Renewal-ETAS (RETAS) Seismic Model

• RETAS is a spatiotemporal marked point-process model that replaces the homogeneous Poisson background with a renewal process based on time elapsed since the last main-shock.
• RETAS enhances earthquake clustering analysis by integrating aftershock triggering from ETAS with a non-Poisson renewal mechanism for main-shock occurrence.
• RETAS employs exact likelihood-based inference, simulation procedures, and robust residual analyses to enable precise parameter estimation and stochastic declustering.

Renewal-ETAS (RETAS) is a spatiotemporal marked point-process model for earthquake occurrence that extends the classical Epidemic-Type Aftershock Sequence (ETAS) framework by replacing the homogeneous Poisson main-shock background with a renewal main-shock arrival process. In RETAS, the main-shock hazard depends on the elapsed time since the most recent main-shock and resets upon the arrival of a new main-shock, while aftershocks remain governed by a self-exciting ETAS-type mechanism. This construction is intended to permit heavier clustering of earthquakes than the spatiotemporal ETAS model of Ogata (1998), while retaining a full space–time–magnitude formulation, exact likelihood-based inference, simulation procedures, and model checking via Rosenblatt residuals (Stindl et al., 2021).

1. Formal definition and model structure

Let $N$ denote the marked spatiotemporal point process of earthquakes, with history

$$\mathcal H_{t-}=\sigma\{(\tau_i,x_i,y_i,m_i)\colon \tau_i<t\}.$$

If $I(t)=\max\{\,i\colon \tau_i<t,\;B_i=0\}$ is the index of the most recent main-shock before $t$, and $$\tilde{\mathcal H}_{t-}=\mathcal H_{t-}\cup\{I(s)\colon s<t\}$$, then the RETAS conditional intensity is

$$\lambda(t,x,y,m\mid\tilde{\mathcal H}_{t-}) =\underbrace{\mu\bigl(t-\tau_{I(t)}\bigr)\,\nu(x,y)}_{\text{renewal main-shock background}} +\sum_{\tau_i<t}\kappa(m_i)\,g\bigl(t-\tau_i\bigr)\,f\bigl(x-x_i,y-y_i\bigr) \quad\times\;J(m)\,.$$

Equivalently, the ground intensity on $[0,T]\times\mathcal S$ is

$$\tilde\lambda_g(t,x,y\mid\tilde{\mathcal H}_{t-}) =\mu\bigl(t-\tau_{I(t)}\bigr)\,\nu(x,y) +\sum_{\tau_i<t}\kappa(m_i)\,g(t-\tau_i)\,f(x-x_i,y-y_i)\,,$$

and $\lambda=\tilde\lambda_g\times J(m)$ (Stindl et al., 2021).

The same product decomposition is emphasized in later work on stochastic declustering, where the conditional intensity is written as

$$\lambda\bigl(t,x,y,m\mid\mathcal H_{t-}\bigr) = \lambda\bigl(t,x,y\mid\tilde{\mathcal H}_{t-}\bigr)\,J(m),$$

with the spatio-temporal component split into a renewal main-shock term and an aggregate aftershock term $$\mathcal H_{t-}=\sigma\{(\tau_i,x_i,y_i,m_i)\colon \tau_i<t\}.$$0 (Stindl et al., 2022).

The principal components are the following. The spatial main-shock density is $$\mathcal H_{t-}=\sigma\{(\tau_i,x_i,y_i,m_i)\colon \tau_i<t\}.$$1, normalized so that $$\mathcal H_{t-}=\sigma\{(\tau_i,x_i,y_i,m_i)\colon \tau_i<t\}.$$2. The aftershock productivity is

$$\mathcal H_{t-}=\sigma\{(\tau_i,x_i,y_i,m_i)\colon \tau_i<t\}.$$3

The temporal aftershock kernel is the Omori-law form

$$\mathcal H_{t-}=\sigma\{(\tau_i,x_i,y_i,m_i)\colon \tau_i<t\}.$$4

The spatial aftershock kernel is $$\mathcal H_{t-}=\sigma\{(\tau_i,x_i,y_i,m_i)\colon \tau_i<t\}.$$5, for example a bivariate normal with standard deviations $$\mathcal H_{t-}=\sigma\{(\tau_i,x_i,y_i,m_i)\colon \tau_i<t\}.$$6. The magnitude density is

$$\mathcal H_{t-}=\sigma\{(\tau_i,x_i,y_i,m_i)\colon \tau_i<t\}.$$7

(Stindl et al., 2021).

