Monte Carlo seismic volcanic envelopes: in need of boundary ...

submitted to Geophys. J. Int.
Monte Carlo seismic volcanic envelopes: in need of boundary
conditions
L. De Siena 1 E. Del Pezzo 2,3 C. Thomas 1 A. Curtis 4 L. Margerin 5
1 University of Münster, Institute for Geophysics, Correnstrasse 24,
Münster, 48149, Germany. (lucadesiena@uni-muenster.de)
2 Istituto Nazionale di Geofisica e Vulcanologia, Department of Naples -
Osservatorio Vesuviano, Via Diocleziano 328, Naples, 80124, Italy.
3 also at Istituto Andaluz de Geofisica, Universidad de Granada, Granada, Spain.
4 School of Geosciences, University of Edinburgh, Edinburgh
Scotland.
5 Ludovic Margerin. Institut de Recherche en Astrophysique et
Plantologie, Université Paul Sabatier / CNRS, 14 Avenue Edouard
Belin, 31400 Toulouse, France.
January 20, 2012
SUMMARY
A deterministic and/or a stochastic technique can be proficiently applied on seismological data
if the limit between coherent and incoherent signals is clearly marked both in time and in fre-
quency. We can exploit the signal to provide adequate images of the Earth only if we know its
degree of coherency. Nevertheless, in volcanic seismic recordings, the mixing between coher-
ent and incoherent signals is continuous. We account for the coherence effects in a volcanic
seismic envelope including in the usual stochastic approach a drastic change in the scatterer
distribution. Even if caused by a deterministic boundary, the effects of this change are still
stochastic in their mathematical and physical form, and can be described by a second set of
transport equations. Coda signals are triggered in our model by three different length scales:

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2 L. De Siena E. Del Pezzo C. Thomas A. Curtis L. Margerin
the mean free path, the transport mean free path, and the distance between the source and a
large scale boundary. Above 10 Hz, we can model our intensity on real envelopes recorded
at Campi Flegrei caldera without the boundary. This as well as the reach of diffusion after
three times the S-wave arrival corroborates the use of the ray approximation and of the coda
normalization method, respectively. Even in this high frequency limit, unexpected large SP
conversions and general broadening are evident in the intermediate coda. Around 1 Hz, the
unexpected broadening affects the whole envelope. Diffraction and interference effects are rel-
evant, while recursive coherent signals arise at intermediate times. We implement a boundary
in the simulation through the definition of a different scattering texture, coincident with the
caldera rim signature at 1.5 km. The boundary acts as a filter between coherent and incoherent
intensities, so that 2D synthetics provide an adequate first order model of the data envelopes
also in the low frequency range. A model based on the theory of inhomogeneous dispersive
turbulence should be implemented to model low frequency envelopes in the whole caldera. We
proved, anyway, that the exclusive application of a deterministic or stochastic approximation
provides unfeasible results, while a mixed approach can better describe the envelope behaviors
in the whole frequency range.
Key words: Radiative Transfer Theory – Large angle scattering – Monte Carlo simulation
1 INTRODUCTION
Radiative transfer theory describes the transport of seismic energy through a scattering medium; it
is one of the most important tools, together with diffusion theory, to synthesize coda intensities, the
main evidence of scattering in the Earth (Wu, 1985). Radiative transfer solutions provide intensity
variations comparable with coda envelopes; these intensity measurements, usually calculated as
the mean square or the root mean square of coda signals, are crucial measurements of the degree
of inhomogeneity of the Earth (Sato & Fehler, 1998). For scalar waves, radiative transfer theory
only provides two exact analytical solutions: the stationary and the 2-D dynamic solutions for
isotropic scattering. Otherwise, radiative transfer equations are solved numerically (i. e. with a
Monte Carlo approach) (Ishimaru, 1997).
A rigorous multiple scattering analytical theory has been developed in the past 20 years to de-

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Monte Carlo seismic envelopes caldera 3
scribe seismic intensities (Zeng et al., 1991; Zeng, 1991, 1993; Sato & Fehler, 1998). In practice,
this theory only provides approximate solutions, particularly for complex volcanic media. Hence,
single scattering and diffusion are used to measure critical scattering parameters, such as the trans-
port mean free path, or the ratio between mean square fluctuations and correlation distance (Sato &
Fehler, 1998; Wegler & Lühr, 2001; Przybilla et al., 2009). Single scattering and diffusion are limit
cases of multiple scattering. When dealing with a tenuous distribution of particles, single scatter-
ing isotropic solutions are used to model energy envelopes at regional scale (Wu, 1985; Przybilla
et al., 2006). The single scattering approximation gives incorrect results in a medium with strong
velocity fluctuations (Sato & Fehler, 1998); diffusion theory is therefore a more suitable candidate
to describe scattering propagation in volcanic areas (Wegler & Lühr, 2001). Nevertheless, for short
source-receiver distances (i. e. in volcano monitoring) diffusion could be an over-estimation of the
incoherence characterizing the seismic wave-field (Yamamoto & Sato, 2010).
The waves scattered by random heterogeneities are never totally incoherent; the coherent field
goes to zero exponentially with time, but is still part of the diffusion solution (Ishimaru, 1997).
The underlying intrinsic coherence in volcanic signals is due to the presence of strong velocity
fluctuations causing multiple scattering, that become important at high frequencies and at short
lapse-times (Sato, 1989). Phenomenologically, these fluctuations broaden in time and lower in
amplitude coda envelopes; both the single scattering model and the diffusion theory cannot explain
these phenomena (Sato & Fehler, 1998; Saito et al., 2005).
The Markov approximation for the parabolic wave equation is the usual approach to account
for broadening and maximum amplitude decay (Sato, 1989; Fehler et al., 2000; Saito et al., 2002).
The parabolic wave equation models the scattering contribution of the long-wavelength spectra. Its
Markov approximation is valid in case of strong forward scattering both for plane and for spher-
ical waves. Markov solutions - also known as Markov envelopes - require the introduction of a
two-frequency mutual coherence function, representing the background correlation of the velocity
field (Sato & Fehler, 1998; Fehler et al., 2000; Saito et al., 2002). The Markov approximation
is used directly or in a finite frequency simulation to synthesize Markov envelopes (Sato et al.,
2004; Saito et al., 2005). Sato et al. (2004) particularly remark that single scattering coefficients

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4 L. De Siena E. Del Pezzo C. Thomas A. Curtis L. Margerin
cannot efficiently reproduce broadening, since they produce large forward scattering. Hence, these
authors introduce the momentum scattering coefficients to describe statistically a medium where
they propagate the Markov solution. Nevertheless, seismic envelopes cannot be entirely repro-
duced without superimposing large-angle scattering at small-scale random heterogeneities, as well
as short-wavelength scattering (Fukushima et al., 2003; Sato et al., 2004; Przybilla et al., 2006).
Except when analytic solutions exist, Monte Carlo numerical simulations of the radiative trans-
fer equations are used to synthesize seismic envelopes (Gusev & Abubakirov, 1987; Hoshiba,
1991; Margerin et al., 1998; Yoshimoto, 2000). Polarization information and broadening of the
envelope are included in the techniques in order to obtain solutions modeled on real envelopes
(Margerin et al., 2000; Przybilla et al., 2006; Przybilla & Korn, 2008). At first aimed at the de-
scription of energy propagation at regional scale, these techniques also provide important results
at teleseismic distances, e. g. through the analysis of PKP precursors (Margerin & Nolet, 2003).
Local scale simulations present different challenges. Margerin & van Tiggelen (2001) and van
Tiggelen et al. (2001) study the coda intensity enhancement due to interference in the near field.
They highlight a reciprocal wave producing constructive interference (coherence) after a transient
regime, likewise a stable spherical source of radius half a wavelength, centered at the seismic
source (Margerin & van Tiggelen, 2001). The theory is valid if the source-receiver distance is
within approximately one elastic wavelength, as can be the case for a monitored volcanic area, but
is unlikely to be observed for earthquake sources due to the broken reciprocity between source and
receiver (van Tiggelen et al., 2001). It is evident that at each scale, and for different waves, Monte
Carlo solutions improve the understanding of the physics underlaying scattering propagation.
We implement a full 2D elastic Monte Carlo simulation of the intensity produced by a point
source in a volcanic medium through a priori definition of the single and momentum scatter-
ing coefficients, using a von Kármán autocorrelation function (Sato et al., 2004; Przybilla et al.,
2006). The coefficients trigger both the probability and the propagation direction after each scat-
tering event. Direct-wave energy reproduction, reach of diffusion, line-of- sight propagation, and
early-coda coherency are investigated comparing the synthetics with the real envelopes recoded
at Campi Flegrei caldera. A large change in the distribution of the scatterers is included to model

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our solutions on real coda envelopes. We include the enhancement of large angle scattering by the
definition of a reverse scattering wave-field (reverse respect to the incident direction) created by
the boundary. Our scope is to obtain a first order model of the real envelopes in order to depict the
mechanism producing broadening in volcanic areas (Yamamoto & Sato, 2010).
2 MONTE CARLO SIMULATION OF MULTIPLE SCATTERING PROPAGATION
Multiple scattering theory describes wave propagation in a tenuous distribution of scatterers via a
single scattering approximation. The single scattering coefficients provide the scattering pattern at
each scattering event; multiple scattering then produces the entire coda envelope. This approach
was applied to synthesize envelopes recorded at long lapse times (late coda) for regional distances
(Sato et al., 2004; Przybilla et al., 2006). Hence, single scattering is the first step for understanding
multiple scattering processes.
We aim to gain an insight into the decay with time of envelopes recorded at local scale and for
volcanic regions. Therefore, we solve the radiative transfer equations by the Monte Carlo method
in the full elastic case, using a single scattering approximation (Margerin et al., 2000; Przybilla
et al., 2006). This is just a basement stone; the small source-receiver distances and the high hetero-
geneity of the velocity wave-field in a volcanic area can break the single scattering approximation,
only plausible at larger scales (Przybilla et al., 2006).
2.1 Statistical description of the medium
A statistical description of the medium through its correlation and spectral aspects is necessary
to describe scattering propagation (Rytov et al., 1987). We consider a crustal area of average
dimension L = 10 km, which can be described as an ensemble of random media, and characterized
by a 2D random velocity field (Figure 1). This field V (x) is described in terms of average wave
velocity V0 and the fractional velocity fluctuation ξ(x), where x is the 2D space coordinate (Sato
& Fehler, 1998):
V (x) = V0{1 + ξ(x)}. (1)

