Geophys. J. Int. (1996) 125,901-911
Multifractal analysis of the spatial distribution of
earthquakes in southern Italy
C. Godano,'t2 M. L. Alonzo' and A. Bottari'
' Istituto Geofisico e Geodetico, Universita' di Messina. via Osseroatorio 4,98100 Messina, Italy
Osservatorio Vesuviano, via Manzoni 249,80123, Napoli, Italy
Accepted 1996 February 2. Received 1996 January 22; in original form 1994 November 12
SUMMARY
We evaluate the complete spectrum of the generalized fractal dimension of the spatial
pattern of microearthquakes in Southern Italy, revealing a multifractal distribution
structure. O u r analysis is focused on the dependence of the multifractal distribution on
the size of the selected area and the kind of seismicity in the area. As the size of the
window varies, we observe that the capacity, information and correlation dimensions
vary significantly, while both d, a n d d - remain unchanged within their errors limits.
We interpret this result in terms of the observation that o u r data are mainly clustered
around a linear fault (the Sisifo fault). When we restrict the selected windows around
the fault, clustering around a line (the fault) is highlighted. The capacity dimension
changes from about 1.8 to about 1.4 a n d the correlation dimension decreases because
we observe in detail the clustering of the seismicity along the fault, which approximates
the maximum intense clustering of the whole data set. Although our results are strongly
influenced by the fact that the d a t a are dominated by the epicentres located on the
fault, we can conclude that multifractal analysis can be a very useful tool to discriminate
the seismicity linked to a particular fault in a given area.
Key words: faulting, fractals, seismicity.
INTRODUCTION
It is widely observed that earthquakes occur around seismic
faults and tend to cluster close to these structures. The fractal
nature of the spatial distribution of earthquakes was first
demonstrated by Kagan & Knopoff (1980) and Sadvskiy et al.
(1984). The fractal structure of the spatial distribution of rock
fractures in a laboratory experiment was also observed by
Hirata, Satoh & Ito (1987). This observation implies that the
complete description of scaling properties of earthquakes can
be carried out by means of the fractal dimension of the spatial
distribution of earthquakes and two other parameters related
to their energy and temporal evolution: the b value of the
Gutenberg-Richter distribution (Gutenberg & Richter 1942;
the physical meaning of this distribution was investigated by
e.g. Kagan & Knopoff 1981; Main & Burton 1984; Scholz
1988; Rundle 1989), and the power-law exponent of the Omori
law (Omori 1894; Utsu 1961). The connection between this
relationship and the fracture mechanics was studied by, among
others, Scholz (1968); Mikumo & Miatake (1979); Yamashita
& Knopoff (1987).
A negative relationship between the correlation dimension
of the spatial distribution of earthquake epicentres and the b
value of the Gutenberg-Richter distribution was found by
Hirata (1989). This relationship is not surprising; in fact, the
0 1996 RAS
b value is related to the fractal dimension of the rupture length
distribution by the relation D,=3b/c, where c is the slope of
the relationship between the log of seismic moment and the
magnitude. It must be noted that this relation was derived by
Aki (1981) using the hypothesis of self-similarity, which means
that the seismic moment is proportional to the rupture length.
Of course, the negative correlation found by Hirata does not
contradict the relation found by Aki, because the fractal
dimension of rupture length and the fractal dimension of the
spatial distribution describe scaling properties of two different
variables that depend on one another in a manner described
by the experimental relation found by Hirata (1989) (for a
more detailed discussion of this dependence see Main 1992).
The negative correlation can easily be explained by taking into
account the fact that the fractal dimension calculated by Hirata
is related to the global degree of spatial clustering. An increase
of the b value means that the relative number of low-energy
earthquakes is increased with respect to the number of highenergy earthquakes. In general this implies that when a seismic
sequence of aftershocks occurs, the global degree of spatial
clustering is increased, causing a decrease of the fractal
dimension.
As is well known, the fractal dimension is not sufficient to
describe all the scaling properties of a measure. In fact, if the
measure does not fill the available space homogeneousiy, we
901
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C . Godano. M . L. Alonzo and A. Bottari
need a complete spectrum of dimensions to characterize the
scaling properties of the measure (Paladin & Vulpiani 1987)
and we call our distribution multifractal.
