3D numerical deformation model of the intrusive event forerunning the 2001 Etna eruption

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3D numerical deformation model of the intrusive event forerunning the 2001 Etna eruption

Gilda Currenti, Ciro del Negro, Gaetana Ganci, Danila Scandura

To cite this version:

Gilda Currenti, Ciro del Negro, Gaetana Ganci, Danila Scandura. 3D numerical deformation model of the intrusive event forerunning the 2001 Etna eruption. Physics of the Earth and Planetary Interiors, Elsevier, 2008, 168 (1-2), pp.88. �10.1016/j.pepi.2008.05.004�. �hal-00532148�

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Accepted Manuscript

Title: 3D numerical deformation model of the intrusive event forerunning the 2001 Etna eruption

Authors: Gilda Currenti, Ciro Del Negro, Gaetana Ganci, Danila Scandura

PII: S0031-9201(08)00089-7

DOI: doi:10.1016/j.pepi.2008.05.004

Reference: PEPI 4940

To appear in:

Physics of the Earth and Planetary Interiors

Received date: 3-8-2007

Revised date: 5-5-2008 Accepted date: 7-5-2008

Please cite this article as: Currenti, G., Del Negro, C., Ganci, G., Scandura, D., 3D numerical deformation model of the intrusive event forerunning the 2001 Etna eruption,

Physics of the Earth and Planetary Interiors (2007), doi:10.1016/j.pepi.2008.05.004

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Accepted Manuscript

3D NUMERICAL DEFORMATION MODEL OF THE INTRUSIVE EVENT 1

FORERUNNING THE 2001 ETNA ERUPTION 2

3

Gilda Currenti¹, Ciro Del Negro¹, Gaetana Ganci^1,2, Danila Scandura^1,3 4

5

1Istituto Nazionale di Geofisica e Vulcanologia, Sezione di Catania, Italy 6

2Dipartimento di Ingegneria Elettrica Elettronica e dei Sistemi, Università di Catania, Italy 7

3Dipartimento di Matematica e Informatica, Università di Catania, Italy 8

9

Abstract 10

3D finite elements models were carried out in order to evaluate the ground deformation produced 11

by the dike intrusion occurred at Etna volcano in the 2001. The Finite Element Method (FEM) 12

allows for considering the medium heterogeneity and the real topography of the volcano. Firstly, 13

we validated the method in a homogeneous elastic half-space and compared the results with those 14

obtained from analytical dislocation models. We performed a convergence analysis to quantify the 15

discretization errors, which are sensitive to the size and quality of the mesh elements and can 16

cause inaccurate numerical solutions. Secondly, several numerical models were conducted to 17

appreciate how the complex distribution of elastic medium parameters and the topography make 18

the numerical results differ from the simple analytical solutions. The numerical model, which 19

account for the real topography of Etna volcano and medium heterogeneity, well match the ground 20

deformation observed at the GPS stations between 11 and 16 July 2001. It appears that 21

topography and heterogeneities provide more details about the ground deformation near the 22

summit crater area that are missed when homogeneous half-space models are used. However, 23

only few discrepancies between the analytical and the numerical models are observed at the GPS 24

stations, which coarsely sample the volcanic edifice far away from the summit craters.

25 26

Key words: finite-element method, ground deformation, medium heterogeneity, Etna volcano 27

28

Introduction 29

Geodetic observations play an important role for the monitoring of the volcanic activity and for the 30

quantitative evaluation of the geophysical processes preceding and accompanying volcanic unrest 31

(Bonaccorso and Davis, 2004). Fundamental insight about volume changes in the magma 32

reservoir and the dynamics of dike intrusion processes is generally obtained using ground 33

deformation (Voight et al.1998; Battaglia et al. 2003; Murase et al. 2006). However, an accurate 34

interpretation of deformation signals is necessary for improving the monitoring of active volcanoes 35

as well as developing a better understanding of the pre-eruptive mechanisms that produce them.

