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Appendix A
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Gravity data processing
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As gravity data come from very different sources, they were homogenised by referring them to
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the IGSN71 network, and then revised in order to avoid inconsistencies and to exclude erroneous
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measurements. After this process, our data set consisted of 1,077 gravity data covering an area of
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20.4 x 21.6 km.
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We computed free air anomaly because the investigated area is almost flat and the surface rocks
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are lithologically rather heterogeneous, making it difficult to determine a reliable average value of
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the density. We subtracted to the anomaly the sea water effect and the regional field. To subtract the
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sea water effect we integrated the off-shore measurements with points extracted from a smaller
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scale bathymetric map, thus obtaining a set of 2,310 points distributed over an area of about 5,500
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km2, large enough to consider border effect negligible over the CF area. Using an approximation of
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the sea floor shape by a triangular mesh, we computed the sea water effect using a density of 1,030
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kg·m-3. This average value results from the analysis of density profiles in the Gulf of Naples
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[Stabile et al., 2007]. As obvious, the correction affects offshore data, but is limited to 10 mGal at
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most. On land, it never exceeds 2 mGal.
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To perform the correction for the regional field, we considered the results of the seismic
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tomography of Judenherc and Zollo [2004]. They interpret the shape of the 5.5 km/s isoline as
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representing the top of the Mesozoic limestone basement that underlie the whole region [Principe et
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al., 1987] at few kilometres depth. This is the most important regional feature of the area. The
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tomography covers an area large enough to include CF and their surroundings both off shore and on
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land, but we extrapolated the border of the basement to match the dimension of the sea floor model.
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The density of the basement was assumed to be 2,600 kg·m-3 according to other gravity
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investigations of the area [Cassano and La Torre, 1987, Berrino et al., 2008]. The computed
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correction is very small because the basement is essentially flat in the area. Never the less, we used
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the known shape of the basement to compute the average density of the overlying rocks. To this
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aim, we optimized the density contrast between the basement and the overlying rocks over the
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whole area covered by the know shape of the basement minimizing the free air anomaly residual.
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The area includes features of the gravity field not directly linked to the shape of the basement (e.g.
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Campi Flegrei caldera low, and Ischia height), so we minimized the L1 norm of the residual. We
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obtain as reference value 2,100 kg·m-3 as a reference value. This result is consistent with the
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average of the values adopted by Cassano and La Torre [1987] for rocks down to 2 km depth.
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These values were calibrated over the density measured on samples extracted from geothermal
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wells.
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3D gravity inversion modelling
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The aim of every gravity survey is to identify the density structure of the investigated area. The
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first step is then to single out a volume within which the sources of anomaly of the gravity field are
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likely to be present. Once this volume is determined, it has to be partitioned into blocks the gravity
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attraction of which can be computed to solve the forward problem. The identified volume will likely
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be characterized by a topography and gravity data are measured at points on it. The positions of data
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points are then samples of the topographic surface. Starting from these samples, it is possible to
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build a triangular mesh representing an approximation of the true surface shape. We accomplish this
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task using the implementation of optimal Delauney triangulation of Watson [1982] that we modified
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to perform additional tasks needed to compute the gravity generated by the forward model. In our
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case, the topography of CF is characterized by very small relieves generated by the volcanic activity
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that only in the northern part of the area become as high as about 200 m.
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If data are scattered the sampling of topography may be very different in different areas. In this
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case it is necessary to add auxiliary points determined from maps or a DEM to improve and
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homogenize the approximation of the topography. Obviously these added points provide more
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details about the shape of the topography, but they do not bear any further information about the
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gravity field.
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Once the mesh is obtained, under each triangle a sequence of triangular prisms, that we called
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column, is built. The uppermost prism has a slanted face following the topography, while the other
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prisms have planar upper and lower faces. The heights of the prisms can be chosen at will and can
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be arranged to have a sequence of regular layers below the base of the topography. The volume
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containing the topography can be instead divided into irregular layers, i.e., containing columns of
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different heights. Assuming that each prism has a given constant density, the gravity attraction of
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the column can be computed analytically at each data point using the method outlined by Okabe
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[1979].
