Time and space analysis of two earthquakes in the Appennines (Italy)

doi:10.4236/ns.2011.39101

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Vol.3, No.9, 768-774 (2011) Natural Science

http://dx.doi.org/10.4236/ns.2011.39101

Time and space analysis of two earthquakes in the

Apennines (Italy)

Michele Caputo1,2, Giovanni Sebastiani3,4*

1Physics Department, Sapienza Università di Roma, Roma, Italy; [email protected]

2Department of Geology and Geophysics, Texas A & M University, College Station, USA;

3Istituto per le Applicazioni del Calcolo “Mauro Picone”, CNR, Roma, Italy;

4Mathematics Department, Sapienza Università di Roma, Roma, Italy; *Corresponding Author: [email protected]

Received 28 July 2011; revised 30 August 2011; accepted 15 September 2011.

ABSTRACT

In this paper, we study two earthquakes: the

April 6th 2009 earthquake of L’Aquila in the re-

gion of Abruzzo (Italy) and the 1997 Colfiorito

earthquake in the regions of Umbria and Marche

(Italy). The data sets of these two earthquakes

were analysed in both time and space domains.

For time domain we used statistical methods

and models both parametric and non-parametric.

Concerning the space domain, we used Mathe-

matical Morphology filters. The time domain

analysis provides evidence of a possible corre-

lation between seismic activities and the tides of

the crust of the Earth. The results obtained show

evidence that the daily number of earthquakes

of the sequences proceeding and following the

April 6th 2009 earthquake of L’Aquila and that of

the sequence following the 1997 Colfiorito earth-

quake have a periodic component of occurrence

with period of about 7 days. It seems that the

maxima of this component occur at a position of

the Moon with respect to the Earth and the Sun

corresponding to approximately 3 days before

the four main Moon phases. The space domain

analysis indicates that the foreshock activity in

both earthquakes is clustered and concentrated.

Furthermore, in each of the two earthquakes the

clusters are located at about 3 kilometers from

the epicentre of the main shock.

Keywords: Earthquakes; Earth Tides; Periodic

Modeling; Spectral Analysis; Mathematical

Morphology; Foreshocks; Aftershocks

1. INTRODUCTION

In this paper we face two issues. The first concerns the

possibility of detecting the effect of the solid Earth tides

for the triggering of earthquakes, which we approach

with the ansatz that it should be easier to detect the effect

of a small perturbation on a system when the system is

unstable. This condition is verified for both foreshock

and aftershock sequences. The second issue focuses to

contribute to find possible forerunners. To this aim, we

analysed the foreshock activity in space domain.

For many decades, especially to the purpose of predi-

cting earthquakes, it has been speculated that the stress

generated by the solid Earth tides may trigger earth-

quakes. From a theoretical point of view the scientists

generally considered that the stress generated by the tides

in the crust of the Earth could be a direct cause of the

triggering, but this is so small that is seemed unlikely

that it may help triggering the earthquakes in a perfectly

elastic or anelastic Earth crust.

The possibility that solid earth tides may accelerate the

preparation process of the earthquakes and possibly trig-

ger them was first suggested long time ago by Schuster

[1,2]. Since then, a large number of attempts to prove the

correlation between the tides and the occurrence of earth-

quakes have been made and a plethora of papers pub-

lished. A good survey of this literature is in the note of

Métivier et al. [3], who also gave a proof of the existence

of this correlation. In fact, they considered a set of 442,

412 earthquakes and observed a clear correlation be-

tween the phase of the diurnal and semi-diurnal solid

Earth tides and the time of occurrence of earthquakes.

These authors show that the correlation is larger with

shallow earthquakes and that earthquakes occur slightly

more often when the Sun and the Moon are in oppo-

sition.

Atef et al. studied a set of 443,428 earthquakes and

found the existence of two promiment spectral peaks

with periods of one and seven days, respectively [4].

They suggested that the cyclic nature of the cultural

noise level leads to the observed periodicities.

M. Caputo et al. / Natural Science 3 (2011) 768-774

769

For the time domain analysis, we note that the sets of

events which we considered is limited in a central Appe-

nines region with linear length of some tens of kilo-

meters. Therefore, the earthquake events in this region

have approximately the same phase in the Earth tide

effect. The spectral analysis of time series resulting from

the observation of components which have the same fre-

quency and different phases may fail to detect signals of

small amplitude for possible interference. In our case,

due to the limited size of the region where each of the

two earthquakes studied occurred, this problem is irre-

levant.

When considering the statistics of earthquakes occurred

in a very limited region and recorded with modern intru-

mentation, including the smallest observable instrumen-

tally, in order to have a reasonable number of events per

time unit, the number of events is usually counted in

events per day, as done here. We search for possible

periodicities in the daily number of earthquakes proceed-

ing and following the April 6th 2009 earthquake of

L’Aquila in the region of Abruzzo (Italy) and also in the

sequence following the 1997 earthquake of Colfiorito in

the regions of Umbria and Marche (Italy).

