# Modeling 3D Free-geometry Volumetric Sources Associated to Geological and Anthropogenic Hazards from Space and Terrestrial Geodetic Data

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Inversion Approach for Deformation and Gravity Changes

#### 2.1. System of Nonlinear Equations

_{l}, y

_{l}, z

_{l}), l = 1,…, n, where gravity changes dg

_{i}, i = 1, …, n

_{g}, and position changes (dx

_{j}, dy

_{j}, dz

_{j}), j = 1,…, n

_{p}, have been observed, nearly simultaneously, but not necessarily in the same locations.

**d**the n

_{g}+n

_{p}observation vector of components dg

_{i}, dx

_{j}, dy

_{j}, dz

_{j}. They try to model this vector by using m buried point sources, located in (X

_{k}, Y

_{k}, Z

_{k}), k = 1, …, m, and characterized by corresponding values for volume v

_{k}, (positive or negative) incremental pressure p

_{k}, and (positive or negative) incremental density ρ

_{k}changes.

**d**as

**d**=

**d**

^{c}+

**ε**

**d**

^{c}represents the n

_{g}+n

_{p}vector of modeled values $d{g}_{i}^{c}$, $d{x}_{j}^{c}$, $d{y}_{j}^{c}$, and $d{z}_{j}^{c}$ for gravity and position changes, and

**ε**is the n

_{g}+n

_{p}vector for residual values coming from inaccuracies in the observation process and also from insufficient model fit [1].

_{k}, Y

_{k}, Z

_{k}; v

_{k}, ρ

_{k}, p

_{k}) within a homogeneous elastic halfspace characterized by a shear modulus μ (given in stress units of pascals, Pa) and a Poisson’s ratio of ν ≅ 0.25.

_{k}and expansion radius within the elastic half-space, described by the Mogi model [31], and (2) the effects produced by the loading of the additional point mass v

_{k}× ρ

_{k}within the elastic half-space, described by the Boussinesq model [55,56].

- Free-air effects, corresponding to the relocation of the benchmarks, due to elevation changes according a free-air vertical gravity gradient (about −290 μGal/m for Campi Flegrei); this effect can be included in the model fit using modeled or observed elevation changes;
- Newtonian effects due to density changes within the original boundaries of the deep bodies;
- Newtonian effects due to mass relocation or change of volume;
- Effects due to mass uplift in the surface corresponding to elevation changes. These effects can be obtained using another vertical gravity gradient, depending on the regional terrain density (similar to the Bouguer correction [57,58,59,60]). Effects 1 and 4 depend on the surface elevation changes and can be combined by using a combined gravity gradient F (about −210 μGal/m);
- Water table effects which correspond to very local and shallow perturbations;

_{0}, dy

_{0}, dz

_{0}, for the position changes (for instance, an unknown change of the leveling origin) and a possible constant term, dg

_{0}, to represent the simplest form of a long-wave component, a global offset value, or a common uncorrected groundwater effect. Then, with the conditions described previously, they adopt the following expressions for the modeled changes [1]:

_{0}is a mean surface gravity value (g

_{0}≈ 9.8 Gal).

_{k}and p

_{k}, but a nonlinear relationship with respect to the geometrical parameters (point sources “filled” with non-zero anomalous density or/and non-zero anomalous pressure). We observe that they calculate the free-air and “Bouguer” effects of the elevation changes as proportional to the modeled values $d{z}_{j}^{c}$, instead of proportional to the observed ones. This allows us to include gravity and elevation changes corresponding to different stations and gives better final results.

_{m}is the uniaxial compaction coefficient. Assuming that displacements at the surface are almost directly proportional to the thickness Δz of the reservoir, the volume integrations for a parallelepiped cell of sides Δx, Δy, Δz and overpressure Δp in equation (2) can be simplified to integration in the horizontal plane only given rise to [5,46]:

_{i}for the displacements in the i-direction are:

#### 2.2. Misfit Conditions

**Q**

_{D}for the gravity and position data dg

_{i}and dx

_{j}, dy

_{j}, dz

_{j}(or ds

_{j}), then the minimization condition for residuals

**ε**, ${\mathit{\epsilon}}^{T}{\mathit{Q}}_{D}^{-1}\mathit{\epsilon}=min$, leads to the maximum likelihood solution. It is also assumed that the data are not correlated, and

**Q**

_{D}is a diagonal matrix of estimated variances corresponding to observation of gravity and position changes. Gravity and elevation observations are independent and, then,

**Q**

_{D}can be decomposed into two covariance matrices

**Q**

_{G}(n

_{g}× n

_{g}) for gravity and

**Q**

_{P}(n

_{p}× n

_{p}) for position changes. Then the minimization condition for observation residuals can be written as

_{g}and n

_{p}vectors for gravity and elevation residuals and θ is a weighting factor for the balance between gravity and deformation fits. This factor is introduced to allow a more versatile fit, making it possible to give optional priority to one kind of observable data over the other. Values for θ can be adopted according to the accuracies of the data sets and considering the approximate equivalence of 1 cm ≈3 μGal.

