1. 1 23
Bulletin of Volcanology
Official Journal of the International
Association of Volcanology and
Chemistry of the Earth`s Interior
(IAVCEI)
ISSN 0258-8900
Volume 74
Number 4
Bull Volcanol (2012) 74:873-880
DOI 10.1007/s00445-011-0570-0
Magma convection and mixing dynamics
as a source of Ultra-Long-Period
oscillations
Antonella Longo, Paolo Papale, Melissa
Vassalli, Gilberto Saccorotti, Chiara
P. Montagna, Andrea Cassioli, Salvatore
Giudice & Enzo Boschi

2. 1 23
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3. RESEARCH ARTICLE
Magma convection and mixing dynamics as a source
of Ultra-Long-Period oscillations
Antonella Longo & Paolo Papale & Melissa Vassalli &
Gilberto Saccorotti & Chiara P. Montagna &
Andrea Cassioli & Salvatore Giudice & Enzo Boschi
Received: 5 May 2011 /Accepted: 14 November 2011 /Published online: 2 December 2011
# Springer-Verlag 2011
Abstract Many volcanic eruptions are shortly preceded by
injection of new magma into a pre-existing, shallow
(<10 km) magma chamber, causing convection and mixing
between the incoming and resident magmas. These process-
es may trigger dyke propagation and further magma rise,
inducing long-term (days to months) volcano deformation,
seismic swarms, gravity anomalies, and changes in the
composition of volcanic plumes and fumaroles, eventually
culminating in an eruption. Although new magma injection
into shallow magma chambers can lead to hazardous event,
such injection is still not systematically detected and recog-
nized. Here, we present the results of numerical simulations
of magma convection and mixing in geometrically complex
magmatic systems, and describe the multiparametric dy-
namics associated with buoyant magma injection. Our
results reveal unexpected pressure trends and pressure oscil-
lations in the Ultra-Long-Period (ULP) range of minutes,
related to the generation of discrete plumes of rising magma.
Very long pressure oscillation wavelengths translate into
comparably ULP ground displacements with amplitudes of
order 10−4
–10−2
m. Thus, new magma injection into magma
chambers beneath volcanoes can be revealed by ULP
ground displacement measured at the surface.
Keywords Magma dynamics . Magma convection . Magma
mixing . ULP ground displacement
Introduction
Various volcanic eruptions have been shortly preceded by
injection of new magma into a magma storage region at
shallow depth (<10 km), triggering mixing and convection a
few hours to weeks before eruption onset (e.g., Bateman 1995;
Folch and Marti 1998; Snyder 2000). Early recognition of
signals from monitoring networks, diagnostic of ongoing
magma convection and mixing in shallow magmatic sys-
tems, is therefore critical for developing reliable early-
warning systems and forecasting of short-term volcanic
hazard.
Although it is a critical precursor of hazardous events,
shallow magma chamber replenishment cannot be directly
observed. Analysis of records from volcano monitoring net-
works can in principle reveal such replenishment, but to-
date there is no established method for identifying the signal
of magma injection.
Recent progress in volcano monitoring worldwide is
providing unprecedented observation of small amplitude
Editorial responsibility: D. Dingwell
Electronic supplementary material The online version of this article
(doi:10.1007/s00445-011-0570-0) contains supplementary material,
which is available to authorized users.
A. Longo (*) :P. Papale :G. Saccorotti :C. P. Montagna :
S. Giudice
Istituto Nazionale di Geofisica e Vulcanologia, Sezione di Pisa,
Via della Faggiola 32,
56126 Pisa, Italy
e-mail: longo@pi.ingv.it
M. Vassalli
School of Geological Sciences, University College Dublin,
Belfield, Dublin 4, Ireland
A. Cassioli
Dipartimento di Sistemi e Informatica, Università di Firenze,
Via di S. Marta 3,
Florence, Italy
E. Boschi
Istituto Nazionale di Geofisica e Vulcanologia,
Via di Vigna Murata 605,
Rome, Italy
Bull Volcanol (2012) 74:873–880
DOI 10.1007/s00445-011-0570-0
Author's personal copy

4. ground oscillations with Ultra-Long-Periods (ULP) of
hundreds of seconds, preceding and accompanying visible
volcanic activity (Voight et al. 2006; Houlì and Montagner
2007; Sanderson et al. 2010). There is a general consensus
on the relevance and potential usefulness of such signals for
understanding the underground magma dynamics and fore-
casting the short-term volcanic hazard. However, the lack of
a generally accepted theoretical background describing the
generation of ULP signals at active volcanoes represents a
severe limitation to both their interpretation and use into
civil protection-oriented procedures. This work provides
that theoretical background, showing through numerical
simulations that magma convection and mixing dynamics
following the arrival of new magma at shallow depths
produce ULP ground displacements observable at the
surface.
