1. Physica A 508 (2018) 305–312
Contents lists available at ScienceDirect
Physica A
journal homepage: www.elsevier.com/locate/physa
New Bayesian frequency–magnitude distribution model for
earthquakes applied in Chile
Ewin Sánchez C. a,b,∗
, Pedro Vega-Jorquera a
a
Departmento de Física y Astronomía, Facultad de Ciencias, Universidad de La Serena, La Serena, Chile
b
Universidad Tecnológica de Chile INACAP, Chile
h i g h l i g h t s
• Seismic frequencies–magnitudes follow a distribution connected to the superstatistics form.
• It is possible to obtain, with a new Frequency–magnitude model, an alternative form for the b Gutenberg–Richter parameter.
• It is possible to obtain, with a new Frequency–magnitude model, excellent goodness-of-fit to samples data from Chilean earthquakes.
a r t i c l e i n f o
Article history:
Received 17 February 2018
Received in revised form 17 May 2018
Available online xxxx
Keywords:
Earthquake
Magnitude–frequency distribution
Bayesian statistics
Superstatistics
a b s t r a c t
We outline a model developed from a Bayesian approach, which can be connected, from the
perspective of Mathai’s pathway model, with the superstatistics representation presented
by Beck and Cohen in 2002, which has been widely applied in non-equilibrium and complex
systems. We evaluated goodness-of-fit to the observed frequency–magnitude distribution
before and after major earthquakes occurred in Chile, from 2010 year to date. This new
model, allows us to obtain an alternative Gutenberg–Richter b parameter, through a wider
range of magnitudes.
© 2018 Elsevier B.V. All rights reserved.
1. Introduction
Since the proposal submitted by Gutenberg and Richter in [1], through the empirical formula:
log N = a − bm (1)
which provides the number of earthquakes N, with magnitude ≥ m, observed in certain region over a certain period of time,
(a and b are positive parameters), many efforts have been made to find a more complete model.
In recent years, Sotolongo-Costa and A. Posadas [2], through fragment size distribution (produced by breakage during
tectonic activities) and the Tsallis statistics, also proposed a relation for frequency–magnitude distribution. Subsequently
other authors, as in [3], have produced new models based on the original works of Sotolongo-Costa and A. Posadas. In
addition, several applications have been made through that perspective, such as [4] and [5], where non-extensive properties
for behavior and evolution of seismic systems are addressed in different regions of our planet.
We derived our model using a Bayesian approach, and this, from the mathematical statistics point of view, can be
connected to the Beck–Cohen formalism (which they named superstatistics [6]), through the Mathai’s pathway model [7]. In
Section 2, we present the fundamental Bayesian framework, and the step-by-step model construction we followed, as well
as posterior goodness-of-fit tests with respect to experimental data. In Section 3, we discuss the implications, analysis and
conclusions.
∗ Corresponding author at: Departmento de Física y Astronomía, Facultad de Ciencias, Universidad de La Serena, La Serena, Chile.
E-mail address: esanchez@userena.cl (E. Sánchez C.).
https://doi.org/10.1016/j.physa.2018.05.119
0378-4371/© 2018 Elsevier B.V. All rights reserved.