**m**is constituted by the values ρ

_{k}, p

_{k}, k = 1, …, m, for the cells of the model and

**Q**

_{M}is a suitable covariance matrix corresponding to the physical configuration of cells and benchmarks. This matrix provides a balanced model, avoiding very shallow solutions. We use a normalizing diagonal matrix

**Q**

_{M}with elements q

_{k}, k = 1, …, m, given for volumes ν

_{k}and distances d

_{jk}as

**Q**

_{M}). It prevents very large fictitious values of mass and/or pressure resulting from a rather poorly determined model (for instance, one that is due to the coupling of some positive and negative sources, aligned stations, peripheral sources).

#### 2.3. Exploration Approach for Solving the System

**d**

^{c}. For the new (k + 1)th step, we try to find and fill a new cell to fit the system:

**d**= f

**d**

^{c}+

**ε**

## 3. Application Results

#### 3.1. Modeling of Campi Flegrei Unrest 1992–2000 Using Deformation and Gravity Changes

^{1/2}, where D is the distance along the line between benchmarks [70], and a final uncertainty of the elevation value d

_{z}at each benchmark result is typically less than 1 cm [74].

^{3}and ±10 MPa as a suitable contrast to obtain extended anomalous bodies [1]. The standard deviation of the inversion residuals is 0.2 cm/yr for the elevation changes, 0.6 mGal/yr for the gravity changes, and about 1.1 cm/yr for the LOS values. Figure 3 shows the 3D source model obtained, and Figure 4 shows some additional views of the inversion residuals for the case of the ascending LOS component.

#### Discussion

#### 3.2. Modeling of Campi Flegrei Unrest 1993–2013 Using Only Displacement Data

^{2}to 250 × 250 km

^{2}[53]. For modern satellite constellations the repeat cycle ranges from 1 day to a few weeks, but the typical repeat cycle for a single satellite mission ranges from 12 to 41 days. Repeatedly acquired SAR data from a single sensor are used to obtain line-of-sight (LOS) time series of ground deformation by applying small baseline subset (SBAS) [86,87,88], persistent scatterer [89] methods, or a combination [90]. The results of these techniques, however, are limited to the time period of the particular data set and do not distinguish horizontal and vertical motion.

#### Discussion

#### 3.3. Real Time Tracking of Magmatic Intrusions During Volcanic Crisis: 2008 May Etna Eruption

#### 3.3.1. Results

^{3}body) to produce a particular magnitude of surface deformation at the current network stations (root mean square, rms, 1 mm) increases with depth and distance to the stations. For deep and distant pressure elements in the inverse model, the relative uncertainty increases with a pattern similar to that of Figure 21a. Figure 21b,c show the 3D uncertainty distribution for vertical and horizontal displacements of the isolated pressure sources. It also is valid for some geometrical components or details of extended sources.

^{3}and a pressure value 1 MPa, required to produce a particular surface deformation level (1 mm, rms value). They can be interpreted as showing the pattern of relative uncertainty for the geometrical aspects (depth and horizontal location) for the inverse model. In addition, the displacement uncertainty pattern is heavily dependent on the pattern of the uncertainty for the source intensity (pressure and volume) of the assumed source element.

^{3}, but assuming pressure values increasing with distance according to Figure 21a. These last model uncertainties show a different pattern, essentially due to geometrical aspects. Further details on uncertainty analysis can be found in the Supplementary material by [17].

#### 3.3.2. Discussion

#### 3.4. Land Subsidence Associated with Overexploitation of Aquifers: Lorca, Spain

^{2}. Historically, piezometric levels were located closely to the land surface, allowing the development of a number of artesian wells and permanent lagoons [109].

^{3}/yr (in 2006) [109,110], which led to a spatially variable continuous piezometric level decrease (at rates within the 0.5–10 m/yr range).

