%Comment on ``Do Earthquakes Exhibit Self-Organized Criticality?'' %
%\'Alvaro Corral %
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%\title{
%Comment on ``Do Earthquakes Exhibit Self-Organized Criticality?''
%}
%\author
%{
%\'Alvaro Corral
%}
%\affiliation{
%Grup de F\'\i sica Estad\'\i stica,
%Departament de F\'\i sica, Facultat de Ci\`encies, 
%Universitat Aut\`onoma de Barcelona,
%%Edifici Cc, 
%E-08193 Bellaterra, Barcelona, Spain 
%}
%\email{alvaro.corral@uab.es}
%\date{\today}

%\begin{abstract}
%{
%...
%}
%\end{abstract}

%\pacs{
%11.30.Er, 12.39.Fe, 11.15.Ha, 11.10.Gh
%}% PACS, the Physics and Astronomy
                             % Classification Scheme.

%\maketitle


%\section{Introduction}

\noindent
{{\bf Comment on ``Do Earthquakes Exhibit Self-Organized Criticality?''}}

\vspace{0.3cm}

In a recent Letter, %%published in this journal, 
Yang, Du, and Ma \cite{Yang}
study the interesting problem 
of the temporal structure of seismicity and its
relation with self-organized criticality (SOC).
Their main finding is that the reshuffling 
%%(or random rearrengement??) 
of earthquake magnitudes
changes the shape of the earthquake recurrence-time 
(or first-return-time)
distribution when the low-magnitude bound, $M_c$,
is raised.
Subsequently, they conclude that 
%%``earthquakes do not happen with completely random magnitudes''
it is not true that {\it an earthquake cannot ``know''
how large it will become}.
First, we show that this important implication is unjustified.



Yang {\it et al.} have in mind a fully uncorrelated
temporal point process with independent magnitudes as a picture
of SOC systems. 
%%(for which spatial degrees of freedom are intentionally ignored).
It is obvious, by construction, 
that %the properties of this model are not altered after
this model is invariant under
random rearrangements of the data; as
Yang {\it et al.} do not find this invariance in Southern California 
seisimicity they claim that  
``earthquakes do not happen with completely random magnitudes''
and therefore they are not a SOC phenomenon.
In fact, {\it the only conclusion that can be drawn from this is
that the seismicity time series is not uncorrelated}, and there
exists some dependence between magnitudes and recurrence times.
[This conclusion can be obtained directly, 
%%without the need of any reshufflings, 
from the fact that a scaling law exists for the 
recurrence-time distributions corresponding to different 
low-magnitude bounds, with a scaling function that is not a decreasing
exponential \cite{Corral04} (characteristic of a Poisson process, the only 
uncorrelated process which verifies a scaling law).]


%%But let us return to the core of Yang {\it et al.}'s results.
The existence of correlations %%in seismicity 
means that,
for a given event $i$, its magnitude $M_i$ may depend on
the magnitude of the previous event, $M_{i-1}$,
as well as on the backwards recurrence time, $T_i=t_i-t_{i-1}$, 
with $t_i$ and $t_{i-1}$ the time of occurrence of both events.
This dependence can be extended to previous magnitudes and 
recurrence times, $T_{i-1}, M_{i-2}, T_{i-2},$ etc.
But further, the recurrence time to the next event, $T_{i+1}$,
may depend on the previous magnitudes, $M_j$ 
and recurrence times $T_j$, $j \le i$.
The reshuffling of magnitudes performed in Ref. \cite{Yang}
breaks (if they exist) the possible correlations of $M_i$ 
with the previous magnitudes, as well as with the previous 
recurrence times, and the correlations of $T_{i+1}$ with the 
previous magnitudes (but not with the previous recurrence times). 
Therefore, any of the influences $M_{i-1}\rightarrow M_i$,
$T_i\rightarrow M_i$, %% or $M_{i-1}\rightarrow T_i$ 
or $M_{i}\rightarrow T_{i+1}$,
may be responsible of Yang {\it et al.}'s results.


