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**Nonlinear Processes in Geophysics**
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**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

- ETHZ, Institute of Geophysics, Swiss Federal Institute of Technology, Zurich, Switzerland

- ETHZ, Institute of Geophysics, Swiss Federal Institute of Technology, Zurich, Switzerland

**Correspondence**: Arnaud Mignan (arnaud.mignan@sed.ethz.ch)

**Correspondence**: Arnaud Mignan (arnaud.mignan@sed.ethz.ch)

Abstract

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The aftershock productivity law is an exponential function of the form
*K*∝exp(*α**M*), 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 Δ*o*_{∗}) for large *M* values.

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Mignan, A.: Utsu aftershock productivity law explained from geometric operations on the permanent static stress field of mainshocks, Nonlin. Processes Geophys., 25, 241–250, https://doi.org/10.5194/npg-25-241-2018, 2018.

1 Introduction

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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 *K* produced by a mainshock of
magnitude *M* as

$$\begin{array}{}\text{(1)}& K\left(M\right)={K}_{\mathrm{0}}\mathrm{exp}\left[\mathit{\alpha}\left(M-{m}_{\mathrm{0}}\right)\right],\end{array}$$

with *m*_{0} the minimum magnitude cutoff (Utsu, 1970b; Ogata, 1988). This
relationship was originally proposed by Utsu (1970a, b) by combining two
other empirical laws, the Gutenberg–Richter relationship (Gutenberg and
Richter, 1944) and Båth's law (Båth, 1965), respectively:

$$\begin{array}{}\text{(2)}& \left\{\begin{array}{l}N\left(\ge m\right)=A\mathrm{exp}\left[-\mathit{\beta}\left(m-{m}_{\mathrm{0}}\right)\right]\\ N\left(\ge M-\mathrm{\Delta}{m}_{\mathrm{B}}\right)=\mathrm{1}\end{array}\right.,\end{array}$$

with *N* the average number of events above magnitude *m*, *A* a seismic activity
constant, *β* the magnitude size ratio (or $b=\mathit{\beta}/\mathrm{ln}\left(\mathrm{10}\right)$ in
base-10 logarithmic scale) and Δ*m*_{B} the magnitude difference
between the mainshock and its largest aftershock, such that

$$\begin{array}{ll}{\displaystyle}K\left(M\right)& {\displaystyle}=N\left(\ge {m}_{\mathrm{0}}\left|M\right.\right)\\ \text{(3)}& {\displaystyle}& {\displaystyle}=\mathrm{exp}\left(-\mathit{\beta}\mathrm{\Delta}{m}_{\mathrm{B}}\right)\mathrm{exp}\left[\mathit{\beta}\left(M-{m}_{\mathrm{0}}\right)\right],\end{array}$$

with ${K}_{\mathrm{0}}=\mathrm{exp}\left(-\mathit{\beta}\mathrm{\Delta}{m}_{\mathrm{B}}\right)$ and
*α*≡*β*. Equation (3) was only implicit in Utsu (1970a) and not
exploited in Utsu (1970b), where *K*_{0} was fitted independently of the value
taken by Båth's parameter Δ*m*_{B}. The *α* value was in
turn decoupled from the *β* value in later studies (e.g., Seif et al., 2017 and references therein).

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
*S* with

$$\begin{array}{}\text{(4)}& S\left(M\right)={\mathrm{10}}^{M-\mathrm{4}}=\mathrm{exp}\left[\mathrm{ln}\left(\mathrm{10}\right)(M-\mathrm{4})\right]\end{array}$$

(Kanamori and Anderson, 1975; Yamanaka and Shimazaki, 1990; Helmstetter,
2003), there is so far no analytical, physical expression of Eq. (1)
available. Although Hainzl et al. (2010) retrieved the exponential behavior
in numerical simulations where aftershocks were produced by the permanent
static stress field of mainshocks of different magnitudes, it remains
unclear how *K*_{0} and *α* relate to the underlying physical
parameters.

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.

