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This article was submitted to Solid Earth Geophysics, a section of the journal Frontiers in Earth Science

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Earthquakes are usually followed by aftershocks. The number of aftershocks—the so-called productivity—depends on the magnitude of the earthquake. In seismic modeling it is usually assumed that the number of aftershocks is approximately the same for earthquakes with the same magnitude. This is one of the key assumptions on which the calculations are based. Although it is known that in reality this number can vary widely, only recently a pattern of such changes, called the earthquake productivity law, has been established. If we consider only direct aftershocks in a fixed magnitude range relative to the magnitude of the main shocks, then their number for a set of earthquakes in some spatiotemporal volume has an exponential distribution form. This means that fewer aftershocks are more likely. The most likely outcome is the complete absence of aftershocks. This pattern is quite counterintuitive, especially when considering aftershocks over a wide range of magnitudes. Here we managed to confirm the fulfillment of the earthquake productivity law for the wide range of magnitudes. For earthquakes of magnitude 6 and higher in the land part of Japan, it is confirmed that the frequency distribution of the number of their direct aftershocks with a minimum magnitude of 5 units less has an exponential shape. In seismicity modeling the validated earthquake productivity law makes it possible to replace the incorrect assumption of constant earthquake productivity with an exponential distribution. The single parameter of this regularity is easily determined from the actual data.

It has recently been found that the number of aftershocks of large earthquakes in the world and the number of direct aftershocks of earthquakes in different regions of the world, considered in a fixed magnitude range relative to the main shock, obeys an exponential distribution (

The exponential shape of the distribution means that the most probable number of direct aftershocks (the mode of the distribution) is 0. This property of earthquake productivity seems so counterintuitive that testing whether this shape of the distribution persists as the range of aftershock magnitudes increases has an important independent meaning. The importance of this verification is also reinforced by the fact that the exponential form of productivity contradicts one of the key elements of the ETAS stochastic model (

The aim of this work is to investigate whether the exponential form of the distribution of earthquake productivity will be preserved with a significant expansion of the range of aftershock magnitudes. This can only be done in a region with a dense network of seismic stations and, at the same time, a high frequency of large earthquakes. We chose the land part of Japan, where the representative magnitude for crustal earthquakes since 2000 is about 1.0, and earthquakes of magnitude 6 and above occur several times a year.

We follow the definition of earthquake productivity adopted by _{
ij
} = _{
j
} − _{
i
} is the interevent time, _{
ij
} the spatial distance between the epicenters, _{
i
} the magnitude of event _{
f
} the fractal dimension of the epicenter distribution and

After introducing the threshold, it turns out that some of the events do not have a “parent” because the proximity to the nearest neighbor exceeds the threshold. We call such events “background”. If the proximity is below the threshold the event has a “parent”. We call such events offsprings. The productivity of an earthquake then is the number of its offsprings. Here, as in (_{
a
} ≥ _{
m
} − Δ_{
m
} is the magnitude of the parent and _{
a
} is the magnitude of the offspring. Note that the magnitude of the offspring may be greater than the magnitude of the parent.

The scheme used here differs from the traditional mainshock-aftershock outline, which usually assumes that aftershocks are weaker than the mainshock. Another difference is that an aftershock sequence consists of a hierarchical tree of parent-offspring sequences with several levels of hierarchy (offspring of offspring, etc.). In the traditional scheme (_{
c
} + Δ_{
c
}.

In this study we use the data of the catalog of the Japan Meteorological Agency (JMA)^{1}

Map of the completeness magnitude _{
c
} for the land part of Japan. The yellow line outlines the area of _{
c
} ≤ 1. Epicenters of the earthquakes (parent events) we used in the analysis of the productivity are shown by circles (5 ≤ _{
w
} = 9.1. Tab at the top shows the frequency-magnitude distribution of earthquakes within the area of _{
c
} ≤ 1.

