The aftershock productivity law is an exponential function of the form

Aftershocks, one of the most studied patterns observed in seismicity, are
characterized by three empirical laws, which are functions of time, such as
the modified Omori law (e.g., Utsu et al., 1995), space (e.g.,
Richards-Dinger et al., 2010; Moradpour et al., 2014) and mainshock
magnitude (Utsu, 1970a, b; Ogata, 1988). The present study focuses on the
latter relationship, i.e., the Utsu aftershock productivity law, which
describes the total number of aftershocks

Although it seems obvious that Eq. (1) can be explained geometrically if the
volume of the aftershock zone is correlated to the mainshock surface area

The aim of the present article is to describe the Utsu aftershock productivity equation (Eq. 1) in terms of a geometrical theory of seismicity coined “solid seismicity”, where the Eq. (4) scaling is parameterized using the solid seismicity postulate (SSP). The SSP has already been shown to effectively explain other empirical laws of both natural and induced seismicity from simple geometric operations on a permanent static stress field (Mignan, 2012, 2016a). The theory is applied here for the first time to describe aftershocks.

“Solid seismicity”, a geometrical theory of seismicity, is based on the following postulate (Mignan et al., 2007; Mignan, 2008, 2012, 2016a).

Solid seismicity postulate:

We mean by “strictly categorized” that any seismicity population is either
part of the background, quiescence or activation regime (or class), with no
other regime or class possible (i.e., a sort of hard labeling). Based on this
postulate, Mignan (2012) demonstrated the power-law behavior of precursory
seismicity in agreement with the observed time-to-failure equation (Varnes,
1989), while Mignan (2016a) demonstrated both the observed parabolic
spatiotemporal front and the linear relationship with injection flow rate of
induced seismicity (Shapiro and Dinske, 2009). It remains unclear whether the
SSP has a physical origin or not. If not, it would still represent a
reasonable approximation of the linear relationship between event production
and static stress field in a simple clock-change model (Hainzl et al., 2010;
Fig. 1a). For the testing of the SSP on the observed spatial distribution of
aftershocks, see Sect. 2.2. The power of Eq. (5) is that it allows seismicity
patterns to be defined in terms of “solids” described by the spatial
envelope

In the case of aftershocks, we define the static stress field of the
mainshock by

Definition of the aftershock solid envelope in a permanent static
stress field:

Possible static stress fields and inferred aftershock spatial
distribution:

For

Geometric origin of the aftershock productivity law:

The aftershock productivity

Here, we predict that the

The SSP predicts a step-like behavior of the aftershock spatial density for
an idealized smooth static stress field (Fig. 4a–b), which is in
disagreement with real aftershock observations. A number of studies have
shown that the spatial linear density distribution of aftershocks

Spatial distribution of aftershocks following the SSP:

In a more realistic setting, the static stress field must be heterogeneous
(due to the occurrence of previous events and other potential stress
perturbations). We therefore simulate the static stress field by adding a
uniform random component bounded over

We consider the case of southern California and extract aftershock sequences
from the relocated earthquake catalog of Hauksson et al. (2012) defined over
the period 1981–2011, using the nearest-neighbor method (Zaliapin et al.,
2008; used with its standard parameters originally calibrated for southern
California, considering only the first aftershock generation). Only events
with magnitudes greater than

Figure 5a represents the spatial linear density distribution of aftershocks

Estimating the solid seismicity parameters from the spatial
distribution of aftershocks:

Comparing Fig. 5a to Fig. 4d suggests that

As for the plateau value

It should be noted that we consider only the first-generation aftershocks to
avoid

The observed number

Aftershock productivity defined as the number of aftershocks

Let us now rewrite the solid seismicity aftershock productivity law (Eq. 12)
by only considering the large

We tested the following piecewise model to identify any break in scaling at
smaller

We now identify whether the lack of break in scaling in aftershock
productivity observed in earthquake catalogues could be an artefact related
to the aftershock selection method. We run epidemic-type aftershock sequence
(ETAS) simulations (Ogata, 1988; Ogata and Zhuang, 2006), with the
seismicity rate

We then select aftershocks from the ETAS simulations with the
nearest-neighbor method. Figure 4d represents the estimated aftershock
productivity, which has lost the break in scaling originally implemented in
the simulations (with an underestimated

One other possible explanation for the lack of scaling break is that our
demonstration assumes moment magnitudes while the southern California
catalogue is in local magnitudes. Deichmann (2017) demonstrated that while

In the present study, a closed-form expression defined from geometric and
static stress parameters was proposed (Eq. 12) to describe the empirical Utsu
aftershock productivity law (Eq. 1). This demonstration is similar to the
previous ones made by the author to explain precursory accelerating
seismicity and induced seismicity (Mignan, 2012, 2016a). In all these
demonstrations, the main physical parameters remain the same, i.e., the
activation density

Although the solid seismicity postulate (Eq. 5) remains to be proven, it is
so far a rather convenient and pragmatic assumption to make to determine the
physical parameters that play a first-order role in the behavior of
seismicity. The similarity of the SSP-simulated and observed values of the
power-law exponent

Analytic geometry, providing both a visual representation and an analytical
expression of the problem at hand (Fig. 3), represents a new approach to try
to better understand the behavior of seismicity. Its current limitation in
the case of aftershock analysis consists of assuming that the static stress
field is radial and described by Eq. (6) (e.g., Dieterich, 1994), which is
likely only valid for mainshocks relieving most of the regional stresses and
with aftershocks occurring on optimally oriented faults (King et al., 1994).
More complex, second-order stress behaviors might explain part of the
scattering observed around Eq. (1) (Fig. 6a), such as overpressure due to
trapped high-pressure gas for example (Miller et al., 2004 – see also
Mignan, 2016a, for an overpressure field due to fluid injection). Other

Finally, the disappearance of the predicted scaling break in the aftershock
productivity law once declustering is applied (Fig. 6) indicates that more
work is required in that domain. Only a declustering technique that does not
dictate a constant scaling at all

The seismicity data used in the present study is published (see Sect. 3.1) and publicly available via the Southern California Earthquake Data Center.

The author declares that he has no conflict of interest.

I thank Nadav Wetzler and two anonymous reviewers, as well as editor Ilya Zaliapin, for their valuable comments. Edited by: Ilya Zaliapin Reviewed by: two anonymous referees