Shape and volume change of pressurized ellipsoidal cavities from deformation and seismic data

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1 JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 4, 22, doi:.29/28j5946, 29 Shape and volume change of pressurized ellipsoidal cavities from deformation and seismic data A. Amoruso and L. Crescentini Received 22 July 28; revised 24 November 28; accepted 3 December 28; published 24 February 29. [] We present exact expressions for the volume change of a pressurized ellipsoidal cavity in an infinite homogeneous elastic medium. The expressions can be used as approximate solutions also for a homogeneous half-space. We show that previously published widely used expressions are correct for spheres but underestimate the ratio of the volume change to the product of pressure and volume in any other case. We discuss the capability to infer the shape of a single ellipsoidal cavity from far-field deformation measurements. Our results indicate that source axis ratios may often be hard to estimate, whereas it may be easier to infer the volume change of the source. We also consider the case of a source region consisting of interconnected pressurized ellipsoidal cavities, neglecting mutually induced stress. If all the cavities share the same shape and orientation, the source is seen in the far field as a single ellipsoidal cavity and it is possible to compute the total volume change from surface displacements. The shape of the apparent single ellipsoid is the same as the shape of the constituting cavities and not of the source region. In any other case a single ellipsoidal cavity might even be unable to give the same surface displacements as the ensemble of cavities. Since sudden volume change of a cavity can generate seismic waves, we discuss the decomposition of the related moment tensor into isotropic, double-couple, and compensated linear-vector dipole force systems in case of magma exchange between two ellipsoidal cavities, giving relations for the moment tensor components. Citation: Amoruso, A., and L. Crescentini (29), Shape and volume change of pressurized ellipsoidal cavities from deformation and seismic data, J. Geophys. Res., 4, 22, doi:.29/28j Introduction Dipartimento di Fisica, Università di Salerno, aronissi (SA), Italy. Copyright 29 by the American Geophysical Union /9/28J5946$9. [2] One of the most promising ways of studying volcanic phenomena is the use of high-precision deformation and gravity data. Critical stages prior to a volcanic eruption are usually investigated in the framework of linear elastostatic analyses developed from the study of volcanic unrest in terms of mechanical models and the overpressure in magma chambers and conduits. Shape and volume (and/or pressure) change of the source are estimated from deformation data. The source is often modeled as a single pressurized ellipsoidal (in most cases simply spheroidal or spherical) cavity. Once the source geometry and volume change are known, the density of the material entering the source may be inferred from gravity data, and this may allow discriminating between deformation episodes due to changes of the hydrothermal system and those due to a strictly magmatic intrusion. [3] Mogi [958] applied solutions for a point spherical pressure source in an infinite elastic medium to surface displacement data to show that, for eruptive activity at Sakurajiima and Kilauea, such observations provided constraints on the reservoir location. Despite the simplicity of the model, surface displacements associated with volcanic activity could be very well explained in terms of a point spherical source at a determined depth in a homogeneous half-space. This model continues to enjoy widespread use and success [e.g., Gottsmann et al., 26a], especially since the sparsity of data available for investigating volcanic activity generally requires that only solutions with the fewest number of parameters can be considered. [4] The elastic field due to an ellipsoidal inclusion in an infinite medium has been treated by Eshelby [957] by integrating distributions of point forces over the surface of the ellipsoid that satisfy boundary conditions. The external displacement field is given by the convolution of a uniform distribution of elementary sources consisting of three double forces acting along the axes of the ellipsoid and the Green s function giving displacement because of double forces of unit intensity. The intensity of the double forces is uniform inside the volume of the ellipsoid. Davis [986] followed Eshelby s approach but used the point force solution for a half-space rather than a full-space (Kelvin force) as the fundamental Green s function. The solution satisfies boundary conditions exactly on the free surface but approximately on the ellipsoid, and is reasonably accurate if the depth to the ellipsoid center is greater than twice its dimension. Davis far-field solution is the sum of displacements from three co-located double forces acting along the 22 of8