2 Physical expression of the aftershock productivity law

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

Solid seismicity postulate: *Seismicity can be strictly categorized into three regimes of constant spatiotemporal densities*
*δ* – *background* *δ*_{0}*, quiescence **δ*_{−} *and activation* *δ*_{+} *(with* ${\mathit{\delta}}_{-}\ll {\mathit{\delta}}_{\mathrm{0}}\ll {\mathit{\delta}}_{+}$*) – occurring respective to the static stress step function:*

$$\begin{array}{}\text{(5)}& \mathit{\delta}\left(\mathit{\sigma}\right)=\left\{\begin{array}{ll}{\mathit{\delta}}_{-},& \mathit{\sigma}<-\mathrm{\Delta}{o}_{\ast}\\ {\mathit{\delta}}_{\mathrm{0}},& \mathit{\sigma}\le \left|\pm \mathrm{\Delta}{o}_{\ast}\right|\\ {\mathit{\delta}}_{+},& \mathit{\sigma}>\mathrm{\Delta}{o}_{\ast}\end{array}\right.,\end{array}$$

*with * *σ* * the static stress (stress unit), * Δ*o*_{∗} * the background stress amplitude range (stress unit), a stress threshold value separating two seismicity regimes, and * *δ*
* the spatial density of events (number of events per unit of volume) per seismicity regime.*

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 ${r}_{\ast}=r\left(\mathit{\sigma}=\pm \mathrm{\Delta}{o}_{\ast}\right)$, where
*r* is the distance from the static stress source (e.g., mainshock rupture)
and *r*_{∗} is the distance *r* at which there is a change of regime
(quiescence–background at $\mathit{\sigma}=-\mathrm{\Delta}{o}_{\ast}$ or
background–activation at $\mathit{\sigma}=\mathrm{\Delta}{o}_{\ast})$. The spatiotemporal
rate of seismicity is then a mathematical expression defined by the density
of events *δ* times the volume characterized by *r*_{∗} (see
previous demonstrations in Mignan et al., 2007 and Mignan, 2011, 2012, 2016a
where simple algebraic expressions were obtained).

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

$$\begin{array}{}\text{(6)}& \mathit{\sigma}\left(r\right)=-\mathrm{\Delta}{\mathit{\sigma}}_{\mathrm{0}}\left[{\left(\mathrm{1}-{\displaystyle \frac{{c}^{\mathrm{3}}}{(r+c{)}^{\mathrm{3}}}}\right)}^{{\scriptscriptstyle \frac{-\mathrm{1}}{\mathrm{2}}}}-\mathrm{1}\right],\end{array}$$

with Δ*σ*_{0} < 0 the mainshock stress drop, *c* the
crack radius and *r* the distance from the crack. Equation (6) is a
simplified representation of stress change from slip on a planar surface in a
homogeneous elastic medium. It takes into account both the square root
singularity at crack tip and the 1∕*r*^{3} falloff at higher distances
(Dieterich, 1994; Fig. 1b). It should be noted that this radial static stress
field does not represent the geometric complexity of Coulomb stress fields
(Fig. 2a). However, we are here only interested in the general behavior of
aftershocks with Eq. (6) retaining the first-order characteristics of this
field (i.e., on-fault seismicity; Fig. 2b), which corresponds to the case
where the mainshock relieves most of the regional stresses and aftershocks
occur on optimally oriented faults. It is also in agreement with
observations, most aftershocks being located on and around the mainshock
fault traces in southern California (Fig. 2c; see Sect. 3). The occasional
cases where aftershocks occur off-fault (e.g., Ross et al., 2017) can be
explained by the mainshock not relieving all of the regional stress (King et
al., 1994; Fig. 2d).

For ${r}_{\ast}=r\left(\mathit{\sigma}=\mathrm{\Delta}{o}_{\ast}\right)$, Eq. (6) yields the aftershock solid envelope of the following form:

$$\begin{array}{}\text{(7)}& {r}_{\ast}\left(c\right)=\left\{{\displaystyle \frac{\mathrm{1}}{{\left[\mathrm{1}-{\left(\mathrm{1}-\frac{\mathrm{\Delta}{\mathit{\sigma}}_{\ast}}{\mathrm{\Delta}{\mathit{\sigma}}_{\mathrm{0}}}\right)}^{-\mathrm{2}}\right]}^{\frac{\mathrm{1}}{\mathrm{3}}}}}-\mathrm{1}\right\}c=Fc\end{array}$$

function of the crack radius *c* and of the ratio between background stress
amplitude range Δ*o*_{∗} and stress drop Δ*σ*_{0} (Fig. 1c). With Δ*σ*_{0} independent of
earthquake size (Kanamori and Anderson, 1975; Abercrombie and Leary, 1993)
and Δ*o*_{∗} assumed constant, *r*_{∗} is directly
proportional to *c* with proportionality constant, or stress factor, *F* (Eq. 7).
Geometrical constraints due to the seismogenic layer width *w*_{0} then yield