36

Volcanic processes are complex geophysical systems and it is difficult to derive analytical 37

deformation models, unless important simplifications and approximations are taken into account 38

(McTigue, 1986; Carbone et al., 2007). The simplest way to model the inflation or deflation of a 39

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magma chamber at depth is a point dilatation source embedded in a homogeneous elastic half- 1

space (Mogi, 1958), which is quite appropriate to approximate spherical over-pressured source 2

when the radius is far smaller than its depth. Finite extended sources have been also analyzed and 3

spherical and ellipsoidal models have been devised to better model pressurized magma chamber 4

(McTigue, 1986; Davis, 1986; Battaglia & Segall, 2004; Yang and Davis, 1986). Instead, dike 5

intrusions and eruptive fissures are usually represented by an opening crack. Tensile dislocation 6

theory has been successfully applied to model rectangular shaped intrusion (Okada, 1992).

7

All these analytical elastic models are quite attractive because of their computational feasibility 8

(Currenti et al., 2007a). However, investigations on the effect of a non-uniform elastic medium are 9

indispensable for describing more realistic models (Currenti et al., 2007b). For the static elastic 10

deformation in a multilayered half-space, there exist a number of semi-analytical solutions 11

(Bonafede et al., 2002; Wang et al., 2003). Moreover, volcanic areas are usually characterized by 12

consistent topographic relief that is responsible for significant effects (Williams & Wadge, 1998).

13

Several studies have been carried out by means of numerical methods to assess the effect of 14

topography on ground deformation generated by magmatic pressurized sources in volcanic areas 15

(Cayol & Cornet, 1998; Williams & Wadge, 2000).

16

In the present work, we applied the 3D FEM (Finite Element Method) to asses the role that medium 17

heterogeneity and topography effects may play in modelling intrusive events at Etna volcano. We 18

focused our attention on modelling the intrusive events forerunning the 2001 eruption. Firstly, the 19

numerical results were compared to the analytical ones computed using the dislocation theory 20

(e.g., Okada, 1992). We assumed a homogenous, isotropic and half-space medium, in order to 21

validate the results of the numerical model with respect to the mesh resolution that strongly affects 22

the solution accuracy. Secondly, a 3D analysis was performed, by introducing the real topography 23

of Mt Etna, in order to examine the effect of the topography on the deformation field. In addition to 24

the topography, the effects of the elastic medium heterogeneities were included considering a 3D 25

complex distribution of medium properties in terms of Young modulus and Poisson ratio inferred 26

from seismic tomography investigations.

27 28

Numerical Model 29

The definition of a FEM problem requires to set up boundary conditions, optimal size of domain, 30

meshing, and the type of the elements that are not known a priori. Therefore, a robust preliminary 31

set-up is needed to obtain realistic results. We carried out the computations with the FE software 32

COMSOL (COMSOL 2004). In order to validate the size of domain and the mesh resolution in the 33

FEM model, we performed several computations and compared the numerical deformation fields 34

with the analytical solution from Okada’s dislocation theory (1992).

35

We considered a 3D FEM model reproducing a rectangular dislocation source in a homogeneous 36

and isotropic half-space. The geometry of the intrusion source is reported in Table I. We assumed 37

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stress-free boundary condition at the upper surface, and zero displacement values at bottom and 1

lateral boundaries. The dislocation source is modelled as a rectangular discontinuity surface by 2

means of “pair elements”. The elements are added in pairs along the surface rupture, where in one 3

element the dislocation +U and in the other −U is assigned. A non null homogeneous dislocation 4

over the rupture area is imposed and an “identity pair” condition along the remaining surface is 5

enforced to assure the continuity of deformation fields on the volume surrounding the dislocation 6

source. The medium is assumed to be elastic with Lame’s coefficients λ=µ, namely Poisson 7

ratio ν =0.25, and Young modulus of 30 GPa. A computational domain of 50x50x35 km is 8

meshed with tetrahedral elements.

9

In the finite element analysis a critical step is the meshing of the computational domain. The 10

number of elements of the mesh largely influences the accuracy of the solution. Starting from a 11

coarse mesh, we incrementally refined the meshed domain increasing the number of nodes. We 12

examined the accuracy of the solution in terms of discretization error due to the meshing procedure 13

to test how the mesh resolution can affect the numerical solution. A convergence analysis was 14

performed to quantify the discretization errors caused by the mesh resolution. Eight different 15

meshes were constructed by refining the domain. The number of nodes was increased to improve 16

the mesh resolution and the solution accuracy (Table II). Besides the number of nodes, also the 17

mesh quality should be taken into account to assure an accurate solution. The mesh not only must 18

properly describe the geometrical boundaries, but it must not have degenerated elements that 19

make the numerical solution instable. Therefore, we defined a factor to track the mesh-quality 20

among the several meshes. For tetrahedral elements, more the element approaches an equilateral 21

tetrahedron, more the numerical solution is stable and accurate (Field, 2000; Edelsbrunner, 2001).

