Elastoplastic modeling of volcano ground deformation

Earth and Planetary Science Letters xxx (2010) xxx–xxx

EPSL-10398; No of Pages 8

Contents lists available at ScienceDirect

Earth and Planetary Science Letters

j ourna l homepage: www.e lsev ie r.com/ locate /eps l

Elasto-plastic modeling of volcano ground deformation

Gilda Currenti, Alessandro Bonaccorso, Ciro Del Negro ⁎, Danila Scandura, Enzo BoschiIstituto Nazionale di Geofisica e Vulcanologia, Sezione di Catania, Italy

⁎ Corresponding author.E-mail address: [email protected] (C. Del Negro).

0012-821X/$ – see front matter © 2010 Elsevier B.V. Adoi:10.1016/j.epsl.2010.05.013

Please cite this article as: Currenti, G., edoi:10.1016/j.epsl.2010.05.013

a b s t r a c t

Article history:Received 12 October 2009Received in revised form 6 May 2010Accepted 12 May 2010Available online xxxx

Editor: L. Stixrude

Keywords:ground deformationelasto-plastic rheologyfinite element methodEtna volcano

Elasto-plastic models for pressure sources in heterogeneous domain were constructed to describe, assess, andinterpret observed deformation in volcanic regions. We used the Finite Element Method (FEM) to simulate thedeformation in a 3D domain partitioned to account for the volcano topography and the heterogeneous materialproperties distribution. Firstly, we evaluated the extent of a heated zone surrounding the magmatic sourcecalculating the temperature distribution by a thermo-mechanical numerical model. Secondly, we includedaround the pressurized magma source an elasto-plastic zone, whose dimension is related to the temperaturedistribution. This elasto-plastic model gave rise to deformation comparable with that obtained from elastic andviscoelastic models, but requiring a geologically satisfactory pressure. We successfully applied the method toreview the ground deformation accompanying the 1993–1997 inflation period on Mt Etna. The modelconsiderably reduces the pressure of a magma chamber to a few tens of MPa to produce the observed surfacedeformation. Results suggest that the approach presented here can lead to more accurate interpretations andinferences in future modeling-based assessments of volcano deformation.

ll rights reserved.

t al., Elasto-plastic modeling of volcano gro

© 2010 Elsevier B.V. All rights reserved.

1. Introduction

When modeling deformation in volcanic regions, the assumptionof elastic rheology can be an over simplification. The elasticapproximation is generally appropriate for small deformations ofcrustal materials with temperatures below the brittle–ductile transi-tion, which is between 600 K and 1000 K, depending principally oncomposition and strain rate. Although elastic behavior well describesthe upper 15 km of the Earth's crust (Ranalli, 1995; Turcotte andSchubert, 2002), in active volcanic areas the variation in brittle–ductile transition may be related to the perturbation in geothermalgradient due to the presence of intrusive bodies or varying saturationstate of fluid-filled fractured rock matrix (Mandal et al., 2007).Materials surrounding long-lived magmatic sources are heatedsignificantly above the brittle–ductile transition and rocks no longerbehave in a purely elastic manner, but permanently deform because ofthe plastic deformation (Fung, 1965; Ranalli, 1995). Therefore, thethermal state makes the elastic approximation inappropriate and cangreatly influence the surface deformation field. Although mechanicaldeformation models based on the assumption of elastic rheology havebeen successfully and widely applied to interpret geodetic dataacquired on several volcanoes (e.g. Walsh and Decker, 1971; Yang etal., 1992; Okada and Yamamoto, 1991; Bonaccorso and Davis, 1999;Currenti et al., 2008a), in many cases elastic models seem unable toreproduce the observed deformation unless unrealistic overpressures

or lower effective rigidity moduli are considered (Newman et al.,2001, 2006; Bonaccorso et al., 2005; Trasatti et al., 2005).

Under the assumption of elastic rheology, the amplitude of thedeformation field for a spheroidal pressure source is linearly related tothe intensity factor V∙ΔP/μ (V=volume, ΔP=pressure change, μ=ri-giditymodulus). Hence, the effect of changes in pressure and in volumeof a spheroidal source on the ground deformation cannot be separatedfrom the estimate of rheology parameters. Because of the close linkbetween crustal rigidity, source pressure and deformation, a lack ofinsight into the rheology contributes to increase the uncertainty onsource volumes and associated pressures (Newman et al., 2001;Bonaccorso et al., 2005). Differences between the static elastic modulusand the dynamic elastic modulus deduced from seismic velocities canfurthermore alter these estimates (Ciccotti and Mulargia, 2004; Chengand Johnston, 1981). Therefore, the correct estimate of the magmaticsource pressure from the ground deformation is still an open issue.

The inclusion of an anelastic rheology also affects the estimate ofpressure change. Particularly, a viscoelastic shell around themagmaticsource requires less overpressure than a purely elastic model toproduce comparable deformation (Dragoni and Magnanensi, 1989;Newman et al., 2001; Newman et al., 2006; Del Negro et al., 2009). Afurther reduction of the overpressure could be obtainedby consideringan elasto-plastic rheology for the material surrounding the magmaticsource. This behavior is expected to enhance deformationwith respectto both the viscoelastic and the elastic rheologies, and hence to requirea geologically satisfactory pressure (Trasatti et al., 2003).

In this work, we evaluated the temperature dependence of theground deformation using a thermo-mechanical numerical model.Finite Element Method (FEM) was used to compute the temperature

und deformation, Earth Planet. Sci. Lett. (2010),

Fig. 1. Examples of common types of hardening functions in the elasto-plastic rheology:A) perfect plastic solid, B) linear strain hardening solid, and C) power-law hardening solid.