A common misconception is that RETAS is merely ETAS with a generic nonstationary background. The defining feature is narrower: the background is specifically a renewal hazard indexed by time since the last main-shock, and the latent main-shock identity $$\mathcal H_{t-}=\sigma\{(\tau_i,x_i,y_i,m_i)\colon \tau_i<t\}.$$8 enters the conditioning explicitly. This is why the model requires an extended history $$\mathcal H_{t-}=\sigma\{(\tau_i,x_i,y_i,m_i)\colon \tau_i<t\}.$$9, not only the ordinary ETAS history.

2. Renewal main-shock process and the reset mechanism

In RETAS, main-shocks form a renewal process with i.i.d. inter-arrival times $I(t)=\max\{\,i\colon \tau_i<t,\;B_i=0\}$0. Their common distribution function, density, survival function, and hazard are

$I(t)=\max\{\,i\colon \tau_i<t,\;B_i=0\}$1

RETAS replaces the constant ETAS background $I(t)=\max\{\,i\colon \tau_i<t,\;B_i=0\}$2 by

$I(t)=\max\{\,i\colon \tau_i<t,\;B_i=0\}$3

so the hazard of the next main-shock depends on the elapsed time since the last main-shock and resets to zero immediately after each main-shock (Stindl et al., 2021).

Two popular inter-arrival choices are given. For the Weibull law,

$I(t)=\max\{\,i\colon \tau_i<t,\;B_i=0\}$4

For the Gamma law,

$I(t)=\max\{\,i\colon \tau_i<t,\;B_i=0\}$5

$I(t)=\max\{\,i\colon \tau_i<t,\;B_i=0\}$6

A later formulation specializes the renewal law to a Gamma inter-arrival density

$I(t)=\max\{\,i\colon \tau_i<t,\;B_i=0\}$7

with hazard $I(t)=\max\{\,i\colon \tau_i<t,\;B_i=0\}$8, where $I(t)=\max\{\,i\colon \tau_i<t,\;B_i=0\}$9 is shape and $t$0 is scale (Stindl et al., 2022).

The limiting relation to ETAS is explicit: when $t$1, RETAS collapses to the spatiotemporal ETAS of Ogata (1998); in the Gamma parameterization, when $t$2, the hazard converges to a constant and recovers the homogeneous Poisson background of the standard ETAS model (Stindl et al., 2021). This suggests that RETAS should be interpreted not as a disjoint alternative to ETAS, but as a nested extension in which ETAS occupies the constant-hazard boundary.

The model is designed to capture non-Poissonian main-shock clustering, including subexponential or over-dispersed inter-arrival times, while preserving long-term stationarity (Stindl et al., 2021). A plausible implication is that RETAS is particularly relevant when the empirical background process appears temporally structured even after ordinary ETAS-type triggering is accounted for.

3. Likelihood, filtering, and parameter estimation

RETAS admits an exact likelihood representation based on latent main-shock indexing. Denote $t$3, and for $t$4 define

$t$5

$t$6

and

$t$7

Let $t$8. Then Theorem 2.1 gives the complete log-likelihood

$t$9

The $$\tilde{\mathcal H}_{t-}=\mathcal H_{t-}\cup\{I(s)\colon s<t\}$$0 satisfy a forward recursion, and one numerically maximizes $$\tilde{\mathcal H}_{t-}=\mathcal H_{t-}\cup\{I(s)\colon s<t\}$$1, for example by quasi-Newton methods, then inverts the Hessian for standard errors. For the space integral one may use one-dimensional or two-dimensional quadrature, or the approximation $$\tilde{\mathcal H}_{t-}=\mathcal H_{t-}\cup\{I(s)\colon s<t\}$$2 when supported (Stindl et al., 2021).