Numerical modeling
The numerical simulations of magma reservoirs and country
rocks solve magma fluid dynamics and rock solid dynamics,
with appropriate conditions at the magma–rocks interface.
The fluid dynamics simulations need robust, stabilized
methods to solve the strongly nonlinear system of conser-
vation equations closed with constitutive equations for the
physical properties of the fluid. In this work, simulations are
performed with GALES (Longo et al. 2006), a stabilized
finite element parallel C++ code solving mass, momentum,
and energy equations for multicomponent homogeneous
gas–liquid (± crystals) mixtures. Code validation analysis
includes several cases from the classical engineering literature,
corresponding to a variety of subsonic to supersonic, one-
component to multicomponent gas–liquid flow regimes.
The multicomponent nature of GALES allows modeling
magma dynamics as they result from the mixing of composi-
tionally diverse magmas. Magma properties and gas–liquid
phase distribution, gas composition, and liquid–gas density
and viscosity, are computed as a function of the local P-T-X
conditions, by employing composition-dependent modeling
and parameterizations (Papale 2001; Papale et al. 2006;
Giordano et al. 2008). Non-Newtonian behavior of multi-
phase magma at high strain rates, which can be relevant
during the rapid ascent of magma along volcanic conduits
and fissures (Caricchi et al. 2007; Costa et al. 2009;
Giordano et al. 2010), is not accounted for.
Determining the time–space-dependent ground displace-
ment requires modeling the magma–rocks boundary condi-
tions and the mechanical response of rocks, the latter
depending on heterogeneous rock properties, presence and
distribution of faults, interfaces, fluids, and volcano topog-
raphy (e.g., O’Brien and Bean 2004). A first-order analysis
performed here assumes magma–rock one-way coupling
and adopts the Green’s functions formulation for a
homogeneous, infinite medium (Aki and Richards 2002).
In this process, we consider as point sources the fluid
dynamics computational grid nodes located at the reservoir
walls. As source time functions, we use the respective
temporal evolutions of magmatic forces computed from
pressures and stresses provided at those nodes by the nu-
merical simulations of magma convection and mixing dy-
namics. Ground displacement at a series of virtual receivers
is finally obtained by integrating, over all sources, the
Green’s functions associated with individual sources.
Such a one-way coupling and homogeneous rocks as-
sumption are a-posteriori justified by the extremely long
wavelength, of the order of hundreds of kilometers, associ-
ated with ULP pressure oscillations.
Numerical simulations
Numerical simulations are performed with reference to the
two volcanoes Etna and Campi Flegrei in Southern Italy.
These volcanoes offer a large literature on eruptive styles
and magma compositions, along with recent reconstructions
of the present state of the magmatic system and character-
istics of country rocks. A range of magma compositions
(reported in Table 1) from basalt (Etna) and basalt to shosh-
onite (Campi Flegrei) is involved. The employed total volatile
contents (Spilliaert et al. 2006; Mangiacapra et al. 2008) span
the range 1.5–3.5 wt.% for H2O and 0.5–2 wt.% for CO2.
Larger volatile contents are carried by the magma coming
from depth, according to open-system degassing at shallow
volcanic depths (Wallace and Anderson 2000). Deep
magmas are therefore expected to be buoyant in shallower,
partially degassed magmas (Longo et al. 2006), giving origin
to processes governed by magma convection and mixing.
The magmatic simulation domains and initial conditions
are shown in Fig. 1, and represent a 2D, Cartesian, simpli-
fied picture consistent with the bulk knowledge from geo-
chemical, petrological, seismological, and geodetical studies
(Carbone et al. 2006; Spilliaert et al. 2006; Aiuppa et al.
2007; Corsaro et al. 2007; Lokmer et al. 2008; Mangiacapra
et al. 2008; Patanè et al. 2008; Zollo et al. 2008; Arienzo et
al. 2010; Di Renzo et al. 2011). Common elements in the
two domains are the presence of one or more chambers as
well as of a feeding dyke and the existence of an initial
gravitationally unstable interface between two compositionally
different magmas.