$$\begin{array}{}\text{(8)}& c\left(M\right)=\left\{\begin{array}{ll}{\left({\displaystyle \frac{S\left(M\right)}{\mathit{\pi}}}\right)}^{{\scriptscriptstyle \frac{\mathrm{1}}{\mathrm{2}}}},& S\left(M\right)\le \mathit{\pi}{w}_{\mathrm{0}}^{\mathrm{2}}\\ {w}_{\mathrm{0}},& S\left(M\right)>\mathit{\pi}{w}_{\mathrm{0}}^{\mathrm{2}}\end{array}\right.,\end{array}$$

with *S* the rupture surface area defined by Eq. (4) and *c* becoming an effective
crack radius (Kanamori and Anderson, 1975; Fig. 1d). Note that the factor of
2 (i.e., using *w*_{0} instead of *w*_{0}∕2) comes from the free surface
effect (e.g., Kanamori and Anderson, 1975; Shaw and Scholz, 2001).

The aftershock productivity *K*(*M*) is then the activation density *δ*_{+}
times the volume *V*_{∗}(*M*) of the aftershock solid. For the case in
which the mainshock relieves most of the regional stress, stresses are
increased all around the rupture (King et al., 1994), which is topologically
identical to stresses increasing radially from the rupture plane (Fig. 2a–b). It follows that the aftershock solid can be represented by a volume
of contour *r*_{∗}(M) from the rupture plane geometric
primitive, i.e., a disk or a rectangle for small and large mainshocks,
respectively. This is illustrated in Fig. 3a–b and can be generalized by

$$\begin{array}{}\text{(9)}& {V}_{\ast}\left(M\right)=\mathrm{2}{r}_{\ast}\left(M\right)S\left(M\right)+{\displaystyle \frac{\mathit{\pi}}{\mathrm{2}}}{r}_{\ast}^{\mathrm{2}}\left(M\right)d,\end{array}$$

where *d* is the distance traveled around the geometric primitive by the
geometric centroid of the semicircle of radius *r*_{∗}(M)
(i.e., Pappus's Centroid Theorem), or

$$\begin{array}{}\text{(10)}& d=\left\{\begin{array}{ll}\mathrm{2}\mathit{\pi}\left(c\left(M\right)+{\displaystyle \frac{\mathrm{4}}{\mathrm{3}\mathit{\pi}}}{r}_{\ast}\left(M\right)\right),& c\left(M\right)+{r}_{\ast}\left(M\right)\le {\displaystyle \frac{{w}_{\mathrm{0}}}{\mathrm{2}}}\\ \mathrm{2}{w}_{\mathrm{0}},& c\left(M\right)+{r}_{\ast}\left(M\right)>{\displaystyle \frac{{w}_{\mathrm{0}}}{\mathrm{2}}}\end{array}\right..\end{array}$$

For the disk, the volume (Eq. 9) corresponds to the sum of a cylinder of
radius *c*(*M*) and height 2*r*_{∗}(M) (first term) and of
half a torus of major radius *c*(*M*) and minus radius *r*_{∗}(M) (second term). For the rectangle, the volume is the sum of a cuboid
of length *l*(*M*) (i.e., rupture length), width *w*_{0} and height 2*r*_{∗}(M) (first term) and of a cylinder of radius *r*_{∗}(M) and height *w*_{0} (second term; see red and orange volumes,
respectively, in Fig. 3a–c). Finally inserting Eqs. (7), (8) and (10) into
Eq. (9), we obtain