A later presentation writes the exact log-likelihood in the usual point-process form,

$$\tilde{\mathcal H}_{t-}=\mathcal H_{t-}\cup\{I(s)\colon s<t\}$$3

and then recovers the same recursive representation using filtered main-shock weights $$\tilde{\mathcal H}_{t-}=\mathcal H_{t-}\cup\{I(s)\colon s<t\}$$4 (Stindl et al., 2022).

The later work also states a ten-parameter core RETAS vector,

$$\tilde{\mathcal H}_{t-}=\mathcal H_{t-}\cup\{I(s)\colon s<t\}$$5

excluding the $$\tilde{\mathcal H}_{t-}=\mathcal H_{t-}\cup\{I(s)\colon s<t\}$$6 of $$\tilde{\mathcal H}_{t-}=\mathcal H_{t-}\cup\{I(s)\colon s<t\}$$7 (Stindl et al., 2022). Although the notation differs slightly across summaries, both formulations describe a likelihood-based parametric fit in which renewal-law parameters, aftershock productivity, temporal decay, spatial dispersion, and magnitude effects are estimated jointly.

The need for the recursive likelihood is substantive rather than merely computational. Because the background hazard is reset by the most recent main-shock, the latent variable $$\tilde{\mathcal H}_{t-}=\mathcal H_{t-}\cup\{I(s)\colon s<t\}$$8 affects all subsequent conditional intensities. Exact inference therefore depends on propagating the filtered probabilities of the current main-shock index rather than treating the background as exogenous.

4. Residual analysis, goodness-of-fit, and simulation

RETAS model checking is based on a sequential application of the Rosenblatt transformation. After fitting, three transformed residual series are computed (Stindl et al., 2021).

The temporal residuals are

$$\tilde{\mathcal H}_{t-}=\mathcal H_{t-}\cup\{I(s)\colon s<t\}$$9

The longitudinal residuals are

$$\lambda(t,x,y,m\mid\tilde{\mathcal H}_{t-}) =\underbrace{\mu\bigl(t-\tau_{I(t)}\bigr)\,\nu(x,y)}_{\text{renewal main-shock background}} +\sum_{\tau_i<t}\kappa(m_i)\,g\bigl(t-\tau_i\bigr)\,f\bigl(x-x_i,y-y_i\bigr) \quad\times\;J(m)\,.$$0

where $$\lambda(t,x,y,m\mid\tilde{\mathcal H}_{t-}) =\underbrace{\mu\bigl(t-\tau_{I(t)}\bigr)\,\nu(x,y)}_{\text{renewal main-shock background}} +\sum_{\tau_i<t}\kappa(m_i)\,g\bigl(t-\tau_i\bigr)\,f\bigl(x-x_i,y-y_i\bigr) \quad\times\;J(m)\,.$$1 is updated via Bayes. The latitudinal residuals are $$\lambda(t,x,y,m\mid\tilde{\mathcal H}_{t-}) =\underbrace{\mu\bigl(t-\tau_{I(t)}\bigr)\,\nu(x,y)}_{\text{renewal main-shock background}} +\sum_{\tau_i<t}\kappa(m_i)\,g\bigl(t-\tau_i\bigr)\,f\bigl(x-x_i,y-y_i\bigr) \quad\times\;J(m)\,.$$2, defined analogously. Under a correct fit, these residuals should be approximately i.i.d. Uniform$$\lambda(t,x,y,m\mid\tilde{\mathcal H}_{t-}) =\underbrace{\mu\bigl(t-\tau_{I(t)}\bigr)\,\nu(x,y)}_{\text{renewal main-shock background}} +\sum_{\tau_i<t}\kappa(m_i)\,g\bigl(t-\tau_i\bigr)\,f\bigl(x-x_i,y-y_i\bigr) \quad\times\;J(m)\,.$$3. Recommended diagnostics are K–S tests for uniformity, Ljung–Box tests for independence, together with QQ-plots and ACFs (Stindl et al., 2021).