Hirata & Imoto (1991) and Hirabayashi, Ito & Yoshii (1992)
showed that the spatial distributions of earthquakes in the
Kanto region of eastern Japan, in California and in Greece
have a multifractal structure. Hirata & Imoto (1991) discuss
in detail the observed values of the dimensions, which range
from a value of 2.2 for the fractal (capacity) dimension to a
value of 1.7 for D,, which indicates the maximum intense
spatial clustering of earthquakes. Using such a result they
suggested that the multifractal structure of spatial distribution
of hypocentres and of energy release could be an indication
that earthquakes are a turbulence phenomenon, so that the /l
model or the random model could be good descriptions of
the earthquake generation process. In the wake of these ideas,
Ito (1992) hypothesized that a modified version of a selforganised critical cellular automaton is a good model for
earthquakes (see also Ito & Matsuzaki 1990).
Eneva (1994) observed that the multifractal distribution of
earthquake epicentres could be influenced by sampling bias.
She found a multifractal distribution, even for a monofractal
set of data generated synthetically. We note, however, that this
result originates from an artefact inserted by Eneva in her
synthetic data, as she eliminated the points that were not close
to real epicentres from the synthetic data set. In other words,
she inserted a multifractal rule into a randomly generated set
of data. O n the other hand, the multifractal structure of the
epicentre distribution can easily be recognized by examining
the data: they do not fill the available space homogeneously,
but the degree of clustering varies locally.
The fractal structure of hypocentre distributions suggests
the idea that DF could be a good precursor parameter for
earthquakes. In fact, changes in spatial clustering degree could
occur before an earthquake (sometimes foreshocks tend to
cluster around the nucleation point of an incoming large
earthquake). Rossi (1990) observed a temporal change in the
fractal dimension of the spatial distribution of hypocentres in
Friuli, Italy, indicating that on an intermediate time-scale DF
decreases before large earthquakes, while the generally
observed decrease of the fractal dimension after the main shock
reflects the high clustering degree of aftershock sequences. A
different result was obtained by De Rubeis et al. (1993), who
found that a maximum value of the fractal dimension precedes
major events and often it sharply decreases just before the
earthquake. This last result seems to be in agreement with that
obtained by Ouchi & Uekawa (1986), who observed a clustering on small scales (7-20 km) just before a strong earthquake, and a clustering on a larger scale (50-100 km) during
the aftershock sequence, while a random distribution seems to
characterize other periods.
In the present paper we calculate the complete spectrum of
dimensions of the spatial distribution of the seismicity of
Southern Italy. Our aim is to investigate the dependence of
the fractal structure on the size of the selected area and on the
kind of seismicity in the area. In fact, if the distribution of
earthquake epicentres is multifractal, we should observe varying scaling properties as we restrict the analysis to windows of
a certain size. This should be a good test for genuine multifractality. We hope that this kind of analysis will be useful in
quantitatively recognizing active seismic faults when they are
not clearly recognizable in any other manner.
DATA
Our initial data set consists of about 400 seismic events
recorded at the local network of the Geophysical Institute of
Messina University during the period 1978-1985. The choice
of this period is linked to the homogeneity of the seismic
network (the number and quality of the seismic stations
changed in 1978 and 1985). Arrival times were integrated with
those of the ING national network and the local network of
the Eolie Islands and Etna volcano. We located the earthquakes
with a standard HYPO 71 program using the velocity model
proposed by Bottari et al. (1979) and shown in Fig. 1. Only
seismic events with more than nine phases were retained for
the location.
After the location we selected earthquakes with a horizontal
error of less than 10 km. As the average depth of the selected
earthquakes (about 300) was 14.95 km and the errors in depth
were too large, we limited our analysis to two dimensions.
However, the seismicity is so shallow (the deepest earthquakes
are located at less than 30 km depth) that the third dimension
is negligible with respect to the other two, thus it should not
influence the results obtained for the 2-D case.
The locations obtained are mainly clustered into three
different zones: Etna volcano, the Eolie Islands (Sisifo fault),
and the Messina Strait. This observation encouraged us to
perform a multifractal analysis; in fact, the multifractal distribution of epicentres can be easily recognized simply by looking
at the location map (Fig. 2), where we clearly observe a
different clustering degree for the different zones.
