Axial Symmetric Crustal Deformation Model for Long Valley Caldera, California

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1 Axial Symmetric Crustal Deformation Model for Long Valley Caldera, California V. Magni 1*, M. Battaglia 1, P. Tizzani 2, A. Manconi 3 and T. Walter 3 1 Dept of Earth Sciences, University of Rome La Sapienza, P.le A. Moro 5, 00185, Roma, Italy. 2 IREA-CNR, Via Diocleziano 328, 80124, Napoli, Italy 3 GFZ Potsdam, D Potsdam *corresponding author; valemagni@libero.it Abstract: Long Valley caldera (California) has experienced since 1978 a significant geological unrest, characterized by recurring earthquake swarms and uplift of the caldera resurgent dome (75 cm in the past 30 years). Many studies suggest that the unrest is caused by a cigar-shaped magma intrusion beneath the resurgent dome at a depth of 7-8 km. Here, we investigate the effects of both vertical and horizontal mechanical discontinuities, and topography, on crustal deformation and stress changes. We use the finite element method to model the surface deformation due to the pressurization of a spheroidal source at depth in an inhomogeneous elastic half-space. The theoretical results are compared with GPS, leveling, EDM and DInSAR measurements. We found that the mechanical discontinuities have a significant influence on the surface deformation; e.g., the presence of a basin amplifies the vertical displacement of 14% compared to the homogeneous model. Keywords: structural mechanics, crustal deformation, source pressurization, mechanical discontinuities. 1. Introduction Long Valley caldera is located on the eastern slope of the Sierra Nevada in central California. The caldera, created by the collapse of the roof of the magma chamber following the climatic eruption of the Bishop tuff (760 ka BP), has an elliptical shape, 17 km x 32 km. A layer more than 2 km thick of tuff, ash fall, rhyolite lavas and fill sediments deposited in the subsiding caldera after the eruption (Hildreth, 2004). Since 1978 Long Valley caldera has entered a period of unrest characterized by recurring earthquakes, emission of carbon dioxide and uplift of the caldera resurgent dome, around 75 cm in the past 30 years (Hill, 2006). Several analytical models suggest that the inflation source beneath the resurgent dome can be described by a prolate spheroid at a depth from 6 to 9 km (Fialko at al., 2001; Battaglia et al., 2003; Langbein et al., 2003). Modeling of gravity residuals is consistent with the intrusion of magma beneath the caldera resurgent dome (Battaglia et al., 2003). We use the finite element method (FEM) to simulate the inflation of the magmatic source beneath the caldera resurgent dome in an inhomogeneous elastic half-space. A load of pressure is applied at the source boundaries to simulate the intrusion of magma. This causes deformation of the half-space and its free surface. The numerical model was implemented using the structural mechanic module of COMSOL ( with the axial symmetry stress-strain option. In this paper, first we validate our model comparing the results with the analytical Yang model (Yang et al., 1988). Then we investigate the effect of topography and both vertical and lateral mechanical discontinuities on crustal deformation. Finally, we compare the models results with leveling, Differential Interferometric Synthetic Array Radar (DInSAR), Electronic Distance Measurements (EDM) and GPS data. We find that the mechanical discontinuities have a significant influence on the surface deformation; e.g., the presence of a basin amplifies the vertical displacement of 14% compared to the homogeneous model. 2. Governing equations A simple model of a volcanic system includes two principal elements: a magma reservoir and a conduit through which the magma may reach the surface. When the volcano is quiescent, the conduit will close allowing a 1

