Magmatic unrest can be successfully monitored and studied from modeling of the induced surface deformation; one limiting factor however is the small number of available magmatic source models. Here we have obtained expressions (quadrupole approximation) for displacements and stresses from the inflation of any pressurized triaxial ellipsoid in an infinite elastic medium. The expressions can be evaluated by combining the effects of seven suitable point sources and are approximately valid also for a heterogeneous half-space. Till now, the only available (approximate or exact) expressions for finite expansion sources referred to spheres, prolate spheroids, and horizontal circular cracks embedded in a homogeneous half-space. Our approach allows to model also oblate spheroids and non-axisymmetric sources, whose effects were previously estimable only in the far field through a moment tensor representation. We also show that when the deformation source is a vertically flattened ellipsoid, the inversion of superficial displacements using a general moment tensor may lead to wrong physical models of the deformation source itself. This may be the case for the Campi Flegrei caldera, as well as for other calderas.
 Uplift and subsidence of the Earth's surface in volcanic areas can often be attributed to inflation or deflation of a cavity, whose morphology and dynamics can be inferred from surface deformation data. The joint inversion of vertical and horizontal displacements is necessary, since the knowledge of one component of the displacement (usually uplift) is generally not sufficient to retrieve the source geometry even for axisymmetric deformation fields [e.g., Dieterich and Decker, 1975]. Moreover, non-axisymmetric deformation fields are often observed, and non-axisymmetric source models have to be used, e.g., in case of dike intrusions. Analytical (approximate or exact) solutions for surface displacements are known only for very small (with respect to depth) cavities, namely a sphere [Mogi, 1958] and a generic triaxial ellipsoid, through its representation in terms of a single moment tensor [Davis, 1986], and for few finite cavity shapes, namely a sphere [McTigue, 1987], a prolate spheroid [Yang et al., 1988], and a circular crack [Fialko et al., 2001].
 Here we present approximate expressions (quadrupole approximation) for computing deformation and stress close to a uniformly pressurized triaxial ellipsoidal cavity, following the approach of Davis , i.e., using Eshelby's results for an infinite medium [Eshelby, 1957], and the half-space point force solution by Mindlin  as the fundamental Green's function. We consider, as a case study, the 1982–1984 Campi Flegrei caldera (Italy) unrest, that might be explained by a sill intrusion [e.g., Woo and Kilburn, 2010]. We show that the moment-tensor approach [Davis, 1986] can lead to incorrect results when inverting ground deformation data generated by a thin ellipsoidal source, to such an extent that retrieved moment eigenvalue ratios can be out of the permitted region for an ellipsoid [Amoruso and Crescentini, 2009] and even negative.
2. Approximate Solutions for Stress and Displacements due to a Pressurized Ellipsoidal Cavity in an Elastic Half-Space
 In what follows, suffixes preceded by a comma denote differentiation and repeating tensor indexes imply summation; coordinates are named using (x1, x2, x3).
 The external displacement field due to a pressurized ellipsoidal cavity in an infinite elastic medium is given by the volume integration over the ellipsoid [Eshelby, 1957; Davis, 1986]
where Uj,ki(r′ − r) is the Green's function describing the displacement in the i direction at r′ due to a pair of opposing forces at position r pointing in the j direction, separated in the k direction (force couple [e.g., Shearer, 2009]). The uniform moment density PjkT (related to the stress-free strain eT which the cavity would undergo if there were no surrounding rock matrix) is diagonal in a coordinate system whose axes are parallel to the principal axes of the ellipsoid; for any other system it can be found by the usual law for transforming tensors. Similar equations also hold for stresses and strains, provided the proper Green's functions are used. Equation (1) is a practically useful approximation also for a semi-infinite medium (x′3 ≤ 0), provided that (i) Uj,ki (r′ − r) are computed for a half-space (e.g., differentiating expressions of Mindlin ), in which case the Green's function does not depend on the difference x′3 − x3 (hence Uj,ki = Uj,ki(x′1 − x1, x′2 − x2, x′3, x3)), and (ii) the size of the ellipsoid is small with respect to its depth [Davis, 1986].
