# Utsu aftershock productivity law explained from geometric operations on the permanent static stress field of mainshocks

## TL;DR

This paper explains the empirical Utsu aftershock productivity law using geometric operations on static stress fields, predicting a break in scaling that may be obscured in observational data.

## Contribution

It introduces a geometric theory of seismicity that derives the exponential productivity law and predicts a scaling break between small and large magnitudes.

## Key findings

- Recover the exponential form of the Utsu law from geometric operations.
- Predict a scaling break between small and large magnitudes.
- Suggest observational artifacts may hide the theoretical scaling break.

## Abstract

The aftershock productivity law, first described by Utsu in 1970, is an exponential function of the form K=K0.exp({\alpha}M) where K is the number of aftershocks, M the mainshock magnitude, and {\alpha} the productivity parameter. The Utsu law remains empirical in nature although it has also been retrieved in static stress simulations. Here, we explain this law based on Solid Seismicity, a geometrical theory of seismicity where seismicity patterns are described by mathematical expressions obtained from geometric operations on a permanent static stress field. We recover the exponential form but with a break in scaling predicted between small and large magnitudes M, with {\alpha}=1.5ln(10) and ln(10), respectively, in agreement with results from previous static stress simulations. We suggest that the lack of break in scaling observed in seismicity catalogues (with {\alpha}=ln(10)) could be an artefact from existing aftershock selection methods, which assume a continuous behavior over the full magnitude range. While the possibility for such an artefact is verified in simulations, the existence of the theoretical kink remains to be proven.

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Source: https://tomesphere.com/paper/1706.06490