As stated in the Introduction, we wish to investigate how
the multifractal spectrum is influenced by the size of the
selected area and by the kind of seismicity in the zone. In
particular, we would like to show that our analysis could be
useful in recognizing seismic faults. Our data set shows the
Sisifo fault as the most active fault in the area, so we separate
our data into five groups, each group more restricted around
the fault than the previous group. Of course, we can easily
select a set of data clearly aligned on the Sisifo fault, and we
do not need the multifractal machinery to isolate such a data
set, but we can use this approach for testing the opportunity
of using multifractal analysis to indicate the seismic activity
associated with a certain fault.
The first window was composed of all available data (303
earthquakes) located in the spatial window shown in Fig. 2,
and the second is a selection of the first limited to the Arc0
Calabro Peloritano and Eolie Islands, composed of 210 seismic
events (Fig. 3). The 199 earthquakes of the third group (Fig. 4)
reduce the second sector to the Messina Strait and Eolie
Islands. The other two windows are selected just around the
Sisifo fault and are composed of 123 and 90 earthquakes,
respectively (Figs 5 and 6 ) . The main difference between the
last two sectors is in the lengths of the smaller sides of the
windows (70.7 and 28.7 km, respectively). The windows are
progressively restricted around the Sisifo fault while trying to
preserve a tectonic homogeneity within the window.
Two other windows were chosen to restrict the analysis to
the Messina Strait area (note that the minimum number of
phases used for the location selection criterion is seven), where
a very dense cluster of events is evident from the location map.
The two windows differ only in their sizes (Figs 7 and 8).
0 1996 RAS, GJI 125, 901-911
Spatial distribution of earthquakes, S. Italy
903
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W
5
!3
a
-20
-30
40
2
3
4
5
6
7
8
9
velocity (MS)
Figure 1. The velocity model used in locating earthquakes.
.
7
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1
Figure 2. Location map of our data set (sector 1). The square indicates the selected area.
0 1996 RAS, GJI 125, 901-911
.
. .
:
. .
!
:
I
-
I
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C . Godano. M . L. Alonzo and A . Bottari
Figure 3. The first selection of the located earthquakes (sector 2).
-
.
. .
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. .
. .
-
- -
-'-
- -
-
-
-
-
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-
.
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-
-
-
-
-
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Figure 4. A narrower window restricted to around the Eolie Islands and the Messina Strait (sector 3).
0 1996 RAS, G J I 125, 901-911
Spatial distribution of earthquakes, S. Italy
Figure 5. A first restriction of the window around the Sisifo fault (sector 4).
I
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Palinu
i
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(0
Figure 6. A narrower window around the Sisifo fault (sector 5 )
0 1996 RAS, G J I 125, 901-911
905
906
C . Godano. M . L. Alonzo and A. Bottari
_ _ -
MCEND
t,S I
I
1
Figure 7. Location map of the Messina Strait area events.
Figure 8. A narrower window for the same data set shown in Fig. 7
0 1996 RAS, GJI 125, 901-911
Spatial distribution of earthquakes, S . Italy
CALCULATING THE MULTIFRACTAL
SPECTRUM: METHOD
As stated in the Introduction, the spatial distribution of
earthquakes can be considered multifractal. In fact, to describe
all the scaling properties, we need a complete set of generalized
dimensions. It is well known that the fractal dimension
describes only the topological scaling properties of a measure.
It can be defined as the capacity measure considering the
number N(1) of hypercubes of edge 1 necessary to cover an
object embedded in a D-dimensional space: N ( l ) a l - D F . In
other words, if we have a probability measure, dp, we can
define a probability density, p i ( l ) = j d p , as the 'mass' of the
hypercube Ai of size 1 with i = 1, 2, ..., N ( I ) . If the distribution
is a uniform one, p ; = I / N ( l ) a l D F , and we shall say that the
object is fractal. On the other hand, if the distribution is
singular with respect to pf, (pi(l)/pf diverges in the limit I-+O),
we have an inhomogeneous fractal (multifractal), and the
probability density will scale as:
(p i ( l ) q ) =
1
pi(l)q+
aF d q
+
1
.
I
,
The d,, are the generalized dimensions, and if they are equal
as q varies we have a homogeneous fractal, while if they vary
as q changes, we have a multifractal (Fig. 9). In the region of
low q the dimensions follow the relation d , , = d o - p q (Paladin
& Vulpiani 1987), from which it is possible to estimate do.
In order to calculate the probability density we can use the
box-counting method, but it requires much computer memory,
so here we use the method of Grassberger & Procaccia (1983).