 Expanding Uj,ki(x′1 − x1, x′2 − x2, x′3, x3) around the ellipsoid center r0 = (x10, x20, x30), we get
Equation (2) is similar to the multipole expansion of the gravitational potential outside a mass distribution. The dipole term is null because of symmetry; Qlm is similar to the usual quadrupole moment, but cannot be made traceless because generally the function PjkT Uj,ki(x′1 − x1, x′2 − x2, x′3, x3) does not satisfy the Laplace's equation with respect to (x1, x2, x3). Qlm is diagonal in a coordinate system whose axes are parallel to the principal axes of the ellipsoid, in which case Qlm = Val2δlm/5, where al is the half-length of the ellipsoid axis which is parallel to the coordinate l-axis, V is the ellipsoid volume, and δlm is the Kronecker delta. In case of a spherical cavity Qlm is isotropic and PjkT Uj,ki(x′1 − x1, x′2 − x2, x′3, x3) satisfies the Laplace's equation; as a consequence, the quadrupole term of the expansion is null.
 Explicit solutions to equation (2) can be obtained, e.g., from Mindlin . The analysis is straightforward but rather tedious, and does not allow to get exact results because equation (2) is approximate ab initio. Moreover, the Green's functions cannot be written explicitly for a heterogeneous half-space. The presence of elastic heterogeneities affects stress and displacements [e.g., Crescentini and Amoruso, 2007], but equations (1) and (2) are still approximately valid provided that the heterogeneities do not alter significantly the uniform pressure condition at the cavity boundary and consequently Eshelby's results remain adequate. A possible approach, alternative to the explicit solutions and that can be implemented also for heterogeneous media, involves the use of seven point sources, of appropriate location and strength (see, e.g., Amoruso et al.  for the Green's functions in a layered half-space). To show that, we re-write equation (2) in the coordinate system whose axes (x1, x2, x3) are parallel to the semi-axes (a1 ≥ a2 ≥ a3 respectively) of the ellipsoid and whose origin is co-located with the ellipsoid center. In this coordinate system, the free surface is not at x3 = 0 and the Green's functions depend separately on the coordinates of the source and the observation point. Thus:
From the finite-difference approximation of the second derivative we get
and similarly for the two terms implying a2 and a3. We finally obtain
 Thus, one point source is located at the ellipsoid center; six half-potency (and opposite in sign) sources are symmetrically distributed along the ellipsoid axes. We have tested the 7-source model in numerous cases against the explicit solution to equation (2), getting a very good agreement (e.g., Figure S1 of the auxiliary material).
3. Model Comparisons
 In this section we show results from the inversion of synthetic surface displacement data, using our approach and other ones. Minimization of the misfit function (mean absolute deviation of residuals) and assessment of parameter uncertainties are performed as described by Amoruso et al. ; here displacements are statistically independent data, all having the same uncertainty.
 As previously mentioned, the quadrupole correction is null for a spherical cavity. Thus, under this approximation a finite sphere and an isotropic point source are equivalent: the more the cavity shape deviates from a sphere, the more important the quadrupole correction is. Like any other approach based on Eshelby's results for an infinite medium [Eshelby, 1957] and the half-space point force solution by Mindlin  as the fundamental Green's function, our approach is applicable in a semi-infinite medium only if the free surface causes negligible deviations from the uniform pressure condition at the cavity boundary. Yang et al.  found that these deviations are smaller than few percent if the ratio γ of the depth to the upper surface of the cavity to its minimum radius of curvature is greater than about 1.5.
 We test the goodness of our approach for finite spherical sources at different depths (0.25 ≤ γ ≤ 3), inverting synthetic superficial displacements computed using the reflection method of McTigue . The fit to data is very good, center depth and volume change (ΔV, computed as by Amoruso and Crescentini ) are retrieved correctly, but the source is seen as a small (point) oblate spheroid because of the free surface effects, which, as expected, decrease as γ increases; the ratio of the polar to the equatorial axes is larger than 0.95 for γ > 1.5 (Figure 1).
 We now compare surface displacements from our approximate finite ellipsoid model (semi-axes a ≥ b ≥ c) with other semi-analytical models for finite-volume and point sources. We compute synthetic superficial displacements for a homogeneous elastic semi-infinite medium, due to a vertical prolate spheroid (forward model A, FMA [Yang et al., 1988]), a horizontal prolate spheroid (FMB), a horizontal circular (radius R) crack [FMC; Fialko et al. 2001]. Synthetics are inverted for our finite-ellipsoid (quadrupole approximation) model (inversion model 1, IM1), a generic moment tensor representing a very small ellipsoid (IM2 [Davis, 1986]), and, in case the retrieved eigenvalue (Mmax ≥ Mint ≥ Mmin) ratios are out of the permitted region [Amoruso and Crescentini, 2009], a very small (point) ellipsoid, represented by a moment tensor whose eigenvalue ratios are constrained inside the permitted region (IM3). Best-fit results (two different source depths, d, for each forward model) are detailed in Table 1. Marginal probabilities show that uncertainties of retrieved parameters are very small (see, e.g., Figure S2). Retrieved parameters from IM1 are always in very good agreement with the true ones, and the spheroidal shape of FMB is correctly found even if all the axis lengths are left free in the inversion. The only (small) discrepancies relate to b/a for FMA, and to a for all models. In case of IM2, eigenvalue ratios are inside the permitted region for FMA and FMB, depth is underestimated for FMA, and the cavity shape is wrong for both FMA and FMB. Eigenvalue ratios are out of the permitted region (the retrieved moment tensor cannot be interpreted in terms of an expanding ellipsoid) for FMC. Using IM3 for FMC, depth is overestimated, but c/b → 0 correctly. As expected, discrepancies reduce when doubling depths. Fit to data is in Figure 2. IM2 usually performs worst, but, given measurement errors in geodetic data, such differences may be hard to distinguish.