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Adding more auxiliary points to the data set improves the approximation of the topography by
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the triangular mesh, but also increases the number of triangles and related columns. To reduce the
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number of parameters and to let them have a more regular shape, the columns have to be grouped.
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To this aim, the area covered by the topography is ideally subdivided into regular cells and the
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triangles belonging to a cell are grouped to form a polyhedron. Then the prisms belonging to a layer
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and to the columns related to grouped triangles are combined into a new block, whose gravity
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attraction is the sum of the attraction of the forming prisms, provided that they all have the same
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density.
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On the basis of the new blocks the kernel matrix A whose elements are the gravity attraction at
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the i-th data point generated by the j-th block, assumed to have unit density, is built. Knowing the
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densities of the blocks, the gravity attraction at data points can be computed by a standard linear
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system of equations.
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The resulting inverse problem is usually ill-posed and solved using the Tichonov regularization
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theory [Tichonov and Arsénine, 1976]. The key point of the theory is that a stable solution of the
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inverse problem can be obtained by imposing that the model has to fit the observed data within both
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a given accuracy δ, usually linked to data and model errors, and an additional constraint. We have
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imposed the minimum weighted norm as the additional constraint on the solution, thus, according to
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the theory, we have minimized the functional
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P=||Am-d||2+||Wm||2
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where m is the vector of the unknowns model parameters, d is the data vector, W is a weighting
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matrix and  is the relative weight between the resulting misfit and the effectiveness of the imposed
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constraint.
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The application of regularization theory to gravimetric and electromagnetic data is extensively
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discussed by Zdhanov [2002]. . The author shows that choosing W as a diagonal matrix which
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elements are the norm of the column vectors of A, the weighting matrix maximizes the sensitivity
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of the model parameters to data changes . He also shows that, for the inverse linear discrete problem
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under consideration, the analytical solution in order to minimize the functional P is
m=(ATA+W2)-1ATd
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. By repeatedly computing this solution for different values of α, the quasi-optimal value of the
parameter can be selected using the method of L-curve [Hansen, 1992].
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To actually set up the inversion, after an extensive series of tests we established the vertical
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parameterization shown in Figure A1. The subaerial part of the model represents one irregular layer
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having bottom at sea level. The part of the model between sea level and the sea floor represents the
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second irregular layer having bottom at 0.2 km b.s.l.. Then a sequence of regular layers is
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established down to 3 km b.s.l. with spacing of 0.4 km, and further down to 4.8 km b.s.l. with
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spacing of 0.6 km, for a total of twelve layers. The resolution is also shown in Figure A1 Taking as
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threshold reference the 3 percent of the maximum value, we observe a satisfied resolution up to a
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depth of about 3 km
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References not cited in the main text
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Hansen, P. C. (1992), Analysis of discrete ill-posed problems by means of the L-curve, SIAM
Review, 34, 561-580.
Okabe, M. (1979), Analytical expression for gravity anomalies due to homogeneous polyhedral
bodies and translation into magnetic anomalies, Geophysics, 44, 730–741.
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Principe, C., M. Rosi, R. Santacroce, and A. Sbrana (1987), Explanatory notes to the geological
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map, in Somma-Vesuvius: edited by R. Santacroce, Quaderni de ‘‘La Ricerca Scientifica’’, 114
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(8) pp. 1–52, CNR, Rome, Italy.
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Stabile, T. A., A. Zollo, M. Vassallo, and G. Iannaccone (2007), Underwater acoustic channel
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properties in the Gulf of Naples and their effects on digital data transmission, Ann. Geophys.,
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50, 313–328.
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Tichonov, A., and V. Arsénine (1976), Méthodes de résolution des problèmes mal posés (french
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translation from russian), MIR, Paris, 202 pp.
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Watson, D. F. (1982), ACORD: Automatic Contouring of Raw Data, Comput. Geosci., 8, 97-101.
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Figure A1 --left: sketch of the basic structure (layer) used to parameterize the density distribution;
right: resolution for each layer of the model. As a reference, a (red) line corresponding to a 3
percent of the maximum value of resolution has been plotted
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