Annals 47, 6, 2004

ANNALS OF GEOPHYSICS, VOL. 47, N. 6, December 2004

Short Note

Space-time combined correlation integral
and earthquake interactions

Patrizia Tosi (1), Valerio De Rubeis (1), Vittorio Loreto (2) and Luciano Pietronero (2)
(1) Istituto Nazionale di Geofisica e Vulcanologia, Roma, Italy

(2) Centro INFM di Meccanica Statistica e Complessità, Dipartimento di Fisica,
Università degli Studi «La Sapienza», Roma, Italy

Abstract
Scale invariant properties of seismicity argue for the presence of complex triggering mechanisms. We propose a
new method, based on the space-time combined generalization of the correlation integral, that leads to a self-con-
sistent visualization and analysis of both spatial and temporal correlations. The analysis was applied on global
medium-high seismicity. Results show that earthquakes do interact even on long distances and are correlated in
time within defined spatial ranges varying over elapsed time. On that base we redefine the aftershock concept.

Key words seismicity - correlation integral - fractal
dimension - clustering

1. Introduction

Seismicity appears to be scale invariant in
many of its aspects. Several papers (Kagan,
1994; Bak et al., 2002; Parson, 2002; Marsan
and Bean, 2003; Corral, 2004) investigate spa
tial and temporal correlations of epicentres, in
volving for example the concepts of Omori law
and fractal dimension. We think that the com
plex phenomenon of seismicity calls for an ap
proach capable of analysing spatial localisation

Mailing address: Dr. Patrizia Tosi, Istituto Nazionale di
Geofisica e Vulcanologia, Via di Vigna Murata 605, 00143
Roma, Italy; e-mail: tosi@ingv.it

and time occurrence in a combined way and
without subjective a priori choices. In this pa
per we introduce a new method of analysis that
leads to a self-consistent analysis and visuali
zation of both spatial and temporal correlations
based on the definition of correlation integral
(Grassberger and Procaccia, 1983).

2. Method

We define the space-time combined correla
tion integral as

2
c ( , ) = $C r τ

(N N - 1)
N -1 N

$ ! ! aΘ r̀ xi - x j j $ Θ `x ti -t j jk
i= 1 j = i + 1

where Θ is the Heaviside step function
( Θ(x) = 0) if x ≤ 0 and Θ(x) = 1 if x > 0) and the

Patrizia Tosi, Valerio De Rubeis, Vittorio Loreto and Luciano Pietronero

sum counts all pairs whose spatial distance
xi - x j # r and whose time interval t i - t j # x.

When applied over all possible values of τ or r,
the well-known correlation integral (Grassberger
and Procaccia, 1983) is returned. It results that
Cc(r, τ) is the generalisation of the correlation in
tegral for a phenomenon that explicates in di
verse dimensions with not comparable measure
ment units. When applied on seismicity Cc(r, τ)
takes into account the distribution of all time in
tervals and epicentral inter-distances between all
pairs of events, irrespective of the relationship
between the main event and any aftershock.

From the space-time combined correlation
integral we define the time correlation dimen
sion and the space correlation dimension for
sets of events within space-time distances r and
τ, respectively as

( , x)2 log C r
( , x) = cD rt 2 log x

and

( , x)2 log C r
( , x) = c .D rs 2 log r

If Cc(r, τ) was a pure power-law in both variables,
then Dt and Ds would correspond to the temporal

and spatial fractal dimensions, respectively. More
generally, the behaviour of Dt and Ds as a function
of r and τ will characterise the clustering features
of earthquakes in space and in time. This method
has been applied to global seismicity to study the
space-temporal correlation between earthquakes
all over the world. Data come from the catalogue
of the National Earthquake Information Center,
USGS, in the time period between 1973 and
2002, with magnitudes mb greater than 5. This
catalogue selection was conditioned by com
pleteness criteria and it presents medium to high
magnitude distribution.

3. Results

The space-time combined correlation inte
gral Cc for global seismicity is represented in
fig. 1 with black contour lines. The local slopes
of this surface in the direction parallel to the
time axis is the time correlation dimension Dt,
plotted in colours as a function of space and
time. The colour coding of each pixel quantifies
the time correlations existing between events
occurring within a given distance and time in
terval. Dt ≅ 1 corresponds to the random occur-

Fig. 1. Space-time combined correlation integral Cc (r, τ) (dark contour lines) and time correlation dimension
Dt (coloured shaded contour) for the catalogue of global seismicity.

Space-time combined correlation integral and earthquake interactions

Fig. 2. Same as fig. 1 for the reshuffled catalogue.

Fig. 3. The limit separating time clustering from time randomness (fixed to Dt = 0.8) as a function of distance
is shown. Points are fitted by the line of equation logr = − 0.55logτ +3.8.

rence of events, while a lesser value of Dt indi
cates time clustering. In fig. 1 two main do
mains appear: one at shorter inter-distances
with low Dt representing time clustering; the
other with Dt ≅ 1 indicating a random time oc
currence of events. The patterns observed in fig.
1 significantly support the hypothesis that
earthquakes are correlated inside some space
temporal ranges. In order to check this hypoth

esis we applied the same analysis to the global
catalogue after a reshuffling procedure. Reshuf
fling consists in mixing the time occurrence of
each event keeping fixed its epicentre coordi
nates. The characteristic of this procedure is
that of maintaining the separate statistical prop
erties of data. The results show (fig. 2) that all
patterns vanish, evidencing constant high val
ues of Dt at all distances and time intervals.