2 G. Currenti et al. / Earth and Planetary Science Letters xxx (2010) xxx–xxx

distribution in the medium generated from a volcano chamber hostingthe magma. We developed an elasto-plastic model in which themagmatic source, embedded in an elastic heterogeneous 3D medium,is surrounded by a region where high temperature induces plasticbehavior. Thismodel was applied to re-analyze the ground deformationaccompanying the1993–1997 inflationperiodonMtEtna that precededthe 2001 and 2002–03 eruptions.

2. Temperature-dependent elasto-plastic numerical model

It is reasonable to assume that rocks near the magma source areconsiderably heated and weakened beyond the brittle-ductile transi-tion temperature,where elasto-plastic rheology ismore appropriate todescribe the mechanical behavior of the rocks. This behavior has alsobeen proven by several laboratory experiments, conducted on rocksunder high temperature and low pressure, which show a strongdecrease in static modulus with increasing temperature (e.g. Rocchi etal., 2002; Tuffen et al., 2008). In order to investigate the effects of theplastic rheology, we used FEM to construct a temperature-dependentelasto-plastic model in the region around the magma chamber. Weused the values of the rheological parameters estimated by Rocchi et al.(2004) and Balme et al. (2004) from experiments under simulatedconditions on actual rocks from Etna. Firstly, we developed atemperature model to derive the rheology behavior of the mediumfrom the computed temperature distribution. Secondly, we con-structed an elasto-plastic deformation model in which the magmaticsource is embedded in an elastic medium and surrounded by atemperature-dependent region of elasto-plastic material. Thirdly, wecompared the elastic and anelastic strain responses of the magmatichost rocks using a simple 3D axi-symmetric model.

2.1. Temperature distribution

To derive the temperature profile, we numerically solved the heatconduction equation:

∇·ðk∇TÞ = −A zð Þ ð1Þ

where T is the temperaturefield, k is thermal conductivity, z is thedepth,and A(z)=ASexp(−z/b) is the crustal volumetric heat production,whereAs is thevolumetric rate of heatproductionat ground surface, andb is a characteristic depth of the order of 10±5 km. Since thedeformation timescales are much shorter than those over which themagma chamber evolution takes place, the temperature distribution,and hence the elasto-plastic volume, can be considered as steady. Asboundary condition at the ground surface, we assumed that the surfaceis kept constant at atmospheric temperature, since the thermalconductivity of the air is much smaller than that of the ground. Weassigned the geothermal temperature values at bottom and lateralboundaries, because they are far enough to not be affected by themagmatic source. A steady-state geothermal profile, which holds for theupper lithosphere, was used (Ranalli, 1995; Turcotte and Schubert,2002):

TðzÞ = Ts +qmzk

� �+

Asb2

k

!1−e−z=b� �

ð2Þ

where Ts is the surface temperature, and qm is theheatflowcoming fromthemantle. A continuous refilling of themagma chamberwas simulatedby setting the temperature on the sourcewall. Physically, this boundarycondition is equivalent to stating that the magma wall acts as a heatingsource (Dragoni et al., 1997; Civetta et al., 2004).

Please cite this article as: Currenti, G., et al., Elasto-plastic modelindoi:10.1016/j.epsl.2010.05.013

2.2. Elasto-plastic rheology

When a rock is strained beyond an elastic limit, Hooke's law nolonger applies. The behavior of rocks beyond their elastic limit is rathercomplicated. Generally, when the stress exceeds a critical value (theyield strength), the material will undergo plastic deformation. Duringan infinitesimal increment of stress, strain changes ɛ can be split upinto elastic ɛe and plastic ɛp components:

dε = dεe + dεp ð3Þ

The plastic strain increments are related to the yield function (oryield surface) F that specifies the stress conditions required for plasticflow and prescribes the relationships among stress componentsduring flow. The yield surface F depends on the state of stress andstrain and on the history of loading. Plastic strain can occur only if thestresses satisfy the general yield criterion:

Fðσ; εp;κÞ = 0 ð4Þ

where κ is a work hardening parameter defining the plasticdeformation history of thematerial. If yielding occurs, we need furtherinformation concerning the increment or rate of deformation in orderto complete the description of the material behavior. Drucker (1951)showed that the plastic strain increment vector must be normal to theyield surface at a regular point. The normality principle leads to theassociated flow rule (i.e., the stress/strain relation):

dεpij = G∂F∂σij

∂F∂σkl

dσkl ð5Þ

where

G = − 1∂F

∂εpmn+ ∂F

∂κ∂κ

∂εpmn

� �∂F

∂σmn

ð6Þ

Different work hardening functions can be assumed on the basis ofstress–strain relationships fitting to experimental measurements onrocks. A few of the more common forms of hardening functions areillustrated in Fig. 1. A perfectly plastic material neglects the effect ofwork hardening and the stress increments lie on the yield surface.

Several yield criteria have been developed to model the mechanicalbehavior of rocks undergoing plastic deformation. Mohr–Coulomb lawand its approximation in 3D case, given by Drucker–Prager failurecriterion, are often used formodeling plastic behavior (Cattin et al., 2005).Davis et al. (1974) showed that a zone around a magma chamber thatexhibits Mohr–Coulomb failure causes the effective pressure source toexpandoutwards including both themagmatic zone and the envelopedoffailed rock. The net result is that for a given surface deformation thestresses aremuch lower than required by elastic models only. Chery et al.(1991) investigated a model in which the upper crust exhibits pressure-dependent plasticity versus the lower crust, which exhibits temperature-dependentviscoplasticitywithavonMisesyield criterion.With increasingtemperature, ductility increases and failure behavior changes from brittle

g of volcano ground deformation, Earth Planet. Sci. Lett. (2010),

Fig. 2. Temperature field distribution assuming a source wall temperature ofTw=1200 K.