22

For each element in the mesh we evaluated its quality with respect to the equilateral tetrahedron 23

as:

24

3 2

)

(A B C D

f V

q= + + + 25 (1)

where V denotes the volume of the tetrahedron, A, B , C, and D are the areas of its faces and 26

3

=216

f is a normalizing coefficient ensuring that the quality of an equilateral tetrahedron is 27

equal to 1. The quality of the worst elements for each mesh is reported in Tab. II. In all the 28

computations the minimum mesh quality was higher than 0.2.

29

To quantify the accuracy of the numerical solutions, a mean misfit function with respect the 30

analytical solution is defined as:

31

∑

=

−

=

∆ ^N

i

AN i FE

i U N

U U

1

/ )

( ⁽²⁾

32

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Accepted Manuscript

where U_i^FE is the computed deformation at the i-th node for the numerical model, U_i^AN is the 1

corresponding result for the analytical model, and N is the total number of nodes in the meshed 2

domain. To provide a reference for the quality of the fit, we have normalized each mean misfit 3

value by the average magnitude of the deformation field:

4

N U U

M

N

i AN

i /

)

(

∑

=

1

5 (3)

The misfit function δ_m =∆

(

U

) /

M

(

U

),

m=x

,

y

,

z, computed for each deformation component, 6

decreases quickly as the mesh quality is improved (Fig. 1). The displacements due to strike-slip, 7

tensile, and thrust faulting in a homogeneous medium are compared to Okada’s dislocation 8

models. The results from the FEM models are generally consistent with those from the dislocation 9

model in a homogeneous half-space as the mesh reaches a number of about 15000 elements. The 10

ratio ∆(U)/M(U) is less than 0.1 for all the displacement components and no significant 11

improvements are obtained when using finer meshes. Therefore, the numerical results are found to 12

be in nearly perfect agreement with the analytical solutions.

13

The domain was proved to be big enough to avoid numerical artefacts in the solution due to the 14

finite boundary condition. The external boundaries of the domain depend on the extent of the 15

displacement components, which is especially related to the depth of the source. We performed 16

several tests for a tensile source, whose geometric parameters as the same as before (Tab. I), and 17

the depth is varied from 1 to 4 km. As the depth of the tensile source increases the deformation 18

components show a greater spatial extent. The deformation field decays to 10% of the maximum 19

value within 10 km from the source when the source is at 1 km depth, within 33 km when the 20

tensile source is at 4 km. The source depth controls the shape and the extent of the deformation 21

field, and so regulates also the computational domain size. As a rule of thumb, the domain size 22

must be at least twice the deformation field extent, or in other words about 40 times the depth of 23

the source to obtain an accurate numerical solution.

24 25

2001 Etna Eruption 26

The numerical modelling procedure is applied to study the ground deformation observed during the 27

Etna 2001 eruption. Late night on 12 July 2001 an intense seismic swarm preceded and 28

accompanied the opening of a system of fractures at the base of the South East (SE) crater and 29

continued without any fall in energy release till the late night of July 16. Some 800 earthquakes 30

were recorded by the early morning of July 13, from a total of 2645 occurring between 12 and 18 31

July (Patanè et al., 2002). In concomitance with the seismic crisis, marked ground deformation (50 32

to 120 mm) were recorded at the GPS permanent stations. Tilt variations were very small in the 33

northern flank, of few µrad in the western flank, and much more marked at stations close to the 34

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fracture field (Bonaccorso et al., 2002). The deformation was mainly cumulated during the late 1

night of July 12 and the following two days. Bonaccorso et al. (2002) modelled the ground 2

deformation changes recorded in the days before the eruption onset (between July 11 and 16) by 3

means of analytical solutions. The tilt and GPS data inversion indicated the response to a tensile 4

mechanism with an opening dislocation of ca. 3.5 m that evidenced an intrusion in the volcano 5

edifice along a ca. N-S direction (Bonaccorso et al., 2002). The location of the tensile source lies 6

within the zone of the seismic swarm that occurred during the magma ascent.