The two domains significantly differ in terms of their
size, geometrical complexity, depth/pressure range,
employed magma compositions and corresponding physico-
chemical properties, and for the resulting density contrast at
the interface between the two magma types (see Fig. 1). An
overpressure corresponding to 1 MPa is imposed as bound-
ary condition at system bottom in the Etna case to include a
forcing component to natural convection as due to thrust
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5. from deep magma. The Campi Flegrei case corresponds
instead to pure natural convection (no overpressure at system
bottom).
In order to reduce the computational challenges, the present
simulations assume constant magma temperature and neglect
dispersed solid phases (crystals). Initial conditions are deter-
mined by lithostatic load at system roof (average rock density
2,500 kg/m3
), zero velocity everywhere, magma-static pres-
sure distribution and corresponding gas–liquid partition
computed with non-ideal composition-dependent modeling
(Papale et al. 2006), also employed in run time to provide
the gas–liquid partition of volatile species H2O and CO2 as a
function of local pressure, temperature, and composition.
Free inflow/outflow of magma is allowed at domain bottom
and no slip conditions are assumed at solid boundaries.
Magma mixing is accounted for by weighting the local
properties on those of the end-member magmas based on
local proportions. This approximates mechanical mingling,
appropriate for the relatively short time scale of the simula-
tions (order of hour) and grid size range (from 0.2 to 10 m,
resulting in order 105
computational elements in each simu-
lation). A time resolution of 0.01 s is employed, after having
verified numerical stability in a preliminary set of test
simulations.
Continuity of pressure and stress is taken as the boundary
condition along the nonmoving magma–rock interface.
Physical properties of rocks are homogeneous averages that
describe the volcanic edifices within the range of considered
depth (<10 km, vp03,000 m/s; vp/vs01/√3, ρ02,500 kg/m3
).
The real characteristics of the magmatic systems and
volcanic edifices, although scarcely known in further detail,
are expected to be substantially more complex. The simu-
lations therefore should be regarded as a first-order charac-
terization of the processes and dynamics associated with the
arrival of buoyant magma into a pre-existing reservoir at
constant fluid system domain volume.
Table 1 Composition of mag-
mas employed in the numerical
simulations
Numbers are wt%. FeO is total
iron as reduced component
Data from Andronico et al.
(2005), Mangiacapra et al.
(2008), and references therein.
H2OT
and CO2
T
indicate
total water and carbon dioxide,
respectively, including the
amount dissolved in the liquid
and that exsolved in the
co-existing gas phase
Simulation Mount Etna Mount Etna Campi Flegrei Campi Flegrei
Magma Shallow (basalt) Deep (basalt) Shallow (shoshonite) Deep (basalt)
SiO2 48.4 47.9 52.5 47.6
TiO2 1.67 1.69 0.85 1.24
Al2O3 17.8 16.9 17.6 15.5
FeO* 10.2 10.5 7.62 8.3
MnO 0.18 0.17 0.12 0.14
MgO 5.53 6.56 3.60 10.0
CaO 10.2 11.1 7.93 11.1
Na2O 3.87 3.31 3.43 2.88
K2O 2.11 1.93 4.28 1.49
H2OT
1.5 3.5 2.0 3.5
CO2
T
0.5 2.0 1.0 2.0
Fig. 1 System definition, and
initial and boundary conditions
for the numerical simulations.
Note the difference in scale of
the two domains. a Mount Etna
case, compositions from the
2002–2003 eruption. The width
of the two shallow dykes is
10 m. b Campi Flegrei case.
Compositions are a shoshonite
(shallow magma) and a basalt
(deep magma). Compositions for
the Mount Etna and Campi Fle-
grei simulations are reported in
Table 1. Total volatile contents
are reported in Fig. 1. Temper-
atures are 1,400 K (Etna)
and 1,433 K (Campi Flegrei)
Bull Volcanol (2012) 74:873–880 875
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6. Results
The numerical results pertaining to magma dynamics are
summarized in the color plots of Fig. 2. Movies showing the
time–space distribution of composition, pressure, and gas
volume fraction are provided as Online Resources 1–6.
Total simulated times correspond to nearly 30 (Etna) and
170 (Campi Flegrei) min of real time.