3G. Currenti et al. / Earth and Planetary Science Letters xxx (2010) xxx–xxx

fracture to ductile flow. The high temperature around a heatedmagmaticsource may lead the rocks behavior toward ductility rather thanbrittleness, and yielding occurs mainly by ductile flow. At this point,there is no longer any pressure dependence of the strength and thebehavior is fully ductile (Scholz, 2002). Therefore, the Mohr–Coulombcriterion may apply to the most of the edifice where brittle failure takesplace, but not to the hot rock surrounding a magma chamber, where vonMises plasticity criterion can better describe the ductile behavior of themedium. The vonMises criterion is usually adopted as a suitable criterionfor theductiledomainof the lithosphere,wherehigh temperatures induceductile deformation and plasticity controls the rock behavior (Cattin et al.,2005). This criterion assumes that isotropic deformation is always relatedelastically to the mean pressure, while the deviatoric strain is elasticallyrelated to deviatoric stress until yielding is not reached, then plastic straintakes place at constant deviatoric stress. The yielding function F= I2′−κ2

dependsonlyon the second invariant I2′=1/2σ ′ijσ′ijof thedeviatoric stressσ′ij=σij−1/3I1δij, while the first invariant of deviatoric stress I1′=σ′kk isidentically zero. For a work hardening material, κ will be allowed tochangewith strain history. For an ideal plasticmaterial obeying vonMisescondition, κ is a constant independent of strain history and it is related tothe yield stress σy of the material in pure shear. Therefore, the materialexhibits linear elasticity as longas I2′≤σy

2/3, andplasticity occurswhen thesecond invariant reaches theyield strengthσy

2 (Fung, 1965;Ranalli, 1995).

2.3. Comparison between elastic and anelastic strain in a 3D axi-symmetricmodel

We developed a 3D axi-symmetric model using FEM in order toassess the effect of rheology on the surface deformation field.Computations were carried out by the commercial software COMSOLMultiphysics, version 3.3 (Comsol, 2006). In such a case a simpler two-dimensional domain can be considered by exploiting the symmetries.The source has a spherical geometry with a radius of 0.5 km and iscentered at 4 km depth. The axi-symmetric model is composed of∼200,000 triangular elements, covering a rectangular half-space thatextends 50 km horizontally from the source centre and 50 km belowthe ground surface. For boundary conditions, the displacements on the

Table 1Thermal model parameters.

Thermal parameters

Ts Surface temperature 300 Kqm Heat flow 0.03 Wm−2

k Thermal conductivity 4 Wm−1 K−1

As Volumetric rate of heat production 2.47×10−6 Wm−3

b Length scale for crustal radioactivedecay

14.170 km

Please cite this article as: Currenti, G., et al., Elasto-plastic modelingdoi:10.1016/j.epsl.2010.05.013

outermost lateral boundaries and on the bottom are fixed to zero,while the boundary at the ground surface is stress-free.

Macroscopic experiments,which investigate themechanical behaviorin triaxial tests at high temperatures, show two principal mechanicaleffects: (i) thermal softening and (ii) thermally enhanced ductility.Therefore, we computed the deformation field for three different cases:(i) elastic, (ii) temperature-dependent elastic and (iii) elasto-plasticrheology. The first model is a three-layered elastic half-space withPoisson ratio ν=0.25. A very soft layer with a Young's modulus of 2 GPawas assumed for the upper part extending from the ground surface to2 km. A Young's modulus of 40 GPa was assumed for the second layerfrom 2 km to 6 km, whereas a Young's modulus of 80 GPa was used forthe third layer from 6 km to 50 km. The second and third models wereconstructed introducing in the layered half-space a shell surrounding themagmatic source. The thickness of the shell is dependent on thetemperature state of the magmatic source. The temperature fielddistribution (Fig. 2) was computed solving the thermal model withvalues of the model parameters reported in Table 1. The temperature onthe magma chamber wall was set to Tw=1200 K. Laboratory experi-ments on basaltic rocks have shown that the elastic modulus decreasessteadily with temperature and ductility is expected above 900 K at adifferential stress of about 10–15 MPa (Rocchi et al., 2004; Dingwell,1998). Etnean rocks remain fully brittle up to 900 K, and above thistemperature the elastic modulus decreases reaching 10% of the originalvalue (Rocchi et al., 2004). Therefore, in the second model the rigiditymodulus varies linearlywith the temperature (18 MPa/K) besides 900 K,and is kept at 10%of the initial values above 1100 K. In the thirdmodel anelasto-plastic behaviorwas supposed in the shell surrounding the source,while the remaining domain was set as elastic. For this third model,starting fromthe temperature distribution,wemodified theproperties ofthemedium through the constitutive equations, allowing the element ofthe computational domain to behave elastically or plastically in functionof the temperature distribution.We associateddifferent rheologies to themedium: (i) elastic behavior where the temperature values are below900 K, and (ii) elasto-plastic behavior above this threshold. To simulatethe ductile behavior of the hot rocks surrounding the magma chamber,we implemented the yield stress/strain laws considering an ideal plasticbehavior obeying to von Mises criterion. The yield strength ofsurrounding rocks was assumed as σy=15 MPa, while the elasticparameters of the medium are those of the first model previouslydescribed. An overpressure of 100 MPa was applied on the source wall.Model results are shown in Fig. 3. The layered elastic model gives a

Fig. 3. Comparison of the deformation expected from a pressurizing (100 MPa)chamber centered at a 4 km depth in a layered medium for different rheologies:A) elastic, B) elastic with varying temperature, C) elasto-plastic.

of volcano ground deformation, Earth Planet. Sci. Lett. (2010),

Table 2Pressure sources from geodetic data inversion at Mt Etna (after Bonforte et al., 2008). All models are based on elastic analytical solutions except Bonaccorso et al. (2005), who alsoused a numerical solution.