7

Although the analytical model provided an acceptable representation of the dyke intrusion occurred 8

on the south flank, it is not able to justify all the details. The simplified assumption of elastic half- 9

space medium disregards effects caused by the topographic relief and the presence of medium 10

heterogeneity, while they could play a role in the modelling procedure. To evaluate how the 11

topography and medium heterogeneity could affect the computed ground deformation, we carried 12

out numerical models including the real topography of the volcano edifice and the medium 13

heterogeneity inferred from seismic tomography study. In the numerical computations we used the 14

source geometry (Tab. III) obtained by the analytical inversion model from Bonaccorso et al.

15

(2002). We evaluate three numerical models in which we considered: (i) an homogeneous elastic 16

medium with a flat surface, (ii) an homogeneous elastic medium with the real topography of Mt 17

Etna, and (iii) an elastic heterogeneous medium with the real topography relief.

18 19

Flat Topography and Homogeneous Medium 20

We performed a 3D FEM model considering a rectangular tensile dislocation embedded in a 21

homogeneous, isotropic and elastic half-space. The medium is assumed to be elastic and 22

poissonian (λ =µ) with a Young modulus of 30 GPa. This numerical model is equivalent to the 23

Okada’s analytical model. Indeed, all the deformation components well match the analytical ones 24

within the computational error, assuring the validity of the domain size, mesh resolution and 25

boundary conditions used in the numerical computations (Fig. 2).

26 27

Real Topography and Homogeneous Medium 28

To examine the effects of topography on the deformation at Mt Etna, we included in the numerical 29

model the real 3D topography of the volcano using a digital elevation model from the 90 m Shuttle 30

Radar Topography Mission (SRTM) data. The domain was modelled using a total of 15585 31

elements and 3288 nodes. Numerical results highlighted the influence of topography on the 32

deformation field. Ground deformation components differ significantly in shape and amplitude from 33

those of the simple model based on the assumption of a homogeneous half-space medium (Fig.

34

3). The discrepancies between the numerical and analytical models are mainly restricted to the 35

volcano summit area where the topography of Mt Etna is most irregular. The symmetry of the 36

solution observed in the analytical solution is lost in the numerical results since the volcano edifice 37

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is rather asymmetric having a strong mass deficit in the eastern sector with respect to the western 1

sector in correspondence of Valle del Bove. The amplitudes of the deformation components are a 2

little smaller because the effective distance between the source and the ground surface is 3

increased. To evaluate the effect of topography, we compare numerical and analytical solutions for 4

source at different depths. We find out that the topography strongly influences the ground 5

deformation especially when the source is quite shallow. . The residuals, the difference between 6

the analytical and the numerical models, gradually increase as the source became shallower (Fig.

7

4). This discrepancy depends also on the horizontal distance between the observation points and 8

the position of the source. We compute the residuals on different grid sizes centred on the tensile 9

dyke extending: (i) 5x5 km, (ii) 10x10 km, (iii) 20x20 km and (iv) 30x30 km. As the grid extends 10

farther from the dyke the numerical solution converges to the analytical one.

11 12

Real Topography and Heterogeneous Medium 13

Finally, we evaluated the deformation pattern caused by a tensile dislocation embedded in a 14

heterogeneous medium. Instead of using a simple multi-layered crustal rigidity model (Currenti et 15

al., 2006), a complex distribution of elastic medium properties was considered. At each node of the 16

mesh different values of Young modulus and Poisson ratio are assigned on the basis of seismic 17

tomography investigations. In this way, smooth elastic medium heterogeneities, instead of sharp 18

boundary layers, may be included that are effectively most likely for many geological settings.

19

Moreover, this procedure allow to better represent the presence of a high rigidity body centered 20

below the SE sector, inferred from recent seismic tomography studies (Chiarabba et al., 2000). We 21

used P-wave and S-wave seismic velocities to derive the elastic medium parameters. Particularly, 22

the Young modulus was estimated by using the following equation (Kearey and Brooks, 1991):

23

( ν ) ρ +

= 2 V

_s²

1 E

(4)

24

where Vs is the seismic shear wave propagation velocity, and ρ the density of the medium which 25

was fixed to 2500 kg/m³. Instead, the values of Poisson ratio were obtained using the equation 26

(Kearey and Brooks, 1991) 27

] 2 ) / ( 2 /[

] 2 ) /

[( ² − ² −

= V_p V_s V_p V_s

ν (5)

