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Volume 25, issue 1 | Copyright
Nonlin. Processes Geophys., 25, 241-250, 2018
© Author(s) 2018. This work is distributed under
the Creative Commons Attribution 4.0 License.

Research article 29 Mar 2018

Research article | 29 Mar 2018

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

Arnaud Mignan Arnaud Mignan
  • ETHZ, Institute of Geophysics, Swiss Federal Institute of Technology, Zurich, Switzerland

Abstract. The aftershock productivity law is an exponential function of the form K ∝ exp(αM), with K being the number of aftershocks triggered by a given mainshock of magnitude M and α ≈ ln(10) being the productivity parameter. This law remains empirical in nature although it has also been retrieved in static stress simulations. Here, we parameterize this law using the solid seismicity postulate (SSP), the basis of a geometrical theory of seismicity where seismicity patterns are described by mathematical expressions obtained from geometric operations on a permanent static stress field. We first test the SSP that relates seismicity density to a static stress step function. We show that it yields a power exponent q = 1.96±0.01 for the power-law spatial linear density distribution of aftershocks, once uniform noise is added to the static stress field, in agreement with observations. We then recover the exponential function of the productivity law with a break in scaling obtained between small and large M, with α = 1.5ln(10) and ln(10), respectively, in agreement with results from previous static stress simulations. Possible biases of aftershock selection, proven to exist in epidemic-type aftershock sequence (ETAS) simulations, may explain the lack of break in scaling observed in seismicity catalogues. The existence of the theoretical kink, however, remains to be proven. Finally, we describe how to estimate the solid seismicity parameters (activation density δ+, aftershock solid envelope r and background stress amplitude range Δo) for large M values.

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Short summary
The Utsu productivity law, one of the main relationships in seismicity statistics, gives the average number of aftershocks produced by a mainshock of a given magnitude. I demonstrate that the law can be formulated in the solid seismicity theory, where it is parameterized in terms of aftershock density within a geometrical solid, constrained by the mainshock size. This suggests that aftershocks can be studied by applying simple rules of analytic geometry on a static stress field.
The Utsu productivity law, one of the main relationships in seismicity statistics, gives the...