The most direct way to test the dependence of a given variable,
in this case $M_i$, with another variable $X$, 
is to measure
the probability density of $X$ conditioned to different values
of $M_i$, $P(X|M_i)$, and compare with the unconditioned probability density
of $X$, $P(X)$. 
%%Equivalently, one may measure the probability density of $X$ conditioned
%%to the values of $M_i$, $P(X|M_i)$, and compare with $P(X)$.
This is what Fig. 1(a) displays, using $X=T_i$ and $X=T_{i+1}$
for the same data as Ref. \cite{Yang},
but restricted to periods of stationary seismicity
(otherwise, for strong aftershock sequences the recurrence times
are shorter and more sensitive to catalog incompleteness). 
As $P(T_i|M_i)$ remains practically unchanged for different sets of values of 
$M_i$, temporal causality leads to the conclusion that $M_i$ is independent
on $T_i$. 
In contrast, $T_{i+1}$ clearly depends on $M_i$, as $P(T_{i+1}|M_i)$ 
changes for different sets of values of $M_i$.
In other words, %as the Omori law suggests, 
{\it the larger the magnitude $M_i$, the shorter 
%%the forward recurrence time $T_{i+1}$, 
the time to the next event $T_{i+1}$, 
but the value of this time has no influence on the magnitude
of the event, $M_{i+1}$}.
On the other hand, 
%a similar analysis to test 
%the dependence of the magnitude on previous magnitudes
%%is analyzed in Fig. 1(b), where 
Fig. 1(b)
shows that $P(M_i|M_{i-1})$ turns out to be indistinguishable
from $P(M_i)$, ensuring the independence of $M_i$ and $M_{i-1}$, $\forall i$
if the $T_i$'s are restricted to be larger than 33 min
(shorter periods of time are not reliable, %%trustable %%%fiable??,
due to data incompleteness).
%We conclude, by extension, that $M_i$ is independent on the previous $M_j$ and $T_j$
%but $M_i$ influences the value of $T_{i+1}$, which is the explanation of 
%Ref \cite{Yang}'s results.
So, {\it when an earthquake starts, its magnitude is undetermined}
(at least from the information available at the catalogs),
whereas the time to the next event decreases when that magnitude 
turns out to be large.


A second, independent point to clarify
is the identification of SOC with the total absence of correlations.
It is true that the BTW sandpile model displays an exponential distribution
of recurrence times, but SOC is much more diverse than the BTW model.
For instance, the Bak-Sneppen model or the Oslo-ricepile model are 
two well recognized examples of SOC with totally different recurrence-time
distributions.
Finally, it is necessary to stress that the concept of SOC 
(as it happens with chaos) does not exclude the possibility
of prediction, as Ref. [17] of Yang {\it et al.} clearly showed.
So, nothing in Ref. \cite{Yang} is against the SOC picture
of earthquakes.

%The Ram\'on y Cajal program and the projects
%BFM2003-06033  (MCyT) and
%2001-SGR-00186 (DGRGC) are acknowledged.




\noindent
\'Alvaro Corral

\noindent
Departament de F\'\i sica, Facultat de Ci\`encies, 
Universitat Aut\`onoma de Barcelona,
E-08193 Bellaterra, Spain 


\today

PACS numbers:
%45.70.Ht,   % avalanches
%05.60.Cd,
%%05.40.-a,   % Fluctuations, random process
%%05.60.-k,   % Transport processes
89.75.Da,    % Complex systems: systems obeying scaling laws
91.30.Dk,    % Seismicity: space and time distribution
%91.30.Px,     % Seismology :[Phenomena related to earthquake prediction]
05.65.+b,    % self-organization % me lo quitan este!!!!!
64.60.Ht   % Dynamic critical phenomena


%%\newpage

\begin{thebibliography}{99} 


\bibitem{Yang}
X. Yang, S. Du, and J. Ma,
%Do Earthquakes Exhibit Self-Organized Criticality?
Phys. Rev. Lett. {\bf 92}, 228501 (2004).


\bibitem{Corral04}
A. Corral,
% Long-Term Clustering, Scaling, and Universality in the Temporal Occurrence of 
% Earthquakes
{Phys. Rev. Lett.} {\bf 92}, 108501 (2004). %;
%cond-mat/0409124.





\end{thebibliography}


%\newpage


\begin{figure}
\centering
%\includegraphics[width=3.5in]{Dtcond_mold_neicL360m50.ps}
%\includegraphics[width=3.5in]{Dtcond_m_neicL360m50.ps}
\includegraphics[width=3.5in]{fig1acomment.ps}
\includegraphics[width=3.5in]{fig1bcomment.ps}
\caption{
(color online)
(a) Probability densities $P(T_i|M_{i-1})$ and $P(T_i|M_{i})$
(shifted %one decade 
upwards) compared to $P(T_i)$.
(b) Probability density $P(M_i|M_{i-1})$
compared to $P(M_{i})$ with $T_i > 2000$ s.
\label{Dtcond}
}
\end{figure} 





\end{document}






cond-mat0412441.extracted_bib
\begin{thebibliography}{99} 


\bibitem{Yang}
X. Yang, S. Du, and J. Ma,
%Do Earthquakes Exhibit Self-Organized Criticality?
Phys. Rev. Lett. {\bf 92}, 228501 (2004).


\bibitem{Corral04}
A. Corral,
% Long-Term Clustering, Scaling, and Universality in the Temporal Occurrence of 
% Earthquakes
{Phys. Rev. Lett.} {\bf 92}, 108501 (2004). %;
%cond-mat/0409124.





\end{thebibliography}