28

where V_p is the seismic P-wave propagation velocity. On the basis of Eqs. 4 and 5, the Young 29

modulus varies from 11.5 GPa to 133 GPa, while the Poisson ratio is in the 0.12-0.32 range. We 30

considered three different media with: (a) homogeneous Young modulus and heterogeneous 31

Poisson ratio, (b) heterogeneous Young modulus and homogeneous Poisson ratio, and (c) 32

heterogeneous Young modulus and heterogeneous Poisson ratio. In the (a) case where only the 33

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Poisson ratio is heterogeneous, the discrepancy with respect to the homogeneous model is 1

negligible. The models with heterogeneous Young modulus are those that engender higher 2

perturbations in the numerical models considering the inhomogeneity of medium properties.

3

Nevertheless, results compared to homogeneous medium model do not reveal significant 4

differences in pattern and intensity of the deformation field (Fig. 5). The influences of the medium 5

heterogeneity seem to affect mainly the horizontal components (Fig. 5 a-b) with respect to the 6

vertical uplift (Fig. 6 c). The discrepancies are on the order of few centimetres. To better appreciate 7

the discrepancies with respect to the analytical results, we compared the displacement field at the 8

locations of GPS continuously running stations. Figure 6 shows the horizontal component of GPS 9

data together with results achieved by the numerical models. The observed vectors are obtained 10

from the interval comparisons 11-16 July before the eruption onset. Discrepancies of at most a few 11

cm between the numerical and analytical models are observed at the GPS stations (Fig. 6). In fact, 12

almost all the GPS stations are located far enough from the summit area and are not significantly 13

affected by the topography. A more general agreement is achieved with respect to the deformation 14

computed from the analytical model at ERDV station. Nevertheless, a higher discrepancy with 15

respect to the observed deformation still remains at the EMFN station on the eastern flank of the 16

volcano and at EGDF on the northern flank. Both the topography and medium heterogeneity 17

included in the numerical model cannot account for the displacements observed by the EMFN and 18

EGDF geodetic stations.

19

Among the considered models, a good match is achieved by the model including only the 20

topography (cyan arrows in Fig, 6). The residuals, the difference between observed field and 21

numerical solutions, show a standard deviation: of 2.3 cm for the analytical model, of 1.7 cm for the 22

numerical model with topography, and of 2.0 cm for the numerical model including the topography 23

and the medium heterogeneity. When the tomography is included in the numerical model the 24

residual increases. The topography is known with higher resolution and accuracy, whereas the 25

medium heterogeneity is estimated from the 3D velocity model. Three factors can alter this 26

estimate: (i) the low spatial resolution of 3 km in the tomography model by Chiarabba et al. (2000), 27

(ii) the variation of 3D velocity model in the analyzed period, (iii) the difference between the static 28

elastic modulus and the dynamic elastic modulus deduced from the P-wave velocity. Recently, 29

Patanè et al. (2006) obtained new 3D velocity models at higher resolution on Mt Etna, detecting 30

significant variations in the elastic parameters during different volcanic cycles. Further comparisons 31

of different 3D velocity models could show up how the accuracy on medium heterogeneity can 32

affect the numerical solutions.

33

It is worth to note that we adopted source parameters from analytical inversion of GPS data by 34

Bonaccorso et al. (2002). To enhance the match between the data and the numerical solution 35

including topography and medium heterogeneity, we performed an exploration through the 36

parameter space varying the width, the depth and the opening of the dyke. The range of variability 37

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of these parameters was also chosen on the basis of the location of the seismic swarm 1

accompanying the dyke intrusion (Fig. 6 Musumeci et al., 2004). The earthquakes were quite 2

shallow from 1000 m asl to -2000 bsl. Starting from the parameter suggested by the analytical 3

inversion (Bonaccorso et al., 2003), we performed a grid search on the numerical model, which 4

includes both the topography and the medium heterogeneity, varying the width within 2000–3000 5

m, the opening within 2-4 m and the depth within 0-1400 m asl. The residuals show a variation 6

range of 2.5 centimeters with a minimum of 1.9 (Fig. 7). The “best” model, associated with the 7

minimum residual, has an opening of the dyke of 2.5 m, a width of 3000 m, and a depth of 1400 m, 8

that well matches the hypocenters of the seismic events occurred during its intrusion (Fig. 7).