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We define R(x), the auto-correlation function (ACF) of the fractional velocity fluctuations between
points of positions x and x’, as:
R(x) ≡ 〈ξ(x+x’)ξ(x’)〉, (2)
and its Fourier transform, the power spectral density function (PSDF), as:
P (k) =
�∞
−∞
R(x) exp(−ikx)dx, (3)
where k is the wave number, and angle brackets stand for ensemble average. The PSDF is the key
function to describe scattering; it must be carefully chosen in order to represent real envelopes.
A power low PSDF corresponds to an exponential or, more generally, a von Kármán-type ACF
(Shapiro & Hubral, 1999). The von Kármán PSDF is:
P (k) =
4πɛ2a2κ (1 + a2k 2 , (4)
) (κ+1)
where a is the correlation length, ɛ 2 ≡ R(0) = 〈ξ(x) 2 〉 is the mean square fluctuation, Γ is the
Gamma function, and κ is the order of the function, which triggers the decay of the PSDF. These
quantities commonly characterize a random distribution of particles; in seismology, an exponential
PSDF is preferable to a Gaussian one in order to depict seismic envelopes. This is particularly
true at local scale, as suggested by well-log and seismic data (Shiomi et al., 1997). Since we
deal with a volcanic medium, we assume that long-wavelength spectra are dominating, causing
broadening, loss of coherence, and decrease in mean field intensity. Multiple scattering simulations
with single scattering coefficients and with an exponential (κ = 0.5) ACF provide envelopes at
regional distance, witch are similar to the real one. To account for the statistical long wavelength
effects we also consider the highest order of the ACF (κ = 1). The PSDFs for both orders are
shown in Figure 2.
2.2 Single scattering coefficients
In seismology, 2D and 3D Monte Carlo solutions reproduce the envelopes of real seismic P- and S-
waves propagating through the Lithosphere both in the acoustic and in the elastic cases (Gusev &
Abubakirov, 1996; Margerin et al., 2000; Przybilla et al., 2009). We simulate the multiply scattered

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wave-field by shooting particles from a point source located in the caldera, whose velocity field is
statistically characterized by a von Kármán PSDF. Collision of the particle with the scatterers are
perfectly elastic; the scattered direction, chosen with a table-look up method, is anisotropic, and is
triggered by the differential single scattering coefficients, equivalent to the differential scattering
cross-sections (Przybilla et al., 2006).
The single differential scattering coefficients are dependent on the von Kármán PSDF as well
as on the scattering patterns, |Xij(θ) 2 |; here θ is the angle respect to the incident direction, while
ij stands for P or S (Sato & Fehler, 1998). The SS differential scattering coefficient gss is given
by:
gss(θ) = k3 s
8π P (2ks sin( θ
2 ))|Xss(θ)| 2 , (5)
where ks is the S-wavenumber, density is correlated to velocity by the factor ν = 0.8 (given by
Birch’s law), and Xss(θ) is defined by:
Xss(θ) = (ν(cos θ − cos 2θ) − 2 cos 2θ). (6)
SS scattering dominates for large lapse-times in the continental Lithosphere - for a complete review
on the variations of the scattering coefficients with space, time, and frequency see Sato & Fehler
(1998). In this paper, we refer to their 2D equivalent, extensively treated by Przybilla et al. (2006).
P- and S-wave mean free paths (lp and ls, respectively) are the average distances traveled by a
particle between two collisions. They are dependent on the single scattering coefficients:
We introduce the notations:
g 0 ij = 1
2π
� 2π
0
g 0 p = g 0 pp + g 0 ps
where the cross-terms satisfies the 2-D reciprocity relation:
gij(θ)dθ. (7)
(8a)
g 0 s = g 0 sp + g 0 ss, (8b)
g 0 ps = ( Vp
Vs
(8c)
) 3
2 g 0 sp, (9)

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8 L. De Siena E. Del Pezzo C. Thomas A. Curtis L. Margerin
and Vp and Vs are P- and S-wave velocities; hence, the mean free paths are:
lp = (g 0 p) −1
(10a)
ls = (g 0 s) −1 ; (10b)
The scattering coefficients depend on the typical correlation distance for a volcanic area (a);
ACF velocity measurements at Campi Flegrei provide a value of a = 0.9 (De Siena et al., 2011).
We assume a mean squared fluctuation ɛ = 0.06%, but a smaller correlation length (a ∼ = 0.5) in
order to account for the highly heterogeneous S-wave velocity field in the caldera (Battaglia et al.,
2008). This assumption is supported by K. & Levander (1992), who found an average correlation
length for the continental crust between 0.2 km and 0.8 km. We also need to lower a to model
both P- and S-wave propagation by using geometrical optics; a larger correlation length leads to a
lS < a, and the wave would be attenuated before going through a single heterogeneity.
The single scattering coefficients numerically calculated for 3 Hz and 18 Hz are shown in
Table 1; a low probability for conversion scattering still exists even at high frequency. We numer-
ically compute the scattering pattern associated with each scattering mode to select the direction
after each scattering event (θ in Figure 1) with a table-look up method. We obtain different tables
of angles for different scattering modes (Table 2). Each of the 201 angles corresponds to an equal
fraction of the total cross section, given by the integral over 2π of the scattering coefficients (Equa-
tion (5)). Table 2 is an example of the angle distribution for single SS and PS scattering, κ = 0.5,
and frequency 3 Hz. Direction is selected by using a random integer number (defined between
1 and 201): it is evident that scattering is most likely in the forward direction for SS scattering,
while the PS (and SP) scattering patterns have their main lobes in transversal directions. Conver-
sion scattering induces a relevant deviation from the incident direction, causing a larger sampling
of the medium (Figure 1).
2.3 Momentum scattering coefficients and transport mean free path
In volcanic regions, multiple scattering at large lapse times is considered as isotropic, even if each
scattering process is not (Yamamoto & Sato, 2010). Some authors prefer to simulate envelopes

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at large lapse times excluding strong forward scattering from the anisotropic single scattering
coefficients. This can be done including the factor (1 − cos θ) in the definition of gij (Gusev &
Abubakirov, 1996; Sato et al., 2004); the SS differential momentum scattering coefficients (gss m )
are therefore defined using Equation (5):
g m ss(θ) = (1 − cos θ) k3 S
8π P (2ks sin θ
2 )|Xss(θ)|, (11)
and represent a background medium rich in short-wavelength inhomogeneities (Table 3).
First introduced by Gusev & Abubakirov (1996) in seismology for characterizing coda exci-
tation at long lapse times, they were then applied by Sato et al. (2004) as background medium
for propagating waves with a diffraction imprinting - named Markov envelopes. In volcanic areas,
these envelopes correlate better than the ones obtained with single scattering coefficients to the
result of the inversion of seismic intensity produced by active sources (Yamamoto & Sato, 2010).
Nevertheless, it is known since the works of Weaver (1990) and Ryzhik et al. (1996) that, in the
elastic case, the inclusion of a factor (1 − cos θ) is not preserving total intensity. Also, the equiva-
lent scale lengths (the transport mean free paths) have more complicated expressions than the ones
obtained from momentum scattering coefficients, at least in the full elastic case.
We introduce the cosine weighted scattering coefficient for the various mode conversions g ∗ ij
(Table 4) as:
g ∗ ij(θ) = 1
� 2π
gij(θ) cos θdθ. (12)
2π 0
Using these expressions, we obtain the transport mean free paths (l ∗ p and l ∗ s, Table 5) in the full
elastic case (Turner, 1998):
l ∗ p =
l ∗ s =
g 0 s − g ∗ ss + g ∗ ps
(g 0 p − g ∗ pp)(g 0 s − g ∗ ss) − g ∗ps g ∗ sp
g 0 p − g ∗ pp + g ∗ sp
(13a)
(g0 p − g∗ pp)(g0 s − g∗ ss) − g ∗ps g∗ . (13b)
sp
In the case of non-preferential scattering (i.e. equal amount of forward and backward scattering),
the transport mean free paths reduce to the mean free paths. The transport mean free paths defined
using the momentum scattering coefficients (l m P and lm S
) result larger for S- than for P- waves
(Table 5). This is unfair, since we expect a greater level of interaction for S-waves, corresponding

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10 L. De Siena E. Del Pezzo C. Thomas A. Curtis L. Margerin
to a fastest reach of diffusion. The opposite happens using the cosine weighted coefficients. Even
if the diffusion constants (D m and D, respectively) are similar at 18 Hz, they differ in the 3 Hz
frequency band. From these considerations as well as for the more sounded connection with the
physics of the problem, we will use l ∗ p,s to investigate diffusion.
2.4 Diffusion
In section 4 we employ the momentum and cosine weighted scattering coefficients to depict the
reflection on a large-scale boundary included in the medium. However the relative transport mean
free paths are the step length of an equivalent diffusion process, hence, a marker of the reach
of diffusion. Most of the techniques employed to measure scattering parameters and to monitor
volcanic area without passive sources only work in the diffusion regime (Wegler, 2003; Brenguier
et al., 2008).
Diffusion represents the scattering radiation field at large lapse times from the source enucle-
ation time. It assumes an energy flux scattered in space almost uniformly to every direction, and
recorded after encountering many particles. A slightly anisotropic angular dependence (like the
one obtained from momentum or cosine weighted scattering coefficients) is necessary to add net
power propagation, still present in the diffusion regime. Diffuse intensity (ID) at distance r can
therefore be written as the sum of the average diffuse intensity (UD(r)) and the diffuse flux vector
(FD(r)), whose direction is given by a unit vector sf:
ID(r, s) ∼ = UD(r) + 3
4π FD(r)sf. (14)
For diffusion to be valid, the second term must be much smaller than the first.
Diffuse energy (ED(r, t)) at distance r in the case of 2D propagation is given by:
ED(r, t) =
exp(− r2
4Dt )
4πDt
Likewise in 3D, 2D diffusion of P- and S-waves requires equipartition to be reached (Hennino
et al., 2001; Margerin, 2006):
Es
Ep
= V 2
p
V 2
s
(15)
, (16)
where Ep and Es are the energies for P- and S- waves. We remind that, in 2D, P-waves can only

