We study the main parameters of earthquakes from the perspective of the first digit phenomenon: the nonuniform probability of the lower first digit different from 0 compared to the higher ones. We found that source parameters like coseismic slip distributions at the fault and coseismic inland displacements show first digit anomaly. We also found the tsunami runups measured after the earthquake to display the phenomenon. Other parameters found to obey first digit anomaly are related to the aftershocks: we show that seismic moment liberation and seismic waiting times also display an anomaly. We explain this finding by invoking a self-organized criticality framework. We demonstrate that critically organized automata show the first digit signature and we interpret this as a possible explanation of the behavior of the studied parameters of the Tohoku earthquake.

With the advent of modern seismological and geodetical instrumentation, the
study of the earthquake process has experienced great advances, much of it
punctuated by the occurrence of giant earthquakes. Since 2004, three of these
spectacular events have occurred, each producing large tsunamis, followed by
human and material losses. These are the 2004 Northern Sumatra

Of the aforementioned events, the Tohoku earthquake is by far the best
recorded, at the local and global level. The Japanese and worldwide effort,
led by universities and public institutions, gathered a great bulk of
information regarding this event, most of it public. We use data and models
of this event to assess and establish a regularity of the source and later
events known as the first digit anomaly

The phenomenon consists in the nonuniform statistical distribution of the
first digit different from 0 present in a – usually large – population of
data

Benford's law probabilities

The first digit phenomenon has received considerable attention

As we pointed out, the first digit phenomenon has recently found some
applications in geophysics.

Leaving aside seismograms, we studied physical parameters closely related to the coseismic and postseismic processes. We choose to review data from the Tohoku earthquake mainly because of data quality. The data used come from direct measurements of earthquake effects and indirect estimations as well. Care was taken in regard to the statistical significance of the samples selected, as we left out interesting data with few samples.

Finite fault model from NEIC

In Table

Coseismic slip distribution

Postseismic slip distribution

As a summary, parameters closely related to the source process display Benford's effect, and those parameters include slip and moment distribution on the fault inverted from seismic data, surface deformation inverted from geodetic data, tsunami heights (possibly related to the source itself) surveyed directly and the GCMT aftershock series' moment release and waiting times.

As has been shown, the first digit anomaly appears in various variables
regarding the process of seismic rupture. The earthquake, now viewed not just
as the slip phase, contains this signature, and it seems natural to search
for a unique mechanism, which could explain the anomaly. Indeed, a model
capable of accounting for global features of earthquakes has already been
proposed, and it is known as self-organized criticality, SOC

A SOC state is a special equilibrium reached by extended systems that are
governed by nonlinear rules generally under dissipative conditions. This
regimen is characterized by power laws and fractal geometries. The existence
of various laws of this type in seismology, Gutenberg–Richer, Omori,
Båth and lately aftershock density distance decay

Selected events of the aftershock series, from 11 March 2012 until
31 January 2012, from the GCMT database

Runup data measured by

Sand pile one-dimensional cellular automaton

We tested two cellular automata, known to present very different behaviors:
the one-dimensional sand pile

In Table

In Table

There are other models that are more akin to model seismicity. One of the
more severe criticisms to the BTW model is the lack of aftershocks, a common
and well-established property of earthquakes. However,

Recent studies on OFC automata

Theoretical power law behavior of a natural system.

BTW two-dimensional cellular automaton

Concerning the actual relationship between earthquakes and SOC, we are
bringing new information to light. What we have learnt is that if SOC is the
underlying mechanism behind the complexity of earthquakes, revealed in power
laws, then its first digit imprint is translated into the main observables of
seismicity like the energy, displacements and tsunami runups. That
is the case of the earthquake source parameters presented. The
aftershocks are an interesting matter as it is believed that the
heterogeneous stress drop at the fault generates barriers, which at the end
generate the complex patterns found in aftershock series, with Omori's law as
one of the main characteristics

How far could this mechanism be pushed? Actually, the spectral analysis at
the core of the criticality (the so-called pink noise fingerprint) offers a
very general explanation. In Fig.

The first digit phenomenon has been taken for a simple mathematical property,
but it has proven to be hard to elucidate the true origins of it

The main properties seem to be (1) the stochastic nature of the earthquake phenomena under study, (2) a scale-independent mechanism, ranging in various orders of magnitude from short-period GPS source inversions to long-period seismic wave imaging, and (3) nonlinear laws of interaction powering the long-range correlations.

We are especially grateful to Armando Cisternas for the critical review of the first drafts. Special thanks go to Lily Seidman who made a thorough review of the writing. We also thank the editor for his many insightful indications. P. A. Toledo and S. R. Riquelme were partially supported by postgraduate Conicyt fellowships. J. A. Campos was partially supported by Fondecyt grant 1130636. Edited by: A. G. Hunt Reviewed by: two anonymous referees