9

However, the best solution does not differ too much with respect to the numerical solution obtained 10

using the dyke parameters inferred from the analytical inversion. Even if the parameters vary in a 11

wide range, the solutions for all the considered models computed at the GPS stations are quite 12

similar. The standard deviation affecting measurements in the horizontal components ranges 13

between 0.2 and 0.5 cm while, over measurements in the vertical component, between 1 and 2 cm 14

(Bonaccorso et al., 2003). Thus, a maximum uncertainty of ± 1 cm and ± 4 cm at the 95%

15

confidence interval can be assumed over the horizontal and vertical component, respectively.

16

Therefore, considering the measurement errors and the computed misfit values, all the numerical 17

models can be assumed as valuable solutions.

18

We also computed the sensitivity of fit to the “best” numerical model for the depth and width 19

parameters (Fig. 8). The sensitivities with respect to these parameters evaluated at the GPS 20

stations are continuous decreasing functions with residuals from 0.2 to 1.6 cm. The low residuals 21

and the sensitivity values point out that it is not possible to better constrain the depth and width 22

parameters of the dyke only on the basis of ground deformation at the available GPS stations. To 23

understand whether this uncertainty is due to the far location of the GPS stations or to the inherent 24

ambiguity of the used model, we recomputed the sensitivities functions over a grid of points 25

extending 10x10 km around the source (Fig. 9). In this case, the depth sensitivity increases, while 26

the width sensitivity is almost unchanged. Therefore, a better coverage of the summit area could 27

enhance the capability to constrain the source depth.

28 29

Conclusions 30

3D finite elements models were carried out in order to evaluate the ground deformation produced 31

by dip-slip, strike-slip and tensile dislocation sources. The numerical results were compared to 32

analytical solutions from Okada’s dislocation theory for validating their accuracy. The meshing 33

procedure was found to be a very delicate procedure in setting up the numerical models. However, 34

the refined mesh, which gives a good compromise between the computational cost and the 35

solution accuracy, is not so computational heavy to run. The FEM technique enabled to consider 36

both the topography and the medium heterogeneity in modeling the intrusive event occurred at Mt 37

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Etna in July 2001. The topography significantly alters the general pattern of the ground deformation 1

especially near the volcano summit. On the contrary, the medium heterogeneity does not strongly 2

affect the expected deformation. Several computations were performed to better understand the 3

effects induced by the numerical model. Although we did not perform a new numerical inversion of 4

the deformation data, we explored the source parameter space with a grid search. In comparison 5

to the analytical solution, the numerical model associated with the minimum residual provides 6

tensile source with a reduced opening and a wider extension which well matches the hypocenters 7

of the seismic swarm occurred during the dyke intrusion. However, this numerical model does not 8

differ too much from the analytical one. The numerical and the analytical solutions are both 9

valuable solutions given the measurement errors and the location of GPS stations. A good match 10

between the observed and the computed deformation was achieved at most stations. An exception 11

is represented by the EMFN station located within the Valle del Bove in the eastern flank, and 12

EGDF station located near the North East Rift. The numerical model is not able to account for the 13

higher discrepancies at the EMFN station. Therefore, the heterogeneity and the topography cannot 14

give an explanation for the displacement observed on the eastern flank. Indeed, the response of 15

the eastern flank of the volcano to the intrusive event is more composite because of the presence 16

of intricate geological structures. The eastern sector is confined in the northern flank by the North- 17

East Rift zone and by the East-West trending left lateral-transtensive Pernicana fault (Fig. 6), while 18

the southern sector is delimited by the Mascalucia-Trecastagni fault system (Rasà et al., 1996).

19

The EGDF and EMFN stations are closer to these geological structures that are not taken into 20

account into the numerical model. A new model should be set up including these discontinuity 21

surfaces to better draw the complex interaction between the magmatic intrusions and the tectonic 22

processes responsible for the kinematics of the eastern flank.

23 24

Acknowledgements 25

This work is supported by the Etna project (DPC-INGV 2004-2006 contract) and the European 26

Commission, 6th Framework Project – ‘VOLUME', Contract No. 08471. This work was developed 27

in the frame of the TecnoLab, the Laboratory for the Technological Advance in Volcano 28

Geophysics organized by DIEES-UNICT and INGV-CT. We thank the editor Keke Zhang and two 29

anonymous reviewers for their useful comments on this paper.