^{2}. A-DInSAR was processed for an extended region with a total area of 170 km

^{2}. In both geometries, ascending and descending, the study area is covered by three bursts of the same swath. A total of 42 Interferometric Wide Swath (IW) SLC images from the Sentinel-1A satellite resulted in 185 interferograms (22 images and 137 interferograms for ascending data, 20 and 48 respectively for descending [5]). The results are shown in Figure 23. Some selected time series are shown in the Supplementary Material by [5] for descending LOS. This is the first DInSAR study of the Alto Guadalentin Basin using two different geometries for the same time period.

#### Discussion and Inversion of the 3D Displacement Field

**A**) LOS A-DInSAR results obtained for descending orbit images, assuming 100% as vertical displacement;

**B**) LOS A-DInSAR results obtained for ascending orbit images, assuming 100% as vertical displacement;

**C**) Up-Down component obtained from the A-DInSAR, combining ascending and descending orbit images results;

**D**) Purely LOS A-DInSAR results obtained for descending orbit images;

**E**) Purely LOS A-DInSAR results obtained for ascending orbit images;

**F**) Purely LOS A-DInSAR results obtained for ascending and descending orbit images;

**G**) Up-Down and E-W A-DInSAR results obtained by combining ascending and descending orbit images results;

**H**) 3D displacements determined using the GNSS surveys results;

**I**) Purely LOS A-DInSAR results obtained for ascending and descending orbit images together with the 3D displacements determined using the GNSS surveys results;

**J**) Up-Down and E-W A-DInSAR results obtained by combining ascending and descending orbit images, together with 3D displacements determined using the GNSS surveys results.

**A**-

**C**are one-dimensional (1D),

**D**–

**G**are 2D (either indirectly by combining Up-Down and E-W in any of the two measured LOS, or directly supplying the values for both displacement components separately),

**H**is a purely 3D dataset, and

**I**and

**J**are a combination of 2D and 3D data (2D+3D data).

**H**) data set, there are just 108 displacement rates, while for the remainder we have thousands of measurements (Table 3).

**A**–

**C**(1D, vertical displacement) and (ii) Cases

**D**–

**J**(2D and 2D+3D). Both sets of results are internally consistent, with respective scattering on the order of 9 and 6% respectively (Table 3).

^{3}). This can have an important effect in predictions of future volume variations and surface displacements.

**C**and the other two cases (

**A**and

**B**). For Case

**C**we obtain an intensity value ~20% greater than the other ones. This data set has been obtained from combining LOS results from ascending and descending orbits and they suffer from the small systematic errors in the methodology [5]. This problem is reflected in two aspects, the poorer misfit values and the appearance of additional sources at the edges of the study area that try to adjust these errors (Figure 25). As a result, it was concluded [5] that it is clearly more appropriate, in agreement with previous works [42], to mitigate this problem by using the ascending and/or descending LOS directly in the inversion procedure. As a result, Case

**C**is not considered further. Cases

**A**and

**B**obtain very consistent results (Table 3), with a dispersion of ~4% for group (i) and a mean intensity value of 40 MPa∙km

^{3}[5].

**G**and

**J**) (Table 3) again provides worst results (higher intensities, more additional sources and poorer misfit values, Figure 25 and Table 3). Case

**H**(only GNSS determined displacements) also provides higher misfits (Table 3) and poorer adjusted geometries (Figure 25h), probably a result of using data with several important limitations [5]. The first one is the reduced number of data points available in relation to the number of unknown parameters (approximately one hundred versus several thousands (Table 3). As previously mentioned, in cases like this, a large number of measurement points are needed to reliably estimate the distribution of reservoir volume changes [5,48]. Also, their limited number produces a poorer spatial distribution than A-DInSAR data and does not cover the entire deformed area (Figure 23). Finally, as described previously, in this reduced data set we have some anomalous results associated to very local effects (Figure 23b and Supplementary Information from [5]). To establish a denser and more extended GNSS network requires additional cost and time for field observation.

**D**–

**F**,

**I**.

## 4. General Summary

- (a)
- It permits simultaneous inversion of the 1D to 3D displacement data coming from different techniques, including terrestrial and space data (GNSS, DInSAR, leveling, etc.);
- (b)
- Non-planar, non-gridded, inexact data can be used;
- (c)
- It allows for objective modelling of two causative structures: Pressure and mass anomalies bodies;
- (d)
- Simple analytical and well-known expressions are used for direct calculation for mass and pressure cells and their aggregation;
- (e)
- This methodology works in a fully 3D context;
- (f)
- There is not additional a priori requirements on the geometry and types of the causative sources;
- (g)
- The method automatically determines the type and number, modeling the geometry, of the causative source structures;
- (h)
- A free geometry of the causative structures is described by aggregation of small elemental cells. Surface deformation can be computed by adding the influence of the small considered prisms. Given that the used direct models are linear and the entire subsurface is assumed to be isotropic, superposition is allowable. Using this assumption, these linear equations permit the computation of surface deformation based on the superposition of many prismatic blocks within a compacting reservoir of any geometric shape [5]. The effect of each cell is computed using point sources [1,5].