Period Source Lat([UTM km])

Long([UTM km])

Depth([km])

ΔP⁎V([N⁎m])

Rigidity([GPa])

Reference Data

Sept 1993–Jul 1994 Mogi 4179 500 3.8 2.19×1017 not reported Puglisi et al. (2001) GPSJun 1993–Oct 1995 Yang 4177.67 499.89 4.8 2.74×1017 30 Lundgren et al. (2003) InSARJul 1996–Jul 1997 Mogi 4180.9 497.61 9.3 13.69×1017 not reported Puglisi and Bonforte (2004) GPSSept 1993–Jul 1997 Mogi 4179.45 496.96 6.8 17.24×1017 30 Palano et al. (2007) GPSSept 1993–Jul 1997 Davis 4177.96 500.7 4 9.6×1017 heterogeneous Bonaccorso et al. (2005) EDM, GPSSept 1994–Sept 1998 Mogi 4181 500 5 2.32×1017 10 Obrizzo et al. (2004) levellingSept 1993–Jul 2000 Mogi 4181.37 496.93 8.1 28.84×1017 30 Palano et al. (2007) GPSJul 1994–Jul 2001 Mogi 4180.5 499 6.2 33.4×1017 not reported Houlie´ et al. (2006) GPS

Table 3Elasto-plastic model parameters. The misfit function is shown in Fig. 4.

Model σy

([MPa])Tw([K])

Tt([K])

χ2 ΔP([MPa])

A 10 1100 900 1.19×104 95B 15 1100 900 1.19×104 98C 20 1100 900 1.19×104 102D 10 1200 800 1.15×104 46E 10 1200 900 1.17×104 69F 10 1100 800 1.16×104 59G 15 1200 800 1.16×104 53H 15 1200 900 1.18×104 75

4 G. Currenti et al. / Earth and Planetary Science Letters xxx (2010) xxx–xxx

vertical uplift above the source center of about 7.5 cm. When the elasticmodulus is decreased with temperature, an enhancement to 9.0 cm isobtained. The elasto-plastic model considerably enhances the grounduplift to a 14.5 cm, which is about twice than that obtained for theheterogeneous elastic model. These results show that the presence of aplastic region can greatly amplify the strain response.

3. Etna application and results

3.1. Etna 1993–97 recharging phase

On Mt Etna, between the 1993 and 1997, geodetic data collected bydifferent monitoring networks (EDM, GPS, and leveling) identified aninflationary phase. This was characterized by a uniform and continuousexpansion of the overall volcano edifice that was not accompanied byeruptive activity (Bonaccorso et al., 2005). The beginning of theinflationary phase was detected from the comparison of syntheticaperture radar (SAR) satellite images taken from 1993 to 1995. Theinversion of interferograms yielded results that were interpreted as aspheroidal magmatic source located at about 5 km b.s.l. (Lundgren et al.,2003). Also leveling data supported the presence of a point-likepressurized source beneath the summit craters at 4.5 km b.s.l. (Obrizzoet al., 2004). A rechargingphasewas alsohighlighted byGPSdata (Puglisiet al., 2001; Puglisi and Bonforte, 2004). Most interpretations of thedeformation on Mt Etna from 1994 to 2004 (Table 2) were based on ahomogeneous elastic half-space model. Despite the different geodeticdata, the methodologies used and the results achieved, the values ofintensity factor V∙ΔP/μ are similar formost of themodels (Table 2), but itis not possible to determine them unambiguously because of the trade-off between the volume and overpressure of the source. Assuming ahomogeneous half-space with an elastic shear modulus of 30 GPa,Lundgren et al. (2003) from SAR images estimated a pressure of about5 MPa for a spheroidal source with a volume of 68.69×109 m3, which isdefinitely toohighas it hasnever been revealedby seismic investigations.Bonaccorso et al. (2005) interpreted the 1993–1997 GPS and EDM datausing an analytical model of an ellipsoidal source with a volume of3×109 m3, an overpressure of 20 MPa and a low homogeneous shearmodulus of 1 GPa. They also evaluated the effect of topography andmedium heterogeneity on the deformation field using a numericalsimulation employing an elastic model. Since the average values of theshear modulus used in the simulation, which had been estimated fromseismic velocity measurements, were larger than those used in thehomogeneous analytical model, the observed displacements in theheterogeneous models were reproduced by increasing the overpressureup to about 300 MPa. The source inferred by the simpler analyticalmodelis equivalent to a lower pressurized magma chamber surrounded by alower rigidity, which can be considered as a sort of “effective” rigidityrequiring the need to use an anelastic rheology (Bonaccorso et al., 2005).Recently, DelNegro et al. (2009) reviewed the1993–1997 inflationphaseon Mt Etna using a 3D temperature-dependent viscoelastic numericalmodel, which allows producing deformation comparable with thoseobtained fromelasticmodelwith a lower pressure of about 200 MPa. The

Please cite this article as: Currenti, G., et al., Elasto-plastic modelindoi:10.1016/j.epsl.2010.05.013

deformation accumulated during the long-lasting inflation phase from1993 to 1997 indicates a permanent deformation of the volcano edifice(Bonforte et al., 2008) and also suggests investigating an elasto-plasticrheology. Therefore, we re-analyzed the ground deformation accompa-nying the 1993–1997 inflation period on Mt Etna assuming a heatedpressurized magma chamber embedded in an elasto-plastic heteroge-neous medium.

3.2. Model application

We adopted the source geometry determined by Bonaccorso et al.(2005), which is an ellipsoid located 4.2 km bsl beneath the centralcraters (latitude 4177.9 UTM km and longitude 500.7 UTM km). Theellipsoid has a semi-major axis of 1854 m and the other two semi-axesof 725 m and 544 m, respectively, with an orientation angle of 124° anda dip angle of 77°. Our model domain is a large volume extending100×100×50 km in order to avoid artifacts in the numerical solutionbecause of the proximity of the boundary. The mesh of the groundsurface was generated using a digital elevation model of Mt Etna fromthe 90 m Shuttle Radar Topography Mission (SRTM) data andbathymetry model from GEBCO database (http://www.gebco.net/).The computational domain was represented by 77,475 arbitrarilydistorted tetrahedral elements connected by 13,634 nodes. The meshresolution is about 100 m around the ellipsoidal source, about 300 m inthe area surrounding the volcano edifice, and decreases to 10 km in thefar field.

