[Paper Review] Comparing different models of aftershock rate decay: the role of catalog incompleteness in the first times after main shock

B. Lolli, Paolo Gasperini|CERN Bulletin|Oct 10, 2005

earthquake and tectonic studiesEarth and Planetary Sciences4 citations

TL;DR

This study evaluates aftershock rate decay models using AICc and BIC criteria across 47 seismic sequences in California and Italy. It finds that the parameter c in Omori-type models is largely an artifact of catalog incompleteness in the first hours after main shocks, not a universal feature of aftershock sequences.

ABSTRACT

We evaluated the efficiency of various models in describing the time decay of aftershock rate of 47 simple sequences occurred in California (37) from 1933 to 2004 and in Italy (10) from 1976 to 2004. We compared the models by the corrected Akaike Information Criterion (AICc) and the Bayesian Information Criterion (BIC), both based on the log-likelihood function but also including a penalty term that takes into account the number of independent observations and of free parameters of each model. We compared the performance of different models by varying the starting time Ts and the minimum magnitude threshold Mmin for each sequence. We found that Omori-type models including parameter c are preferable to those not including it, only for short Ts and low Mmin while the latters generally perform better than the formers for Ts longer than a few hours and Mmin larger than the main shock magnitude Mm minus 3 units. This clearly indicates that a value of parameter c different from zero does not represent a general property of aftershock sequences in California and Italy but it is very likely induced in most cases by catalog incompleteness in the first times after the main shock. We also considered other models of aftershock decay proposed in the literature: the Stretched Exponential Law in two forms (including and not including a time shift) and the band Limited Power Law (LPL). We found that such models perform worse than the Modified Omori Model (MOM) and other Omori-type models for the large majority of sequences, although for LPL, the relatively short duration of the analyzed sequences (one year) might also contribute to its poor performance.

Motivation & Objective

• To assess the performance of various aftershock rate decay models in describing temporal decay following mainshocks.
• To investigate how catalog incompleteness in the initial hours after a mainshock affects model selection and parameter estimation.
• To determine whether the presence of parameter c in Omori-type models reflects a true physical process or is an artifact of data limitations.
• To compare the Modified Omori Model (MOM) and alternative models (e.g., Stretched Exponential, LPL) using information criteria across diverse seismic sequences.
• To evaluate the sensitivity of model performance to variations in starting time (Ts) and magnitude threshold (Mmin).

Proposed method

• Applied corrected Akaike Information Criterion (AICc) and Bayesian Information Criterion (BIC) to compare model fit, penalizing for number of parameters and observations.
• Evaluated 47 aftershock sequences (37 in California, 10 in Italy) from 1933–2004 with varying Ts (starting time after mainshock) and Mmin (minimum magnitude threshold).
• Tested multiple models: Modified Omori Model (MOM), standard Omori-type models (with/without c), Stretched Exponential (with/without time shift), and Band-Limited Power Law (LPL).
• Used log-likelihood functions to quantify model fit, adjusting for model complexity via AICc and BIC penalties.
• Systematically varied Ts and Mmin to assess how data completeness affects model preference and parameter stability.
• Focused on sequences with a maximum duration of one year to limit the influence of long-term decay behavior on model evaluation.

Experimental results

Research questions

• RQ1 Does the inclusion of parameter c in Omori-type models represent a general property of aftershock sequences or is it an artifact of data incompleteness?
• RQ2 How does the choice of starting time (Ts) and magnitude threshold (Mmin) affect the relative performance of different aftershock decay models?
• RQ3 How do alternative models (e.g., Stretched Exponential, LPL) compare to the Modified Omori Model in fitting aftershock rate decay?
• RQ4 Is the observed preference for models with parameter c statistically robust, or does it depend on early-time data incompleteness?
• RQ5 To what extent does the short duration of the analyzed sequences (one year) affect the performance of models like LPL?

Key findings

• Models including parameter c in the Omori-type formulation perform better than those without c only when Ts is short (a few hours) and Mmin is low.
• For longer Ts (beyond a few hours) and higher Mmin (Mmin ≥ Mm − 3), models without parameter c outperform those with c, indicating c is not a universal feature.
• The presence of a non-zero c is strongly linked to catalog incompleteness in the first hours after the mainshock, not a physical property of aftershock sequences.
• The Modified Omori Model (MOM) and other Omori-type models consistently outperform the Stretched Exponential and LPL models across the majority of sequences.
• The LPL model performs poorly, likely due to the limited duration of the sequences (one year), which may not capture its full decay behavior.
• Information criteria (AICc and BIC) provide a robust framework for model comparison, especially when accounting for data completeness and model complexity.