We assume that the crust of the Earth, in the seismic

regions, has a large number of small faults with different

orientations [5]. For instance, the faults with area A100

are 5000 times more numerous than those with area A [6].

This is supported by the observations of Wallace in

California [7]. There is therefore a non negligible pro-

bability that the tidal stress field meets a fault with the

appropriate orientation to be effected in the direction to

increase the existing tectonic field. If the stress acting on

this fault is near the value which may trigger an

earthquake then the tidal field may possibly trigger the

earthquake.

2. METHODS OF ANALYSIS

2.1. Time Sequence Analysis

We performed two kinds of analysis of the measured

time sequences: parametric and non-parametric, as des-

cribed below.

Let us first consider the measured foreshock sequence



M of the number of events happened during each day

i, from day

b until day

e, within a given spatial

region. This sequence was described here mathematically

by means of a parametric mixture model with an unkown

number of Lorentian components















 (1)

First, we determined the number p of components of

the model. This was done by testing the hypothesis that

there is a significant positive difference of the derivative

discrete approximation

  





2* 2

fff

iii

NNN



 . The

distribution of this quantity under the null hypothesis of

no difference was approximated by the empirical

distribution of the same quantity in a time interval of

days before

b, where there is no apparent activity. The

values of the parameters in Eq.1 were found by mini-

mizing the sum of squared differences between the mea-

sured and the theoretical values

 



eff

ib MN



The period of the sequence was estimated here on the

basis of the average difference









 be-

tween consecutive peak locations 1kk



,

=1, ,1.kp





The measured sequence



M of the number of daily

events after the main shock from day a

b to day a

was analysed both parametrically and non-parametrically

to detect the presence of periodicities. In both cases, first

a parametric power-law function was used for modeling

the number of daily events



NCit





 (2)

where 0

t, C and



are real parameters.

Also here the minimization of the sum of squared

differences between



M and



N was adopted to

estimate the model parameters. Based on the estimated

model and the measured sequence, their residual diffe-

rence sequence



R was computed. The residual diffe-

rence sequence was from one side analysed parametri-

cally by modelling



R as sum of a single component

sinusoidal model and zero mean uncorrelated errors;

amplitude, frequency and phase were estimated by mini-

mizing the sum of squared errors.

The non-parametric analysis of



R was performed

as follows. In 1987, Jaynes [8] showed that the mode of

the periodogram, i.e. the mode of





, time average

of the squared absolute value of the discrete Fourier

transform of a sampled signal, introduced by Schuster

about eighty years before, is a sufficient statistics for the

frequency of a single component periodic signal. Closely

related to the periodogram is the power spectral density

of a stationary stochastic process



Rt , i.e. the Fourier

transform of its autocorrelation function

 

=e d

PERtRt







 

 





 (3)

where we assumed that the process auto-correlation fun-

ction only depends on the time lag



, i.e.













=ERtRt













. The relation between the pe-

riodogram and the power spectral density is

 

,EC P







 (4)

when the time window T goes to. Instead of

M. Caputo et al. / Natural Science 3 (2011) 768-774

770

obtaining the periodogram by calculating the discrete

Fourier transform of data, we have preferred to use the

(sample) power spectral density. This is motivated by the

fact that we could easily cope also with the case where the

residual sequence is modeled by a sum of pure periodic

signals with stochastic amplitudes and phases. Then, to

make consistent the periodogram, it will be sufficient to

weight properly the auto-correlation when calculating its

discrete Fourier transform. The auto-correlation k



lag =1, ,=1

aaa

kneb was estimated by





 







eaaaaa

iik

keaaa

RRRR











(5)

where



R denotes the sample mean and the sequence

was periodically extended from the original one:

 

 (6)

=1,, a

in. The power spectral density was then calcu-

lated for a finite number of frequencies by means of the

discrete Fourier transform of the sequence k



=1,, a

kn.

2.2. Spatial Analysis

Our focus here is to study the distribution of epicentres.

Among the many different approaches available (see e.g.

Caputo and Postpischl 1974, [9]), to have a more rigo-

rous definition of the clustering, we adopted the one of

Mathematical Morphology [10]. This approach was deve-

loped in the applied field of Mineralogy and is based on

topological and algebraic properties of images. In parti-

cular, we used a basic operator, i.e. the opening. This

consists of the application in sequence of first the erosion

operator followed by the one of dilation. We first dis-

cretized the geographical region in a regular lattice





,,,=1,,ij ijm. Let ij

N be the total number of

events happened in the cell





,ij within a given time

interval. We reduced the data into binary codes, in which

1 indicates that there is at least one event in the cell and 0

otherwise. Concerning the application of the erosion

operator, given the binary image ij

N, a spatial cell that

is qualified 1, is requalified zero when at least one of its

eight neighbouring cells is not qualified 1. Instead, the

dilation operation consists of qualifying 1 the neigh-

bouring cells of each cell qualified 1. The opening opera-

tion produces an image with smooth contours and it elimi-

nates isolated cells qualified 1 or small clusters of them.