They found the following:
[C(I,q ) is the statistical moment of order q of the correlation
integral], where 0 is the Heaviside function, n is the number
of points, and the modulus is the Euclidean one. If we call p(q)
\
multifractal
_ _ _ homogeneous fractal
-
9
Figure 9. A typical pattern of the dimension spectra in the case of a
homogeneous fractal (dashed line) and an inhomogeneous fractal
(solid line).
0 1996 RAS, GJI 125, 901-911
907
the slope of C(1, q ) versus I on a log-log scale (which can be
calculated by a simple linear fit), it is possible evaluate the
d,,,:
The slope has to be determined in the scaling region where
the probability distribution is a power law.
The most interesting dimensions are the fractal dimension,
the correlation dimension, and the information dimension,
which are obtained, respectively, for q= -1, 0 and 1. The
physical meaning of the first is obviously a capacity measure,
the information dimension is linked to the scaling rate, while
the correlation dimension is related to the global clustering
degree (Paladin & Vulpiani 1987). Two other interesting
quantities are d , and d-,, which indicate, respectively, the
most and the least intensive clustering of the distribution.
RESULTS A N D DISCUSSION
The most relevant problem in determining the fractal structure
is to establish the scaling region of the correlation integral
C(t, 4). The lowest limit is determined by the location error;
in our case the maximum error was 10 km, while the average
error was about 5 km. We decided to select as the lower limit
of the scaling region a value in between the average and the
maximum error. The uppermost limit of the scaling region was
obviously decided on the basis of saturation (80 km in our
case). All the lines provide a very good linear fit (Figs IOa-d
show some examples for sectors 2 and 5 for q = 1 and q =
kO.4) with a correlation coefficient >0.95 and very low errors,
except for the negative values of q in sectors 1 and 2 (Figs
lla-g). The large error in these sectors could be due to
boundary effects. For negative q we are studying the sparser
seismic events, which have a distance from the border of less
than 80 km, so we may be underestimating the probability
density at high value of 1.
To avoid such a problem we have calculated the density
function considered in the Grassberger & Procaccia correlation
integral, using the distances between earthquakes of the whole
catalogue (sector 1) and the events of the analysed sectors (the
narrower windows). Of course, a suitable normalization was
performed for each window. Obviously, when we analyse the
sector 1 data the border effects cannot be eliminated due to
the absence of external earthquakes. We note that such a
procedure allows a significant reduction of border effects only
for the narrowest windows, while such effects are only
smoothed for sectors 2 and 3.
Figs ll(a)-(e) show the dimension spectra for the different
sectors (excluding the two from the Messina Strait area). It is
evident that there are no significant differences in either d - ,
or d , in all sectors, while there are significant changes (about
15 per cent) in the fractal, information and correlation dimensions when we pass from sectors 1, 2 and 3 to sectors 4 and 5.
As can be seen from Table 1, the fractal dimension is close to
2 for sectors 1, 2 and 3, where we have a sparse seismicity
throughout, while it decreases to about 1.4 for the other two
sectors, where the seismicity is clustered around the Sisifo
fault. Similarly, the information and correlation dimensions
also decrease. In the second case this diminishing is related to
the increase of global spatial clustering in sectors 4 and 5 due
to the fact that our analysis is, for these sectors, performed on
908
C. Godano, M . L. Alonzo and A. Bottari
SECTOR 5
@+?Q
I
10
100
L
IM.0
7
SECTOR2
3,
~
0 q=+l
10
1
1
100
10
L
Figure 10. The correlation integrals for: (a) sector 2, q= kO.4 (b) sector 5, q= kO.4; (c) sector 2, q=
Table 1. The most significant dimension values for the different sectors. do is the fractal dimension, d, is the information dimension, d, is
the correlation dimension, d-, is related to the less intense clustering
and d , is related to the most intense clustering.
dii
di
d?