Initially, the interface between the two magma types is
perturbed by buoyancy and, for Mount Etna, by pressure
forces. After about 20 s (Etna) and 70 s (Campi Flegrei) a
plume of buoyant magma starts to develop. The density
difference between the head of the plume and the surround-
ing magma progressively increases as the plume moves up
towards regions characterized by lower pressure. As a con-
sequence, plume buoyancy progressively increases, enhanc-
ing plume expansion and acceleration. The vertical motion
of the rising plume is disturbed by the formation of a series
of vortexes that favor further magma mixing. Subsequent
plumes give origin to small buoyant magma batches inter-
acting with each other, further enhancing mixing between
the incoming magma and the magma originally residing in
the chamber.
In the Etna case, plume rise and expansion cause magma
pressure increase in the chamber up to a maximum of
1.5 MPa after about 60 s (Fig. 3). Sinking of dense magma
into the feeding dyke, favored by chamber pressure in-
crease, reduces the mass in the chamber and consequently
the local pressure. Intense mixing of volatile-rich and
volatile-poor magmas takes place in the feeding dyke, where
compression by the sinking magma results in magma pres-
sure increase. As a consequence of mixing at dyke level, no
pure deep magma component enters the chamber after a
very short initial phase. Alternating phases dominated by
buoyancy and sinking at chamber inlet result in ULP pres-
sure fluctuations with a period of about 110 s and amplitude
decreasing with time (Fig. 3). The progressive substitution
of the original denser magma by lighter buoyant magma
leads to overall chamber pressure decrease, by a maximum
of −2 MPa after about 25 min.
The simulation pertaining to Campi Flegrei (Figs. 2 and 3)
shows similar general dynamics, with discrete buoyant
batches of volatile-rich, lighter-mixed magma rising and
expanding into the chamber, where further mixing with the
resident magma takes place. The pressure change after the
first 30 min is about 1 order of magnitude smaller than in the
Etna case, likely due to much larger chamber size therefore
more mass of new magma required to cause comparable
pressure changes. However, once a substantial pressure
increase is produced, ULP oscillations appear, this time with
longer period of about 330 s. The amplitude of pressure
oscillations is 1 order of magnitude lower than for the Etna
case, with no damping emerging up to the maximum simu-
lated time. The overall pressure change in the chamber is
about −0.5 MPa, about 1/4 of that for the Etna case in less
than 1/6 time. A trend towards overall pressure decrease
emerges, superimposed to an extremely long-period oscilla-
tion over a time scale comparable with that of the simulation.
Over the nearly 3 h of simulated time a volatile-rich mixture of
basalt and shoshonite accumulates close to the chamber roof,
giving origin to a stratified reservoir.
The elastodynamic simulations reveal that the computed
ULP pressure oscillations, originated by the ingression of
buoyant magma in the magma chamber, translate into com-
parably ULP ground displacement dynamics with ampli-
tudes of millimeter (Campi Flegrei) to micrometer order
(Etna). Figure 4 shows such ground oscillations, as they
would be recorded by instruments having cutoff periods of
Fig. 2 Simulated dynamics at
three different times. a–c Mount
Etna case, d–f Campi Flegrei
case. The color plots represent-
ing the evolution of magma
composition refer to the areas
identified by the boxes on the
left. Note (from comparison
with Fig. 1) that the scale of
the two domains is very
different. Movies showing the
time evolution of composition,
pressure, and gas volume
fraction for both the Mount
Etna and Campi Flegrei cases
are given in Online Resources
from 1 to 6
876 Bull Volcanol (2012) 74:873–880
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7. 500, 200, and 50 s. It must be recalled in fact that any
instrument measuring ground displacement can only record
movements occurring over a certain frequency range, acting
therefore as a filter. A cutoff period of 50 s characterizes
most classical broadband seismometers, whereas oscillations
with longer periods can be detected by dilatometer, Global
Positioning System (GPS), or tiltmeter networks (e.g., Lay
and Wallace 1995). Accordingly, Fig. 4 shows that ULP
ground movements like those predicted by the present
modeling could not be detected by classical broadband seis-
mometers (although more recent seismometers extend their
working range up to 100–200 s periods; Havskov and Alguacil
2004), while they are in principle visible in the records from
other instruments, especially borehole dilatometers charac-
terized by high signal-to-noise ratio.