3. RESULTS

3.1. Data Description

We analysed two data sets, both recorded by the Isti-

tuto Nazionale Geofisica e Vulcanologia (I.N.G.V.).

The first data set refers to the earthquake in L’Aquila

(Italy) with main shock in 6 April 2009. The sequence of

events corresponds to a spatial region composed by three

circles. The circle centres are the epicentres of the three

events with magnitude larger than 5. These three events

happened in days 6, 7 and 9 April 2009, and the epicen-

tres longitude and latitude were (+13.334, +42.334),

( 13.464,42.275)



 and ( 13.343,42.484) , respecti-

vely. The diameter of the three circles was about 33 km.

We computed the sequence of the daily number of events

happened in the spatial region described above. The first

day was 1 December 2008 and the total number of days

was 298. In Figure 1, the sequence of the daily number

of events is shown. The spatial distribution of the total

number of events in the time interval starting from day

52 to day 80 corresponding to the foreshock activity is

shown in Figure 2. We used a 64 64 lattice of cells

each of about 0.8 km length.

Figure 1. L’Aquila: the daily number of events.

Figure 2. Aquila: the spatial distribution of epicen-

tres during foreshock activity. The image is 64 ×

64 with a pixel length of about 0.8 kilometers. The

North direction is ↑.

M. Caputo et al. / Natural Science 3 (2011) 768-774

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The second one refers to the earthquake in Colfiorito

(Perugia, Italy) with main activity in September-October

1997. Also here, we computed the sequence of the daily

number of events happened in the spatial region des-

cribed below. The first day was 26 August 1997 and the

total number of days was 81. We analysed a sequence of

events within a circular region with centre the epicentre

of the event with maximum magnitude, that happened in

26 September 1997 and whose longitude and latitude

were ( 12.891,43.023) . The diameter of the circle

was about 50 km. We show the sequence of the daily

number of events in Figure 3. The spatial distribution of

the total number of events in the time interval starting

from day 7 to day 25 corresponding to the foreshock

activity is shown in Figure 4. Also here we used a

64 64 lattice of cells each of about 0.8 km length.

Figure 3. Colfiorito: the daily number of events.

Figure 4. Colfiorito: the spatial distribution of

epicentres during foreshock activity. The image is

64 × 64 with a pixel length of about 0.8 kilo-

meters. The North direction is ↑.

3.2. L’Aquila Earthquake Data Analysis

We first analysed the measured foreshock sequence.

Here the analysis was based on a parametric modeling,

as described in Section 2. The result is shown in Figure

5. In the same figure, the estimated mixture model is

shown. We note the clear presence of four peaks sepa-

rated in average by a time interval of 7.1 days.

We turn now to the analysis of the sequence after the

main shock. This was performed based on either the total

after main shock sequence or a final part of it. In Figure

6, the total after main shock sequence is shown superim-

posed to the power law fitting. The residual difference

between the measured and theoretical sequences is plo-

tted in Figure 7, and its power spectral density is shown

is Figure 8. We note the main peak at a frequency corre-

Figure 5. L’Aquila: the daily number of events in the foreshock

sequence.

Figure 6. L’Aquila: the daily number of events in the after

main shock sequence.

M. Caputo et al. / Natural Science 3 (2011) 768-774

772

Figure 7. L’Aquila: the residual daily number of events in the

after main shock sequence.

Figure 8. L’Aquila: the power spectral density of the residual

daily number of events in the after main shock sequence.

sponding to a period of 7.1 days. The procedure was also

applied to a final part of the after main shock sequence.

The difference between the measured and the theoretical

sequences appears in Figure 9 superimposed to a fitting

by a pure sinusoidal parametric model. The period of the

estimated sinusoidal model was 7.1 days. The non-para-

metric analysis of the residual sequence in Figure 9, by

means of the power spectral density is shown is Figure

10. The maximum peak noted in the last figure corre-

sponds to a period of 7.1 days. We note that the optimi-

zation of the main peak intensity of the power spectral

density with respect to perturbations of the location of

the last sequence datum, corresponded to a sequence

length of 49 days, which a multiple of seven.

We note that the peaks in the foreshock sequence of

Figure 5 correspond to Fridays, while those in the after-

Figure 9. L’Aquila: the residual daily number of events in a

final part of the after main shock sequence.