2 22 22 axis of the ellipsoid and has been widely used to interpret surface deformations in volcanic areas [e.g., Langbein et al., 995; onaccorso et al., 25]. This solution has also been incorporated into studies with layered media [e.g., Amoruso et al., 27, 28]. Far-field deformation from a pressurized ellipsoidal cavity is thus the same as from a moment tensor, whose eigenvalues are proportional to the product of the source pressure and volume and depend on the axis ratios; the normalized eigenvectors of the moment tensor represent the directions of the ellipsoid axes. [5] Yang et al. [988] gave exact analytic expressions for the deformation field resulting from inflation of a finite prolate spheroidal cavity in an infinite elastic medium, and approximate solutions for a dipping spheroid in an elastic half-space. They have shown that the model is satisfactory accurate if the minimum radius of curvature of the upper surface is less than or equal to its depth beneath the free surface. Comparison of surface displacements generated by the point [Davis, 986] and finite [Yang et al., 988] models gives good agreement, provided this criterion is satisfied. Differently from the point model, the finite model can be used to estimate the distribution of stresses within the volcano in the near field of the source. The finite model has been used by Pritchard and Simons [24], onaccorso et al. [25] and Gottsmann et al. [26b] among others. [6] Fialko et al. [2] derived exact expressions for vertical and horizontal displacements of the free surface of a half-space from a horizontal circular crack in a semi-infinite elastic solid, and calculated surface displacements for the special case of a uniformly pressurized crack. The pressurizedcrack model has been used by attaglia et al. [26] and Gottsmann et al. [26b] among others. Crescentini and Amoruso [27], and Amoruso et al. [27, 28] have used a suitable regular distribution of point cracks over the pressurized-crack plane to approximate surface displacements and gravity changes due to a horizontal circular crack embedded in a layered half-space. [7] When modeling the source as a pressurized ellipsoid, inversion of measured surface displacements allows to estimate the product between the source volume (V) and overpressure (P). The volume change of the source (DV) is of great importance, e.g., for estimating the intrusion density from gravity data, but its link to VP is not trivial. Here we derive exact expressions for the volume change of a pressurized ellipsoidal cavity, using expressions for the elastic field due to an ellipsoidal inclusion in an infinite medium given by Eshelby [957]. As pointed out by Davis [986], Eshelby s expressions are approximately valid also in a half-space, provided that the appropriate Green s functions are used. [8] We find that the volume change is proportional to the sum of the intensities of the double forces constituting the elementary sources into which the ellipsoidal source can be decomposed, but the proportionality factor depends on the shape of the ellipsoidal cavity (i.e., its axis ratios). Similarly, we find that the volume change is proportional to the trace of the moment tensor that represents the far-field displacements from the ellipsoidal source, but once again the proportionality factor depends on the shape of the ellipsoidal cavity. As an example, we consider the source of the 97 deformation at Kilauea Volcano, Hawaii, as modeled by Davis [986]. We additionally discuss the case of interconnected equally pressurized ellipsoidal cavities. [9] As regards seismic records, focal mechanisms of earthquakes recorded in volcanic and geothermal areas sometimes show strong departures from pure double-couple ones. The moment tensor can be decomposed into isotropic (ISO), double-couple (DC) and compensated linear-vector dipole (CLVD) force systems, e.g., using the method of Knopoff and Randall [97]. The ISO component is attributed to a spherical magma oscillation, the DC component to shear faulting, and the CLVD to rapid movement of magmatic fluid [e.g., Julian et al., 998]. Here we show that the moment tensor of earthquakes generated by a sudden magma exchange between two ellipsoidal cavities (e.g., because of the breaking of a barrier) can include all the three (ISO, DC, and CLVD) components. As an example, we consider three deep low-frequency earthquakes occurred beneath Iwate volcano, Japan, in 998 and 999, as modeled by Nakamichi et al. [23]. 