$$\begin{array}{}\text{(11)}& K\left(M\right)={\mathit{\delta}}_{+}\left\{\begin{array}{l}\left[{\displaystyle \frac{\mathrm{2}F}{\sqrt{\mathit{\pi}}}}+{F}^{\mathrm{2}}\sqrt{\mathit{\pi}}\left(\mathrm{1}+{\displaystyle \frac{\mathrm{4}}{\mathrm{3}\mathit{\pi}}}F\right)\right]{S}^{{\scriptscriptstyle \frac{\mathrm{3}}{\mathrm{2}}}}\left(M\right),\\ \phantom{\rule{1em}{0ex}}S\left(M\right)\le {\left({\displaystyle \frac{{w}_{\mathrm{0}}\sqrt{\mathit{\pi}}}{\mathrm{2}(\mathrm{1}+F)}}\right)}^{\mathrm{2}}\\ {\displaystyle \frac{\mathrm{2}F}{\sqrt{\mathit{\pi}}}}{S}^{{\scriptscriptstyle \frac{\mathrm{3}}{\mathrm{2}}}}\left(M\right)+{F}^{\mathrm{2}}{w}_{\mathrm{0}}S\left(M\right),\\ \phantom{\rule{1em}{0ex}}{\left({\displaystyle \frac{{w}_{\mathrm{0}}\sqrt{\mathit{\pi}}}{\mathrm{2}(\mathrm{1}+F)}}\right)}^{\mathrm{2}}<S\left(M\right)\le \mathit{\pi}{w}_{\mathrm{0}}^{\mathrm{2}}\\ \mathrm{2}F{w}_{\mathrm{0}}S\left(M\right)+\mathit{\pi}{F}^{\mathrm{2}}{w}_{\mathrm{0}}^{\mathrm{3}},\\ \phantom{\rule{1em}{0ex}}S\left(M\right)>\mathit{\pi}{w}_{\mathrm{0}}^{\mathrm{2}}\end{array}\right.\end{array}$$

which is represented in Fig. 3d. Considering the two main regimes only (small versus large mainshocks) and inserting Eq. (4) into (11), we get

$$\begin{array}{ll}\text{(12)}& {\displaystyle}& {\displaystyle}K\left(M\right)={\displaystyle}& {\displaystyle}{\mathit{\delta}}_{+}\left\{\begin{array}{l}\left[{\displaystyle \frac{\mathrm{2}F}{\sqrt{\mathit{\pi}}}}+{F}^{\mathrm{2}}\sqrt{\mathit{\pi}}\left(\mathrm{1}+{\displaystyle \frac{\mathrm{4}}{\mathrm{3}\mathit{\pi}}}F\right)\right]\mathrm{exp}\left[{\displaystyle \frac{\mathrm{3}\mathrm{ln}\left(\mathrm{10}\right)}{\mathrm{2}}}\left(M-\mathrm{4}\right)\right],\\ \phantom{\rule{1em}{0ex}}\text{small}M\\ \mathrm{2}F{w}_{\mathrm{0}}\mathrm{exp}\left[\mathrm{ln}\left(\mathrm{10}\right)\left(M-\mathrm{4}\right)\right]+\mathit{\pi}{F}^{\mathrm{2}}{w}_{\mathrm{0}}^{\mathrm{3}},\\ \phantom{\rule{1em}{0ex}}\text{large}M\end{array}\right.\end{array}$$

which is a closed-form expression of the same form as the original Utsu
productivity law (Eq. 1). Note that *K* and *δ*_{+} are both, implicitly,
functions of the selected minimum aftershock magnitude threshold *m*_{0}.

Here, we predict that the *α* value decreases from $\mathrm{3}\mathrm{ln}\left(\mathrm{10}\right)/\mathrm{2}\approx \mathrm{3.45}$ to ln (10)≈2.30 when switching regime from small to large
mainshocks (or from 1.5 to 1 in a base-10 logarithmic scale). It should be
noted that Hainzl et al. (2010) observed the same break in scaling in static
stress transfer simulations, which corroborates our analytical findings.
Hainzl et al. (2010) simulated aftershocks using the clock-change model where
events were advanced in time by the static stress change produced by a
mainshock in a 3-D medium. They explained the scaling break
observed in simulation as a transition from 3-D to 2-D scaling regime when
the mainshock rupture dimension approached *w*_{0}, which is compatible with
the present demonstration. For large *M*, the scaling is fundamentally the
same as in Eq. (4). Since that relation also explains the slope of the
Gutenberg–Richter law (see physical explanation given by Kanamori and
Anderson, 1975), it follows that *α*≡*β*, which is also in
agreement with the original formulation of Utsu (1970a, b; Eq. 3).