2. 306 E. Sánchez C., P. Vega-Jorquera / Physica A 508 (2018) 305–312
2. Fit model
2.1. Bayesian framework
Marginalization is a very important technique applied in the Bayesian inference process, by use of which we may obtain
an unconditional model for a random observable variable X, which depends of a parameter Θ.
Marginalization can be realized assuming Θ itself as a random variable, which can take values Θ = {θ0, θ1, . . . ., θm},
with the probability distribution given by a function P(Θ).
Each of the joint probabilities we know are given by the product rule,
P(xi, θj) = P(xi|θj) P(θj) (2)
Thus, in general, for X and Θ continuous random variables, and given our knowledge I of the system, marginal distribution
P(X) is given by
P(X) =
∫
P(X|Θ, I) P(Θ|I) dΘ (3)
This marginalization incorporates the uncertainty associated with assigning a specific value to the Θ parameter, as well
as the intrinsic uncertainty present in the observed system’s properties.
2.2. Model construction
Take a geographic region that has some seismic activity, where the magnitude m is a scalar variable related to the energy
released in each seismic event.
We use an expression that is compatible with the Gutenberg–Richter relation of magnitudes (pertinence can be seen
in [8]), and assume that it is suitable for the considered geographic region,
f (m) = e−β(m−m0)
(4)
where m0 is the minimum threshold for the magnitudes considered. The β parameter must take a positive value, greater
than zero, and its relation to the Gutenberg–Richter parameter b is:
b =
β
ln 10
(5)
This parameter b, of the Gutenberg–Richter magnitude–frequency relationship, has been the subject of many studies,
since it is associated with the intensity of the seismic activity observed in the region of interest.
We want to determine, by marginalization, the unconditional frequency distribution of magnitudes m, considering the
information I we have about this type of system.
Note that if the system information is given by I0, then
P(m|I0) = ⟨δ
(
m(
−→x ) − m
)
⟩I0
(6)
where
−→x is the vector of all physical parameters hidden at the time we observe the system, which are involved in the
manifestation of observed seismic activity. By not knowing all of them, makes it impossible for us to accurately predict
the values of m each time an event occurs. It could be parameters that account for the physical process of elastic strain
accumulation and the triggering mechanism, or other processes involved in mechanical stresses due to plate movements
(e.g. [9,10]). Thus,
P(m|I0) = f
(
m(
−→x )
)
∫
d
−→x δ
(
m(
−→x ) − m
)
(7)
that is, P(m|I0) = f
(
m(
−→x )
)
Ω(m), where Ω(m) must be the density of states.
Since (4) has a similar form to the Boltzmann distribution (as it can also be seen in [11] within a context of earthquake
statistics), and also, the magnitude m is related to the release of energy, then we propose Ω ∼ (m − m0)
3n
2
−1
to obtain a
normalized conditional density function of m given β, so we have
p(m|β) =
e−β (m−m0)
(m − m0)
3
2
n−1
β
3
2
n
Γ (3
2
n)
(8)
where n is related to the system’s degree of freedom. On the other hand, we hypothesize that β has a gamma distribution
(relevance of the chosen distribution can be seen in [12,13]). Thus,

3. E. Sánchez C., P. Vega-Jorquera / Physica A 508 (2018) 305–312 307
p(β) =
λ−c
Γ (c)
βc−1
e− β
λ (9)
with λ and c as positive parameters.
We can write the joint probability of magnitude m and the parameter β, given I,
p(m, β|I) = P(m|β, I)P(β|I) (10)
Assuming β as a random variable, marginalization can be done, then (I is implicit):
p(m) =
∫
p(m|β)p(β)dβ (11)
that is
p(m) =
λ
3
2
n
Γ (c + 3
2
n)
Γ (3
2
n)Γ (c)
(m − m0)
3
2
n−1
[
1 + (m − m0)λ
]−c− 3
2
n
(12)
The cumulative distribution function is:
F(m ≥ m0) =
∫ m
m0
p(m′
) dm′
(13)
Explicitly:
F(m ≥ m0) =
2
3
λ
3
2
n
(m − m0)
3
2
n
2F1
(
[3
2
n, c + 3
2
n]; [1 + 3
2
n]; −(m − m0)λ
)
Γ (c + 3
2
n)
nΓ (c)Γ (3
2
n)
(14)
2.3. Performing the cumulative function distribution
In order to check our model, following an analogous way to other authors, we evaluate (13), instead of (12), with seismic
magnitudes data registered in the territory of Chile, around the strongest events (according to moment magnitude) from
2010 to date.
The main events, which make up each sample, are the following:
• Cauquenes 2010 (−36.290, −73.239), 8.8 magnitude, occurred on February 27, 2010
• Tirúa 2011 (−38.350, −73.27), 7.0 magnitude, occurred on January 02, 2011
• Constitución 2012 (−35.12, −72.13), 7.0 magnitude, occurred on March 25, 2012
• Vallenar 2013 (−28.06, −70.84), 6.8 magnitude, occurred on January 30, 2013
• Iquique 2014 (−19.63, −70.86), 8.2 magnitude, occurred on April 01, 2014
• Isla de Pascua 2014 (−32.11, −110.77), 7.1 magnitude, occurred on October 08, 2014
• Coquimbo 2015 (−31.535, −71.919), 8.4 magnitude, occurred on September 16, 2015
Data, obtained from the National Seismological Center of the University of Chile, were divided into two groups: the
previous ones, and those subsequent to the main event.
Figures below, show the fitting of cumulative normalized function F(m), given by (13), on the magnitudes m registered
during one month (with m ≥ m0), and can be seen the corresponding fitting parameters; sum of squared errors (SSE),
adjusted R-squared (ARS), and the root mean square deviation (RMSE).
Fig. 1. Cauquenes 2010, before the main event. m0=2.1, SSE=0.008055, ARS=0.9983, RMSE=0.01475.

