Sources responsible for volcanic unrest produce characteristic surface deformation. Given a sufficient number of distributed observation points, inversion is the preferred procedure for retrieving the source parameters of location and volume or pressure change. Most often the solutions have been for point sources embedded in a homogeneous half-space. Recent work indicates that layered structures, particularly those with soft superficial layers, significantly perturb the deformation pattern compared with that for the homogeneous medium. We apply the methods of L. Crescentini and A. Amoruso to data for the most recent mini-uplift in the Campi Flegrei caldera and show that models using a homogeneous medium cannot adequately fit all the data. Incorporating a layered structure appropriate for Campi Flegrei allows a significantly better fit, avoiding characteristic discrepancies which are revealed by a synthetic test. Failure to use such structure results in incorrect source parameters, possibly leading to misleading geophysical interpretations.
 The Campi Flegrei (CF) caldera is a volcano-tectonic depression, between Naples and the volcanic islands of Ischia and Procida. It is a highly populated area (about 400000 people) located 15 km west of Naples inside the Campanian Plain, a graben-like structure at the eastern margin of the Tyrrhenian Sea. The caldera is a nested and resurgent structure created by a subsidence of the CF area generally ascribed to several collapses during the last 40000 years.
 The last eruption occurred in 1538 at western periphery of CF and created a spatter cone (Monte Nuovo, about 130 m). Since the last eruption, like many calderas, the CF caldera suffered notable unrest episodes, including large ground deformations [Newhall and Dzurisin, 1988], seismic swarms and increases in the degassing activity [Barberi et al., 1984]. Its bradyseismic (slow ground movement) activity during the last 2000 years is spectacularly recorded by the peculiar geographic setting and the presence of Roman ruins: the caldera is partially submerged, so sea level provides a natural reference level for relative ground movements, and marine deposits on Roman ruins and historical documents have been studied to reveal large, secular subsidence, on which are superposed very fast uplifts, not always giving rise to an eruption [e.g., Dvorak and Mastrolorenzo, 1991].
 The caldera has been generally subsiding (at about 1.5 cm per year) from 1538 till 1969. A substantial ground uplift, more than 1 m of deformation, occurred in the period 1969–1972 and, after a small subsidence of about 30 cm after 1972, a very strong uplift occurred in the period 1982–1984 (about 1.8 m), with subsequent partial recovery. Superposed on the still continuing subsidence are some short uplift phases (miniuplifts during 1989, 1994, 2000, 2004–2006); ground level remains about 2.5 m above pre-1970 levels at the town of Pozzuoli [e.g., Troise et al., 2007].
 Inversion of ground deformation data is a powerful tool to infer features of the source responsible for such volcanic unrest, whether at CF or elsewhere. The source is usually modelled as an expanding (or contracting) vertical spheroid embedded in a homogeneous half-space. If the source dimensions are small with respect to depth, it can be approximated by a point source. In this case the displacement pattern depends on the source aspect ratio (ratio of polar to equatorial radii), but the ratio of horizontal to vertical surface displacement (H/V) at all surface location is always equal to the ratio of epicentral distance to source depth (r/d). Thus, if the epicentral distance is known, in principle the displacement vector of one bench mark is sufficient to infer source depth.
 The effects of crustal layering on ground displacements and gravity changes due to expanding sources have been recently investigated [e.g., Battaglia and Segall, 2004; Crescentini and Amoruso, 2007; Manconi et al., 2007]. In particular, Crescentini and Amoruso  have shown that the presence of soft superficial layers affects the deformation pattern (giving an apparent shallower source if layering is not taken into account), H/V ratios (no longer simply r/d), and the subsurface mass redistribution effects on gravity changes. Consequently H/V ratios are not such simple indicators as in the case of a homogeneous half-space.
 Here we show that, at least for large contrasts among layer properties, neglecting layering results in misestimates of the source parameters, and additionally it is not possible to obtain simultaneously a good fit to both horizontal displacements and H/V ratios.
 These effects are quite evident in the data for the 2004–2006 unrest of the CF caldera. Leveling data are available for all recent events. Precision leveling surveys have been performed since 1969, and continuous GPS (CGPS) data are available since year 2000 [Pingue et al., 2006; Troise et al., 2007].