d-,
d-
SEC I
172iOl.5
1.49f0.07
SEC 2
1.8Oi-O.li
1.44frO.lI
1.28k0.07
I 2 1 k007
2.01 k0.60
0.97i0.07
1.62f0.92
0.9820.07
SEC 3
1.73i0.17
1.42k0.11
1.20f0.08
2.20f0.21
0.97+0.07
SEC 4
1.49f0.20
1.19k0.10
1.05fOO8
2.19f0.28
0.91 i 0 . 0 7
SEC 5
1.39iO.II
1.17+0.07
1.03fr0.06
2.0620.30
0.89k0.07
narrower windows around the fault, so that the global clustering is very close to the maximum intense
clustering (d,
The Messina Strait area requires a separate discussion
because, as shown in Figs ll(f)-(g), seismic events here are
space filling. In fact, the spectrum of dimensions for these data
exhibits a homogeneous fractal behaviour and the generalized
dimensions can be considered as equal within error. Such a
result does not change as the window is restricted. Before
concluding, we note that the fractal dimension (2.3 for both
the windows in this area) is greater than the topological
dimension. This may suggest that we have overestimated the
k 0; (d) sector 5, q = L- 1.0
dimensions, although they remain equal to the topological
dimension within error.
CONCLUSIONS
We have calculated the multifractal spectra for the spatial
distribution of earthquakes in southern Italy for five windows
of different sizes around the Sisifo fault (see Figs 2-6) and for
two other windows in the Messina Strait area (Figs 7-8). Our
result for the first five windows show that there are no
significant changes in either d-, or d,, which take a value of
about 0.95 and 2.0, respectively, for all sectors. Such values
indicate that the least intense clustering tends to fill the
embedding space, while the most intense clustering tends to
define a line that can be identified as the Sisifo fault.
The fractal dimensions of sectors 4 and 5 are smaller by
about 18 per cent (1.7% 1.45) compared to the other three
windows. This result indicates that the data set has the
'capacity' to fill almost all the space for the larger sectors,
while earthquake epicentres tend to cover only the fault line
for the narrower windows. This is typical of multifractal
distributions; in fact, for such distributions, the scaling proper0 1996 RAS, G J I 125, 901-911
Spatial distribution of earthquakes, S. Italy
-2
4
0
2
4
6
6
SECTOR 3
-&
a
i
I .6
T
909
SECTOR 4
1
7--1----6
9
4
-2
0
2
4
6
9
Figure 11. The dimension spectra for (a) sector 1, (b) sector 2, (c) sector 3, (d) sector 4, (e) sector 5, ( f ) Messina Strait area (larger window) and
(g) Messina Strait area (narrower window).
ties are a function of the point where the analysis is performed.
The observed variations of fractal and correlation dimensions
confirm that our results are due to genuine multifractality.
The close agreement of the correlation dimension to d , for
sectors 4 and 5 indicates that the global clustering degree of
seismic events in these windows is almost equal to the most
intense clustering.
The monofractality observed in the Messina Strait area
could be due to its geological structure, which is dominated
by the presence of a graben with randomly distributed faults.
The monofractal distribution is confirmed by the independence
of the result with respect to the window size.
The difference observed between the seismicity of the Sisifo
fault (multifractally distributed) and that of the Messina Strait
area (monofractally distributed) confirms our hypothesis that
0 1996 RAS, GJI 125, 901-911
the multifractal analysis of earthquake spatial distributions is
a good tool for characterizing seismic faults. Where the seismicity is dominated by a single fault or a network of faults,
the structure of the earthquake density is multifractal due to
the inhomogeneous spatial clustering of earthquakes around
faults, whilst where a dominant system of faults is absent a
random or monofractal distribution will be observed. The
values of fractal dimensions confirm this hypothesis; in fact,
we found that seismicity tends to fill an almost linear space
when we restrict the window size around the Sisifo fault. In
the Messina Strait area the fractal dimension indicates that
data are overfilling the available space.
In concluding, regarding the Sisifo fault seismicity, we note
that our data set is not the best for identification of seismic
structures associated with randomly distributed seismicity
910
C . Godano, M . L. Alonzo and A. Bottari
I (d
- __ -
i
2.4
__
-~-
SECTOR 5
i
MESSINA STRAIT 1
2f
2.0
1.5
1.2
4
10
4
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-2
2
4
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i
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6
9
Figure 11. (Continued.)
because the random part of epicentre distribution is not
significant with respect to the clustering of epicentres around
the fault. Anyway, the closeness of d2 and d, is so clear when
we select narrow windows around the fault that we can say
that the suggested method is valid.
ACKNOWLEDGMENTS
We would like to thank the technical staff of the Istituto
Geofisico e Geodetic0 of the Messina University very much
for their assistance in locating earthquakes.
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