4. 308 E. Sánchez C., P. Vega-Jorquera / Physica A 508 (2018) 305–312
Fig. 2. Cauquenes 2010, after the main event. m0=2.0, SSE=0.006105, ARS=0.9989, RMSE=0.01285.
Fig. 3. Tirúa 2011, before the main event. m0=2.0, SSE=0.007461, ARS=0.9985, RMSE=0.0142.
Fig. 4. Tirúa 2011, after the main event. m0=2.0, SSE=0.001717, ARS=0.9997, RMSE=0.006813.
Fig. 5. Constitución 2012, before the main event. m0=2.0, SSE=0.01135, ARS=0.9978, RMSE=0.01751.

5. E. Sánchez C., P. Vega-Jorquera / Physica A 508 (2018) 305–312 309
Fig. 6. Constitución 2012, after the main event. m0=2.0, SSE=0.008373, ARS=0.9985, RMSE=0.01504.
To say in addition, that goodness-of-fit was tested using the least absolute residuals (LAR) method, where extremes have
less influence on the fit [14]. This method minimizes possible instrumental inaccuracies in measuring extreme magnitudes
(lower/higher).
The values obtained are shown in Table 1:
Table 1
Fit parameters.
Reference n c λ
Fig. 1 Cauquenes 1.969 57.65 0.0335
Fig. 2 Cauquenes 4.002 22.81 0.1452
Fig. 3 Tirúa 2.140 90.58 0.0268
Fig. 4 Tirúa 2.744 48.56 0.0604
Fig. 5 Constitución 4.381 10.85 0.4497
Fig. 6 Constitución 5.854 13.71 0.4896
Fig. 7 Vallenar 2.390 53.77 0.0626
Fig. 8 Vallenar 2.813 27.89 0.1232
Fig. 9 Iquique 2.393 40.70 0.0690
Fig. 10 Iquique 3.151 31.24 0.1044
Fig. 11 Isla de Pascua 2.333 52.18 0.0594
Fig. 12 Isla de Pascua 4.900 13.89 0.3934
Fig. 13 Coquimbo 4.057 12.96 0.3663
Fig. 14 Coquimbo 6.042 8.322 0.7563
3. Final remarks
The Pathway theoretical model, representing a general functional form containing a broad family of densities, was
presented by A.M. Mathai in [15], which in its scalar version is:
f (x) = ςxγ −1
[1 − b(1 − a)xδ
]
1
1−a (15)
with b > 0, δ > 0, γ > 0 y 1 − b(1 − a)xδ
> 0.
Parameter a is the pathway parameter, and ς the normalizing constant.
For a < 1, we obtain a generalized type-1 beta distribution, whilst a > 1 leads to a generalized type-2 beta distribution,
and for a −→ 1 a generalized gamma model can be found.
Mathai A.M., Haubold H.J and Tsallis C., have analyzed in [16], the connection between Pathway model and nonextensive
statistics and Beck–Cohen Superstatistics, presenting the xγ −1
factor as the respective density of states of expression (15),
from where nonextensive statistics is a particular case for γ = 1, x > 0, and one of the forms of Beck–Cohen superstatistics
is a special case for γ = 1, x > 0 and a > 1.
A general form of the Beck–Cohen probability density function was presented in [7,17] from a Bayesian perspective, and
previously, in [18], F. Sattin demonstrated that the classical Beck–Cohen function [6] has a solid construction from a Bayesian
framework too.
The aforementioned generalized superstatistics expression has the form:
f (x) =
δΓ (
γ +ρ
δ
)(x − x0)γ −1
[
1 + (x−x0)δ
h
]−(
γ +ρ
δ )
h
γ
δ Γ (
γ
δ
)Γ (
ρ
δ
)
(16)
for x ≥ 0, γ > 0, ρ > 0, δ > 0, h > 0.
From this perspective, our model (12) belongs to the generalized type-2 beta distribution, and, in particular, can be seen
as a generalized superstatistics form.