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Camacho, A.G.; González, P.J.; Fernández, J.; Berrino, G. Simultaneous inversion of surface deformation and gravity changes by means of extended bodies with a free geometry: Application to deforming calderas. J. Geophys. Res.
**2011**, 116, B10401. [Google Scholar] [CrossRef] - Dzurisin, D. Volcano Deformation: Geodetic Monitoring Techniques; Springer-Praxis Books in Geophysical Sciences. Praxis Publishing Ltd.: Chichester, UK, 2007; p. 441. [Google Scholar]
- Herring, T. Treatise on Geophysics, Geodesy, 1st ed.; Elsevier: Amsterdam, The Netherlands, 2009; Volume 3, p. 446. [Google Scholar]
- Fernández, J.; Pepe, A.; Poland, M.P.; Sigmundsson, F. Volcano geodesy: Recent developments and future challenges. J. Volcanol. Geotherm. Res.
**2017**, 344, 1–12. [Google Scholar] [CrossRef] - Fernández, J.; Prieto, J.F.; Escayo, J.; Camacho, A.G.; Luzón, F.; Tiampo, K.F.; Palano, M.; Abajo, T.; Pérez, E.; Velasco, J.; et al. Modeling the two-and three-dimensional displacement field in Lorca, Spain, subsidence and the global implications. Sci. Rep.
**2018**, 8, 14782. [Google Scholar] [CrossRef] [PubMed] - Huang, Y.; Yu, M.; Xu, Q.; Sawada, K.; Moriguchi, S.; Yashima, A.; Liu, C.; Xue, L. InSAR-derived digital elevation models for terrain change analysis of earthquake-triggered flow-like kandslides bases on ALOS/PALSAR imagery. Environ. Earth Sci.
**2015**, 73, 7661–7668. [Google Scholar] [CrossRef] - Yu, M.; Huang, Y.; Zhou, J.; Mao, L. Modeling of landslide topography based on micro-unmanned aerial vehicle photography and structure-from-motion. Environ. Earth Sci.
**2017**, 76, 520. [Google Scholar] [CrossRef] - Biggs, J.; Ebmeier, S.K.; Aspinall, W.P.; Lu, Z.; Pritchard, M.E.; Sparks, R.S.J.; Mather, T.A. Global link between deformation and volcanic eruption quantified by satellite imagery. Nat. Commun.
**2014**, 5, 3471. [Google Scholar] [CrossRef] [PubMed] - Biggs, J.; Pritchard, M.E. Global volcano monitoring: What does it mean when volcanoes deform? Elements
**2017**, 13, 17–22. [Google Scholar] [CrossRef] - Tralli, D.M.; Blom, R.G.; Zlotnicki, V.; Donnellan, A.; Evans, D.L. Satellite remote sensing of earthquake, volcano, flood, landslide and coastal inundation hazards. ISPRS J. Photogramm. Remote Sens.
**2005**, 59, 185–198. [Google Scholar] [CrossRef] - Bilham, R. The seismic future of cities. Bull. Earthq. Eng.
**2009**, 7, 839–887. [Google Scholar] [CrossRef] [Green Version] - González, P.J. Medida y Caracterización de Deformaciones Usando Técnicas Geodésicas y de Teledetección. Aplicación en Volcanología y Sismotectónica. Ph.D. Thesis, Universidad Complutense de Madrid, Madrid, Spain, 2010. (In Spanish). [Google Scholar]
- Fernández, J.; González, P.J.; Camacho, A.G.; Prieto, J.F.; Bru, G. An overview of geodetic volcano research in the Canary Islands. Pure Appl. Geophys.
**2015**, 172, 3189–3228. [Google Scholar] [CrossRef] - Terranova, C.; Ventura, G.; Vilardo, G. Multiple causes of ground deformation in the Napoli metropolitan area (Italy) from integrated persistent scatterers DinSAR, geological, hydrological, and urban infrastructure data. Earth-Sci. Rev.