We performed eight models to investigate how the grounddeformation is affected by the rheology parameters (elastic moduli,yield strength, and temperature threshold) and thermal state (sourcewall temperature). The elastic moduli were fixed to those inferred fromseismic tomography (Currenti et al., 2007, 2008b), whereas differentyield strength values were used in accordance with laboratorymeasurements on basalts from Etna (Rocchi et al., 2004; Balme et al.,2004). A vertical geothermal gradient of 22 K/km was assumed for theareas surrounding the volcano edifice in agreement with the temper-ature measurements carried out in deep boreholes (AGIP, 1977). Thesource wall temperature Tw was chosen to range between 1100 and1200 K (Corsaro and Pompilio, 2004). The effect of the transition

g of volcano ground deformation, Earth Planet. Sci. Lett. (2010),

Fig. 4. Chi-square values obtained varying the overpressure from 0 to 120 MPa fordifferent parameters of elasto-plastic models (see Table 3). The chi-square value for theelastic model using a pressure of about 300 MPa (Bonaccorso et al., 2005) is alsoreported (horizontal thick line).

5G. Currenti et al. / Earth and Planetary Science Letters xxx (2010) xxx–xxx

temperature Tt,which controls if thebehavior of themedium is elastic orelasto-plastic, was investigated in a range from 800 to 900 K wherelaboratory experiments showed ductile deformation failure (Rocchi etal., 2004). The parameters used in all models are reported in Table 3. Foreach model, we performed different simulations varying the magmaoverpressure from 1 MPa to 120 MPa (with a step of 1 MPa) andcomputed the chi-square value χ2 as:

χ2 =∑ðUobs

x −Ucalcx Þ + ∑ðUobs

y −Ucalcy Þ

σ2H

+∑ðUobs

z −Ucalcz Þ

σ2V

ð7Þ

where Uobs and Ucalc are the observed and computed displacementsrespectively, and σ is the standard deviation of the measurements(Bonaccorso et al., 2005). The standard deviation affecting measure-ments ranges between 2 and 3 mm in the horizontal components andbetween 5 and 6 mm in the vertical component. The χ2 values areshown in Fig. 4 for all the models and the pressure changes

Fig. 5. Vertical uplifts for the elasto-plastic models D (gray lines) and G (black lines) atOBS (solid lines), TDF (dotted lines), ESLN (dashed lines) stations (see Fig. 7 for thepositions) for increasing source pressure.

Please cite this article as: Currenti, G., et al., Elasto-plastic modelingdoi:10.1016/j.epsl.2010.05.013

corresponding to the minimum χ2 that best fits the geodetic data arereported in Table 3. The pressure change estimates range between46MPa (model D) and 102 MPa (model C). In all the models, as thepressure increases, the deformation starts to grow linearly as far as theyield condition is not satisfied. When the elastic limit is reached, theground deformation increases more rapidly since plastic deformationprevails. In themodelswith lowervalues of theyield strength (A,D, E, F),the medium fails plastically at lower values of the pressure change andthe ground deformation is enhanced with respect to the other models(Fig. 5). Therefore, thepressure estimates are strongly dependent on themodel parameters. The value of pressure, which gives the minimum χ2

value, is tuned by the yield strength. The higher the yield strength, thehigher the pressure required to obtain the same amount of deformation.Besides the yield strength, another sensitive parameter is the transitiontemperature. As the threshold temperature decreases, the volumeparticipating to the elasto-plastic flow increases and gives morecontribution to the plastic deformation. A variation of 100 K in thetemperature is sufficient to vary the estimated pressure from 46MPa(model D) to 69 MPa (model E) when the other parameters are keptconstant.

Further simulations were performed assuming a decrease inrigidity modulus with increasing temperature for the models A, B,and C. Following the results of laboratory tests on basalt rock samplesfrom Etna (Rocchi et al., 2004), we assumed that the elastic modulusdecreases steadily at 900 K reaching 10% of the original values at1100 K. The comparison among the different models shows that thedependence of the elastic modulus on temperature does not affect thesimulation results very much (Fig. 6). A slight difference is obtained atlower overpressure when the deformation is almost elastic. When theplastic deformation prevails on the elastic deformation, it becomes thedominant process and the effect of variations in the elastic modulusdue to temperature is negligible.

Theminimumχ2 values of the elasto-plastic models are comparablewith each other and less than the χ2 value of the elastic model (Fig. 4).On the basis of the χ2 values, all the considered models provide similarfits to the data with fairly similar pressure changes, which are wellbelow the value obtained using an elastic rheology. The minimum χ2 isobtained by the model D (Fig. 4), which also requires the minimumpressure change (46 MPa). The fit to the data is slightly improved from1.4×104 for the elastic model to 1.1×104 for the elasto-plastic model(Fig. 7). It is worth noting that the elasto-plastic model uses a pressurechange of only 46 MPa, whereas the elastic model needs an unrealistic

Fig. 6. Chi-square values obtained varying the overpressure from 0 to 120 MPa fordifferent elasto-plastic models: (i) A, B, and C (continuous lines) simulations withrigidity modulus independent on temperature, (ii) AEt, BEt, CEt (dotted lines)simulations using decreasing rigidity modulus with increasing temperature.

of volcano ground deformation, Earth Planet. Sci. Lett. (2010),

Fig. 7. Comparison between GPS observed (black) and computed deformation during the 1993–1997 period. The numerical computations are performed assuming a heterogeneouselastic medium (red, after Bonaccorso et al., 2005) and the elasto-plastic model D (blue). Horizontal displacements (top) and vertical displacements (bottom) are calculated at GPSstations. Assuming the same volume and rigidity, the elastic model requires an overpressure of ≈300 MPa, whereas the elasto-plastic model reaches similar deformation with anoverpressure of ≈46 MPa.