2. Volume Change for One Pressurized Ellipsoidal Cavity [] Eshelby [957] obtained the elastic field due to an ellipsoidal inclusion in an infinite medium by integrating over the surface of the ellipsoid distributions of point forces that satisfy boundary conditions on that surface. When considering an elastic half-space, the problem is complicated by the necessity to satisfy boundary conditions on two surfaces, the ellipsoid and the free surface [Davis, 986]. It can be solved using a two-step approach: () replacing the ellipsoidal cavity with the material external to it [as in the work by Eshelby, 957] and (2) finding the distribution of half-space point forces applied to its surface so that pressure P is exerted on the surrounding rock matrix and boundary conditions on the free surface are satisfied [Davis, 986]. The solution satisfies boundary conditions exactly on the free surface but approximately on the ellipsoid, and is reasonably accurate if the depth to the ellipsoid center is greater than twice its dimension. [] Using Eshelby s notation, e T ij (with i, j =,2,3) indicates the stress-free strain which the ellipsoidal inclusion would undergo if there were no matrix, e C ij the constrained strain of both the inclusion and the matrix, and p ij I the stress inside the inclusion. We have [Eshelby, 957] p I ij ¼ 2m ec ij e T ij þ ld ij e C e T where l and m are the Lamé parameters of the rock matrix, e C = e C kk, e T = e T kk, and repeating tensor indexes imply I summation. For a pressurized cavity, p ij = d ij P, and 3P ¼ p I kk ¼ 3l e C e T þ 2m e C e T ¼ 3k e C e T where k = l +2m/3 is the bulk modulus. [2] The volume change of the ellipsoidal inclusion (DV) is given by DV = Ve C. In order to derive an expression for ðþ ð2þ 2of8

3 22 22 for DV in the work by Tiampo et al. [2] always underestimates the volume change. The underestimate increases as the shortest axis of the ellipsoid becomes smaller and smaller with respect to the other two axes. The line b/a = in Figure relates to oblate spheroids, whose end points are a sphere (c/b = ) and a penny-shaped crack (c/b! ). The line c/b = relates to prolate spheroids, whose end points are a sphere (b/a = ) and a cigar-like source (b/a! ). Numerical values of R V for a spheroid are plotted in Figure 2, as a function of the spheroid aspect ratio (ratio of polar to equatorial radius). [5] The relation DV = 3PV/(4m) had been used to estimate the volume change of prolate spheroidal sources in various volcanic areas, like Long Valley Caldera [e.g., attaglia et al., 23; Newman et al., 26] and Campi Flegrei Caldera [attaglia et al., 26]. Rivalta and Segall [28] used DV =3PV/(4m) to model the effects of magma compressibility. The relevance of the misestimate is of course dependent on the ellipsoid axis ratios. Figure. Ratio R V of the volume change for an ellipsoidal cavity to the volume change for a spherical cavity, as a function of b/a (intermediate to major semiaxis of the ellipsoid) and c/b (minor to intermediate semiaxis of the ellipsoid); n =.25. The two cavities share the same value of the product of pressure and volume. The grey rectangle indicates ranges of the axis ratios consistent with moment tensor eigenvalues for the 97 deformation at Kilauea Volcano, Hawaii, as modeled by Davis [986] (see text and Figure 3 for details). DV, we put ij =2m e T ij + ld ij e T and = kk =3ke T, substitute it into equation (2), and finally obtain DV ¼ VP 3k P 3 2n ¼ 2þ ð nþ V P m P 3 ð3þ 3. Source Geometry From Far-Field Displacements [6] In case the reference axes (x, y, z) are aligned with the ellipsoid axes, Davis [986] showed that far-field deformation from a pressurized ellipsoidal cavity is the same as deformation from the moment tensor M ¼ P T a P T b P T c where P T a =, P T b = 22, P T c = 33. Davis [986] gave expressions for computing P T a /P, P T b /P, and P T c /P (and, consequently, also /P) as a function of the ellipsoid axis