Simulation is also part of the RETAS methodology. The algorithm proceeds by first generating main-shock arrival times $$\lambda(t,x,y,m\mid\tilde{\mathcal H}_{t-}) =\underbrace{\mu\bigl(t-\tau_{I(t)}\bigr)\,\nu(x,y)}_{\text{renewal main-shock background}} +\sum_{\tau_i<t}\kappa(m_i)\,g\bigl(t-\tau_i\bigr)\,f\bigl(x-x_i,y-y_i\bigr) \quad\times\;J(m)\,.$$4 from the renewal law $$\lambda(t,x,y,m\mid\tilde{\mathcal H}_{t-}) =\underbrace{\mu\bigl(t-\tau_{I(t)}\bigr)\,\nu(x,y)}_{\text{renewal main-shock background}} +\sum_{\tau_i<t}\kappa(m_i)\,g\bigl(t-\tau_i\bigr)\,f\bigl(x-x_i,y-y_i\bigr) \quad\times\;J(m)\,.$$5, then assigning each main-shock a location $$\lambda(t,x,y,m\mid\tilde{\mathcal H}_{t-}) =\underbrace{\mu\bigl(t-\tau_{I(t)}\bigr)\,\nu(x,y)}_{\text{renewal main-shock background}} +\sum_{\tau_i<t}\kappa(m_i)\,g\bigl(t-\tau_i\bigr)\,f\bigl(x-x_i,y-y_i\bigr) \quad\times\;J(m)\,.$$6 and magnitude $$\lambda(t,x,y,m\mid\tilde{\mathcal H}_{t-}) =\underbrace{\mu\bigl(t-\tau_{I(t)}\bigr)\,\nu(x,y)}_{\text{renewal main-shock background}} +\sum_{\tau_i<t}\kappa(m_i)\,g\bigl(t-\tau_i\bigr)\,f\bigl(x-x_i,y-y_i\bigr) \quad\times\;J(m)\,.$$7. After that, the process branches generationally: for each earthquake in generation $$\lambda(t,x,y,m\mid\tilde{\mathcal H}_{t-}) =\underbrace{\mu\bigl(t-\tau_{I(t)}\bigr)\,\nu(x,y)}_{\text{renewal main-shock background}} +\sum_{\tau_i<t}\kappa(m_i)\,g\bigl(t-\tau_i\bigr)\,f\bigl(x-x_i,y-y_i\bigr) \quad\times\;J(m)\,.$$8, draw

$$\lambda(t,x,y,m\mid\tilde{\mathcal H}_{t-}) =\underbrace{\mu\bigl(t-\tau_{I(t)}\bigr)\,\nu(x,y)}_{\text{renewal main-shock background}} +\sum_{\tau_i<t}\kappa(m_i)\,g\bigl(t-\tau_i\bigr)\,f\bigl(x-x_i,y-y_i\bigr) \quad\times\;J(m)\,.$$9

then for each candidate offspring sample $[0,T]\times\mathcal S$0, add a spatial displacement from $[0,T]\times\mathcal S$1, reject points outside the observation window or spatial domain, and sample the new magnitude from $[0,T]\times\mathcal S$2. The procedure repeats until a generation is empty (Stindl et al., 2021).

This simulation scheme was developed to validate both the likelihood evaluation algorithm and the goodness-of-fit test procedure (Stindl et al., 2021). It also clarifies an important structural property: RETAS is not a purely temporal renewal process augmented with independent aftershocks, but a branching spatiotemporal process whose immigrant arrivals themselves obey renewal dynamics.