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 *ρ* is
well represented by a power law, expressed as

$$\begin{array}{}\text{(13)}& \mathit{\rho}\left(r\right)\propto {r}^{-q},\end{array}$$

with *r* the distance from the mainshock and *q* the power-law exponent. This
parameter ranges over $\mathrm{1.3}\le q\le \mathrm{2.5}$ (Felzer and Brodsky, 2006; Lipiello
et al., 2009; Marsan and Lengliné, 2010; Richards-Dinger et al., 2010;
Shearer, 2012; Gu et al., 2013; Moradpour et al., 2014; van der Elst and
Shaw, 2015). Although Felzer and Brodsky (2006) suggested a dynamic stress
origin for aftershocks, their results were later questioned by
Richards-Dinger et al. (2010). Most of the studies cited above suggest that
the *q* value is explained from a static stress process. As for the examples
of aftershocks shown to be dynamically triggered (e.g., Fan and Shearer,
2016), they are too few to alter the aftershock productivity law and too
remote to be consistently defined as aftershocks in cluster methods.

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 $\pm \mathrm{\Delta}{o}_{\ast}$ following
Mignan (2011) (see also King and Bowman, 2003). Note that any deviation above
Δ*o*_{∗} would be flattened to Δ*o*_{∗} over time by
temporal diffusion (the so-called “historical ghost static stress field” in
Mignan, 2016a). Figure 4c shows the resulting stress field and Fig. 4d the
predicted aftershock spatial density. Adding uniform noise blurs the contour
of the aftershock solid, switching the aftershock spatial density from a step
function (Fig. 4b) to a power law (Fig. 4d). We fit Eq. (13) to the simulated
data using the maximum likelihood estimation (MLE) method with ${r}_{min}={r}_{\ast}$ (Clauset et al., 2009) and find $q=\mathrm{1.96}\pm \mathrm{0.01}$, in agreement
with the aftershock literature. This result alone is, however, insufficient to
prove the validity of the SSP.

3 Observations and model fitting

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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 *m*_{0}=2.0 are considered (a conservative
estimate following results of Tormann et al., 2014; saturation effects
immediately after the mainshock are negligible when considering entire
aftershock sequences; Helmstetter et al., 2005).

Figure 5a represents the spatial linear density distribution of aftershocks
*ρ*(*r*) for the four largest strike-slip mainshocks in southern
California: 1987 *M*=6.6 Superstition Hills, 1992 *M*=7.3 Landers, 1999
*M*=7.1 Hector Mine and 2010 *M*=7.2 El Mayor. The distance between
mainshock and aftershocks is calculated as
$r=\sqrt{(x-{x}_{\mathrm{0}}{)}^{\mathrm{2}}+(y-{y}_{\mathrm{0}}{)}^{\mathrm{2}}}$, with (*x*, *y*) the aftershock
coordinates and (*x*_{0}, *y*_{0}) the coordinates of the nearest point to
the mainshock fault rupture (as depicted in Fig. 2c). The dashed black lines
shown in Fig. 5a are visual guides to *q*=1.96, showing that the SSP is
compatible with real aftershock observations.

Comparing Fig. 5a to Fig. 4d suggests that *r*_{∗} can be roughly
estimated from the spatial linear density plot, being the maximum distance
*r* at which the plateau ends, here leading to ${r}_{\ast}\approx $ 1 km. This
parameter is constant for different large *M* values since both *w*_{0} and
Δ*σ*_{0} are constant while Δ*σ*_{∗} is also
a priori a constant. We can then estimate the ratio $\mathrm{\Delta}{\mathit{\sigma}}_{\ast}/\mathrm{\Delta}{\mathit{\sigma}}_{\mathrm{0}}$ from Eq. (7). However, the result is ambiguous due to
uncertainties in the width *w*_{0}. For ${w}_{\mathrm{0}}=\left\{\mathrm{5},\mathrm{10},\mathrm{15}\right\}$ km, we get $\mathrm{\Delta}{\mathit{\sigma}}_{\ast}/\mathrm{\Delta}{\mathit{\sigma}}_{\mathrm{0}}=\left\{-\mathrm{0.54},-\mathrm{1.01},-\mathrm{1.38}\right\}$.

As for the plateau value *ρ* (*r* < *r*_{∗}), it provides an
estimate of the aftershock activation density *δ*_{+}, with

$$\begin{array}{}\text{(14)}& {\mathit{\delta}}_{+}={\displaystyle \frac{\mathit{\rho}\left(M,r<{r}_{\ast}\right)}{\mathrm{exp}\left[\mathrm{ln}\left(\mathrm{10}\right)(M-\mathrm{4})\right]}}\end{array}$$

a volumetric density, i.e., the linear density *ρ* normalized by the
mainshock rupture area (Eq. 4). Due to the fluctuations in *ρ* (*r* < *r*_{∗}), *δ*_{+} will be estimated from the
productivity law instead (see Sect. 3.3), and *ρ* (*r* < *r*_{∗}) will then be estimated from Eq. (14) (horizontal dashed colored lines), as
detailed below.