30 31

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33

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Williams, C. A. and Wadge, G., 2000. An accurate and efficient method for including the effects of 1

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Figure Captions 1

2

Figure 1. Misfit values between numerical and analytical solutions for all the displacement 3

components. Dip-slip (a), strike-slip (b) and tensile (c) dislocation sources are considered.

4

Figure 2. Deformation field due to a tensile dislocation in a homogeneous, isotropic and elastic 5

half-space: (a) eastern component, (b) northern component, (c) vertical uplift. Contour lines are at 6

0.05 m.

7

Figure 3. Deformation components caused by the 2001 intrusive dike in a homogeneous, isotropic 8

and elastic medium with the real topography of Mt Etna: (a) eastern component, (b) northern 9

component, (c) vertical uplift. Contour lines are at 0.05 m.

10

Figure 4. Comparisons between the analytical and numerical solutions including the Mt Etna 11

topography. The computations were performed for different depths of the source and varying the 12

extent of the grid point on which the misfit is evaluated.

13

Figure 5. Deformation fields of the tensile crack in a heterogeneous medium with the real 14

topography of Mt Etna: (a) eastern component, (b) northern component, (c) vertical uplift. Contour 15

lines are at 0.05 m.

16

Figure 6. Observed and expected horizontal displacements at the GPS stations between 11 and 17

16 July 2001. The vertical uplift, computed by the “best” numerical model using real topography 18

and medium heterogeneity, is shown. The black circles represent the epicentres of the seismic 19

event accompanying the dyke intrusion (after Musumeci at el., 2004).

20

Figure 7. Residuals between the observed ground deformation and numerical solutions. The circle 21

represent the numerical model with source parameters coming from the analytical inversion, while 22

the square is the “best” numerical model (with minimum residual).

23

Figure 8. Sensitivity of fit to depth and width parameters of the intrusive dyke. The residuals on the 24

x-component (circle), the y-component (square), and vertical uplift (triangle) are computed at the 25

locations of GPS stations.

26

x-component (circle), the y-component (square), and vertical uplift (triangle) are computed over a 28

grid extending 10x10 km at the ground surface around the source.

29 30 31 32 33 34 35 36 37

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Accepted Manuscript

Tables 1

2

Table I. Geometrical parameters of the dislocation source.

3 4

Length:

L [m]

Width:

W [m]

Inclination angle:

δ[degree]

Strike angle:

θ[degree]

Depth of top:

d [m]

Dislocation:

B r

[m]

1000 1000 90° 0° 1000 1

5 6 7 8

Table II. Statistics for the eight simulations in which the mesh parameters are changed.

9 10

Simulation # 1 # 2 # 3 # 4 # 5 # 6 # 7 # 8

Mesh vertices 2228 2513 2570 3195 3977 5101 8591 36196 Elements

(Tetrahedral) 9858 11236 11505 14663 18888 25141 45021 205700 Minimum

element quality 0.3544 0.3083 0.3441 0.3109 0.3417 0.2960 0.3050 0.2454 11

12

Table III. Parameters of the intrusive dike forerunning the 2001 eruption (Bonaccorso et al., 2002).

13 14

Tensile dislocation: B

r [m]

Length:

L [m]

Width:

W [m]

Depth of top:

d [m]

Inclination angle:

δ [degree]

Strike angle:

θ[degree]

3.5 2200 2300 1400 asl 90° 7°

15 16 17 18 19 20 21

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1 2 3

4 5

Figure 6. Observed and expected horizontal displacements at the GPS stations between 11 and 6

16 July 2001. The vertical uplift, computed by the “best” numerical model using real topography 7

and medium heterogeneity, is shown. The black circles represent the epicentres of the seismic 8

event accompanying the dyke intrusion (after Musumeci at el., 2004).

9

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Accepted Manuscript

1 2

Figure 7. Residuals between the observed ground deformation and numerical solutions. The circle 3

represent the numerical model with source parameters coming from the analytical inversion, while 4

the square is the “best” numerical model (with minimum residual).

5 6 7 8 9

11 10

x-component (circle), the y-component (square), and vertical uplift (triangle) are computed at the 13

locations of GPS stations.

14 15 16

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Accepted Manuscript

1 2

x-component (circle), the y-component (square), and vertical uplift (triangle) are computed over a 4

grid extending 10x10 km at the ground surface around the source.

5 6