6 G. Currenti et al. / Earth and Planetary Science Letters xxx (2010) xxx–xxx

pressure of about 300 MPa to get comparable deformation. The elasticmodel with a 46 MPa of pressure change would have caused grounduplifts of few centimeters, whereas the elasto-plastic model reachestens of centimeters (Fig. 8). Interpretation of long-term deformation iscomplicated by the coarse spatial resolution of the available geodeticdata, and the contribution ofmore than onemechanism to the observedground displacements. A significant contribution to the horizontaldisplacements on the easternmost stations (MIL and GIA), affected by

Please cite this article as: Currenti, G., et al., Elasto-plastic modelindoi:10.1016/j.epsl.2010.05.013

flank instability, could be given by the effect of the sliding of the volcanoeastern flank (Bonaccorso et al., 2006), which could also alter theestimate of the source parameters.

It is worth noting that our numerical models disregard the stressinduced by the topographic load. In the near-surface region, topographyloading can make the stress state different from the lithostatic stressstate (zero deviatoric stress), usually assumed for half-spacemodel. Dueto the topography, themedium is initially in a non-zero deviatoric stress

g of volcano ground deformation, Earth Planet. Sci. Lett. (2010),

Fig. 8. Comparison between the vertical uplift along the AB profile (Fig. 7) for the elastic(dashed line) and elasto-plastic model D (solid line) using a pressure change of 46 MPa.

7G. Currenti et al. / Earth and Planetary Science Letters xxx (2010) xxx–xxx

state and the elastic limit could even be reached with a lower sourceoverpressure.

4. Discussion and conclusions

Geodetic observations are useful to discriminate between differentdeformation models (elastic, viscoelastic, and elasto-plastic) and mayprovide valuable constraints on the rheological and mechanicalparameters of the medium. The elastic models usually require eitheroverpressures of several hundred MPa or exceedingly high sourcevolumes to justify the observed ground deformation. High overpres-sure values are unrealistic as the induced stresses would be so highthat rocks would fracture because of the low tensile strength ofcommon solid rocks (Balme et al, 2004; Haimson and Rummel, 1982;Schultz, 1995; Amadei and Stephansson, 1997).

We showed that the overpressure can be lowered to geologicallysatisfying values if elasto-plastic behavior is taken into account. Wetested the elasto-plastic behavior for the recharging phase occurredbetween 1993 and 1997 on Etna volcano. Despite the clear evidence ofthe overall volcano edifice expansion identified at Mt Etna by differentgeodetic data (InSAR, leveling, GPS and EDM), the estimate of sourceparameters is still under debate. The high overpressure foreseen by theelastic models is incompatible with the low levels of both volcanicactivity and seismicity occurring in the analyzed period (Patanè et al.,2003). The low seismicity associated with the recharging phase in thetime interval 1993–1997 would support the hypothesis that thevolume surrounding the pressurizing source could mainly have failedby ductility rather than brittleness. Although the summit craters werecontinuously degassing between March 1993 and July 1995, no freshmagma reached the surface (Allard et al., 2006). Afterwards, a series ofparoxysmal eruptions occurred at the summit craters (Bocca Nuovaand North-East craters), whereas the South-East crater became activein the autumnof 1996 after 5 years of repose. This volcanic activitywasverymodest and only in themiddle of 1998 significantly resumedwitha series of lava fountaining from the summit craters (Allard et al.,2006). A shallowbroad region of lowQphotfluids has been recognizedon the west of an high rigidity body beneath the South-East flank ofEtna volcano (Martinez-Arevalo et al., 2005; De Gori et al., 2005;Patanè et al., 2006), which concurs with the location of the estimatedellipsoidal source inferred by Bonaccorso et al. (2005) and used in theelasto-plasticmodel in this work. During the last decades, this area hasbeen a preferential pathway of magma rising and a region ofintermediate magma storage (Bonforte et al., 2008), which could

Please cite this article as: Currenti, G., et al., Elasto-plastic modelingdoi:10.1016/j.epsl.2010.05.013

have perturbed the geothermal gradient and, hence, the mechanicalbehavior of the surrounding rocks. Recently, Bonforte et al. (2008)showed that the deformation observed during the long-lastinginflation phase was not recovered in the following eruptive period,indicating a permanent deformation of the volcano. The deformationseems to be mainly accumulated following a non-elastic behavior. Allthese evidences point to the elasto-plastic rheology as the moreprobable behavior to perform more realistic numerical simulations.Results suggest that the integration of thematerial rheology variationsinto volcano deformation modeling is a critical and necessaryadvancement toward more reliable predictions for eruptive activity.

Acknowledgments

This studywas undertakenwith financial support from the V3-LAVAandV4-FLANKprojects (DPC-INGV2007–2009 contract). Thisworkwasdeveloped in the frame of the TecnoLab, the Laboratory for theTechnological Advance in Volcano Geophysics organized by INGV-CTand DIEES-UNICT. We thank the Editor Lars P. Stixrude and theanonymous referees who provided constructive comments for improv-ing the manuscript.

References

AGIP, S.p.A., 1977. Temperature sotterranee. Grafiche Fili Brugora, Milano.Allard, P., Behncke, B., D'Amico, S., Neri, M., Gambino, S., 2006. Mount Etna 1993–2005:

Anatomy of an evolving eruptive cycle. Earth Sci. Rev. 78, 85–114.Amadei, B.,Stephansson, O., 1997. Rock Stress and its Measurement. Chapman and Hall, London.