The map of the completeness magnitude _{
c
} is constructed as follows: at each point, events are selected from the magnitude intervals [_{
M
}] in a circle with radius ^{
pM
} km, left end of the magnitude interval varies from 0.5 to 3. The _{
M
} is the length of the segment of the frequency-magnitude distribution where we check a linear shape of the distribution. The exponent _{
M
}] over the entire territory. The completeness magnitude _{
c
} is defined as the minimum value of _{0} ≥ _{1}10^{(b−δ)Δm
}, where _{
M
}], _{0} number of events with magnitude _{1} number of events with magnitude

High resolution of the _{
c
}-value is achieved through the determination of the smallest space-magnitude scale in which the Gutenberg-Richter law is verified. The multiscale procedure isolates the magnitude range that meets the best local seismicity and local record capacity. Here we use the values of the parameters _{
M
} = 1, _{
c
} is assigned to the centers of the circles located on a grid of 0.1 × 0.1°.

For further analysis, we chose the _{
c
} = 1 completeness level (yellow outline in _{
c
} ≥ 1 region closely matches the _{
c
} ≥ 1.7 completeness magnitude region found by the JMA for the whole period in the earthquake catalog. The _{
c
} ≥ 1 region closely matches the _{
c
} ≥ 1.7 completeness magnitude region declared by the JMA for the earthquakes with focal depth ^{2}
_{
c
} estimates is explained by the difference in focal depth of earthquakes. The level of registration is better for the most shallow seismicity

For the territory under consideration, estimates were made of the values of the proximity function parameters (1): _{
f
} = 1.68 and log_{10}
_{0} = −1.46. _{
f
} is determined by _{0} threshold was determined using the method from (

For each earthquake with _{
m
} ≥ 6.0, we calculated the productivity: the number of offsprings with magnitude _{
a
} ≥ _{
m
} − Δ

Cumulative frequency-productivity graphs for parent earthquakes with _{
m
} ≥ 6 and offspring events with _{
a
} ≥ _{
m
} − 5.

Thus, we may conclude that the exponential form of the distribution of Δ

In the presented analysis, the earthquake productivity was not separated according to different levels of the hierarchy due to small number of _{
m
} ≥ 6.0 events. Using worlwide stastistics of productivity, it was shown by _{
m
} ≥ 5.0 and Δ

Cumulative frequency-productivity graphs for parent earthquakes with _{
m
} ≥ 5 and offspring events with _{
a
} ≥ _{
m
} − 4: all earthquakes and separately for 8 highest hierarchy levels. Tab at the top shows the estimates of Λ_{4} and its standard errors calculated by bootstrap method.

We verified whether the observed exponential distribution of productivity is a property of the data, or, alternatively, the result of the choice of the proximity function. We selected only background earthquakes (events of the hierarchy level 0) from the catalog and, assuming _{0} =

Productivity distribution for background events (blue histogram) compared to productivity distribution for clustered events (red histogram).

The main result obtained here—the exponential form of the distribution of the Δ_{
m
} ≥ 6 and Δ_{
M
} = 5, the productivity reaches 3,000, but the total number of parent events is only 56. Under these conditions, the probability of realizing the productivity value exactly 0 is small:

It is usually assumed that the productivity of earthquakes depends mainly on their magnitude. This, in particular, is used in the ETAS (

In a nearest neighbor schema, each parent can have multiple offsprings, but each offspring can only have one parent. This makes the productivity averaging procedure meaningful, since in such a procedure each offspring is taken into account only once. We have also shown that productivity depends little on the level of the hierarchy. This means that there is no need to distinguish between main shocks and aftershocks during averaging. Δ

Δ_{ΔM
} is a parameter.

It is easy to show (_{ΔM
} is the average value of _{ΔM
} parameter and its estimate. The clustering factor is a convenient and a simple parameter to characterize the productivity in a set of earthquakes, for example, earthquakes in a certain space-time volume.

Publicly available datasets were analyzed in this study. This data can be found here:

All authors listed have made a substantial, direct, and intellectual contribution to the work and approved it for publication.

The study was supported by a grant from the Russian Science Foundation (project No. 20-17-00180).

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors, and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.

The authors thank the Japan Meteorological Agency for providing the Seismological Bulletin of Japan.

The Supplementary Material for this article can be found online at:

Japan Meteorological Agency, The Seismological Bulletin of Japan. (2022).

Japan Meteorological Agency, User’s guide for The Seismological Bulletin of Japan. (2022)