ratios, in terms of elliptic integrals. From equation (3), volume change depends on the trace m = V(P T a + P T b + P T c ), which is rotational invariant and can be calculated in any A ð5þ where n =.5 (3k 2m)/(3k + m) is the Poisson s ratio. [3] The approach followed by Tiampo et al. [2] is very similar to ours as regards equation (2), but then they used the relation 3e C = e T ( + n)/( n), which is valid for spheres only [Eshelby, 957], and obtained DV = 3PV/(4m). As a consequence this last expression is correct for spheres, but incorrect for ellipsoids. [4] We now consider the ratio R V of the volume changes for an ellipsoidal (DV e ) and spherical (DV s ) cavity sharing the same VP value. DV e DV s ¼ 2 3 2n þ n P 3 ð4þ This ratio measures how much easier an ellipsoid can expand with respect to a sphere and depends on the ellipsoid semiaxis (a b c) ratios; the way of calculating /P is specified later in the text. Figure (where n =.25) shows that R V is > for any nonspherical ellipsoid, and depends on c/b much more than on b/a. In other words, the expression Figure 2. Ratio R V of the volume change for a spheroidal cavity to the volume change for a spherical cavity, as a function of the aspect ratio (ratio of polar to equatorial radii) of the spheroidal cavity; n =.25. The two cavities share the same value of the product of pressure and volume. 3of8

4 22 22 Figure 3. Contour plot showing the ellipsoid semiaxis (a b c) ratios as a function of the moment tensor eigenvalue (m max m int m min ) ratios; n =.25. The moment tensor can be obtained from farfield deformation measurements. Dots indicate eigenvalue ratios from 3 random realizations of the eigenvalues from three independent Gaussian distributions related to the 97 deformation at Kilauea Volcano, Hawaii, as modeled by Davis [986] (means, Davis best-fit values; standard deviations, half Davis s stated errors). reference system, or, in other words, for any orientation of the cavity axes. [7] If two of the axes are equally long (i.e., the ellipsoid is a spheroid), the eigenvalues of the moment tensor can be computed in terms of elementary functions (for details, see the work of Eshelby [957]). We obtain approximate results for very small or very large axis ratio using appropriate Taylor expansions of equations (3.5) and (3.6) in the work by Eshelby [957]. [8] For a very thin spheroid (a = b c) we get Thus P T a P ¼ 2n P ¼ PT a P þ PT b P þ PT c P From equation (3) we get ¼ 5 4n 2n ðþ ð2þ P T c P ¼ a c 4 ð nþ 2 pð 2nÞ ð6þ DV ¼ VP m : ð3þ Thus From equation (3) P T a P ¼ a c 4nð nþ pð 2nÞ P ¼ PT a P þ PT b P þ PT c P ¼ a c DV ¼ VP ð 2nÞ a 4 n 2 ð Þ 2m ð þ nþ c pð 2nÞ a3 ð nþ P m 4 ð n 2 Þ pð 2nÞ where the last equality strictly holds for c/a!. This result is in full agreement with what obtained by Fialko et al. [2] for a finite circular crack of radius a at large depths, i.e., when it approximates a very thin spheroid. [9] For a very thin cigar-shaped spheroid (a b = c) we get P T c P ¼ 2 n 2n ð7þ ð8þ ð9þ ðþ [2] Finally, for a sphere (a = b = c) we get and consequently Pa T P ¼ 3 n 2 2n DV ¼ VP ð 2nÞ 3 3 n 2m ð þ nþ 2 2n 3 ¼ 3 VP 4 m ð4þ ð5þ as expected. [2] Inversion of far-field deformation data allows retrieval of the eigenvalues (m max m int m min ) and eigenvectors of M. Figure 3 shows how the shape of the ellipsoidal source is related to them, when n =.25. Acceptable eigenvalue ratios are confined to a small region. If m max /m int > 2, then c/b (here <.4) may be usually evaluated from far-field deformation data but b/a is practically undefined since even small changes of m int /m min can cause large changes of b/a.if m max /m int < 2, then both c/b (here >.3) and b/a may be generally evaluated. Since /P depends on m max /m int and m int /m min, it is also possible to get DV from the trace of the moment tensor (m) and the ratios m max /m int, m int /m min (see Figure 4, n =.25). 4of8