Amadei, B., Stephansson, O., 1997. Rock stress and its measurements. Chapman & Hall,London, p. 490.

Martinez-Arevalo, C., Patanè, D., Rietbrock, A., Ibanez, J.M., 2005. The intrusive processleading to the Mt. Etna 2001 flank eruption: Constraints from 3-D attenuationtomography. Geophys. Res. Lett. 32, L21309. doi:10.1029/2005GL023736.

Balme, M.R., Rocchi, V., Jones, C., Sammonds, P.R., Meredith, P.G., Boon, S., 2004. Fracturetoughness measurements on igneous rocks using a high-pressure, high-temperaturerock fracture mechanics cell. J. Volcanol. Geoth. Res. 132, 159–172.

Bonaccorso, A., Davis, P.M., 1999. Models of ground deformationfrom vertical volcanicconduits with application to eruptions ofMount St. Helens andMount Etna. Geophys.Res. L ett. 104 (B5), 10531–10542.

Bonaccorso, A., Cianetti, S., Giunchi, C., Trasatti, E., Bonafede,M., Boschi, E., 2005. Analyticaland 3D numerical modeling of Mt. Etna (Italy) volcano inflation. Geophys. J. Int. 163,852–862.

Bonaccorso, A., Bonforte, A., Guglielmino, F., Palano, M., Puglisi, G., 2006. Compositeground deformation pattern forerunning the 20042005 Mount Etna eruption.J. Geophys. Res. 111, B12207. doi:10.1029/2005JB004206.

Bonforte, A., Bonaccorso, A., Guglielmino, F., Palano, M., Puglisi, G., 2008. Feeding systemand magma storage beneath Mt Etna as revealed by recent inflation/deflation cycles.J. Geophys. Res. 113, B05406. doi:10.1029/2007JB005334.

Cattin, R., Cecile Doubre, C., de Chabalier, J.B., King, G., Vigny, C., Avouac, J.P., Jean-ClaudeRuegg, J.C., 2005. Numerical modelling of quaternary deformation and post-riftingdisplacement in the Asal–Ghoubbet rift (Djibouti, Africa). Earth Planet. Sci. Lett. 239,352–367. doi:10.1016/j.epsl.2005.07.028.Chiarabba, C., Amato, A., Boschi, E., Barberi,F., 2000. Recent seismicity and tomographic modeling of the Mount Etna plumbingsystem. J. Geophys. Res. 105, 10923–10938.

Comsol Multiphysics 3.3, 2006. Comsol AB, Stockholm, Sweden.Ciccotti, M., Mulargia, F., 2004. Differences between static and dynamic elastic moduli

of a typical seismogenic rock. Geophys. J. Int. 157, 474–477.Cheng, C.H., Johnston, D.H., 1981. Dynamic and static moduli. Geophys. Res. Lett. 8 (1),

39–42.Chery, J., Bonneville, A., Vilotte, J.P., Yuen, D., 1991. Numerical modelling of caldera,

dynamical behaviour. Geophys. J. Int. 105 (2), 365–379. doi:10.1016/j.epsl.2005.07.028.

Civetta, L., D'Antonio, M., De Lorenzo, S., Di Renzo, V., Gasparini, P., 2004. Thermal andgeochemical constraints on the ‘deep’magmatic structure ofMt. Vesuvius. J. Volcanol.Geotherm. Res. 133, 1–12.

Corsaro, R.A., Pompilio, M., 2004. Magma dynamics in the shallow plumbing system ofMt. Etna as recorded by compositional variations in volcanics of recent summitactivity (1995–1999). J. Volcanol. Geotherm. Res. 137, 55–71. doi:10.1016/j.jvolgeores.2004.05.008.

Currenti, G., Del Negro, C., Ganci, G., 2007. Modeling of ground deformation and gravityfields using finite element method: an application to Etna volcano. Geophys. J. Int..doi:10.1111/j.1365-246X.2007.03380.x

Currenti, G., Del Negro, C., Ganci, G., Scandura, D., 2008a. 3D numerical deformationmodel of the intrusive event forerunning the 2001 Etna eruption. Phys. EarthPlanet. Inter.. doi:10.1016/j.pepi.2008.05.004

Currenti, G., Del Negro, C., Ganci, G., Williams, C., 2008b. Static stress changes inducedby the magmatic intrusions during the 2002–2003 Etna eruption. J. Geophys. Res.113, B10206. doi:10.1029/2007JB005301.

of volcano ground deformation, Earth Planet. Sci. Lett. (2010),

8 G. Currenti et al. / Earth and Planetary Science Letters xxx (2010) xxx–xxx

Davis, P.M., Hastie, L.M., Stacey, F.D., 1974. Stresses within an active volcano withparticular reference to Kilauea. Tectonophysics 22, 355–362.

De Gori, P., Chiarabba, C., Patanè, D., 2005. Qp structure of Mt Etna: constraints for thephysics of the plumbing system. J. Geophys. Res. 110. doi:10.1029/2003JB002875.

Del Negro, C., Currenti, G., Scandura, D., 2009. Temperature-dependent viscoelasticmodeling of ground deformation: application to Etna volcano during the 1993–1997inflation period. Phys. Earth Planet. In. 172, 299–309. doi:10.1016/j.pepi.2008.10.019.

Dingwell, D.B., 1998. The glass transition in hydrous granitic melts. Phys. Earth Planet.Inter. 107, 1–8.

Dragoni, M., Harabaglia, P., Mongelli, F., 1997. Stress field at a transcurrent plate boundaryin the presence of frictional heat production at depth. Pure Appl. Geophys. 150,181–201.

Dragoni, M., Magnanensi, C., 1989. Displacement and stress produced by a pressurized,spherical magma chamber, surrounded by a viscoelastic shell. Phys. Earth Planet.Inter. 56, 316–328.