Discussion and conclusions
The simulation results show that the evolution of the mag-
matic pressure depends nonlinearly on the characteristics of
the simulated system, varying from point to point in re-
sponse to both general and local dynamics. Pressure varia-
tions over the simulation time scale do not constitute the
focus of this work, since their evaluation requires longer
computing times. We note however that in both simulation
cases presented here, and over the maximum simulated
times, the overall pressure in the shallow system decreases
as a consequence of partial replacement by lighter magma.
This is true even for the Etna case, for which a 1 MPa
overpressure at system bottom has been imposed in the
simulations as a forcing component to magma flow.
Fig. 3 Pressure variations as a function of time. The location of the
reported pressure variations is indicated by the red points in the left
schemes of Fig. 1. a Mount Etna case, b Campi Flegrei case. In both
panels, the upper diagram shows the difference between the local
pressure at current time and at time zero, while the bottom diagram
shows the same quantity after subtraction of a detrending function (red
curve in the upper diagrams)
Fig. 4 Computed time series of
ground displacements
(topographic effects neglected).
Left panels Mount Etna case.
From top to bottom, high-pass
filtering at corner periods of
500, 200, and 50 s. The data
were detrended before filtering,
demeaned and tapered using a
Tukey window to alleviate
border effects. The synthetic
recordings are from a virtual
receiver located on a flat
surface 2.5 km east of the
surface projection of the central
chamber point. Right panels
Campi Flegrei case, same as
above with virtual receiver
located 4 km east of the
chamber axis
Bull Volcanol (2012) 74:873–880 877
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8. Pressure decrease is a counterintuitive consequence of
new magma injection and appears to relate to reduced over-
all magma density in the constant volume chamber. The
density decrease in fact reduces both the hydrostatic and
thermodynamic pressure contributions, the former related to
the weight of the magmatic column above any given point,
the latter related to an equation of state for the multiphase
multicomponent magma. The density decrease is in turn a
consequence of injection of lighter magma and sinking of
denser magma towards deeper system regions. Gas exsolution
and expansion upon ascent through the shallow chamber at
one hand tends to increase pressure (and consequently favors
dense magma sinking into the feeding dyke) at the other hand
contributes further to overall density decrease.
It is relevant to note that an elastic response of reservoir
walls, not included in the fluid dynamics simulations, may
act as a buffer, reducing the extent of overall pressure
change without however changing its sign. We stress how-
ever that pressure decrease at shallow level is not proposed
here as an unavoidable consequence of magma chamber
recharge by buoyant magma. In fact, the above described
processes controlled by light magma injection and dense
magma sinking are highly nonlinear and a general pressure
balance cannot be established on the basis of only two
simulations performed here. It is well possible that other
system conditions (e.g., different geometries, different com-
positions involved, different efficiency of magma exchange
at dyke level, etc.) may result in different pressure trends.
This subject forms part of ongoing investigation that will be
presented elsewhere.
The most remarkable feature and the outstanding result
emerging from the present simulations is represented by ULP
pressure oscillations accompanying magma convection and
mixing (see Fig. 3 and Online Resources 7–17 for pressure
oscillations computed at several different points in the
computational domain). Such oscillations are a consequence
of the complex dynamics characterized by buoyant plume
ascent and expansion, local vortex formation, and dense
magma sinking. The patterns of ULP pressure oscillations are
different from point to point in the magmatic domain, depend-
ing in a complex way on both global and local dynamics.
Although desirable, a parameterization of the different con-
tributes that concur to determine the period of pressure oscil-
lations is not possible at the moment, since it requires a
substantially larger number of simulations in a comparably
large range of system conditions.
When integrated along the entire boundaries of the sim-
ulated domain and transported to the Earth’s surface, com-
plex patterns of ground motion dominated by ULP
frequencies emerge (Fig. 4). Although heterogeneities in
country rocks have been neglected in the present analysis,
they are not expected to affect significantly the patterns of
ground motion over the frequency of the computed ULP
displacements. Oscillating ground motion with a period of,
say, 100 s corresponds in fact to wavelengths of order
100 km. Heterogeneities over a spatial scale of 1 km or less
have therefore negligible effects at such wavelengths.