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Monte Carlo seismic envelopes caldera 11
couple with SV-waves. Hence, from previous expressions of the diffusivity of elastic waves (D) in
3-D (Weaver, 1990; Ryzhik et al., 1996), we deduce the following 2D diffusivity:
D =
1
1 + 2γ2 [Vpl ∗ p
2 + γ2Vsl ∗ s
2
], (17)
where γ = Vp
Vs . The diffusivity (Dm and D), the mean free paths, and the transport mean free
paths are reported in Table 5 for different frequencies and κ = 0.5. Figure 3 shows the diffuse
intensities given by Equation 15, at a distance of 4 km, and using the diffusivity D. At both
frequencies, the solutions decay exponentially after a few seconds; they are compared with the
synthetic intensities, obtained through the Monte Carlo simulation for a single source in section
2.5. Before this comparison, anyway, some considerations are necessary
The time soil at which diffusion is reached is dependent on topography (Sanchez-Sesma,
1993). We concentrate anyway on area almost devoid of topography; if compared with volcanic
cones, where topography-effects may induce a faster diffusion reach (Wegler, 2003) or coda-
localization (Aki & Ferrazzini, 2000), the envelopes provided by the scattering coefficients could
diverge from the recorded envelopes. Two assumptions must also be made before setting the time
soil at which diffusion is reached. The first is that, relative to scattering, absorption is low (Marg-
erin et al., 2001). This is obviously unrealistic in a volcanic area, where intrinsic losses due to
the presence of fluids and/or melt are of importance, even if scattering is usually predominant
(Del Pezzo, 2008); nevertheless, we prefer to concentrate on the loss by scattering, before gaining
an insight into intrinsic losses. The second assumption is that strong variations in the density of
scatterers occur only after the time required to cross one l ∗ p,s. We will point out this last factor in
section 4.
2.5 Setting of the single scattering model
We follow a 2D scattering scheme valid for both P- and S-waves, similar to the ones of Margerin
et al. (2000) and Przybilla et al. (2006); for P-waves, propagation is shown in Figure 1. A volcano-
tectonic source emits most of its energy around 3 Hz (Chouet, 2003); hence, we consider a 3 Hz
Ricker point source located near the center of the caldera. The impulsive source radiates totally

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12 L. De Siena E. Del Pezzo C. Thomas A. Curtis L. Margerin
coherent P- and S-wave energies at t = t 0 p and t = t 0 s, respectively. Wave propagation is studied
using the scattering pattern defined in section 2.2 for different frequency bands (3 Hz and 18 Hz).
The direct intensities (I d p,s(t)) are recorded at a receiver-distance rd and at the P- and S-wave
arrival times (tp,s) as:
I d p,s(t) =
1 rd
exp(− ), (18)
2πrdc0δt lp,s
where δt = 0.01s is the time step of the simulation, and lp,s stands for the P- or S- mean free path.
To account for the source function, we assumed a source intensity at the source (I 0 ) sum of the P-
and S-waves intensities:
I 0 (t) = 1
2π δ(t − t0 p,s), (19)
where t 0 p,s are the P- and S-wave enucleation times. This is just an approximation, since the
Green function at the receiver presents tales (THIS LAST ASSERTION MUST BE BETTER
EXPLAINED IN THE NEXT DRAFT!!!!) To account for the real source shape we apply a con-
volution operator (Ri(t, f)), dependent on time t and frequency f, and correspondent to a 3 Hz
Ricker wavelet, to the whole envelope (Ryan, 1994):
Ri(t, f) = (1 − 2πf 2 t 2 ) exp(−π 2 f 2 t 2 ), (20)
with breadth (B(f), the time interval between the center of each of its 2 characteristics side lobes):
�
(6)
B(f) = = 0.26s (21)
πf
This convolution marks the source effects on the Monte Carlo solutions at every lapse time.
We simulate the length of a random walk by an exponential probability low, dependent on the
mean free paths (Przybilla et al., 2006). The distance between collisions (dd, Figure 1) is:
dd = 1
lp,s
exp(− r
lp,s
). (22)
The (eventual) change in polarization and the direction after scattering are triggered by the single
scattering coefficients and the relative scattering patterns (Figure 1, Tables 1, and Table 2). The
scattering coefficients are a direct measure of the probability of a PP, SS, PS, or SP scattering
(Przybilla et al., 2006). A random number defines the polarization; a second one determines the
direction after scattering (angle θ in Figure 1) using the lookup tables (e. g. Table 2).

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After each collision, the intensity scattered at a receiver at distance rsd (Isd) is recorded by
using the joint probability of detecting a particle of given polarization (Margerin et al., 2000):
rsd exp(− lP,S
Isd = P (w, a, k, θsd, κ) ∗ )
. (23)
The probability P is dependent on the recorded mode w (PP, SS ,PS, or SP), on θsd, the angle
between the propagation direction and the scatterer- receiver direction, and on the order of the von
Kármán function , κ. The dependency of P on θsd expresses the intrinsic forward anisotropy of
the problem. If a particle is shot towards East at the source, the receivers in the Eastern part record
intensities much higher than the ones in the Western part, since it is extremely unlikely for the
particle to scatter backwards.
The product of the P (or S) wave-number (kp,s = k) and the correlation length a is always larger
than 1. The total number of propagated particles is N = 10 6 . The simulation can be repeated for
a number of sources equivalent to the one considered in section 3. The location of the sources
varies randomly in a block of 1 km side, to account for the different hypocenters. The final image
is the average of the intensities produced by the different sources. The code is parallelized using
OpenMP (for the detection at different receivers) and MPI (for the simulation at different sources).
This is of extreme importance at 18 Hz, due to the reduced mean free path, producing a large
number of collisions.
In Figure 3a (18 Hz) and Figure 3b (3 Hz) we show the synthetic single scattering intensities
obtained for a single source, κ = 0.5, and at a source-station distanceof 4 km. Diffusion solutions
obtained by using Equations (13), (15), and (17) for κ = 0.5 and κ = 1 (dotted line and plus
line in Figure 3a,b) have the same magnitude order and follow the same exponential decay after
approximately 3tS; they result indistinguishable in logarithmic scale. We check in section 3.1 the
correspondence with real data envelopes recorded at a ray-distance of 4 km.
We have now a reference model to interpret real seismic envelopes recorded in the caldera.
Our aim is to understand the degree of coherency still present in early coda in different frequency
bands, and to highlight the actual lapse-time at which diffusion is reached. Our assumption is
rsd

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14 L. De Siena E. Del Pezzo C. Thomas A. Curtis L. Margerin
that, in a volcanic environment, an unbounded model is unreliable, especially if it is exclusively
triggered by a set of single scattering linear coefficients.
3 DATA
We use the high frequency velocity recordings of 11 low period three-component seismic stations
installed by the University of Wisconsin inside the Campi Flegrei caldera during the 1982-1984
seismic crisis (e. g. Pujol & Aster (1990)). In this period, a large number of low magnitude volcano-
tectonic earthquakes was located in a cube of 1 km side in the center of the caldera (G. et al., 1999).
The seismicity was relocated using non-linear algorithms, while the corresponding waveforms
were used by different authors to obtain velocity, attenuation, and scattering images of the caldera
(De Lorenzo et al., 2001; Zollo et al., 2002; Tramelli et al., 2006; Battaglia et al., 2006; De Siena
et al., 2010).
We create a data set of 212 waveforms produced by 59 earthquakes located in the 1 km side
cube. For each event-station pair we trace rays with the ray-bending approach of Block (1991) in
the 3D S-wave velocity structure of Battaglia et al. (2008). We then perform a 3D rotation of the
seismic waveforms in the path and transverse directions, employing both vertical and horizontal
recordings, and compute the root-mean-square (RMS) envelopes of the seismograms in the path
and transverse directions (Sato & Fehler, 1998). The transverse envelopes are the average of the
two tangential components. Finally, we shift each envelope to its S-wave arrival, and sum the traces
at each station in order to form an average envelope-per-station, or array envelope.
We first consider the normalized array envelopes obtained after filtering in the 18 Hz and 3 Hz
frequency bands, and recorded at 4 km distance (Figure 4a,b). In Figures 5-8, the corresponding
station is labeled as W02; we remark that this is not the station nearest to the epicenter area,
evidenced with a X in the center of the caldera, North of Pozzuoli. In the same figures, we show
the trace of the caldera rim at 2 km depth as a broad gray circumference; this signature is modeled
on the velocity and scattering tomographic images of Battaglia et al. (2008) and Tramelli et al.
(2006).