It should be noted that we consider only the first-generation aftershocks to
avoid *ρ* heterogeneities from secondary aftershock clusters occurring
off-fault. An example of such heterogeneity and anisotropy is illustrated by
the Landers–Big Bear case (Fig. 2c; dotted colored curve in Fig. 5a). Those
cases are not systematic and therefore not considered in the aftershock
productivity law. However, they are also due to static stress changes (e.g.,
King et al., 1994) with the anisotropic effects explainable by solid
seismicity through the concept of historical ghost static stress field
(Mignan, 2016a).

The observed number *n* of aftershocks of magnitude *m*≥*m*_{0} produced by
a mainshock of magnitude *M* (for a total of *N* mainshocks) in southern
California is shown in Figs. 5b (for large *M*≥6) and 6a (for the full
range *M*≥*m*_{0}). We fit Eq. (1) to the data using the MLE method with
the log-likelihood function

$$\begin{array}{ll}{\displaystyle}& {\displaystyle}\mathrm{LL}\left(\mathit{\theta};X=\left\{{n}_{i};i=\mathrm{1},\mathrm{\dots},N\right\}\right)=\\ \text{(15)}& {\displaystyle}& {\displaystyle}\phantom{\rule{1em}{0ex}}{\sum}_{i=\mathrm{1}}^{N}\left[{n}_{i}\mathrm{ln}\left[{K}_{i}\left(\mathit{\theta}\right)\right]-{K}_{i}\left(\mathit{\theta}\right)-\mathrm{ln}\left({n}_{i}\mathrm{!}\right)\right]\end{array}$$

for a Poisson process, representing the stochasticity of the count *K* of
aftershocks produced by a mainshock at any given time. Inserting Eq. (1) in
Eq. (15) yields

$$\begin{array}{ll}{\displaystyle}& {\displaystyle}\mathrm{LL}\left(\mathit{\theta}=\left\{{K}_{\mathrm{0}},\mathit{\alpha}\right\};X\right)=\mathrm{ln}\left({K}_{\mathrm{0}}\right){\sum}_{i=\mathrm{1}}^{N}{n}_{i}\\ {\displaystyle}& {\displaystyle}\phantom{\rule{1em}{0ex}}+\mathit{\alpha}{\sum}_{i=\mathrm{1}}^{N}\left[{n}_{i}\left({M}_{i}-{m}_{\mathrm{0}}\right)\right]-{K}_{\mathrm{0}}{\sum}_{i=\mathrm{1}}^{N}\mathrm{exp}\left[\mathit{\alpha}\left({M}_{i}-{m}_{\mathrm{0}}\right)\right]\\ \text{(16)}& {\displaystyle}& {\displaystyle}\phantom{\rule{1em}{0ex}}-{\sum}_{i=\mathrm{1}}^{N}\mathrm{ln}\left({n}_{i}\mathrm{!}\right)\end{array}$$

(note that the last term can be set to 0 during LL maximization). For southern
California, we obtain *α*_{MLE}=2.32 (1.01 in log_{10} scale)
and *K*_{0}=0.025 when considering large (*M*≥6) mainshocks only to
avoid the issues of scaling break and data dispersion at lower magnitudes.
This result, represented by the black solid line in Fig. 5b, is in
agreement with previous studies in the same region (e.g., Helmstetter, 2003;
Helmstetter et al., 2005; Zaliapin and Ben-Zion, 2013; Seif et al., 2017)
and with $\mathit{\alpha}=\mathrm{ln}\left(\mathrm{10}\right)\approx \mathrm{2.30}$ predicted for large mainshocks
in solid seismicity (Eq. 12). Moreover we find a bulk *β*_{MLE}=2.34 (1.02 in log_{10} scale) (Aki, 1965), in agreement with *α*≡*β*.