Drucker, D.C., 1951. A more fundamental approach to plastic stress strain relations.Proc. 1st. U.S. Natl. Congress Appl. Mech. 487–491.

Fung, Y.C., 1965. Foundations of solid Mechanics. Prentice-Hall, Englewood Cliffs.Haimson, B.C., Rummel, F., 1982. Hydrofracturing stress measurements in the Iceland

research drill hole at Reydarfjördur Iceland. J. Geophys. Res. 87, 6631–6649.Houlie´, N., Briole, P., Bonforte, A., Puglisi, G., 2006. Large scale ground deformation of Etna

observed by GPS between 1994 and 2001. Geophys. Res. Lett. 33, L02309 . doi:10.1029/2005GL024414.

Lundgren, P., Berardino, P., Coltelli, M., Fornaro, G., Lanari, R., Puglisi, G., Sansosti, E.,Tesauro, M., 2003. Coupled magma chamber inflation and sector collapse slipobserved with synthetic aperture radar interferometry on Mt. Etna volcano.J. Geophys. Res. 108 (B5), 2247. doi:10.1029/2001JB000657.

Mandal, P., Chadha, R.K., Raju, I.P., Kumar, N., Satyamurty, C., Narsaiah, R., Maji, A., 2007.Coulomb static stress variations in the Kachchh, Gujarat, India: Implications for theoccurrences of two recent earthquakes (Mw=5.6) in the 2001 Bhuj earthquakeregion. Geophys. J. Int. doi:10.1111/j.1365-246X.2006.03301.x.

Newman, A.V., Dixon, T., Ofoegbu, G., Dixon, J., 2001. Geodetic data constraints on recentactivity at LongValley Caldera, California: evidence for viscoelastic rheology. J. Volcan.Geoth. Res. 105, 183–206.

Newman, A.V., Dixon, T.H., Gourmelen, N., 2006. A four-dimensional viscoelasticdeformation model for Long Valley Caldera, California, between 1995 and 2000.J. Volcanol. Geotherm. Res. 150, 244–269.

Obrizzo, F., Pingue, F., Troise, C., DeNatale, G., 2004. Bayesian inversion of 1994–98 verticaldisplacements at Mt Etna; evidence for magma intrusion. Geophys. J. Int. 157 (2),935–946. doi:10.1111/j.1365- 246X.2004.02160.x.

Please cite this article as: Currenti, G., et al., Elasto-plastic modelindoi:10.1016/j.epsl.2010.05.013

Okada, Y., Yamamoto, Y., 1991. Dyke intrusion model for the 1989 seismovolcanicactivity of Off Ito, Central Japan. J. Geophys. Res. 96, 10361–10376.

Palano, M., Puglisi, G., Gresta, S., 2007. Ground deformation at Mt. Etna: a jointinterpretation of GPS and InSAR data from 1993 to 2000. Boll. Geofis. Teor. Appl. 48(2), 81–98.

Patanè, D., De Gori, P., Chiarabba, C., Bonaccorso, A., 2003. Magma ascent and thepressurization of Mount Etna's volcanic system. Science 299, 2061–2063.

Patanè, D., Barberi, G., Cocina, O., De Gori, P., Chiarabba, C., 2006. Time-resolved seismictomography detects magma intrusions at mount etna. Science 313, 821–823.

Puglisi, G., Bonforte, A., Maugeri, S.R., 2001. Ground deformation patterns on Mt. Etna,between 1992 and 1994, inferred from GPS data. Bull. Volcanol. 62, 371–384.

Puglisi, G., Bonforte, A., 2004. Dynamics of Mount Etna volcano inferred from static andkinematic GPS measurements. J. Geophys. Res. 109, B11404. doi:10.1029/2003JB002878.

Ranalli, G., 1995. Rheology of the Earth. Chapman and Hall, London, p. 413.Rocchi, V., Sammonds, P.R., Kilburn, C.R.J., 2004. Fracturing of Etnean andVesuvian rocks at

high temperatures and low pressures. J. Volcanol. Geotherm. Res. 132, 137–157.Rocchi, V., Sammonds, P.R., Kilburn, C.R.J., 2002. Flow and fracture maps for basaltic

rock deformation at high temperatures. J. Volcanol. Geotherm. Res. 120, 25–42.Scholz, C.H., 2002. The mechanics of earthquakes and faulting. Cambridge University

Press, Cambridge, p. 471.Schultz, R.A., 1995. Limits of strength and deformation properties of jointed basaltic

rock masses. Rock Mechanics and Rock Engineering 28, 1–15.Trasatti, E., Giunchi, C., Bonafede, M., 2003. Effects of topography and rheological layering

on ground deformation in volcanic regions. J.Volcanol. Geotherm. Res. 122, 89–110.Trasatti, E., Giunchi, C., Bonafede, M., 2005. Structural and rheological constraints on

source depth and overpressure estimates at the Campi Flegrei caldera, Italy. J. Volc.Geoth. Res. 144, 105–118.

Tuffen, H., Smith, R., Sammonds, P.R., 2008. Evidence for seismogenic fracture of silicicmagma. Nature 453 (7194), 511–515. doi:10.1038/nature06989.

Turcotte, D.L., Schubert, G., 2002. Geodynamics — applications of continuum physics togeological problems, 2nd edition. Wiley, New York. pp. 528.

Walsh, J.B., Decker, R.W., 1971. Surface deformation associatedwith volcanism. J. Geophys.Res. 76, 3291–3302.

Yang, X., Davis, P.M., Delaney, P.T., Okamura, A.T., 1992. Geodetic analysis of dike intrusionand motion of the magma reservoir beneath the summit of Kilauea volcano, Hawaii:1970–1985. J. Geophys. Res. 97, 3305–3324.

g of volcano ground deformation, Earth Planet. Sci. Lett. (2010),