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3.1 Envelopes at 4 km ray-distance
Monte Carlo seismic envelopes caldera 15
At 18 Hz, the synthetic coda envelopes obtained via the single scattering coefficients and the array
coda envelopes at station W02 are in good agreement (Figure 4a). Respect to the correspond-
ing single source synthetic (Figure 3a), the direct wave and the coda of the array synthetic are
broadened due to the differences in source locations. The diffusion time is marked at 3tS: after
this limit, late coda intensities are of the same order of diffusion intensities, supporting the use of
coda techniques. On the other hand, between 3 s and 10 s, real intensities are twice the synthetic
ones. Single scattering seems a good starting point to synthesize envelopes in this frequency band.
Anyway, the cause of the large intermediate-time intensities must be investigated to synthesize the
complete envelope.
Single scattering array synthetics do not reproduce real intensities at 3 Hz (Figure 4b). The
variable source location does not model the broadening of coda waves nor for early nor for late
coda. Data and synthetic coda present a common behavior only after 9ts. The ratio between the
direct wave and the early coda intensities is not respected as well as the phase of the conversions,
absent in the synthetics. The broadening of the synthetic envelope is even smaller than at 18 Hz,
mainly due to the difference in mean free paths (Table 5): a longer mean free path implies a smaller
number of interaction, and less energy arriving to the station.
There are many possible explanations to this disagreement. First we are not considering a 3D
model, while every tomography performed in the area evidences the 3D nature of both the attenua-
tion and scattering wave-fields (Tramelli et al., 2006; Battaglia et al., 2008; De Siena et al., 2010).
The disagreement could be due e. g. to horizontal and vertical interfaces, like the one characteriz-
ing the whole caldera at 3 km depth(e. g. Zollo et al. (2008)). We could also model the envelope
using momentum scattering coefficients, as done e.g. by Yamamoto & Sato (2010). This provides
a greater degree of conversion, and an increase of backward scattering, therefore new phases aris-
ing in the intermediate coda. A direct application of momentum scattering coefficient is anyway
physically unfair. First, we should have more likely a boundary, causing the passage to a different
scattering regime. In addition, the associated transport mean free paths (Table 5) represent more
a measurement of diffusion than a characteristic scattering length. Finally, momentum scattering

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16 L. De Siena E. Del Pezzo C. Thomas A. Curtis L. Margerin
coefficients completely disregard forward propagation. This is good if we model a trapping bound-
ary; dissipation is anyway always present in real envelopes and a small amount of forward intensity
is necessary to simulate a not conservative medium.
3.2 Envelopes in the caldera
We will investigate the anomalous behavior of intermediate coda at 18 Hz, and of the whole enve-
lope at 3 Hz, including the effect of deterministic boundaries in the transport theory. Namely we
assume that these effects are produced by a passage between two scattering regime. This change
in the scattering texture represents anomalies of dimensions comparable with the wavelength, and
enhances large angle scattering. Nevertheless, before dealing with the new theory and the corre-
sponding synthetics, we will discuss the data envelopes in the whole caldera, to get a better insight
into the scattering propagation at different distances.
3.2.1 Envelopes at 18 Hz
In Figure 5 (radial) and Figure 6 (transverse), we image the normalized envelopes recorded at
each station after filtering in the 18 Hz frequency band. Each envelope is the average of the ones
produced by the 59 earthquakes in the X region. In this frequency band, we expect strong forward
SS scattering (Table 2), producing a relatively high and thin peak for the S-direct arrival, with a
smooth exponential decrease at larger lapse times. In terms of line-of-sight propagation (a topic
treated in section 3.3) we expect the degree of coherence in the coda to be lower at 18 Hz than
at 3 Hz; this is a consequence of the shorter mean free paths (Table 5) corresponding to a greater
number of collisions.
Most of the stations in the rest of the caldera present a clear direct impulsive onset. This is
followed in many cases by a second coherent phase of relatively large intensity in the early coda
(W21, W17, and, more evidently, W11 and W20). This high intensity is particularly evident on
both transverse and radial envelopes in the western and central quadrants (Figure 6, e.g. station
W20). Eastern stations present instead a larger incoherent broadening (e.g. compare W03 and

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Monte Carlo seismic envelopes caldera 17
W04), and a single coherent intensity is hardly distinguishable. Going farther from the epicenter
(e.g. stations W3) coda broadening increases almost everywhere.
3.2.2 Envelopes at 3 Hz
In Figure 7 and Figure 8, we show the normalized radial and transverse envelopes calculated in
the 3 Hz frequency band. We remark the differences between these envelopes, the ones calculated
at 18 Hz, and the synthetic modeling (Figures 4-6). Most stations in proximity of the epicenter
present a distinct direct peak (W17, W21, mainly W2), usually followed by a second peak of
equal or larger intensity (e. g. station W11). The envelopes recorded North, West, and South of the
epicenter (stations W4, W11, W20, and W21) present strong correlations; this common behavior
proves that a similar scattering medium is sampled by the wave-fields. Pulse broadening is large
everywhere, but it increases towards East (stations W2, W3, W15), at the expenses of correlation
with the other stations. The center of the caldera looks like a binary medium, as evidenced e.g.
by scattering tomography (De Siena et al., 2011). As the largest ray distances, coherent phases
usually disappear, and the envelopes present noise-like shape (stations W9, W10, and W14) with
an exponential decrease at lapse time as large as 6 ∗ tS. The highest peak is also as late as 3 ∗ tS
(e.g W5), hardly discernible from the early coda constant trend. Station W9, having a high direct
coherent peak at 18 Hz, is now totally noise-like, with almost no decrease with increasing lapse-
time.
3.2.3 Single earthquake envelopes
We evidence the interference effects in coda envelopes considering the propagation of the wave-
field produced from a single earthquake, and recorded at three stations East of the epicenter (Figure
9). The earthquake was recorded on the first of April 1984; both the earthquake magnitude (M = 2)
and the source mechanism (shown in Figure 9) were calculated by Zollo & Bernard (1993). Hence,
we remove these source effects from the unfiltered envelopes calculated at three different stations
(W17, W2, and W15), disposed at increasing distances on a continuous curve. We compute the

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18 L. De Siena E. Del Pezzo C. Thomas A. Curtis L. Margerin
S-wave travel times by using the same ray-bending approach used for array analysis (Thurber &
Eberhart–Phillips, 1999).
With increasing distance, the scattered intensity progressively increases at the expense of the
direct intensity, and the maximum intensity is evident at larger lapse-times. Hence, the envelopes
are clearly affected by broadening; we define its duration as the pulse broadening td, the lag time
between the S wave onset and the time when the envelope decays to the half of the maximum
amplitude (Saito et al., 2005). td is shown in Figure 9 as a horizontal broad gray segment on each
envelope; during propagation, it is always of the same order of the travel time, in contradiction with
the low-angle scattering approximation (Gusev & Abubakirov, 1996). Filtering at 18 Hz produces
more compact envelopes (Figure 5-6), and td is usually less than the travel time; this is never true
at 3 Hz.
This results correlate with the difficulty found in imaging attenuation and scattering at fre-
quencies lower than 6 Hz (Tramelli et al., 2006; De Siena et al., 2010). The broadened envelopes
recorded at Campi Flegrei are an evidence of a much more complicated scattering pattern than
the one described by a single scattering approximation. The shape of the envelopes of Figure 9,
particularly of the early coda is more similar to the one produced by a closed cavity (Derode et al.,
2003), and requires at least a change in the scattering properties of the medium.
3.3 Line-of-sight propagation
Data prove that high intensity coherent signals trigger the envelopes at intermediate times. The un-
usual broadening characterizes envelopes recorded in volcanic areas (Sato & Fehler, 1998; Saito
et al., 2005). The high intensity intermediate-time conversions question the opportunity of describ-
ing waveform propagation by using the usual multiple scattering techniques. A description in term
of line-of-sight propagation provides important markers to discern the latent degree of coherence
in local volcanic recordings. To find the cause of this unexpected coherence is the aim of this
description.
The total intensity < I > we calculate as a seismic envelope represents the magnitude of the
seismic wave-field, a mixture of coherent and incoherent intensities (I c and I i , Wu (1985)). We

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Monte Carlo seismic envelopes caldera 19
consider a wave incident on the medium characterized by the ACF and PSDF described in section
2.1. The field recorded at a receiver consists of the sum of the average (coherent) field (φ c 0 = 〈φ〉)
and the fluctuating (incoherent) field (φ f
0) (Ishimaru, 1997). The coherent S-wave intensity can be
expressed as:
Ic = exp(−g 0 sz − bz) (24)
where b is the absorption coefficient and g 0 s is given by Equation (8). Hence, the incoherent inten-
sity is:
Ii = exp(−bz)[1 − exp(−g 0 sz)]. (25)
We start neglecting any intensity contribution coming from the region outside of the medium
dimension, L (Figure 1).
Apart for intrinsic attenuation, the quantity triggering the ratio between coherent and incoher-
ent intensity is the optical distance ζ 0 = g 0 s ∗ z = z/ls, where ls is the mean free path. If ζ 0 1. Already at a
distance of 4 km, the average ray-distance for the Campi Flegrei caldera (De Siena et al., 2010), we
obtain ζ 0 > 1. The incoherent intensity is dominant over the coherent one; after the direct arrival
we obtain a field dominated by incoherent intensities, where coherent ones are still present. In a
way similar to laser propagation, 18 Hz direct waves are able to image the Earth structure crossed
by the ray, since only a small amount of energy is lost by backscattering. The ray approximation
is fulfilled at first order. After the direct wave, anyway, the field is mainly incoherent; no coherent
phases, except for the low-intensity phases due to conversion, are present in the envelopes. The
same calculation for frequency 3 Hz gives an optical distance of ζ 0 = 0.16. In this case, coherent
and incoherent intensities have comparable magnitudes (Ishimaru, 1997), and we expect coherent
phases in the data envelopes.
In a complicated volcanic medium like the one under study blobs of finite dimension diffract
in a conical sector the incident wave. At ray distance rd (e.g. Figure 1), the diffracted wave has a

