \beta>0,\; m>m_0,$$2 bits for $$v_\beta(m)=\beta \exp[-\beta(m-m_0)], \qquad \beta>0,\; m>m_0,$$3, and $$v_\beta(m)=\beta \exp[-\beta(m-m_0)], \qquad \beta>0,\; m>m_0,$$4 and $$v_\beta(m)=\beta \exp[-\beta(m-m_0)], \qquad \beta>0,\; m>m_0,$$5 bits for $$v_\beta(m)=\beta \exp[-\beta(m-m_0)], \qquad \beta>0,\; m>m_0,$$6. Fitted aftershock-decay exponents remain close to Omori–Utsu behavior, with $$v_\beta(m)=\beta \exp[-\beta(m-m_0)], \qquad \beta>0,\; m>m_0,$$7 for $$v_\beta(m)=\beta \exp[-\beta(m-m_0)], \qquad \beta>0,\; m>m_0,$$8 and $$v_\beta(m)=\beta \exp[-\beta(m-m_0)], \qquad \beta>0,\; m>m_0,$$9 for $n$00. The paper concludes that stronger non-Poisson clustering produces more catalogue information and that routine declustering may remove information useful for forecasting and nowcasting (Baughman et al., 2023).

Other studies frame temporal misspecification as a short-term versus long-term bias. In the Southern California CL-ETAS study, ETAS is said to underestimate short-term seismicity and overestimate long-term seismicity. Forecasts are evaluated over 1 day, 15 days, and 30 days after the 1999 Hector Mine, 2010 Baja California, and 2019 Ridgecrest earthquakes. ETAS, ConvLSTM, and CL-ETAS are tested by the Number Test, Magnitude Test, Spatial Test, and Pseudo-likelihood Test. The paper reports that CL-ETAS passes all four tests at 1 day, 15 days, and 30 days; ConvLSTM passes the N-test only at 1 day and the M-test only marginally at each horizon; ETAS fails the N- and M-tests in all horizons. The interpretation is that the hybrid model remedies the short-term underprediction and long-term overprediction behavior of ETAS by combining ETAS-generated forecast structure with deep spatio-temporal learning (Zhang et al., 2023).

Adaptive background modeling provides yet another route to temporal scaling. In DGP-ETAS, the triggering law

$n$01

is kept intact, but the background rate $n$02 is replaced by a hierarchical deep Gaussian process, with positivity enforced through constructions such as

$n$03

Inference uses a Metropolis-within-Gibbs scheme with Elliptical Slice Sampling for white-noise vectors, HMC for scalar and hyperparameters, and exact sampling of branching variables $n$04. On synthetic catalogues, the two-layer DGP better captures the abrupt edges of a square-wave background rate and learns that the characteristic timescale shortens near the transitions. On real data, the model finds a low and nearly constant background for the 2019 Ridgecrest sequence, but a sharply varying, multiscale background for the 2016–2019 Cahuilla swarm, including two distinct phases of aseismic forcing (Muir et al., 2023).

These results support a broad conclusion: time-scaled ETAS is not limited to rescaling elapsed time inside the Omori kernel. It also includes methods that preserve or recover temporal information embedded in clustering, correct short/long horizon bias, and infer background rates whose effective time scale changes over the sequence. A plausible implication is that the principal controversy around time scaling is not whether ETAS should remain self-exciting, but which components of the model—coordinate time, productivity, cluster recursion, or background forcing—should carry the temporal heterogeneity that real catalogues exhibit.