Patrizia Tosi, Valerio De Rubeis, Vittorio Loreto and Luciano Pietronero

To delineate a limit of the clustering resulting
from fig. 1, all the points with Dt = 0.8 were plot
ted on a separate figure (fig. 3). It is interesting to
see that, within the temporal ranges shown, the
clustering boundary can well be approximated by
a straight line on this log-log plot, thus indicating

a power-law behaviour. Using the least squares
fitting we obtained logr = − 0.55logτ +3.8. This
relation places a strong constraint on time rela
tions among events, evidencing how distance
plays a dynamic role. In particular, the relation
can be read as defining a temporal correlation

Fig. 4. Space-time combined correlation integral Cc (r, τ) (dark contour lines) and space correlation dimension
Ds (coloured shaded contour) for the catalogue of global seismicity.

Fig. 5. Same as fig. 4 for the reshuffled catalogue.

Space-time combined correlation integral and earthquake interactions

reaching long distances that quickly shrinks
over time following a power-law. For relatively
short spatial ranges (around 100 km) events are
time clustered and correlated for long time in
tervals (around 3 years). Over longer distances
time correlation lasts for a short period (less
than 30 days for 1000 km).

The local slopes of the surface of the space
time combined correlation integral, in the direc
tion parallel to the space axis, correspond to the
space correlation dimension Ds, plotted in
colours in fig. 4 as a function of space and time.
The colour coding of each pixel quantifies the
space clustering existing between events occur
ring within a given distance and time interval.
Ds ≅ 2 identifies a random distribution of earth
quakes, Ds ≅ 1 indicates that epicentres tend to
dispose along lines and Ds < 1 corresponds to
space clustering. Even in this plot different do
mains are easily recognised. At short distances, a
high space correlation dimension domain is clear
ly separated from space clustering ( 0 < Ds < 1)
that is present at greater distances: both condi
tions last for inter-time up to 100 days. The dis
appearance of clustering with time leaves room
to a general Ds ≅ 1, interpreted as the activity of
seismicity on plate boundaries. Even in this
case we tested the goodness of the results ap
plying the same analysis on the reshuffled cata
logue. The resulting plot in fig. 5 shows that the
single statistical properties of data are not suffi
cient to produce the clustering domains appear

ing in the combined approach of fig. 4, but a re-
al connection between space and time is needed.

Localisation errors certainly plays an im
portant role at short spatial ranges, generating
high values of Ds (fig. 4), but it appears that the
area with high space correlation dimension is
evolving with time calling for the presence of a
physical process. In particular, plotting in fig. 6
the points with Ds = 1, chosen as the limit sepa
rating random behaviour from space clustering,
it appear that in the shown ranges they follow a
straight line. Fitting with least squares we ob
tained the relation logr = 0.1logτ +1.2. The sep
aration line defines an area around each epicen
tre, slowly growing in time, within which seis
mic events are randomly distributed.

3. Discussion

Figures 1 and 4 show that earthquakes are
connected with each other in a non trivial way
and that a dynamic interaction appears when
space and time are analysed together with the
combined correlation integral. The results reveal
a statistical property of the global seismicity of
medium-high magnitude, but interpreting them
as an average behaviour of events after the oc
currence of each earthquake of magnitude
greater than 5, a possible scenario appears. The
term ‘aftershock’ can be redefined on the basis of
our findings: aftershocks are all earthquakes con-

Fig. 6. The limit separating spatial clustering from spatial randomness (fixed to Ds = 1.0) as a function of time
is shown. Points are fitted by the line of equation logr = 0.1logτ +1.2.

Patrizia Tosi, Valerio De Rubeis, Vittorio Loreto and Luciano Pietronero

nected to one reference event preceding them, as
revealed by their temporal correlation (low Dt).
In this sense all earthquakes occurred at a dis
tance from reference event less than the radius r,
defined by the relation logr = − 0.55logτ +3.8.
(where τ is the elapsed time, fig. 3), are after
shocks of that event. This aftershock region
reaches long distances from the reference
‘main’ event, but it quickly shrinks over time. If
compared to an homogeneous time distribution,
this area of influence can be interpreted as a re
gion of modified probability of earthquake time
occurrence. The epicentres of these connected
events tend to cluster in space, apart from the
near field (an area of radius 10-20 km), where
earthquakes are randomly placed. Even this
near field aftershock region has a dynamic
boundary, increasing slowly in size according
to the equation logr = 0.1logτ +1.2 (fig. 6). This
result is in agreement with other authors (Taji
ma and Kanamori, 1985; Marsan et al., 2000;
Helmstetter, 2003; Huc and Main, 2003) who
found a migration of aftershocks, defined with
classic methods, away from a main shock. This
migration is described in terms of a law
r ( ) + t

H , where d td t r ( ) is the mean distance be
tween main event and aftershocks occurring af
ter time t, with an exponent H < 0.5 correspon
ding to a sub-diffusive process.

4. Conclusions

In summary, we have introduced a new sta
tistical tool, the combined space-time correla
tion integral, which allows us to perform a si
multaneous and self-consistent investigation of
the correlation properties of earthquakes. This
tool leads to the discovery, visualization and
deep analysis of the complex interrelationships
existing between the spatial distribution of epi
centers and their occurrence in time. The analy

sis performed on the worldwide seismicity cat
alogue and the corresponding reshuffled cata
logue, strongly suggests that earthquakes of
medium-high magnitude do interact with each
other. This result led to a new definition of af
tershocks, as all earthquakes with non-random
occurrence with respect to the reference ‘main’
event, without considering their magnitude. Fi
nally the analysis revealed how the aftershock
region modifies over elapsed time.

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(received September 1, 2004;

accepted October 1, 2004)