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20 L. De Siena E. Del Pezzo C. Thomas A. Curtis L. Margerin
finite spread, a measure of pulse broadening given by sp ∼ = 3/(g 0 s ∗ rd) (Ishimaru, 1997); as long
as the spread is much less than the shadow of the blob, diffraction effects are small. Nevertheless,
assuming rd = 4 km we obtain sp ∼ = 19 at 3 Hz, and the effect of diffraction is dominant. At
18 Hz the same calculation leads to sp ∼ = .6: as expected, diffraction effects are less important,
even if still present, for high frequency waves. Incoherent intensity will be assigned from now on
to the single scattering solutions. The coherent phases still present after the direct arrival require
a change in the scattering texture of the medium, by the inclusion of one - or more - boundary
conditions in the medium.
4 LARGE-ANGLE SCATTERING FROM RANDOMLY DISTRIBUTED
SCATTERERS: INCLUDING DETERMINISTIC BACKSCATTERING FROM
BOUNDARY
Attenuation and scattering image Campi Flegrei caldera as a binary medium (e.g. Figure 5), where
scattering intensity at large frequencies is predominant in its center and borders (Tramelli et al.,
2006; De Siena et al., 2011). The presence of a rim surrounding the caldera is a feasible cause for
an enhancement of large angle scattering. This inclusion could obviously produce the deterministic
effect of a second wave-field directly produced at the boundary. In Figure X we show the synthetic
seismic envelopes obtained adding the reflection of each particle on the rim using Snell laws.
It is evident that these intensities are not relevant respect to the single scattering solution: this
is expected, since each reflection happens at a different point of the rim, and the corresponding
intensities are various order of magnitude less than the incoherent ones .
We propose a second mechanism which could better explain the broadened recorded en-
velopes: the enhancement of the large angle scattering due to a drastic change in the scattering
regime, where the caldera rim acts as a slab of randomly distributed scatterers (Ishimaru, 1997).
As a first approximation, we discuss the case of normal incidence of the particle on the rim - which
is a good approximation for a source in the center, and strong forward anisotropy - and we neglect
the dimension of the scatterers. We also assume that the total intensities follow the equation of

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transfer:
and continue to disregard phases.
Monte Carlo seismic envelopes caldera 21
d
ds I(r, s) = −ρσtI(r, s) + ρσt
�
p(s, s
4π 4π
′ )I(r, s ′ )dω ′ , (26)
Following Ishimaru (1977), we can apply the concept of zonation, where the passage from a
zone of anisotropic forward scattering to one of large angle backward scattering is totally defined
on the base of the path traveled by the particles. Namely, until each particle path is below a given
soil, forward single scattering is still dominant. The inclusion of a boundary allows the separation
of the total intensity in forward (I+) and backward (I−) intensities:
⎧
⎪⎨ I+(r, s) s · z > 0
I(r, s) =
⎪⎩ I−(r, s) s · z < 0
This is a major change respect to single scattering solutions, where only conversion scattering
could drastically change the direction from the incident one. We start considering this as the unique
boundary in our model, namely, an interface separating the caldera - from the unbounded region
outside of it (Figure Y).
Equation (26) becomes a set of two equations. For the forward intensity:
d
ds I+(r, s) = −ρσtI+(r, s) + ρσt
�
p(s, s
4π
′ )I+(r, s ′ )dω ′ + ρσt
�
4π
+2π
p(s, s
+2π
′ )I−(r, s ′ )dω ′
where +2π and −2π signs stand for integrations for s’ in the ranges s ′ · z > 0 and s ′ · z < 0.
Similarly, for s · z < 0:
d
ds I−(r, s) = −ρσtI−(r, s) + ρσt
4π
�
+2π
p(s, s ′ )I−(r, s ′ )dω ′ + ρσt
4π
�
p(s, s
+2π
′ )I+(r, s ′ )dω ′
Forward and backward intensities are connected; in the high frequency case, multiple scattering
forward intensity should be dominating, with a small contribution coming from backward. Instead,
in presence of a slab separating two media having different scattering features - deterministic in
nature - equation of transport - which is instead statistical - can be transformed.
We highlighted the small differences caused by a deterministic reflection on the boundary in
Figure X. With the new approach, each time the propagation distance of a particle overcomes the
source-station boundary, we use the transport equation for I− given by (29). The forward intensity
(27)
(28)
(29)

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22 L. De Siena E. Del Pezzo C. Thomas A. Curtis L. Margerin
I+ in last factor is given by the equation (28) computed at the previous scattering event, without
the contribution of backward intensity. Afterwards, the propagation will still be modeled by using
the transport equation in the form of Equations (27), (28), and (29), with the forward intensity.
4.1 The complete large-angle bounded Monte Carlo simulation
The problem of modeling intermediate coda intensities for a caldera is approached constraining
part of the medium with a circular boundary condition: this is similar to including a circular rim of
ray 5 km, and center at the epicenter in the single scattering simulation. The stations can be inside
the rim-boundary, in the proximity (on) the rim-boundary, or far-outside of it.
We can both:
• model a totally-dissipating boundary, where I− is not created after crossing the boundary,
• model a mixed reflective-dissipating boundary, and consider the I− contribution after crossing
the boundary.
The presence of a totally-dissipative boundary does not produce a backward intensity: its effect
is to dissipate the forward intensity only, and the equation of transport must not be updated. We
model dissipation by using momentum scattering coefficients and transport mean free paths, in-
stead of completely disregarding forward intensity after the boundary: this will have consequences
on the reach of diffusion and on the correlation of late coda synthetics with data.
In the case of a mixed reflective and dissipating boundary, the interface does not produce a
deterministic wave-field - an effective reflection phase at each scattering triggered by Snell’s laws.
Instead, each time the particle-source distance becomes larger than the source-boundary distance,
a source producing energy in a random backward direction on the boundary itself is added to the
intensities recorded at the different stations. The backward angle is triggered by the momentum
scattering coefficients - since they exclude the forward direction. Afterwards, Equations (27) - (29)
trigger propagation.
We can start considering the iteration solution n - eventually, the solution at scattering event n
- after crossing the boundary. From this point, Equation (29) rules backward intensity, and its first

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Monte Carlo seismic envelopes caldera 23
contribution I− is dependent on the (n − 1)th iteration solution of I+ - I+(n−1):
d
ds I−(r, s) = −ρσtI−(r, s) + ρσt
�
p(s, s
4π
′ )I−(r, s ′ )dω ′ + ρσt
�
4π
+2π
p(s, s
+2π
′ )I+(n−1)(r, s ′ )dω ′ .
For the forward first iteration after the boundary, I+(n−1), we can assume I−n−1 = I−0 = 0 in
Equation (28) - since backward propagation starts only after crossing the slab - and we continue
to consider the usual transport equation:
d
ds I+n(r, s) = −ρσtI+n(r, s) + ρσt
4π
�
(30)
p(s, s
+2π
′ )I+n(r, s ′ )dω ′ + 0. (31)
Nevertheless, we model dissipation using the transport mean free path, and the contribution of I+
after the boundary mainly affects late coda.
The first backward intensity recorded results dependent on forward intensity: this last can be
considered as the source of backward intensity rewriting Equation (30) as:
d
ds I−(r, s) + ρσtI−(r, s) − ρσt
�
p(s, s
4π
′ )I−(r, s ′ )dω ′ = ρσt
�
4π
+2π
p(s, s
+2π
′ )I+(n−1)(r, s ′ )dω ′ .
This corresponds to the single scattering solution for a statistical source created by the boundary.
The contribution of the simple boundary can be calculated as the probability of recording the
backscattered intensity at the n forward-intensity scattering. We rewrite the detected intensity of
equation (23) as forward intensity at the n scattering (I n+
sd ):
I n+
sd = KP (w, a, k, θsd,
rsd exp(−
κ) ∗
rsd
(32)
lP,S )
, (33)
where the different factors are still dependent on the single scattering coefficients. After crossing
the boundary, we add a second intensity (I (n−1) −sd):
I −
sd = KP ∗ (w, a, k, θsd,
rsd exp(−
κ) ∗
rsd
lP,S )
, (34)
Our first hypothesis is that P* is still dependent on single scattering coefficients after the inter-
action. This corresponds to a single scattering solution - an intensity that propagates in the same
medium, but with a large angle deviation respect to I+ (Figure X). Nevertheless the particles -
which have a certain size distribution and some absorption - produce a multiple scattering - not a
single scattering - intensity. Physically multiple scattering backscattering intensity is not depen-

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24 L. De Siena E. Del Pezzo C. Thomas A. Curtis L. Margerin
dent anymore on the scattering cross-section - therefore on the single scattering coefficients - but
on the power absorbed by the particle, the absorption cross-section σa (Ishimaru, 1997). The dif-
ference can be understood considering a slab of particles having a different scattering regime - not
a simple boundary separating the caldera from an unbounded region.
4.2 Slab
Ishimaru (1997) formulates and gives analytic solutions to the previous intensity equations for a
wave-field perpendicularly incident on a slab - therefore on a region having a finite dimension
d - in the small angle approximation (Figure Z). Introducing longitudinal (z) and transverse (ρ)
coordinates, and rewriting numerical density as ρn:
∂
∂z I−(z, ρ, s) + s · ∇tI−(z, ρ, s) = −ρnσtI−(z, ρ, s) + ρnσt
4π
where
Q = ρnσt
4π
�
� � +∞
−∞
p(s − s ′ )I−(z, ρ, s ′ )dω ′ + Q.
(35)
p(s − s
+2π
′ )I+(z, ρ, s ′ )dω ′ . (36)
Q is the source term for the backward simulation - like the one due to a simple boundary - but now
a second intensity dependent on d must be added at the receiver.
The analytic solution can be obtained if we consider that:
p(s − s ′ ) = σb
σt
, (37)
where σb is the backscattering cross section. We can also write I+(z) as the attenuated intensity at
the boundary for a plane wave having intensity I0 at the source:
I+(z) = I0 exp(−ρnσaB), (38)
where B is the source-boundary distance. Substituting Equations (37) and (38) in Equation (36)
we obtain the new source term:
Q = ρnσb
4π I0 exp(−ρnσaB) exp(−ρnσaz). (39)