Let us now rewrite the solid seismicity aftershock productivity law (Eq. 12)
by only considering the large *M* case and injecting ${r}_{\ast}=F{w}_{\mathrm{0}}$ (by
combining Eqs. 7–8). We get

$$\begin{array}{}\text{(17)}& K\left(M>{M}_{\mathrm{break}}\right)={\mathit{\delta}}_{+}\left\{\mathrm{2}{r}_{\ast}\mathrm{exp}\left[\mathrm{ln}\left(\mathrm{10}\right)(M-\mathrm{4})\right]+\mathit{\pi}{r}_{\ast}^{\mathrm{2}}{w}_{\mathrm{0}}\right\}.\end{array}$$

The role of *w*_{0} is illustrated in Fig. 5b for different values (dashed
and dotted curves) and shown to be insignificant for large *M* values.
Therefore Eq. (17) can be approximated to

$$\begin{array}{}\text{(18)}& K\left(M>{M}_{\mathrm{break}}\right)\approx \mathrm{2}{\mathit{\delta}}_{+}{r}_{\ast}\mathrm{exp}\left[ln\left(\mathrm{10}\right)(M-\mathrm{4})\right].\end{array}$$

By analogy with Eq. (1), we get

$$\begin{array}{}\text{(19)}& {\mathit{\delta}}_{+}={\displaystyle \frac{{K}_{\mathrm{0}}\mathrm{exp}\left[\mathrm{ln}\left(\mathrm{10}\right)\left(\mathrm{4}-{m}_{\mathrm{0}}\right)\right]}{\mathrm{2}{r}_{\ast}}}.\end{array}$$

With ${r}_{\ast}\approx \mathrm{1}$ km estimated from *ρ*(*r*) (Sect. 3.2) and
*K*_{0}=0.025, we obtain ${\mathit{\delta}}_{+}=\mathrm{1.23}$ events km^{−3} for *m*_{0}=2. We then get back the plateau *ρ* (*r* < *r*_{∗}) for
different *M* values from Eq. (14), as shown in Fig. 5a (horizontal dashed
colored lines). Although based on limited data, this result suggests that the
activation parameter *δ*_{+} is constant (at least for large *M*) in
southern California. Note that if *ρ* (*r* < *r*_{∗}) was
well constrained, it could have been estimated jointly with *r*_{∗} from
Fig. 5a to predict the aftershock productivity law of Fig. 5b without further
fitting required (hence removing *K*_{0} from the equation, *K*_{0} having no
physical meaning in solid seismicity).

4 Role of aftershock selection on productivity scaling break

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We tested the following piecewise model to identify any break in scaling at
smaller *M*, as predicted by Eq. (12):

$$\begin{array}{ll}\text{(20)}& {\displaystyle}& {\displaystyle}K\left(M\right)={\displaystyle}& {\displaystyle}\left\{\begin{array}{l}{K}_{\mathrm{0}}{\displaystyle \frac{\mathrm{exp}\left[\mathrm{ln}\left(\mathrm{10}\right)({M}_{\mathrm{break}}-{m}_{\mathrm{0}})\right]}{\mathrm{exp}\left[\frac{\mathrm{3}}{\mathrm{2}}\mathrm{ln}\left(\mathrm{10}\right)({M}_{\mathrm{break}}-{m}_{\mathrm{0}})\right]}}\mathrm{exp}\left[{\displaystyle \frac{\mathrm{3}}{\mathrm{2}}}\mathrm{ln}\left(\mathrm{10}\right)(M-{m}_{\mathrm{0}})\right],\\ \phantom{\rule{1em}{0ex}}M\le {M}_{\mathrm{break}}\\ {K}_{\mathrm{0}}\mathrm{exp}\left[\mathrm{ln}\left(\mathrm{10}\right)(M-{m}_{\mathrm{0}})\right],\\ \phantom{\rule{1em}{0ex}}M>{M}_{\mathrm{break}}\end{array}\right.\end{array}$$

but with the best MLE result obtained for *M*_{break}=*m*_{0}, suggesting
no break in scaling in the aftershock productivity data, as observed in
Fig. 6a. Final parameter estimates are *α*_{MLE}=1.95 (0.85 in
log_{10} scale) and *K*_{0}=0.141 for the full mainshock magnitude
range *M*≥*m*_{0} (dotted line), subject to high scattering at low *M*
values.