5. Stochastic declustering and semi-parametric estimation

A central development after the original RETAS formulation is a stochastic declustering method tailored to the model’s latent main-shock structure. The problem is that the declustering algorithm available for ETAS is not directly applicable to RETAS, because the renewal background induces dependence on the most recent main-shock index and hence on the full catalog structure (Stindl et al., 2022).

The proposed solution computes smoothed probabilities conditional on all available information in the catalog. Specifically,

$[0,T]\times\mathcal S$3

via a two-pass forward–filter, backward–smooth algorithm. The forward pass computes the filtered $[0,T]\times\mathcal S$4. The backward pass defines weights

$[0,T]\times\mathcal S$5

with recursion

$[0,T]\times\mathcal S$6

These are combined into smoothed index probabilities

$[0,T]\times\mathcal S$7

and then into smoothed main-shock and parent probabilities $[0,T]\times\mathcal S$8 and $[0,T]\times\mathcal S$9 (Stindl et al., 2022).

Because $$\tilde\lambda_g(t,x,y\mid\tilde{\mathcal H}_{t-}) =\mu\bigl(t-\tau_{I(t)}\bigr)\,\nu(x,y) +\sum_{\tau_i<t}\kappa(m_i)\,g(t-\tau_i)\,f(x-x_i,y-y_i)\,,$$0 is generally unknown, the model can be estimated semi-parametrically by iterating between likelihood maximization and weighted kernel-density estimation of the main-shock spatial intensity: $$\tilde\lambda_g(t,x,y\mid\tilde{\mathcal H}_{t-}) =\mu\bigl(t-\tau_{I(t)}\bigr)\,\nu(x,y) +\sum_{\tau_i<t}\kappa(m_i)\,g(t-\tau_i)\,f(x-x_i,y-y_i)\,,$$1 where $$\tilde\lambda_g(t,x,y\mid\tilde{\mathcal H}_{t-}) =\mu\bigl(t-\tau_{I(t)}\bigr)\,\nu(x,y) +\sum_{\tau_i<t}\kappa(m_i)\,g(t-\tau_i)\,f(x-x_i,y-y_i)\,,$$2 is a $$\tilde\lambda_g(t,x,y\mid\tilde{\mathcal H}_{t-}) =\mu\bigl(t-\tau_{I(t)}\bigr)\,\nu(x,y) +\sum_{\tau_i<t}\kappa(m_i)\,g(t-\tau_i)\,f(x-x_i,y-y_i)\,,$$3 bandwidth matrix and $$\tilde\lambda_g(t,x,y\mid\tilde{\mathcal H}_{t-}) =\mu\bigl(t-\tau_{I(t)}\bigr)\,\nu(x,y) +\sum_{\tau_i<t}\kappa(m_i)\,g(t-\tau_i)\,f(x-x_i,y-y_i)\,,$$4 is a bivariate kernel such as a Gaussian kernel. The iterative procedure is:

1. Fix $$\tilde\lambda_g(t,x,y\mid\tilde{\mathcal H}_{t-}) =\mu\bigl(t-\tau_{I(t)}\bigr)\,\nu(x,y) +\sum_{\tau_i<t}\kappa(m_i)\,g(t-\tau_i)\,f(x-x_i,y-y_i)\,,$$5, maximize $$\tilde\lambda_g(t,x,y\mid\tilde{\mathcal H}_{t-}) =\mu\bigl(t-\tau_{I(t)}\bigr)\,\nu(x,y) +\sum_{\tau_i<t}\kappa(m_i)\,g(t-\tau_i)\,f(x-x_i,y-y_i)\,,$$6 to obtain $$\tilde\lambda_g(t,x,y\mid\tilde{\mathcal H}_{t-}) =\mu\bigl(t-\tau_{I(t)}\bigr)\,\nu(x,y) +\sum_{\tau_i<t}\kappa(m_i)\,g(t-\tau_i)\,f(x-x_i,y-y_i)\,,$$7.
2. Compute $$\tilde\lambda_g(t,x,y\mid\tilde{\mathcal H}_{t-}) =\mu\bigl(t-\tau_{I(t)}\bigr)\,\nu(x,y) +\sum_{\tau_i<t}\kappa(m_i)\,g(t-\tau_i)\,f(x-x_i,y-y_i)\,,$$8, then re-estimate $$\tilde\lambda_g(t,x,y\mid\tilde{\mathcal H}_{t-}) =\mu\bigl(t-\tau_{I(t)}\bigr)\,\nu(x,y) +\sum_{\tau_i<t}\kappa(m_i)\,g(t-\tau_i)\,f(x-x_i,y-y_i)\,,$$9.
3. Iterate until $\lambda=\tilde\lambda_g\times J(m)$0.