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Monte Carlo seismic envelopes caldera 25
The analytical solution of Equation (35) with the new source term is given by (Ishimaru, 1997):
I−(z, s) = ρnσbI0d [1 − exp(−2ρnσad)]
exp(−ρnσaB). (40)
4π 2ρnσad
The single scattering backward solution is instead:
I f
−(z, s) = ρnσbI0d
4π
[1 − exp(−2ρnσtd)]
2ρnσtd
exp(−ρnσaB). (41)
This relations are applicable in the range θ > (kp,sl T P,S )− 1: for S-waves this corresponds to θ >??
or phenomena like backscattering enhancement should be modeled (Margerin & van Tiggelen,
2001; van Tiggelen et al., 2001). Nevertheless, these phenomena are excluded from our tractation,
since our source-detection distance is always larger than one elastic wavelength at both frequen-
cies.
It is evident that, in presence of a multiple scattering regime, backward scattering from single
scattering solution underestimates real intensity, and that absorption must be added to get the
correct ones.
5 RESULTS
5.1 Results at 18 Hz
The number of interaction before the boundary is large in this case, and the forward intensity
lost after crossing it is low. Dissipation after the boundary is modeled by using the momentum
scattering coefficients and transport mean free path - which are of the order of more than 100
km (Table 4). At 18 Hz (Figure 10a) this is not equivalent to completely disregarding scattering
propagation after the boundary. Most of the scattered energy is produced before: nevertheless, late
coda intensities increase due to the contribution of the momentum scattering coefficients (gray
dotted line). The ratio between direct and coda in real data can only be reached summing these to
the single scattering intensities (black dotted line).
The presence of a dissipative boundary causing large angle scattering does not produce the
arrival of a second intensity - or, better, a general broadening of the envelope in the intermediate
coda (Figure 10a). This happens considering a reflecting boundary - Figure 10b.

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26 L. De Siena E. Del Pezzo C. Thomas A. Curtis L. Margerin
NOT PRESENT IN FIGURE 12!!!
5.2 Results at 3 Hz
The result of the modeling with a dispersive caldera at 3 Hz is shown in Figure 11a, for the station
at 4 km inside the rim. The inclusion of momentum scattering dissipation allows fora relevant
increase of the late coda: this confirms their importance for modeling low-frequency coda even for
volcano-tectonic earthquakes (Sato et al., 2004).
THIS SIMULATION MUST BE UPDATED!!!
The peak near the S-direct arrival is due to the contribution of I n+ in equation (33). The
envelope modeled without considering I n+ after the crossing the boundary is shown in Figure Yb.
STILL TO BE DONE!!!!
A second wave-field at time almost equal to twice the travel time of the direct wave is created
by the interaction with the boundary.
6 DISCUSSION
Attenuation tomography has been performed with the ray method, and suggests the reliability of
a ray approximation at these frequencies (De Siena et al., 2010). At 3 Hz the average wavelength
is of the order of the correlation length of the area, as estimated by De Siena et al. (2011). This
creates large instabilities in the seismic coda, even at large lapse-time, which do not allow reliable
attenuation and scattering images (Tramelli et al., 2006; De Siena et al., 2010). These anomalies
are evidently produced by the interaction of the wave-field with the medium in the Mie scattering
regime: they can therefore provide new hints on the structure of the caldera.
The forward intensity after crossing the boundary is ruled by momentum scattering coeffi-
cients, in the case of dissipating caldera. It is therefore quickly lost due to the dimension of the
transport mean free path (Tables 4). We do not change the propagation reference system to pre-
serve total intensity. Backward intensity (I − ) is still ruled by the mean free path: what makes the
difference between the two frequency ranges is the ratio between this last quantity (around .16 km
for 18 Hz and 5 km for 3Hz) and the source-boundary distance (5 km). At 18 Hz the interaction

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Monte Carlo seismic envelopes caldera 27
with the boundary happens after several mean free paths, and the wave-field is triggered by the
scattering regime inside the caldera. At 3 Hz the interaction with the boundary is continuous, and
the wave-field suffers too many interactions to become diffusive at the expected time-soil. In both
cases the effect of a boundary is seen clearly at intermediate times: at 3 Hz anyway if affects also
the late coda, forbidding the diffusion reach.
Synthetic and data examples show that, if a deterministic approach provides unrealistic mod-
els of waveform envelopes, a totally stochastic one is also unreliable, if boundary conditions are
not taken into account. As the frequency increases, momentum scattering coefficients are able to
model late coda intensities. Diffusion solutions prove the efficiency of diffusion-derived methods
- as coda-normalization - for times greater than 3tS in this frequency range. At intermediate times
anyway, the effect of the boundary is also of importance in this frequency range, while a determin-
istic location of larger scale structure by a direct application of Snell laws results unfeasible.
At 3 Hz the passage to a different scattering regime and of the boundary is highlighted in the
data by the recursive coherence at almost constant in the intermediate and late coda. We model
these evidences introducing three spatial quantities: mean and transport mean free path, and dis-
tance between source and boundary. The correlation between synthetic and data envelopes is strik-
ing, even in the 2D simulation. This is actually not surprising: the rim considered in the simulation
can be easily transformed in an horizontal interface - as the one generally present under Campi
Flegrei (Zollo et al., 2008; Battaglia et al., 2008; De Siena et al., 2010). The only parameter to
provide results the source-boundary distance, both in the vertical and horizontal directions.
De Siena et al. (2010) and De Siena et al. (2011) recently highlighted the differences in at-
tenuation and scattering characterizing the eastern and western part of the Campi Flegrei caldera.
The eastern part is dominated by intrinsic attenuation, with its center approximately located in
the zone of Solfatara. A seismic interface is located below Campi Flegrei at around 3 km depth,
as evidenced by reflection seismology (Zollo et al., 2008); attenuation tomography reveals that
this interface is broken by an high attenuation anomaly in its center (black region in Figures 5-8).
In the central and western parts, scattering attenuation is much stronger than intrinsic (De Siena
et al., 2011). We expect that the differences between real and synthetic envelopes could be reduced

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28 L. De Siena E. Del Pezzo C. Thomas A. Curtis L. Margerin
modeling the deterministic effect of these scattering anomalies, and that, contrary to what happens
for high frequency late coda, first-order coda decrease might be heavily influenced by large-angle
scattering at lower frequencies
7 CONCLUSIONS
We demonstrate by means of Monte Carlo simulation and array processing of real envelopes that
a statistical approach cannot deal with the description of whole seismic envelopes in an active
caldera, unless deterministic interfaces are included into the scattering medium. We sketched this
interfaces starting from a simplistic model - mainly introducing a circular boundary, as well as
a linear drastic change into the scattering properties of the caldera. The effect is anyway statis-
tical and can be dealt with transport theory - namely with the inclusion of large-angle scattering
enhanced by interfaces, and careful use of dissipation. We implemented in a single scattering sim-
ulation both effects through the direct application of backward transport theory (Ishimaru, 1997).
We define the scattered intensities through the definition of first-order and momentum scattering
coefficients - this last are used to account for the dissipation outside the caldera rim.
This model is still applicable when the fractional fluctuation are small, and successful for
higher-frequency bands because of self-similarity of the media spectra; however, as noted, the
characteristic time of the Markov approximation increases to the order of the travel time even at 18
Hz, testifying not only the inadequacy of single scattering solutions, but also of the diffusion model
at lower frequencies. The intermediate envelopes in the frequency ranges investigated cannot be
modeled without the addition of deterministic effects, causing enhanced large angle intensities.
The passage to a complete multiple scattering synthetic model in this interval results necessary.
The Markov approximation has been so far the main road to add diffraction effects to the
single scattered wave-field in order to describe envelope broadening and maximum amplitude
decay. We approach the problem including coherency information directly into the Monte Carlo
simulation, without the requirement for additional propagators. The envelope demonstrate the need
for a momentum scattering coefficient to model late coda envelopes recorded in an active caldera,
and the persistence of the coherency for the first seconds after the P- and S-wave onsets. Diffusion

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Monte Carlo seismic envelopes caldera 29
is usually reached after a lapse time of 10 s (3 times the S-wave travel time) at 4 km for 18 Hz, as
demonstrated by the comparison of synthetics, real data, and diffusion solutions.
Modeling a volcano-tectonic earthquake envelope - namely, the energy-with-time produced
by an impulsive source - even at small source receiver distance, requires a great computational
effort. It can lead to direct measures of the attenuation properties of the medium, and, at a second
step, to their imaging through tomography (Tramelli et al., 2006; De Siena et al., 2010). These are
important issues for the assessment of volcanic risk and volcanic eruption monitoring; they give
the possibility to locate melt and feeding systems trough their effect on seismic waveforms. In the
future, the effect of boundaries and slab will have to be added to usual tomography techniques, to
get a major insight into the effect of interference on attenuation and scattering images.
The interference effect of a second wave-field can be used as marker to understand complex
processes which could have consequences on the application of coda wave interferometry (Snieder
et al., 2002). Anyway, a complete Markov FD simulation will be necessary, not to rely on char-
acteristic time to account for diffraction effects. In addition, Monte Carlo modeling with complex
deterministic scattering pattern - e. g. circular and elliptic ring - could give new hint on the gener-
ation of this interference effects for real calderas.
Finally, the observation of a dissipative range at long ray distances, and the strong increase of
coherent intensity caused by larger scale anomalies suggest the application of turbulence theory
to the modeling of coda envelopes. The equations modeling the wave intensity for a medium
characterized by 2 different scales of heterogeneities - namely eddys and blobs - could provide an
unprecedented view on the topic of volcanic coda wave modeling.
ACKNOWLEDGMENTS
The HPC-EUROPA project provided the necessary resources for the development of the program:
We thank the whole staff at EPCC (Edinburgh Parallel Computing Center) in Edinburgh, and
particularly Dr. Adam Carter, who were fundamental for the parallelization of the code.