6. 310 E. Sánchez C., P. Vega-Jorquera / Physica A 508 (2018) 305–312
Fig. 7. Vallenar 2013, before the main event. m0=2.1, SSE=0.002415, ARS=0.9995, RMSE=0.008078.
Fig. 8. Vallenar 2013, after the main event. m0=2.1, SSE=0.002972, ARS=0.9994, RMSE=0.008962.
On the other hand, let us consider our parameters model. It can be seen, that it is possible to obtain with (5), values for
the Gutenberg–Richter b parameter through the average β0 of distribution (9), which is given by β0 = cλ, where those
parameters, c and λ, we propose to take them from the values provided by the fitting of expression (14).
In this way, we found that b increases after each main event considered, which is when the intensity of the seismic
activity also increases, as also happens using the Gutenberg–Richter magnitude–frequency relationship, as described by
other authors [19].
In Table 2 is shown b (before) and b′
(after) founded:
Table 2
b values, before and after of main event.
Event b b′
Cauquenes 2010 0.8380 1.4384
Tirúa 2011 1.0551 1.2746
Constitución 2012 2.1190 2.9152
Vallenar 2013 1.4625 1.4923
Iquique 2014 1.3454 1.4164
Isla de Pascua 2014 1.0180 2.3731
Coquimbo 2015 2.0617 2.7334
Additionally, a possible interpretation for the n parameter, according to our approach, is that it could possibly be related
to the system’s degrees of freedom, via energy releases during the time interval considered. According to the above, we could
relate the value of n with, for example, some parameter linked to active hypocenters, which could give us information about
the mechanism that generates earthquakes during the seismic activity. In fact, goodness-of-fit shows that n increases after
each main event.
Thus, the analytical connection between the model (12), presented in this paper, and the superstatistics representation,
in addition to the appropriateness of the error indexes (SSE, ARS and RMSE) delivered by the model fitting to selected data
samples, provide a solid support to the ideas expressed previously, in this section.

7. E. Sánchez C., P. Vega-Jorquera / Physica A 508 (2018) 305–312 311
Fig. 9. Iquique 2014, before the main event. m0=2.0, SSE=0.002915, ARS=0.9994, RMSE=0.008877.
Fig. 10. Iquique 2014, after the main event. m0=2.0, SSE=0.003519, ARS=0.9993, RMSE=0.009753.
Fig. 11. Isla de Pascua 2014, before the main event. m0=2.2, SSE=0.002245, ARS=0.9995, RMSE=0.00779.
Fig. 12. Isla de Pascua 2014, after the main event. m0=2.0, SSE=0.01034, ARS=0.9982, RMSE=0.01672.

8. 312 E. Sánchez C., P. Vega-Jorquera / Physica A 508 (2018) 305–312
Fig. 13. Coquimbo 2015, before the main event. m0=2.0, SSE=0.002035, ARS=0.9996, RMSE=0.007417.
Fig. 14. Coquimbo 2015, after the main event. m0=2.2, SSE=0.01544, ARS=0.9972, RMSE=0.02043.
Acknowledgments
E. Sánchez would like to recognize and show appreciation of the valuable comments and suggestions made by Sergio
Davis I. (CCHEN). Thanks to the Direction of Research of Development of the University of La Serena (DIDULS), and the
Post-graduate program of the Department of Physics and Astronomy of University of La Serena.
Data were obtained from the National Seismological Center of the University of Chile (http://www.sismologia.cl).
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