The above discussion may suggest that ULP pressure
oscillations in the magmatic body translate into ground
oscillations with same or similar waveforms. However, even
when looking at the 500 s low pass-filtered ground displace-
ment waveforms in the upper diagrams of Fig. 4, that is not
the case. The computed ground displacement waveforms are
in fact substantially more complex than the pressure oscil-
lations that generate them. The reason for such complex
relationships rests in the fact that ground displacement
waveforms result from the interactions between pressure
waves generated from many sources and having different
waveforms; even more importantly, pressure waves are gen-
erated at different times along the fluid–solid domain bound-
ary, as a response to the internal time–space-dependent
dynamics affecting the magmatic body. The general conclu-
sion is therefore that spatially extended magmatic bodies
undergoing pressure fluctuations and representing a source
for ground displacements should not be regarded as lumped
bodies undergoing overall pressure increase or decrease. A
corollary is that full-waveform inversion of ground displace-
ment signals, if it neglects the time–space-dependent magma
dynamics as do inversions performed at volcanoes, could lead
to misleading results, since those waveforms critically depend
on the internal dynamics of the fluid system.
ULP ground oscillations are proposed here as a diagnos-
tic product of deep magma convection. Due to either intrin-
sic technological limitation or coarse sampling rates, ULP
ground displacements with frequency range 10−2
–10−3
Hz
like those from the present simulations are still seldom
measured at real volcanoes. In the large majority of cases,
in fact, the employed instruments do not cover such a
frequency range (Scarpa 2001) or do not have enough
temporal resolution. This is true at Mount Etna and Campi
Flegrei where broadband seismic instruments have a cutoff
frequency corresponding to about 40 s and long baseline
tiltmeter data are recovered with a frequency of one every
10 min (Mount Etna; Bonaccorso et al. 2002; Cannata et al.
2009) or have just been set up and are still in the experi-
mental phase (Campi Flegrei; R. Scarpa, personal commu-
nication). The resolving capability of GPS networks is of
order several millimeters or larger (Bonaccorso et al. 2002),
still too low to detect ground oscillations with amplitude
comparable to that emerging from the present simulations.
Additionally, GPS data are commonly averaged over hours
or tens of hours to obtain a picture of the long-term volcano
deformation (Bonaccorso et al. 2002).
Identification of sustained oscillations with small ampli-
tude like those emerging from numerical modeling, requires
on the one hand the deployment of near-field advanced
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9. instrumentation represented by arrays of very broadband
seismometers, borehole strain meters, long baseline tilt-
meters, and GPS networks; on the other hand, it needs the
development and application of sophisticated data analyses
able to separate the contributions of the spatially coherent
signals from those due to noise. Although systematic
collection and analysis of signals in the frequency range
10−2
–10−3
Hz is still not carried out (Scarpa 2001), recent
advances in volcano monitoring, and particularly the deploy-
ment of very broadband instruments at a growing number of
volcanoes throughout the world, are increasingly revealing
ULP ground oscillations. At Soufrière Hills, Montserrat (West
Indies), dilatometric data show oscillations with periods
around 103
s, following phases of dome collapse (Voight et
al. 2006). ULP ground motion with periods in the range
30–600 s was recorded at Santiaguito volcano, Guatemala,
during a 3.5 days broadband survey (Sanderson et al. 2010).
Ground oscillations in the frequency range 10−2
–10−3
Hz
were measured at Piton de la Fournaise volcano, Réunion
Island, and interpreted as due to pressure oscillations in a
shallow magma chamber as a consequence of injection of
magma of deeper provenance (Houlì and Montagner 2007).
The numerical simulations presented here show that con-
vection dynamics taking place in magma reservoirs are
expected to produce pressure oscillations and ground motion
in the ULP frequency range. This paper provides therefore the
physical background for the interpretation of emerging ULP
signals at active volcanoes, at the same time offering a strong
motivation for reexamining the existing seismic, strain, tilt,
and high-rate GPS records worldwide for the presence of ULP
signals. The results of highly nonlinear magmatic processes
simulated here, and the records from the most recent and
advanced broadband instruments deployed at volcanoes,
concur to demonstrate that new and vital information on the
underground volcano dynamics can be gained through exten-
sive and systematic exploration of the ultralow frequency
domain of geophysical signals.
Acknowledgments This work has been performed in the frame of
Projects FIRB RBAU01M72W and RBPR05B2ZJ; and Projects
INGV-DPC 2004–2006 V3_2, and 2007–2009 V1 and V4.
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