Table 1. Inversions of Superficial Displacements From Forward Models FMA, FMB, and FMC
Fixed in the inversion.
Eigenvalue ratios incompatible with a point ellipsoidal source.
 The Campi Flegrei (CF) caldera is located in a densely populated area close to Naples (Southern Italy) renowned as a site of continual slow vertical movements. For at least 2000 years the caldera is slowly deflating, but major episodes of rapid uplifts occurred in the early 1500s (up to 8 m, culminating in the Monte Nuovo eruption in 1538) in 1969–1973 and in 1982–1984. These last two episodes have resulted in a net uplift of 3.5 m, but did not lead to an eruption. Geodetic data for the 1982–1984 unrest have been studied extensively [e.g., Woo and Kilburn, 2010, and references therein], but its source (recently modelled through a horizontal crack or a generic moment tensor) is still debated.
 Here we get clues on how deformation from a horizontal circular crack, possible source of both the major 1982–1984 unrest [Amoruso et al., 2008] and the minor 2004–2006 uplift [Amoruso et al., 2007], would be interpreted by using different source models. We use the approach of Fialko et al.  to generate synthetic data sets (FMCCF) related to the 1982–1984 levelling and EDM benchmark geometry (Figure S3). Then we give somewhat arbitrary (but reasonable, see Amoruso et al. ) errors to synthetic uplifts (1 cm) and distance changes (3 mm), and invert them (IM1, IM2, IM3). IM1 gives correct results for all model parameters (see Table 2), apart from a slight underestimate of R; IM2 gives a moment tensor whose eigenvalues are out of the permitted region for an ellipsoid and overestimates d by about 600 m; IM3 overestimates d by about 400 m, but gives c/b → 0 correctly. Fit to data is excellent for all the inversion models.
Table 2. Inversions of Superficial Displacements From Forward Models FMCCF and FMDsa
 As a subsequent step, we relax the axisymmetric assumption [e.g., Woo and Kilburn, 2010] and generate synthetic leveling and EDM data for finite triaxial ellipsoids (FMDs, differing in the source dimensions), elongated in the NW-SE direction like the main resurgent block of the CF caldera [e.g., Arienzo et al., 2010]. Synthetics are computed using our quadrupole model because no other semi-analytical method is available for this kind of source, and previously shown comparisons validate our approach. The minimum axis of the ellipsoid is vertical and the maximum axis strikes 120°. Results are summarized in Table 2. When source dimensions are not small with respect to depth (FMD1 and FMD2), we obtain Mmax ≫ Mint and Mmin < 0, and depth is largely overestimated. The retrieved moment eigenvalues cannot represent a pressurized ellipsoidal cavity, and any attempt to interpret the moment tensor would lead to physical models different from the forward one. Also IM3 is not able to invert data correctly. Even if the source dimensions are small with respect to depth, but still the ellipsoid is vertically flattened (FMD3), eigenvalue ratios are out of the permitted region; but in this case IM3 inverts data correctly. When the source dimensions are small with respect to depth and the ellipsoid is not vertically flattened (FMD4) both IM2 and IM3 succeed in inverting data correctly. Fit to data is always excellent, as evident from Figure 3, where the worst cases (FMD1, FMD2) are shown.
 Accurate inversion of real data requires the use of trade-off curves to balance the different data sets [e.g., Amoruso et al., 2008], information criteria to select the best model [e.g., Amoruso and Crescentini, 2007], and a more realistic representation of the medium; here we show that these criteria are not sufficient when inverting superficial displacements using a general moment tensor when a vertically flattened ellipsoidal source is involved. This may be the case for CF, and may be true as well for other calderas.
 We thank Noel Gourmelen for helpful review comments. 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.