dorsal/arxiv
View SchemaAnomalous Power Law Distribution of Total Lifetimes of Branching Processes Relevant to Earthquakes
| Authors | A. Saichev, D. Sornette |
|---|---|
| Categories | |
| ArXiv ID | physics/0404019 |
| URL | https://arxiv.org/abs/physics/0404019 |
| DOI | 10.1103/PhysRevE.70.046123 |
| Journal | Phys. Rev. E 70, 046123 (2004) |
Abstract
We consider a branching model of triggered seismicity, the ETAS (epidemic-type aftershock sequence) model which assumes that each earthquake can trigger other earthquakes (``aftershocks''). An aftershock sequence results in this model from the cascade of aftershocks of each past earthquake. Due to the large fluctuations of the number of aftershocks triggered directly by any earthquake (``productivity'' or ``fertility''), there is a large variability of the total number of aftershocks from one sequence to another, for the same mainshock magnitude. We study the regime where the distribution of fertilities $\mu$ is characterized by a power law $\sim 1/\mu^{1+\gamma}$ and the bare Omori law for the memory of previous triggering mothers decays slowly as $\sim 1/t^{1+\theta}$, with $0 < \theta <1$ relevant for earthquakes. Using the tool of generating probability functions and a quasistatic approximation which is shown to be exact asymptotically for large durations, we show that the density distribution of total aftershock lifetimes scales as $\sim 1/t^{1+\theta/\gamma}$ when the average branching ratio is critical ($n=1$). The coefficient $1<\gamma = b/\alpha<2$ quantifies the interplay between the exponent $b \approx 1$ of the Gutenberg-Richter magnitude distribution $ \sim 10^{-bm}$ and the increase $\sim 10^{\alpha m}$ of the number of aftershocks with the mainshock magnitude $m$ (productivity) with $\alpha \approx 0.8$. More generally, our results apply to any stochastic branching process with a power-law distribution of offsprings per mother and a long memory.
{
"annotation_id": "5b12ad07-3f85-479c-a8a0-283dab783a0f",
"date_created": "2026-03-02T18:00:49.389000Z",
"date_modified": "2026-03-02T18:00:49.389000Z",
"file_hash": "9a4ea03a3cef701ada125bfa3c38d0303a556de3804254124ae421ef39a2567c",
"private": false,
"record": {
"abstract": "We consider a branching model of triggered seismicity, the ETAS\n(epidemic-type aftershock sequence) model which assumes that each earthquake\ncan trigger other earthquakes (``aftershocks\u0027\u0027). An aftershock sequence results\nin this model from the cascade of aftershocks of each past earthquake. Due to\nthe large fluctuations of the number of aftershocks triggered directly by any\nearthquake (``productivity\u0027\u0027 or ``fertility\u0027\u0027), there is a large variability of\nthe total number of aftershocks from one sequence to another, for the same\nmainshock magnitude. We study the regime where the distribution of fertilities\n$\\mu$ is characterized by a power law $\\sim 1/\\mu^{1+\\gamma}$ and the bare\nOmori law for the memory of previous triggering mothers decays slowly as $\\sim\n1/t^{1+\\theta}$, with $0 \u003c \\theta \u003c1$ relevant for earthquakes. Using the tool\nof generating probability functions and a quasistatic approximation which is\nshown to be exact asymptotically for large durations, we show that the density\ndistribution of total aftershock lifetimes scales as $\\sim\n1/t^{1+\\theta/\\gamma}$ when the average branching ratio is critical ($n=1$).\nThe coefficient $1\u003c\\gamma = b/\\alpha\u003c2$ quantifies the interplay between the\nexponent $b \\approx 1$ of the Gutenberg-Richter magnitude distribution $ \\sim\n10^{-bm}$ and the increase $\\sim 10^{\\alpha m}$ of the number of aftershocks\nwith the mainshock magnitude $m$ (productivity) with $\\alpha \\approx 0.8$. More\ngenerally, our results apply to any stochastic branching process with a\npower-law distribution of offsprings per mother and a long memory.",
"arxiv_id": "physics/0404019",
"authors": [
"A. Saichev",
"D. Sornette"
],
"categories": [
"physics.geo-ph",
"physics.gen-ph"
],
"doi": "10.1103/PhysRevE.70.046123",
"journal_ref": "Phys. Rev. E 70, 046123 (2004)",
"title": "Anomalous Power Law Distribution of Total Lifetimes of Branching Processes Relevant to Earthquakes",
"url": "https://arxiv.org/abs/physics/0404019"
},
"schema_id": "dorsal/arxiv",
"source": {
"execution_id": "225b6bee-a5aa-441b-9525-974d2c47ecc3",
"id": "arXiv Dataset IDs",
"type": "Model",
"variant": "snapshot-2026-03-01",
"version": "0.1.0"
},
"user_id": 1000002
}