5 22 22 Figure 4. Contour plot showing mdv/m as a function of the moment tensor eigenvalue (m max m int m min ) ratios; n =.25. The moment tensor can be obtained from far-field deformation measurements. Dots indicate eigenvalue ratios from 3 random realizations of the eigenvalues from three independent Gaussian distributions related to the 97 deformation at Kilauea Volcano, Hawaii, as modeled by Davis [986] (means, Davis best-fit values; standard deviations, half Davis s stated errors). [22] For the mere sake of showing how our results can be used, we consider the source of the 97 deformation at Kilauea Volcano, Hawaii, as modeled by Davis [986]. After inversion of horizontal and vertical displacements, the eigenvalues of the moment tensor obtained by Davis are (35.9 ± 2.5) 6 m Nm, (24.8 ±.7) 6 m Nm, and (9.2 ±.4) 6 m Nm. Even if we can assume that eigenvalues from data inversion are normally distributed around real values, ratios m max /m int and m int /m min are not. Thus for the sake of generality we follow a Monte Carlo approach to give an idea of the confidence region for b/a and c/b. We generate 3 random realizations of the moment tensor eigenvalues from three independent normal (Gaussian) distributions (means, Davis best-fit values; standard deviations, half Davis s stated uncertainties) and compute m max /m int and m int /m min for each realization. We do not take into account possible correlations between eigenvalues, because they are not discussed in the work by Davis [986]. Values of m max /m int and m int /m min from synthetic eigenvalues are shown in Figures 3 and 4 (dots). The alternative method of using the uncertainty ellipse obtained from the covariance matrix and appropriate Taylor expansions for the moments of functions of random variables is acceptable only if eigenvalue relative uncertainties are small. Such a condition is satisfied in case of Davis data, and consequently the point cloud closely resembles the uncertainty ellipse of m max /m int and m int /m min. Figure 3 indicates that Davis solution is consistent with c/b.7 and b/a <.5. Resulting confidence region is roughly sketched by the grey rectangle in Figure. As regards the volume change, Figure 4 gives mdv/m.25. For m 8 6 m Nm, we get DV 7 m 3. At least in this case, DV is better constrained than the ellipsoid shape. 4. Volume Change for an Ensemble of Equally Pressurized Ellipsoidal Cavities [23] We now consider a source region including N ellipsoidal cavities. If they are interconnected by narrow fluidfilled conduits (whose contribution to the total volume change is negligible) we can assume that the ellipsoidal cavities share the same pressure. Each cavity induces stress at the surface of nearby cavities, thus modifying boundary conditions with respect to a single cavity. A similar effect occurs for an ellipsoidal cavity embedded in a half-space: as pointed out by Davis [986] the half-space Green s functions introduce stresses at the ellipsoid surface corresponding to image sources on the far side of the free surface. These stresses decay as /(distance 3 ) and numerical computations suggest that neglecting induced stresses is quantitatively applicable if depth to radius ratios are greater than 2 (for a discussion, see the work of Davis [986]). Therefore we treat an ensemble of equally pressurized ellipsoidal cavities as if each cavity were isolated, assuming that the distance between cavity surfaces is generally larger than twice the cavity size. [24] Far-field deformation due to cavity k is equivalent to that of the moment tensor (M k ) ij = V k (p k T ) ij. Ratios among eigenvalues are related to the cavity shape, and normalized eigenvectors give the cavity orientation. As a first approximation, moment tensors associated with the cavities inside the volume source can be considered co-located. Overall far-field deformation is thus equivalent to that of the moment tensor M ij ¼ P XN V k k¼ h k ij =P [25] Overall volume change is given by DV ¼ XN k¼ DV k 2n ¼ 2þ ð nþ X N i¼ 2n VP ¼ 2þ ð nþ m P k V k m P 3 X N k¼ V k k PV 3! i : ð6þ ð7þ where p k T =(p k T ) ii refers to the k-th cavity and V = P N k¼ V k. [26] If all the cavities share the same shape and orientation (even if volumes are different) we get M ij = VP ( ij/p). 5of8

6 22 22 The ensemble of cavities is consequently seen as a single ellipsoidal cavity sharing the same shape and orientation, and equation (3) can be used to estimate the volume change of the source. However, the shape of the apparent single ellipsoidal cavity does not necessarily reflect the shape of the source region. [27] If all the cavities share the same shape but not the same orientation, their shape is not given by the eigenvalues of M ij. The ensemble of cavities is consequently not seen as a single ellipsoidal cavity sharing the same shape. Eigenvalue ratios can even be out of the permitted region for a single ellipsoidal cavity (see Figure 3). However, from the constancy of the ratio /P for all the cavities, we get m ¼ M ii ¼ XN k¼ ¼ VP pt P V k P pt k P ð8þ and VP can be obtained