Bandwidth selection is performed by minimizing the corrected Akaike information criterion

$\lambda=\tilde\lambda_g\times J(m)$1

with the effective number of parameters taken as the KDE degrees of freedom

$\lambda=\tilde\lambda_g\times J(m)$2

and $\lambda=\tilde\lambda_g\times J(m)$3 (Stindl et al., 2022).

This framework makes declustering fully probabilistic rather than threshold-based. A plausible implication is that it is better aligned with the latent-branching interpretation of point-process seismicity models than deterministic labels of “background” and “aftershock.”

6. Empirical behavior, model comparison, and practical use

RETAS is presented as affording additional flexibility relative to classical spatiotemporal ETAS and as having the potential for superior modeling and forecasting of seismicity across catalogs with distinctly different seismic activity (Stindl et al., 2021). The comparison is operationalized through likelihood fit, information criteria, goodness-of-fit diagnostics, and declustering performance.

Several comparative facts are stated directly. When the estimated renewal parameter indicates departure from the Poisson case, empirical analyses report substantial AIC reductions and better Rosenblatt goodness-of-fit statistics on catalogs including New Zealand, Japan, Chile, and China (Stindl et al., 2021). Practical recommendations are correspondingly explicit: fit both ETAS and RETAS, compare AIC and goodness-of-fit; if $\lambda=\tilde\lambda_g\times J(m)$4 is close to $\lambda=\tilde\lambda_g\times J(m)$5 and there is no goodness-of-fit gain, the simpler ETAS model suffices (Stindl et al., 2021).

Simulation evidence for stochastic declustering is reported in detail. Over $\lambda=\tilde\lambda_g\times J(m)$6 simulated catalogs, when the true $\lambda=\tilde\lambda_g\times J(m)$7 is known, the maximum-likelihood estimator $\lambda=\tilde\lambda_g\times J(m)$8 is nearly unbiased with correct standard errors and $\lambda=\tilde\lambda_g\times J(m)$9 coverage. When $$\lambda\bigl(t,x,y,m\mid\mathcal H_{t-}\bigr) = \lambda\bigl(t,x,y\mid\tilde{\mathcal H}_{t-}\bigr)\,J(m),$$0 is estimated, $$\lambda\bigl(t,x,y,m\mid\mathcal H_{t-}\bigr) = \lambda\bigl(t,x,y\mid\tilde{\mathcal H}_{t-}\bigr)\,J(m),$$1 selects a bandwidth near the default plug-in scale $$\lambda\bigl(t,x,y,m\mid\mathcal H_{t-}\bigr) = \lambda\bigl(t,x,y\mid\tilde{\mathcal H}_{t-}\bigr)\,J(m),$$2, and $$\lambda\bigl(t,x,y,m\mid\mathcal H_{t-}\bigr) = \lambda\bigl(t,x,y\mid\tilde{\mathcal H}_{t-}\bigr)\,J(m),$$3 remains accurate. For strong renewal $$\lambda\bigl(t,x,y,m\mid\mathcal H_{t-}\bigr) = \lambda\bigl(t,x,y\mid\tilde{\mathcal H}_{t-}\bigr)\,J(m),$$4, RETAS main-shock classification attains $$\lambda\bigl(t,x,y,m\mid\mathcal H_{t-}\bigr) = \lambda\bigl(t,x,y\mid\tilde{\mathcal H}_{t-}\bigr)\,J(m),$$5 versus ETAS $$\lambda\bigl(t,x,y,m\mid\mathcal H_{t-}\bigr) = \lambda\bigl(t,x,y\mid\tilde{\mathcal H}_{t-}\bigr)\,J(m),$$6, and overall branching-structure recovery is approximately $$\lambda\bigl(t,x,y,m\mid\mathcal H_{t-}\bigr) = \lambda\bigl(t,x,y\mid\tilde{\mathcal H}_{t-}\bigr)\,J(m),$$7 correct for RETAS versus $$\lambda\bigl(t,x,y,m\mid\mathcal H_{t-}\bigr) = \lambda\bigl(t,x,y\mid\tilde{\mathcal H}_{t-}\bigr)\,J(m),$$8 for ETAS (Stindl et al., 2022).