2 pressure build up in the reservoir. Yang et al (1988) formulated a model for the pressurization of a finite cigar-shaped magma body (Figure 1). They found an approximate, but quite accurate solution, for a dipping prolate ellipsoid in a homogeneous elastic half-space using a halfspace double force and center of dilatation solution. Minor corrections to Yang et al (1988) s analytical model are reported in Newman et al. (2006). geometric aspect ratio A = ba between the semi-major axis a and the semi-minor axis b, the source location ( x0, y0, d ), the dip angle θ and the azimuth angle φ measured clockwise from the positive North direction. 3. Data The seismic activity and evident crustal deformation at Long Valley caldera at the beginning of the 80s prompted the U.S. Geological Surface to intensify the monitoring in the area (Hill, 2006). Here, we examine the phase of unrest and we compare the results of the model with two-color EDM, leveling/gps and DInSAR data. 3.1 Two-color EDM Figure 1. Geometry of a spheroidal source. While the expression in the near field are quite complex, in the far field the expressions for the surface deformation for a vertical prolate ellipsoid become relatively simple u z * ( ν) P ab νp d P = d μ R R * ( ν) P ab νp d P ur = r μ R R (1) (2) where v is the Poisson s ratio, µ is the shear modulus, d is the source depth, r is the radial distance from the surface projection of the source center to the bench mark, a and b are respectively the semi-major and semi-minor axis * of the ellipsoid, P and P are proportional to the pressure change Δ P (see page 4250 of Yang et al., 1988) and R 2 = r 2 + d 2. The expression for the volume change of a pressurized spheroid is 2 Δ V = πab Δ P μ (Tiampo et al., 2000). The general solution for a prolate spheroid depends on seven parameters (Figure 1): Δ P, the Horizontal deformation is monitored by measuring changes in the baseline lengths of the two-color EDM network (Langbein et al., 1993). Baseline lengths were measured from several times weekly to several times yearly from January 1984 through the fall of 2006 (quake.wr.usgs.gov/research/deformation/twocol or/longvalley.html). In total there are 34 baselines with measurements that span the interval (Figure 2). Of these, 13 use CASA as their common end point and 21 involve moving the instrument to other locations. The instrument used for these measures has a precision characterized by σ 2 =(a 2 +b 2 L 2 ) where a=0.3 mm, b=0.12 ppm of the baseline length L (Langbein et al., 1993). 3.2 Uplift from leveling and GPS Leveling, in which orthometric height differences between stations are measured with a precise optical-level, has been used to measure vertical deformation along the 65-km-long line along Hwy 365 from Tom's Place to Lee Vining and along several other routes within the caldera. Complete leveling of the caldera occurred each summer from 1982 to 1986, and in 1988 and The random (or white noise) standard error for height changes differences between benchmarks computed from first-order leveling is 0.7 mm/km 1/2, or differencing the results of two surveys (Battaglia et al., 2008) 2

3 σ ( m km )= (3) The latest complete leveling survey was in Battaglia et al. (2003) surveyed 44 of the existing leveling monuments in Long Valley in July 1999 using dual frequency GPS receivers to bring up to date the direct measurement of vertical deformation within the whole caldera. Vertical displacement in the caldera can be obtained by differencing GPS and leveling heights. Heights obtained from GPS are typically expressed as elevations above an ellipsoidal model of the Earth, and not the Earth s geoid. As a result, GPS ellipsoidal heights are not directly comparable with heights above mean sea level determined by leveling surveys. The conversion from ellipsoid to orthometric heights requires a geoid height model that relates the local ellipsoid to the local geoid (Battaglia et al., 2008). spanning the time interval from June 1992 to August 2000 (Tizzani et al., 2007). The ERS 1/2 satellite data were processed by using the SBAS- InSAR algorithm (Berardino et al., 2002). This approach allows us to detect earth surface displacements and to analyze their temporal evolution by generating mean deformation velocity maps and time series, projected along the radar line of sight (LOS). The InSAR measurements have a spatial resolution of ~ 100 m with an accuracy of about 2 mm/year for the deformation velocity and 10 mm for surface displacements. Casu et al. (2006) compared the deformation data obtained by DInSAR with leveling and GPS data in the same area. They found that the standard deviation on the single data of a displacement time series is about 5 mm. 4. Long Valley caldera model To explain the crustal deformation at Long Valley caldera, many authors used spherical or ellipsoidal sources (Newman et al., 2006; Battaglia et al., 2003; Langbein et al., 2003; Fialko et al., 2001;). Here we use a vertical prolate spheroid with a aspect ratio of 0.7. The source location is E and N (UTM coordinates) and the depth is 7 km. This choice is based on existing analytical results (e.g., Battaglia et al., 2003). Velocities of seismic P and S waves (Vp and Vs) are used to calculate the density and mechanical properties of the half-space (Oppenheimer et al., 1993). The empirical relation between the density ρ and Vp is (Brocher, 2005): ρ = Vp (4) Figure 2. Long Valley caldera map. White square are some of EDM benchmark InSAR To measure the deformation of the entire caldera floor and its surroundings, we analyzed a data set composed by 21 descending orbit SAR images (Track 485, Frame 2845), acquired by the European Space Agency ERS-1/2 satellites The density obtained by Eq. (4) can be used to calculate the two Lamé constants (λ and µ) and the Young s modulus E: λ = ρ(vp 2 2Vs 2 ) (5) E = μ = Vs 2 ρ (6) μ(3λ + 2μ) λ + μ (7) 3