from m if the cavity shape is assumed a priori. In this case we can use the relation 2n DV ¼ m 3VP 2þ ð nþ ð Þ ð9þ to infer the total volume change. [28] The relation m = P N k¼ m k and equation (9) still hold if the cavities have different shape and orientation, but it is impossible to infer the volume change of an ensemble of ellipsoidal cavities from M ij, if, as usual, PV cannot be estimated independently. Also the presence of double-couple sources does not affect the validity of equation (9), since related moments are trace free. 5. Moment Tensor Decomposition [29] We now consider the case of two coupled spheroidal (a = b 6¼ c) cavities. We neglect stress induced at the surface of each cavity by the other one, and treat each cavity as isolated. In other words, we assume that the distance between cavity surfaces is larger than cavity size. For some reason (e.g., fluid migration) volume of both cavities (DV and DV 2 ) change. Total volume change DV = DV + DV 2 can be 6¼. Symmetry axes of both cavities and 2 are supposed parallel to the z axis of the coordinate system. [3] From equations (3) and (5), the moment tensor is M ¼ 2 þ n 2n m DV þ 2 þ n 2n m DV 2 =P 3 2 =P 2 3 Pa T =P Pa T =P A Pc T =P P2a T =P 2 P2a T =P 2 A P2c T =P 2 ð2þ where subscripts and 2 refer to the two cavities respectively. [3] The moment tensor can be decomposed into isotropic (ISO), double-couple (DC) and compensated linear-vector dipole (CLVD) force systems (M = M ISO + M DC + M CLVD ). We perform the decomposition using the method of Knopoff and Randall [97], which makes the major axis of the CLVD coincide with the corresponding axis of the DC. We obtain M ISO ¼ 2 þ n 3 2n m DV =P =P 3 þ DV 2 2 =P 2 DV 2 =P 2 3 A M CLVD ¼ 4 3 M DC A þ n P T 2n m DV a =P Pc T =P =P 3 þ DV 2 DV P T 2a =P 2 P T 2c =P 2 2 =P 2 3 ð2þ ð22þ =2 =2 A ð23þ [32] In this case, the moment tensor has null DC component. [33] If the symmetry axes of cavities and 2 are parallel to the z axis and y axis of the coordinate system respectively, from equations (3) and (5) the moment tensor is M ¼ 2 þ n 2n m DV þ 2 þ n 2n m DV 2 =P 3 2 =P 2 3 Pa T =P Pa T =P A Pc T =P P2a T =P 2 P2c T =P 2 A P2a T =P 2 ð24þ where once again subscripts and 2 refer to the two cavities respectively. [34] M ISO is given again by equation (2). oth M DC and M CLVD are now nonnull, and may be computed as indicated by Knopoff and Randall [97]. From the decomposition algorithm it follows that M DC 33 and M CLVD 33 share the same sign. [35] Sudden volume change of the two cavities (e.g., because of the breaking of a barrier) can generate seismic waves. Focal mechanisms of earthquakes recorded in volcanic and geothermal areas sometimes show strong departures from pure DC ones. For the mere sake of showing how our results can be used to interpret non-dc earthquakes, we consider three deep low-frequency earthquakes occurred beneath Iwate volcano, Japan, in 998 and 999. The three events (DLFA-2, DLFA-3 and DLFA-5) are the only ones having an isotropic component percentage larger than 5%, 6of8

7 22 22 moment tensor decomposition is by no means univocal, Figure 5 evokes the possible existence of a quasi-spherical chamber, suffering magma injection from elongated dikes (DLFA-3 and DLFA-5) and ejection into sill-like openings (DLFA-2). If this picture is correct, from equation (2) magma injection into the chamber would be about 9 m 3 for DLFA-3 and 5 m 3 for DLFA-5 (m =.7 Pa; Nakamichi et al. [23]). Magma ejection from the chamber would be about 5 m 3 for DLFA-2. Using the isotropic component only, Nakamichi et al. [23] obtained.45 m 3 and.8 m 3 for a crack and a general explosion or implosion, respectively, for DLFA-2; the same approach gives.46 m 3 and.8 m 3 for DLFA-3, and.57 m 3 and.32 m 3 for DLFA-5. Figure 5. Symbols: (top) Normalized (with respect to the sum of the absolute values of the three moment tensor components) CLVD and ISO components of events DLFA- 2 (diamonds), (middle) DLFA-3 (squares), and (bottom) DLFA-5 (triangles). Lines: isolines of the normalized CLVD and ISO components in case of magma migration inside pairs of spheroidal cavities (i.e., DV 2 = DV ) having mutually perpendicular symmetry axes. Isolines have been generated using DV < and refers to fixed values of c /a (dashed lines) and c 2 /a 2 (solid lines). among those analyzed by Nakamichi et al. [23]. From Table of Nakamichi et al. [23], moment tensor eigenvalues are:.39 Nm,.98 Nm,.69 Nm for DLFA-2;.54 Nm,.2 Nm, 2.5 Nm for