 We invert all horizontal and vertical components of the CGPS data as well leveling data (all from published plots) to retrieve the causative source of the 2004–2006 unrest. We show that only by taking into account crustal layering it is possible to have a good fit of all deformation data and hence obtain a robust estimate of source parameters.
2. Modelling Surface Deformation
 Following the approach in the work by Davis  far-field deformation due to inflation of a mass-less pressurized vertical spheroid is obtained by a weighted combination of an isotropic point source (IPS) and a compensated linear vertical dipole (CLVD). Weights depend on the source aspect ratio. We compute Green's functions for ground displacements due to an IPS and a CLVD using code from Wang et al. .
 To determine parameters for the layered model we use the vertical profile of density from a recent 1D Vp model [Judenherc and Zollo, 2004] and the Nafe-Drake curve. Following Trasatti et al.  we take into account the drained response of the medium (more appropriate for studying slow ground deformation) by setting the Poisson's ratio to 0.25 at all depths. Note that the layered model used in this and the previous work is determined from seismic data; no deformation data were used to obtain the model parameters which are not adjusted to improve the fit in any of the inversions. Adopted values of Vp, density and rigidity for selected depths are listed in Table 1. Values for intermediate depths are obtained by linear interpolation between adjacent listed depths.
Table 1. Campi Flegrei Multilayered Model Used in This Work
 We assume that the source is either a point spheroid or a finite (horizontal) crack, and is identified by its horizontal projection (x, y), depth, potency (i. e. volume change), aspect ratio (point source) or radius (finite crack). The problem of computing deformation due to a finite horizontal crack embedded in a homogeneous half-space has been solved by Fialko et al. ; no solution is available for a horizontally layered elastic medium, but a point crack (PC) is a good approximation of a finite crack (FC) if source depth to radius exceeds 5. A regular distribution of PCs over the FC mid-plane gives the same ground displacements in a homogeneous half-space as those computed using the code by Fialko et al.  if source depth to radius exceeds 0.8 [Crescentini and Amoruso, 2007]. Vertical opening of each PC is proportional to 1 − (r/R)4, where r is distance from the crack axis and R is the crack radius. PC spacing is (source depth)/10. We also approximate the FC with the same PC distribution in the layered half-space.
 Data inversion leads to minimizing a cost function which measures the disagreement between model and observations for different model parameters. We use two different cost functions, namely the mean squared deviation of residuals (chi-square fitting, , appropriate for normally distributed errors) and the mean absolute deviation of residuals (, appropriate for two-sided-exponentially distributed errors and commonly used for robust fitting) [e.g., Amoruso et al., 2002]:
where j = 1, …, M indicates different data sets, wj is the weight of each data set in the cost function, i = 1, …, Nj indicates rotated independent data (xi) in each data set [Amoruso and Crescentini, 2007], fi(a) is model prediction of xi, and σi is uncertainty of xi. Here we use wj = 1 for each data set.
 Cost function minimization is obtained using Adaptive Simulating Annealing (ASA) [Ingber, 1993] (finding the best global fit of a non-linear non-convex cost-function over a D-dimensional space) or Neighbourhood Algorithm (NA) [Sambridge, 1999a] (generating ensembles of models which preferentially sample the good data fitting regions of the parameter space, rather than seeking a single optimal model). Assessment of parameter uncertainties is performed using NA-Bayes (NAB) [Sambridge, 1999b] (making quantitative inferences from the entire ensemble produced by any direct search method and allowing measures of resolution and trade-offs, within a Bayesian framework) and bootstrapping [e.g., Amoruso et al., 2005] (the best fitting technique is applied to a large number of synthetic data sets, generated from the actual data set using random resampling with replacement).
Figure 1 serves to illustrate the difficulties of attempting to model displacements observed on the surface of a layered half-space if the layering is ignored (i. e. using a homogeneous half-space). We generate displacements and H/V ratios due to a point crack (depth 4000 m) in our layered medium (our synthetic data) and then attempt to fit those data with sources (point source or extended crack) in a homogeneous half-space. If horizontal displacements are well matched, predicted H/V ratios are too small, while fitting H/V values results in discrepancies in the horizontal displacement pattern, either in shape or in position of the maximum (too close to the source axis). We use inversions as briefly described later in the text.