4 Eq. (7) allows us to know the dynamic Young s modulus at different depths. The static Young s modulus E is obtained from empirical relations between applied pressure (depth) and elastic parameters for crystalline rocks or tuffs (Cheng et al., 1981). We create two functions in Comsol that linearly interpolate the values of static Young s modulus E for different materials. In this way we can simulate the caldera basin (domain 2), which has a Young s modulus different from the surrounding material (domain 1); see Table 1. Boundary conditions let the surface free and do not allow any normal strain at the side and the bottom of the half-space (roller condition). The pressurization on the source is simulated applying a normal load force at the walls of the cavity. Depth (km) Vp (km/s) E1 (Gpa) E2 (Gpa) Table 1. Young s modulus calculated from Vp for a crystalline rock (E1) and a tuff (E2). Figure 3. Quadrilateral mesh in the whole half-space (a) and near the source (b). The image c shows the variation of E inside the basin and outside. The mesh is quadrilateral and it is finer close to the source, where the deformation is bigger, and at the surface, in order to have a good resolution where the displacements of the model is compared with the real data (Figure 3). We used Comsol Script to set up an inverse problem where all the parameters are fixed (source geometry and location and mechanical properties of the half-space) except for the pressure applied on the cavity. The root mean square penalty function (RMS) is used to compare the results obtained with different pressure values. For a spheroidal source at 7 km of depth the value of ΔP/µ that fits better the leveling, DInsar and EDM data is 0.004, with a RMS=0.03 (Figure 4 and 5). 4

5 Figure 4. Model compared to leveling and InSAR data. The best fit is obtained with a ΔP/µ= considered the topography of Long Valley caldera, with the resurgent dome above the magmatic source 400 m higher then the caldera floor. Model 3 has three layers with different Young s modulus: in the first one, at the surface, E=75 GPa, in the second one E=175 GPa and in the remaining half-space E=125 GPa. The average of the Young s modulus of the first two layers is 125 GPa. Model 4 simulates the caldera structure as a basin, softer (E=75 GPa) then the surrounding material (E=125 GPa). In all the models the values of the mechanical constants close to the source are the same, hence the ratio ΔP/µ is always Figure 5. Model compared to EDM data This value is consistent with other analytical and numerical models in the literature (Fialko et al., 2001; Battaglia et al., 2003; Langbein et al., 2003). 5. Modeling results Figure 6 shows the geometries and the mechanical properties of the axial symmetric models. In each model the source is a spheroidal cavity of major semiaxis of 1.5 km and minor semiaxis of 1 km at depth of 7 km. The pressure applied at the source is always the same. For each model we calculate the vertical and radial displacement. The half-space extends 60 km horizontally from the source center and 100 km below the surface. Model 1 and 2 are homogeneous elastic halfspace with a Young s modulus of 125 GPa and a Poisson s ratio of In Model 2 we Figure 6. Geometry and mechanical properties of Model 1, 2, 3, 4. Colors represent different values of Young s Modulus: E=125 GPa (green), E=175 GPa (red) and E=75 GPa (blue). To validate the model we compared the results of Model 1 with the analytical solution of Yang et al. (1988) in a homogeneous elastic halfspace. Displacements are normalized by the maximum vertical displacement of the analytical solution, whereas radial distances are normalized by the magma chamber depth ( x d ). The very low difference both for vertical (1.1%) and radial (1.5%) displacements in the first 20 km of radial distance from the source shows the goodness of the numerical model (Figure 7). The presence of topography in Model 2 causes a smaller uplift above the source (9.4%), where the resurgent dome is, and a maximum difference of radial displacement of 6.7%. Model 3 shows that a softer layer at surface can amplify the uplift (4%) and have also effects on the radial displacement (6.3%). 5