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

$$\begin{array}{}\text{(21)}& \left\{\begin{array}{l}\mathit{\lambda}\left(t,x,y\right)=\mathit{\mu}\left(t,x,y\right)+{\sum}_{i:{t}_{j}<t}K\left({M}_{i}\right)\\ \phantom{\rule{1em}{0ex}}f(t-{t}_{i})g\left(x-{x}_{i},y-{y}_{i}\left|{M}_{i}\right.\right)\\ f\left(t\right)={c}^{p-\mathrm{1}}(p-\mathrm{1})(t+c{)}^{-p}\\ g\left(x,y\left|M\right.\right)={\displaystyle \frac{\mathrm{1}}{\mathit{\pi}}}{\left(d{e}^{\mathit{\gamma}\left(M-{m}_{\mathrm{0}}\right)}\right)}^{q-\mathrm{1}}\\ \phantom{\rule{1em}{0ex}}{\left({x}^{\mathrm{2}}+{y}^{\mathrm{2}}+d{e}^{\mathit{\gamma}\left(M-{m}_{\mathrm{0}}\right)}\right)}^{-q}(q-\mathrm{1}).\end{array}\right.\end{array}$$

Aftershock sequences are defined by power laws, both in time and space (for
an alternative temporal function, see Mignan, 2015, 2016b; the spatial
power-law distribution is in agreement with solid seismicity in the case of a
heterogeneous static stress field – see Sect. 2.2). The southern
California background seismicity, as defined by the nearest-neighbor method
(with same *t*, *x*, *y* and *m*), is denoted by *μ*. We fix the ETAS parameters to $\mathit{\theta}=\left\{c=\mathrm{0.011}\right.$ day, *p*=1.08, *d*=0.0019 km^{2}, *q*=1.47, *γ*=2.01, *β*=2.29, *K*_{0}=0.08},
following the fitting results of Seif et al. (2017) for the southern
California relocated catalog and *m*_{0}=2 (see their Table 1). However, we
define the productivity function *K*(*M*) from Eq. (20) with *M*_{break}=5. Examples of ETAS simulations are shown in Fig. 6b for comparison with
the observed southern California time series. Figure 6c allows us to verify
that the simulated aftershock productivity is kinked at *M*_{break},
as defined by Eq. (20).

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 *α*_{MLE}=2.07 as
observed in the real case for *M*≥*m*_{0}). Note that a similar result is
obtained when using a windowing method (Gardner and Knopoff, 1974). This
demonstrates that the theoretical break in scaling predicted in the
aftershock productivity law can be lost in observations due to an aftershock
selection bias, all declustering techniques assuming continuity over the
entire magnitude range. While such a bias is possible, it does not prove
that the break in scaling exists. The fact that a similar break in scaling
was obtained in independent Coulomb stress simulations (Hainzl et al., 2010),
however, provides high confidence in our results.

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
*M*_{L}∝*M*_{w} at large *M*, *M*_{L}∝1.5*M*_{w} at smaller *M*
values. This could in theory cancel the kink in real data. However, the
scaling break predicted by Deichmann (2017) occurs at several magnitude
units below the geometric scaling break expected by solid seismicity,
invalidating this second option for mid-range magnitudes *M*.

5 Conclusions

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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 *δ*_{+} (also *δ*_{−} and *δ*_{0}), the
background stress amplitude range Δ*o*_{∗} and the solid envelope
*r*_{∗} which describes the geometry of the “seismicity solid”
(Fig. 3a–b). Further studies will be needed to evaluate whether the
*δ*_{+} and Δ*o*_{∗} parameters are universal or
region-specific and if the same values apply to different types of seismicity
at a same location.

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 *q* of the aftershock spatial density distribution shows
that the SSP is consistent with large aftershock observations once uniform
noise is added to the stress field (Figs. 4d–5a). The impact of other types
of noise on *q* has yet to be investigated. The SSP is also complementary to
the more common simulations of static stress loading (King and Bowman, 2003)
and static stress triggering (Hainzl et al., 2010).

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
*σ*(*r*) formulations could be tested in the future, the only constraint
on generating so-called seismicity solids being the use of the postulated
static stress step function of Eq. (5) (i.e., the solid seismicity
postulate).

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 *M* will be able to identify whether a scaling
break really exists or not.

Data availability

Back to toptop
Data availability.

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

Competing interests

Back to toptop
Competing interests.

The author declares that he has no conflict of interest.

Acknowledgements

Back to toptop
Acknowledgements.

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

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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...

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