On the New Zealand catalog for 1980–2020 with $$\lambda\bigl(t,x,y,m\mid\mathcal H_{t-}\bigr) = \lambda\bigl(t,x,y\mid\tilde{\mathcal H}_{t-}\bigr)\,J(m),$$9 and $$\mathcal H_{t-}=\sigma\{(\tau_i,x_i,y_i,m_i)\colon \tau_i<t\}.$$00, the optimal smoothing is $$\mathcal H_{t-}=\sigma\{(\tau_i,x_i,y_i,m_i)\colon \tau_i<t\}.$$01. Reported estimates are

$$\mathcal H_{t-}=\sigma\{(\tau_i,x_i,y_i,m_i)\colon \tau_i<t\}.$$02

$$\mathcal H_{t-}=\sigma\{(\tau_i,x_i,y_i,m_i)\colon \tau_i<t\}.$$03

The renewal main-shock mean interval is approximately $$\mathcal H_{t-}=\sigma\{(\tau_i,x_i,y_i,m_i)\colon \tau_i<t\}.$$04 days, and $$\mathcal H_{t-}=\sigma\{(\tau_i,x_i,y_i,m_i)\colon \tau_i<t\}.$$05 strongly rejects a Poisson background. The weighted KDE $$\mathcal H_{t-}=\sigma\{(\tau_i,x_i,y_i,m_i)\colon \tau_i<t\}.$$06 traces known fault lines; approximately $$\mathcal H_{t-}=\sigma\{(\tau_i,x_i,y_i,m_i)\colon \tau_i<t\}.$$07 aftershock productivity at $$\mathcal H_{t-}=\sigma\{(\tau_i,x_i,y_i,m_i)\colon \tau_i<t\}.$$08 rises super-linearly with magnitude; and RETAS identifies major multi-generation clusters such as the 1995 East Cape and 2010 Darfield sequences, yielding more plausible declustering than ETAS (Stindl et al., 2022).

The applied uses emphasized in the summaries are improved seismicity forecasting, stochastic declustering, and real-time updating of forecasts because the conditional hazard $$\mathcal H_{t-}=\sigma\{(\tau_i,x_i,y_i,m_i)\colon \tau_i<t\}.$$09 resets after each main-shock (Stindl et al., 2021). The implementation guidance is likewise concrete: use efficient one-dimensional quadrature in polar coordinates when possible, exploit the recursive likelihood algorithm of Theorem 2.1, and validate fitted models via Rosenblatt residuals (Stindl et al., 2021).

A recurrent misunderstanding is that a better declustering result automatically implies a universally better forecasting model. The reported results support improved fit and more plausible branch reconstruction when main-shock inter-arrival times deviate from Poisson, but they also state a model-selection criterion under which ordinary ETAS remains adequate when the estimated renewal parameter is near the Poisson boundary.