Figure 10. L’Aquila: the power spectral density of the residual

daily number of events in a final part of the after main shock

sequence.

shock of Figure 9 are located on Sundays. Furthermore,

the corresponding position of the Moon with respect to

the Earth and the Sun was approximately 3 days before

the four main Moon phases (data from US Naval Obser-

vatory, Astronomical Application Department).

We show now the results of the spatial analysis. We

recall Figure 2, where the total counts of events in the

foreshock period from day 52 to day 80 is shown. In Fi-

gure 11, we represent the counts ij

N only in the cells

where the Mathematical Morphology opened image is

qualified 1. The image clearly shows the presence of two

clusters close to each other with a maximum linear di-

mension of the largest one of about 5 Kilometers. The

isolated pixel we also put in the figure corresponds to the

epicentre of the main shock. We notice that this epicentre

M. Caputo et al. / Natural Science 3 (2011) 768-774

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Figure 11. Aquila: the spatial distribution of epicentres during

foreshock activity after the application of Mathematical Mor-

phology filtering. The image is 64 × 64 with a pixel length of

about 0.8 kilometers. The North direction is ↑.

Figure 12. Colfiorito: the residual daily number of events in a

final part of the after main shock sequence.

is about 3 kilometers from the closest cluster.

3.3. Colfiorito Earthquake Data Analysis

Also here, we adopted a power-law model for the after

main shock sequence starting from the relevant shock at

day 51. The difference between the measured and the

theoretical sequences is plotted in Figure 12 superim-

posed to a pure sinusoidal model fit. The period of the

estimated sinusoidal model is 6.7 days. The result of the

non-parametric analysis by the power spectral density is

shown is Figure 13. Here, it is evident a peak at a fre-

quency corresponding to a period of 7.0 days. The opti-

mization with respect to the location of the last datum as

before now corresponds to a a sequence length of 28

days, which again is a multiple of seven.

We note that the peaks in the aftershock of Figure 13

are located on Mondays. Also here, the corresponding

position of the Moon with respect to the Earth and the

Sun was approximately 3 days before the four main Moon

phases (data from US Naval Observatory, Astronomical

Application Department).

Figure 14, shows the counts ij

N only in the cells

where the Mathematical Morphology opened image is

qualified 1. The image clearly shows the presence of one

Figure 13. Colfiorito: the power spectral density of the residual

daily number of events in a final part of the after main shock

sequence.

Figure 14. Colfiorito: the spatial distribution of epi-

centres during foreshock activity after the applica-

tion of Mathematical Morphology filtering. The image

is 64 × 64 with a pixel length of about 0.8 kilo-

meters. The North direction is ↑.

M. Caputo et al. / Natural Science 3 (2011) 768-774

774

cluster with a maximum linear dimension of about 5

Kilometers. Also here, the isolated pixel we also put in

the figure corresponds to the epicentre of the main shock,

at a distance of about 3 kilometers from the cluster.

4. CONCLUSIONS

The two different sets of aftershocks and a set of fore-

shocks indicate a possible correlation between the time

sequence of events and the stress field generated by the

solid Earth tides. The sequences of aftershocks obviously

occurred in an unstable stress field due to the previous

major shock which destabilized the region, while the

sequence of foreshocks occurred in increasing stress field.

The possible correlation between tides and earthquakes

therefore concerns two different circumstances and could

be associated with different physical conditions although

the formal causes of the correlations are the same. It is to

be noted that the 7 days periodicity has been found in

both: foreshocks and aftershocks sequences and using

different methodologies. The two regions where the

earthquakes occurred are separated by about 50 kilo-

meters and belong to the same geologic formation.

Concerning the possibility that the cause of the 7 days

periodicity be due to the weekly noise generated by the

human activity in the working days we note that, in both

cases, after the major shock the human activity was con-

tinuous without respect to the festivities; in particular

was continuous the activity of heavy working machines

and of transportations. Furthermore, we checked that the

maxima of the peaks of the three sequences of above

occurred in three different days of the week.

We turn now to the possbile influence of the position

of the Moon with respect to the Earth and the Sun. We

note that the peaks in the three sequences of above,

different to each other for period of the year and/or geo-

graphical location of earthquake, occurred approximately

three days before the four main Moon phases.

Finally we provide some comments on the results of

the spatial analysis. To find evidence of a possible fore-

runner, we focus our attention on the foreshock sequence.

Here, the application of the Mathematical Morphology

opening operator provides a clear picture of the clus-

tering of the sismic events. We notice that in both cases

of the L’Aquila and Colfiorito earthquakes, the epicentre

of the main shock is at distance of only about 3 kilometer

from the closest cluster.

5. ACKNOWLEDGEMENTS

We thanks Dr. Aladino Govoni of the I.N.G.V. for providing the data

and for useful discussions and suggestions.

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