DLFA-3;.7 Nm,.33 Nm,. Nm for DLFA-5. [36] Normalized (with respect to the sum of the absolute values of the three moment tensor components) CLVD and ISO components of each event are shown in Figure 5 (symbols), where we also plot isolines of the normalized CLVD and ISO components in case of magma migration inside pairs of spheroidal cavities (i.e., DV 2 = DV ) having mutually perpendicular symmetry axes. Isolines have been generated using DV < and refers to fixed values of c /a (dashed lines) and c 2 /a 2 (solid lines). Thus magma movement is from the cavity whose aspect ratio is given by the dashed line to the cavity whose aspect ratio is given by the solid line. We use two panels for each event because two different paired cavity configuration is consistent with each event. Although the interpretation of the 6. Conclusions [37] We have derived expressions for the volume change of a pressurized ellipsoidal cavity, using expressions for the elastic field due to an ellipsoidal inclusion in an infinite medium given by Eshelby [957]. As pointed out by Davis [986], Eshelby s expressions are approximately valid also in a half-space, provided that the appropriate Green s functions are used, and the far-field deformation from a pressurized ellipsoidal cavity is the same as from a moment tensor, whose eigenvalues are proportional to the product of the ellipsoid pressure and volume and depend on the axis ratios; the normalized eigenvectors of the moment tensor represent the directions of the axes of the ellipsoid. [38] We find that only for a given shape of the ellipsoidal cavity (i.e., for given axis ratios) the volume change is proportional to the sum of the intensities of the double forces constituting the elementary sources into which the ellipsoidal source can be decomposed. We also find that only for a given shape of the ellipsoidal cavity, is the volume change proportional to the trace of the moment tensor that represents the far-field displacements from the ellipsoidal source. For a given moment tensor trace, the volume change can vary up to a factor of two, depending on the moment tensor eigenvalue ratios. [39] While the dependence of the volume change on the moment tensor eigenvalue ratios generally allows an estimate of DV from far-field deformation data, retrieval of the cavity shape (ellipsoid axis ratios) may be not robust if the ratio of the minor to intermediate axes is small (<.4). [4] Source properties retrieved from far-field deformation data are less reliable for cases of interconnected equally pressurized ellipsoidal cavities. Neglecting stress induced at the surface of each cavity by nearby ones, we find that () if all the cavities share the same shape and orientation, but have different volumes, the ensemble of cavities is seen in the far field as a single ellipsoidal cavity, the shape of the apparent single ellipsoidal cavity is the same as the shape of the constituting cavities and not of the source region, and it is possible to compute the total volume change from surface displacements; (2) if all the cavities share the same shape but have different orientations, a single ellipsoidal cavity might even be unable to give the same surface displacements as the ensemble of cavities and it is possible to compute the total volume change only if the cavity shape is assumed a priori; and (3) if the cavities are different both in shape and orientation, it is possible to compute the total 7of8

8 22 22 volume change only if the product of pressure and total volume is assumed a priori. [4] Volume change due to sudden magma movement between paired cavities can originate seismic waves. Source moment tensor can be decomposed into isotropic, doublecouple, and compensated linear-vector dipole force systems. We give relations for the moment tensor components and show that volume-change values obtained from the isotropic component only can be much smaller than those really involved in the magma exchange process. Our approach might improve modeling of non-dc earthquakes in volcanic and geothermal areas. [42] Acknowledgments. This research has benefited from funding provided by the Italian Presidenza del Consiglio dei Ministri-Dipartimento della Protezione Civile (DPC). Scientific papers funded by DPC do not represent its official opinion and policies. We are grateful to A. T. Linde for useful comments. References Amoruso, A., L. Crescentini, A. T. Linde, I. S. Sacks, R. 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