 For the 2004–2006 CF ground uplift, we invert jointly precision leveling and CGPS data obtained from plots by Del Gaudio et al.  and Troise et al. . Leveling surveys used here were carried out in November, 2004, and December, 2006. CGPS time series give displacements with respect to QUAR reference benchmark [Pingue et al., 2006]; here we use inferred displacements for about the same period as the leveling surveys. Benchmark positions have been obtained from “The Campi Flegrei Caldera GIS Database” on line at http://ipf.ov.ingv.it/cf_gis.html and are shown in Figure 2. Leveling data errors are treated as by Amoruso and Crescentini ; plots in the work by Del Gaudio et al.  give a measurement error of about 0.7 mm/, and we estimated uncorrelated uncertainties to be about 2 mm on the basis of best fit misfits (for a discussion, see Amoruso and Crescentini ). From plotted measurement errors and uncertainty on background trend, we assigned 2-mm error to horizontal CGPS displacements and 5-mm error to vertical CGPS displacements, but we have also performed inversions using doubled errors and obtain almost identical results. Here we show results obtained using NA and NAB for conciseness. ASA and bootstrapping give very similar results.
 For both a homogeneous half-space and the layered one, the misfit for an extended crack is lower than for a point source, regardless of its aspect ratio. Thus we show results obtained inverting deformation data for a finite crack only. Figures 3 and 4compare CGPS and leveling data with calculated model values for the best fit (-misfit) extended crack embedded in a homogeneous or layered half-space. Consistent with results for the synthetic test shown in Figure 1, the homogeneous half-space model allows a good fit to levelings and the shape of the horizontal displacement pattern, but is unable to fit the H/V ratios. However, using the layered half-space model allows a good fit to all available deformation data. As a consequence, a misfit reduction of 10% (-norm) or 15% (-norm) is obtained for the layered half-space with respect to a homogeneous half-space. Misfit reduction is mainly due to improved matching of the CGPS horizontal displacements whose misfit reduction is about 30% for both norms.
 Marginal 1D probability density functions (PDFs) for an extended crack embedded in the layered half-space (computed using NAB) show that all parameters are well-resolved, although 2D PDFs show the existence of weak trade-offs (Figures S1 and S2). The source responsible for the CF 2004–2006 uplift is an extended crack, about 3500 m in depth, having a radius of about 2500 m (see Figure 1 for its position in the CF area). Potency is about 1.4 × 106 m3.
3. Discussion and Conclusions
 We have applied the procedures developed by Crescentini and Amoruso  to all available deformation data for the recent (2004–2006) mini-uplift in the CF caldera. As anticipated in that previous work, we find that use of a layered half-space model allows a significant improvement in our ability to match the observations provided of course that the layered model is appropriate (here based on seismically derived estimates of the P wave speed for the crust). Synthetic tests reveal characteristic discrepancies between best fit models embedded in a homogeneous half-space compared with the synthetic data generated by a source in a layered medium. That source models in a homogeneous half-space exhibit similar misfits with the real data for CF provides confidence that the approach adopted here (based on Crescentini and Amoruso ) is indeed valid. Additionally a study of the large uplift episode 1982–1984 (A. Amoruso et al., manuscript in preparation, 2007) demands a source with essentially the same location as that found here.
 These studies are at some variance with work by Troise et al.  in which a freely slipping ring fault (in a homogeneous half-space) is invoked. While the idea of a ring fault is intuitively reasonable, there is no direct evidence to support ring fault contribution to deformation in CF nor would one expect a freely slipping boundary condition to apply.
 We conclude that there is no requirement to introduce such additional sources in order to interpret CF deformation data since, by using a realistic layered half-space with properties constrained by local seismic data, we are able to fit all the deformation data with a horizontal crack source. A future study will reveal whether a deeper supply reservoir (as suggested by the work of Okada  for Uzu volcano in Japan) is required by the data.
 This work was funded by the Italian Dipartimento della Protezione Civile in the frame of the 2004–2006 Agreement with Istituto Nazionale di Geofisica e Vulcanologia (INGV) and FP6 EU-Volume project.