6 The presence of a basin 4 km deep, simulating the Bishop tuff layer, has an evident effect especially on the vertical displacement (17.8%). Figure 7. Models compared with Yang analytical solution. Red line is the Yang displacement and blue line is the FEM displacement. 6. Summary and conclusions In this paper we use the finite element method to simulate the surface deformation at Long Valley caldera due to a pressurization of a magmatic source at depth. We approximate the source as a vertical prolate spheroid in a heterogeneous elastic half-space. We infer the mechanical properties of the crust from seismic velocities Vp and Vs, through empirical relations. The model includes the presence of the caldera basin, which is softer then the surrounding material. Comparing the model displacements with leveling/gps, DInSAR and EDM data, we obtained a best fit between experimental measurements and theoretical results for a ΔP/µ=0.004, which is in agreement with previous studies (e.g., Newman et al., 2006). Mechanical discontinuities are introduced in our study using a step by step approach. First, we validate our numerical model. The homogeneous Model 1 has a good agreement with the analytical solution by Yang et al (1988), hence the model geometry, the boundary settings and the mesh are suitable to implement our conceptual model of Long Valley caldera. Then, we investigate the effect of caldera topography on deformation (Model 2). Our results indicate that its influence is less than 10% and can probably be neglected as first order effect. Model 3 shows that vertical mechanical heterogeneities may have a stronger influence on the surface displacement. A shallow softer layer will amplify the uplift and the radial displacement by 4% and 6.4%, compared to the homogenous half-space (Model 1). In Model 4, we include the presence of a basin to simulate both horizontal and vertical heterogeneities. The basin amplifies the surface deformation by 17.8% for the uplift and 11% for the radial displacement compared to the homogeneous (Model 1) and by 13.8% and 4.7% with respect to Model 3 (vertical mechanical discontinuities only). These results show that the inappropriate use and interpretation of a homogeneous half-space model could lead to underestimate the ground deformation and erroneously constrain the source of geological unrest. 7. References 1. Battaglia, M., Segall, P., Murray, J., Cervelli, P., Langbein, J., The mechanics of unrest of Long Valley caldera, California: 1.Modelingthe geometry of the source using GPS, leveling and two-color EDM data, J. of Volcanol. and Geotherm. Res., 127, (2003) 2. Battaglia, M., Dzurisin, D., Langbein, J., Svarc, J., Hill, D.P., Converting NAD83 GPS heights into NAVD88 elevations with LVGEOID, a hybrid geoid height model for the Long Valley volcanic region, California: U.S. Geological Survey Scientific Investigation Report , 32 p. (2008). 2. Berardino, P., Fornaro, G., Lanari, R., Sansosti, E., A new algorithm for surface deformation monitoring based on Small Baseline Differential SAR Interferograms, IEEE Transactions on Geoscience and Remote Sensing, 40 (11), (2002) 3. Brocher, T., Empirical relations between elastic wavespeeds and density in the earth s crust, Bull. of Seismol. Soc. Of America, 95 (6), (2005) 4. Casu, F., Manzo, M., Lanari, R., A quantitative assessment of the SBAS algorithm performance for surface deformation retrieval from DInSAR data, Remote Sensing of Environment, 102, (2006) 5. Cheng, C.H., Jhonston, D.H., Dynamic and static moduli, Geophysic. Res. Lett., 8 (1), (1981) 6

7 6. Fialko, Y., Simons, M., Khazan, Y., Finite source modeling of magmatic unrest in Socorro, New Mexico, and Long Valley, California, Geophysic. J. Int., 146, (2001) 7. Hildreth, W., Volcanological perspectives on Long Valley, Mammoth Mountain and Mono Craters: several contiguous but discrete systems, J. of Volcanol. and Geotherm. Ress., 136, (2004) 8. Hill, D. P., Unrest in Long Valley Caldera, California, , Geological Society, London, 269, 1-24 (2006) 9. Langbein, J., Deformation of the Long Valley caldera, California: inferences from measurements from 1988 to 2001, J. of Volcanol. and Geotherm. Res., 127, (2003) 10. Langbein, J., Dzurisin, D., Marshall, G., Stein, R., Rundle, J., Shallow and peripheral volcanic sources of inflation revealed by modeling two color geodimeter and leveling data from Long Valley Caldera, California, , J. Geopyhis. Res., 100, (1995) 11. Langbein, J., Hill,D.P., Parker, P.N., Wilkinson, S.K., An episode of reinflationof the Long Valley Caldera, eastern-california, , J. Geopyhis. Res., 98, (1993) 12. McTigue, D.F., Elastic stress and deformation near a finite spherical magma body; resolution of the point source paradox, Journal of Geophysical Research, 92, (1987) 13. Newman, A.V., Dixon, T.H., Gourmelen, N., A four dimensional visco-elastic deformation model for Long Valley caldera, California, between 1995 and 2000, J. of Volcanol. Geotherm. Res., 150, (2006) 14. Oppenheimer, D., Klein, F., Eaton, J., Lester, F., The Northern California Seismic Network Bulletin January-December (1992) (on line at Tiampo, K.F., Rundle, J.B., Fernandez, J., Langbein, J., Spherical and ellipsoidal volcanic sources at Long Valley caldera, California, using a genetic algorithm inversion technique, J. of Volcanol. and Geotherm. Res., 102, (2000) 16. Tizzani, P., Berardino, P., Casu F., Euillades,P., Manzo, M., Ricciardi, G.P., Zeni, G., Lanari, R., Deformation of Long Valley caldera and Mono Basin, Caifornia, investigated with the SBAS-InSAR approach, Remote Sensing of Enviroment, 108, (2007) 17. Yang, X., Davis, P.M., Dieterich, J.H., Deformation for inflation of a dipping finite prolate spheroid in an elastic half-space as a model for volcanic stressing, J. of Geophys. Res., 93, (1988) 8. Acknowledgements V. Magni and M. Battaglia have been partially funded by the USGS Volcano Hazard Team under Contract 08WRSA0249. M. Battaglia is funded by the Rientro dei Cervelli program of the Italian Ministry of Research and University. Comments by D.P. Hill and M. Mangan helped to improve the manuscript. 7