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30 L. De Siena E. Del Pezzo C. Thomas A. Curtis L. Margerin
References
Aki, K. & Ferrazzini, V., 2000. Seismic monitoring and modeling of an active volcano for pre-
diction, Journal of Geophysical Research, 105, 16617–16640.
Battaglia, J., Troise, C., Obrizzo, F., Pingue, F., & Natale, G. D., 2006. Evidence of fluid migra-
tion as the source of deformation at campi flegrei caldera (italy), Geophysical Research Letters.,
33(L01307), doi:10.1029/2005GL024904.
Battaglia, J., Zollo, A., Virieux, J., & Dello Iacono, D., 2008. Merging active and passive data sets
in traveltime tomography: the case study of Campi Flegrei caldera (Southern Italy), Geophysical
Prospecting, 56, 555–573.
Block, L. V., 1991. Joint hypocenter–velocity inversion of local earthquake arrival time data in
two geothermal regions, Ph.d. dissertation, Massachusetts Institute of Technology, Cambridge.
Brenguier, F., Shapiro, N. M., Campillo, M., Ferrazzini, V., Duputel, Z., Coutant, O., & Nerces-
sian, A., 2008. Towards forecasting volcanic eruptions using seismic noise, Nature Geoscience,
January, doi :10.1038/ngeo104.
Chouet, B., 2003. Volcano seismology, PAGEOPH, 160, 739–788.
De Lorenzo, S., Zollo, A., & Mongelli, F., 2001. Source parameters and three-dimensional at-
tenuation structure from the inversion of microearthquake pulse width data: Qp imaging and
inferences on the thermal state of the Campi Flegrei caldera (southern Italy), Journal of Geo-
physical Research, 106, 16265–16286.
De Siena, L., Del Pezzo, E., & Bianco, F., 2010. Campi Flegrei seismic attenuation image:
evidences of gas resevoirs, hydrotermal basins and feeding systems, Journal of Geophysical
Research, 115(B0), 9312–9329.
De Siena, L., Del Pezzo, E., & Bianco, F., 2011. A scattering image of Campi Flegrei from the
autocorrelation functions of velocity tomograms, Geophysical Journal International, 184(3),
1304–1310.
Del Pezzo, E., 2008. Seismic wave scattering in volcanoes, in Earth Heterogeneity and Scattering
Effects of Seismic Waves, vol. 50 of Advances in Geophysics, chap. 13, pp. 353–369, Elsevier,
Amsterdam.

694
695
696
697
698
699
700
701
702
703
704
705
706
707
708
709
710
711
712
713
714
715
716
717
718
719
720
721
Monte Carlo seismic envelopes caldera 31
Derode, A., Larose, E., Tanter, M., de Rosny, J., Tourin, A., Campillo, M., & Fink, M., 2003.
Recovering the green’s function from field-field correlations in an open scattering medium (l),
Journal of the Acoustic Society of America, 113(6), 2973–2976.
Fehler, M. C., Sato, H., & Huang, L. J., 2000. Envelope broadening of outgoing waves in 2d
random media: a comparison between the markov approximation and numerical simulations,
Bulletin of the Seismological Society of America, 90, 914–928.
Fukushima, Y., Nishizawa, O., Sato, H., & Ohtake, M., 2003. Laboratory study on scattering
characteristics of shear waves in rock samples, Bulletin of the Seismological Society of America,
93(1), 253–263.
G., O., Civetta, L., Gaudio, C. D., de Vita, S., Vito, M. A. D., Isaia, R., Petrazzuoli, S. M., Ric-
ciardi, G., & Ricco, C., 1999. Short-term ground deformations and seismicity in the resurgent
Campi Flegrei caldera (Italy): an example of active block-resurgence in a densely populated
area, in Volcanism in the Campi Flegrei, vol. 91, pp. 415 – 451, eds G., O., Civetta, L., &
Valentine, G. A., Elsevier.
Gusev, A. A. & Abubakirov, I. R., 1987. Monte-carlo simulation of record envelope of a near
earthquake, Physics of the Earth and Planetary Interiors, 49, 30–36.
Gusev, A. A. & Abubakirov, I. R., 1996. Simulated envelopes of non-isotropically scattered body
waves as compared to observed ones: Another manifestation of fractal heterogeneity, Geophys-
ical Journal International, 127, 49–60.
Hennino, R., Tregueres, N., Shapiro, N. M., Margerin, L., van Tiggelen, B. A., & Weaver, R. L.,
2001. Observation of Equipartition of Seismic Waves, Physical Review Letters, 86(15), 3447–
3450.
Hoshiba, M., 1991. Simulation of multiple–scattered coda wave excitation based on the energy
conservation law, Physics of the Earth and Planetary Interiors, 67, 123–136.
Ishimaru, A., 1977. Theory and Application of Wave Propagation and Scattering in Random
Media, Proceedings of the IEEE, 65(7), 1030 – 1061.
Ishimaru, A., 1997. Wave Propagation and Scattering in Random Media, The IEEE/OUP Series
on Electromagnetic Wave Theory, Oxford University Press and IEEE Press, Oxford, 2nd edn.

722
723
724
725
726
727
728
729
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742
743
744
745
746
747
748
749
32 L. De Siena E. Del Pezzo C. Thomas A. Curtis L. Margerin
K., H. & Levander, A. R., 1992. A stochastic view of lower crustal fabric based on evidence from
the Ivrea zone, Geophysical Research Letters, 19, 1153 – 1156.
Margerin, L., 2006. Attenuation, transport and diffusion of scalar waves in textured media,
Tectonophysics, 416, 229–244.
Margerin, L. & Nolet, G., 2003. Multiple scattering of high-frequency seismic waves in the deep
Earth: Modeling and numerical examples, Journal of Geophysical Research, 108(B5), 2234–
2249.
Margerin, L. & van Tiggelen, M. C. B. A., 2001. Coherent backscattering of acoustic waves in
the near field, Geophysical Journal International, 145, 593–603.
Margerin, L., Campillo, M., & Tiggelen, B. V., 1998. Radiative transfer and diffusion of waves in
a layered medium: New insight into coda Q, Geophysical Journal International, 134, 596–612.
Margerin, L., Campillo, M., & van Tiggelen, B. A., 2000. Monte Carlo simulation of multiple
scattering of elastic waves, Journal of Geophysical Research, 105(B4), 7873–7892.
Margerin, L., van Tiggelen, B. A., & Campillo, M., 2001. Effect of absorption on energy parti-
tioning of elastic waves in the seismic coda, Bullettin of the Seismological Society of America,
91, 624 – 627.
Przybilla, J. & Korn, M., 2008. Monte carlo simulation of radiative energy transfer in continu-
ous elastic random media - three-component envelopes and numerical validation, Geophysical
Journal International, 173, 566–576.
Przybilla, J., Korn, M., & Wegler, U., 2006. Radiative transfer of elastic waves versus finite
difference simulations in two-dimensional random media, Journal of Geophysical Research,
111(B0), 4305–4317.
Przybilla, J., Wegler, U., & Korn, M., 2009. Estimation of crustal scattering parameters with
elastic radiative transfer theory, Journal of Geophysical Research, 111(B0), 4305–4317.
Pujol, W. S. & Aster, R., 1990. Joint hypocentral determination and the detection of low-velocity
anomalies. an example from the phlegrean fields earthquakes, Bulletin of the Seismological
Society of America, 80(1), 129–139.
Ryan, H., 1994. Ricker, Ormsby, Klauder, Butterworth - A Choice of Wavelets, Canadian Society

750
751
752
753
754
755
756
757
758
759
760
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762
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764
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767
768
769
770
771
772
773
774
775
776
777
of Exploration Geophysicist Recorder, 19(7), 8 – 9.
Monte Carlo seismic envelopes caldera 33
Rytov, S. M., Kravtsov, Y. A., & Tatarskii, V. I., 1987. Principles of Statistical Radiophysics, vol.
4, Wave Propagation Through Random Media, Springer-Verlag, New York.
Ryzhik, L., Papanicolaou, G., & Keller, J. B., 1996. Transport equations for elastic and other
waves in random media, Wave motion, 24, 327 – 370.
Saito, T., Sato, H., & Ohtake, M., 2002. Envelope broadening of spherically outgoing waves in
three-dimensional random media having power law spectra, Journal of Geophysical Research,
107(B5), 2089–2103.
Saito, T., Sato, H., Ohtake, M., & Obara, K., 2005. Unified explanation of envelope broadening
and maximumamplitude decay of high–frequency seismograms based on the envelope simula-
tion using the markov approximation: Forearc side of the volcanic front in northeastern honshu,
japan, Journal of Geophysical Research, 110(B0), 1304.
Sanchez-Sesma, F., 1993. Topographic effects for incident P, SV and Raileigh waves, Tectono-
physics, 218(1–3), 113–125.
Sato, H., 1989. Broadening of seismogram envelopes in the randomly inhomogeneous litho-
sphere based on the parabolic approximation: southeastern honshu, japan, Journal of Geophys-
ical Research, 94, 17735–17747.
Sato, H. & Fehler, M. C., 1998. Seismic Wave Propagation and Scattering in the heterogeneous
Earth, Springer and Verlag, New York, USA.
Sato, H., Fehler, M., & Saito, T., 2004. Hybrid synthesis of scalar wave envelopes in two-
dimensional random media having rich short-wavelength spectra, Journal of Geophysical Re-
search, 109(B0), 6303–6314.
Shapiro, S. A. & Hubral, P., 1999. Elastic Waves in Random Media, Fundamentals of Seismic
Stratigraphic Filtering, Springer, Berlin, Germany.
Shiomi, K., Sato, H., & Ohtake, M., 1997. Broad–band power–law spectra of well–log data in
japan, Geophysical Journal International, 130, 57–64.
Snieder, R., Grêt, A., , Doumba, H., & Scales, J., 2002. Coda wave interferometry for estimating
nonlinear behavior in seismic velocity, Science, 295, 455–473.

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803
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805
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Thurber, C. H. & Eberhart–Phillips, D., 1999. Local earthquake tomography with flexible grid-
ding, Computers and Geosciences, 25, 809–818.
Tramelli, A., Pezzo, E. D., Bianco, F., & Boschi, E., 2006. 3d scattering image of the campi
flegrei caldera (southern italy). new hints on the position of the old caldera rim, Physics of the
Earth and Planetary Interiors, 155, 269–280.
Turner, J. A., 1998. Scattering and Diffusion of Seismic Waves, Bullettin of the Seismological
Society, 88(1), 276–283.
van Tiggelen, B. A., Margerin, L., & Campillo, M., 2001. Coherent backscattering of elastic
waves: Specific role of source, polarization, and nearfield, Journal of the Acoustic Society of
America, 110(3), 1291–1298.
Weaver, R. L., 1990. Diffusivity of ultrasound in polycrystals, Journal of the Mechanics and
Physics of Solids, 1(38), 55–86.
Wegler, U., 2003. Analysis of Multiple Scattering at Vesuvius Volcano, Italy, using Data of the
TomoVes active seismic experiment, Journal of Volcanology and Geothermal Research, 128,
45–63.
Wegler, U. & Lühr, B. G., 2001. Scattering behaviour at merapi volcano (java) revealed from an
active seismic experiment, Geophysical Journal International, 145, 579–592.
Wu, R. S., 1985. Multiple scattering and energy transfer of seismic waves - separation of sact-
tering effect for intrinsic attenuation, i, Geophysical Journal of the Royal Astronomical Society,
82, 57–80.
Yamamoto, M. & Sato, H., 2010. Multiple scattering and mode conversion revealed by an active
seismic experiment at asama volcano, japan, Journal of Geophysical Research, 115(B0), 7304–
7317.
Yoshimoto, K., 2000. Monte carlo simulation of seismogram envelopes in scattering media,
Journal of Geophysical Research, B3(105), 6153–6162.
Zeng, Y., 1991. Compact solution for multiple scattered wave energy in time domain, Bulletin of
the Seismological Society of America, 81, 1022–1029.
Zeng, Y., 1993. Theory of scattered p– and s–wave energy in a random isotropic scattering

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807
808
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827
828
829
830
831
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medium, Bulletin of the Seismological Society of America, 83, 1264–1276.
Zeng, Y., Su, F., & Aki, K., 1991. Scattering wave energy propagation in a random isotropic
scattering medium i. theory, Journal of Geophysical Research, 96, 607–619.
Zollo, A. & Bernard, P., 1993. Fault mechanisms from near-source data: joint inversion of S
polarizations and P polarities Seismic Evidence for a Low–Velocity Zone in the Upper Crust
Beneath Mount Vesuvius, Geophysical Journal International, 104, 441–451.
Zollo, A., D’Auria, L., Matteis, R. D., andJ. Virieux, A. H., & Gasparini, P., 2002. Bayesian
estimation of 2-d p-velocity models from active seismic arrival time data: imaging of the shallow
structure of mt vesuvius (southern italy), Geophysical Journal International, 151, 566–582.
Zollo, A., Maercklin, N., Vassallo, M., Dello Iacono, D., Virieux, J., & Gasparini, P., 2008. Seis-
mic reflections reveal a massive melt layer feeding Campi Flegrei caldera, Geophysical Re-
search Letters, 35(L112306), doi:10.1029/2008GL034242.
8 FIGURE LEGENDS
Figure 1: Sketch of the single scattering simulation for a P-wave source (big black square) produc-
ing direct intensity (Id) and scattered intensities (Isd) in a medium of average velocity V0. At each
collision on scatterers (small black squares), the scattered intensity is recorded at each station (tri-
angles). This intensity is dependent both on the scattering angle, θsd, and on the scatterer-receiver
distance, rsd. rd is the source-station distance, while dd is the distance between scatterers. A fi-
nite probability of PS (and SP) conversion exists; in this case, a PS conversion deviates strongly
deviates the particle from its incidence direction.
Figure 2: Power spectral density functions (PSDFs) of two-dimensional von Kármántype ran-
dom media used in this study.
Figure 3: Time dependence of the diffusion equations for κ = 0.5 (dotted line) κ = 1 (plus
line) at 18 Hz (a) and 3 Hz (b) over-imposed on the result of the first-order Monte-Carlo simulation
for a single source (black continuous line).
Figure 4: Transverse envelopes (gray line) recorded at station W02 in the 18 Hz (a) and 3Hz

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36 L. De Siena E. Del Pezzo C. Thomas A. Curtis L. Margerin
(b) frequency bands are compared with the corresponding single scattering synthetics (black gray
line).
Figure 5: Normalized radial envelopes obtained at 18 Hz in the Campi Flegrei caldera. The
X represents the area of the epicenters (a block of 1 km side). The results of the separation of
intrinsic and scattering attenuation performed by De Siena et al. (2011) is over-imposed in the
center. Light gray and black represent high scattering attenuation and high intrinsic attenuation,
respectively. The dotted broad circumference is a simplified model of the caldera rim at 1.5 km
deduced by scattering and velocity images (Tramelli et al., 2006; Battaglia et al., 2008).
Figure 6: Same as Figure 5 for the transverse component.
Figure 7: Same as Figure 5 for the 3 Hz frequency band.
Figure 8: Same as Figure 5 for the transverse component and the 3 Hz frequency band.
Figure 9: Map of the Campi Flegrei caldera with three unfiltered envelopes recorded at stations
W17, W02, and W15. The epicenter is in the 1 km block of Figures 5-8; its focal mechanism was
used to correct the envelope intensities, using the results of Zollo & Bernard (1993). The gray
circle (caldera rim) is also shown.
Figure 10: Sketch of the simulation with caldera rim as boundary.
Figure 11: Envelope recorded at station W02 with the result of the simulation at 18 Hz with a
dissipative caldera (continuous black line). Intensities produced by propagation inside the caldera
rim are shown with a black dotted line. After the reaching the boundary, the total intensity is
triggered by the transport mean free path (gray dotted line).

Figure 1.
Monte Carlo seismic envelopes caldera 37

38 L. De Siena E. Del Pezzo C. Thomas A. Curtis L. Margerin
Figure 2.
Table 1. Single scattering coefficients at different frequencies and for different von Kármán orders
κ = 0.5
κ = 1
3 Hz 18 Hz
gss 0.03819 1.61805
gpp 0.00776 0.31248
gps 0.00205 0.00233
gsp 0.00090 0.00102
gss 0.06334 2.56472
gpp 0.01260 0.49416
gps 0.00109 0.00022
gsp 0.00048 0.00010

Intensity
Intensity
x 10 -4
3
2
1
0
x 10 -5
4
3
2
1
0
5
5
10 15
Time (s)
10 15
Time (s)
Figure 3.
Monte Carlo seismic envelopes caldera 39
20
20
25
25
a)
30
b)
30

40 L. De Siena E. Del Pezzo C. Thomas A. Curtis L. Margerin
Figure 4.

Figure 5.
Monte Carlo seismic envelopes caldera 41

42 L. De Siena E. Del Pezzo C. Thomas A. Curtis L. Margerin
Figure 6.

Figure 7.
Monte Carlo seismic envelopes caldera 43

44 L. De Siena E. Del Pezzo C. Thomas A. Curtis L. Margerin
Figure 8.
Figure 9.

Figure 10.
Figure 11.
Monte Carlo seismic envelopes caldera 45

46 L. De Siena E. Del Pezzo C. Thomas A. Curtis L. Margerin
Table 2. Angular distribution of SS (first column) and PS (second) scattering at 3 Hz. The angle is randomly
selected, and defines the deviation from the incident direction after scattering.
Random number SS angle (rad) PS angle (rad)
1 0.0010 0.1274
2 0.0020 0.1626
3 0.0030 0.1880
4 0.0040 0.2088
5 0.0050 0.2227
6 0.0060 0.2432
7 0.0072 0.2582
... ... ...
97 0.5998 2.6570
98 0.7798 2.7222
99 1.2875 2.8124
100 2.2496 3.1378
101 3.1411 3.1415
102 4.0336 3.1454
103 4.9957 3.4708
104 5.5034 3.5610
105 5.6802 3.6262
... ... ...
106 6.0260 6.0250
195 6.2772 6.0400
196 6.2782 6.0605
197 6.2792 6.0744
198 6.2802 6.0952
199 6.2812 6.1206
200 6.2822 6.1558
201 6.2832 6.2832

Monte Carlo seismic envelopes caldera 47
Table 3. Momentum scattering coefficients at different frequencies and for different von Kármán orders
κ = 0.5
κ = 1
3 Hz 18 Hz
g m ss 0.00222 0.00370
g m pp 0.00180 0.00335
g m ps 0.00136 0.00143
g m sp 0.00060 0.00063
g m ss 0.00103 0.00136
g m pp 0.00111 0.00139
g m ps 0.00046 0.00008
g m sp 0.00020 0.00004
Table 4. Cosine weighted scattering coefficients at different frequencies and for different von Kármán orders
κ = 0.5
κ = 1
3 Hz 18 Hz
gss 0.03610 1.61305
gpp 0.00704 0.31007
gps 0.00119 0.00140
gsp 0.00052 0.00061
gss 0.06186 2.56211
gpp 0.01185 0.49266
gps 0.00072 0.00015
gsp 0.00031 0.00007
Table 5. Mean and transport mean free paths, and diffusion constants obtained at 3 Hz and 18 Hz for κ = .5.
(km) 3 Hz 18 Hz
lp 101.94 3.18
ls 25.58 0.62
l m p 316.02 208.98
l m s 354.99 231.12
l ∗ p 545.17 268.02
l ∗ s 427.98 193.62
D m 404.50 262.77
D 559.50 262.47