**2015**, 146, 105–119. [Google Scholar] [CrossRef] - Sparks, R.S.J.; Biggs, J.; Neuberg, J.W. Monitoring volcanoes. Science
**2012**, 335, 1310–1311. [Google Scholar] [CrossRef] [PubMed] - Sigmundsson, F.; Hreinsdóttir, S.; Hooper, A.; Árnadóttir, T.; Pedersen, R.; Roberts, M.J.; Óskarsson, N.; Auriac, A.; Decriem, J.; Einarsson, P.; et al. Intrusion triggering of the 2010 Eyjafjallajökull explosive eruption. Nature
**2010**, 468, 426–430. [Google Scholar] [CrossRef] [PubMed] - Cannavò, F.; Camacho, A.G.; González, P.J.; Mattia, M.; Puglisi, G.; Fernández, J. Real time tracking of magmatic intrusions by means of ground deformation modeling during volcanic crises. Sci. Rep.
**2015**, 5, 10970. [Google Scholar] [CrossRef] [PubMed] - Galloway, D.; Jones, D.R.; Ingebritsen, S. Land Subsidence in the United States; US Geological Survey Circular: Reston, VA, USA, 1999. [Google Scholar]
- Dokka, R.K. Modern-day tectonic subsidence in coastal Louisiana. Geology
**2006**, 34, 281–284. [Google Scholar] [CrossRef] - Amelung, F.; Galloway, D.L.; Bell, J.W.; Zebker, H.A.; Laczniak, R.J. Sensing the ups and downs of Las Vegas: InSAR reveals structural control of land subsidence and aquifer-system deformation. Geology
**1999**, 27, 483–486. [Google Scholar] [CrossRef] - González, P.J.; Fernández, J. Drought-driven transient aquifer compaction imaged using multitemporal satellite radar interferometry. Geology
**2011**, 39, 551–554. [Google Scholar] [CrossRef] - Boní, R.; Herrera, G.; Meisina, C.; Notti, D.; Béjar-Pizarro, M.; Zucca, F.; González, P.J.; Palano, M.; Tomás, R.; Fernández, J.; et al. Twenty-year advanced DInSAR analysis of severe land subsidence: The Alto Guadalentín basin (Spain) case study. Eng. Geol.
**2015**, 198, 40–52. [Google Scholar] [CrossRef] - Samsonov, S.V.; Tiampo, K.F.; Feng, W. Fast subsidence in downtown of Seattle observed with satellite radar. Remote Sens. Appl. Soc. Environ.
**2016**, 4, 179–187. [Google Scholar] [CrossRef] - Bru, G.; González, P.J.; Mateos, R.M.; Roldán, F.J.; Herrera, G.; Béjar-Pizarro, M.; Fernández, J. A-DInSAR monitoring of landslide and subsidence activity: A case of urban damage in arcos de la frontera, Spain. Remote Sens.
**2017**, 9, 787. [Google Scholar] [CrossRef] - Gourmelen, N.; Amelung, F.; Casu, F.; Manzo, M.; Lanari, R. Mining-related ground deformation in Crescent Valley, Nevada: Implications for sparse GPS networks. Geophys. Res. Lett.
**2007**, 34, L09309. [Google Scholar] [CrossRef] - Ma, C.; Cheng, X.; Yang, Y.; Zhang, X.; Guo, Z.; Zou, Y. Investigation on mining subsidence based on multi-temporal InSAR and time-series analysis of the small baseline subset—case study of working faces 22201-1/2 in Bu’ertai mine, shendong coalfield. Remote Sens.
**2016**, 8, 951. [Google Scholar] [CrossRef] - Lisowski, M. Analytical volcano deformation source parameters in volcano deformation: New geodetic monitoring techniques. In Volcano Deformation: Geodetic Monitoring Techniques; Dzurisin, D., Ed.; Springer-Verlag: Berlin/Heidelberg, Germany, 2007; pp. 279–304. [Google Scholar]
- Segall, P. Earthquake and Volcano Deformation; Princeton University Press: Princeton, NJ, USA; Oxford, UK, 2010; p. 432. ISBN 978-0-691-13302-7. [Google Scholar]
- Rymer, H.; Williams-Jones, G. Volcanic eruption prediction: Magma chamber physics from gravity and deformation measurements. Geophys. Res. Lett.
**2000**, 27, 16–2389. [Google Scholar] [CrossRef] - Fernández, J.; Tiampo, K.F.; Jentzsch, G.; Charco, M.; Rundle, J.B. Inflation or deflation? New results for Mayon volcano applying elastic-gravitational modeling. Geophys. Res. Lett.
**2001**, 28, 2349–2352. [Google Scholar] [CrossRef] - Masterlark, T. Magma intrusion and deformation predictions: Sensitivities to the Mogi assumptions. J. Geophys. Res.
**2007**, 112, B06419. [Google Scholar] [CrossRef] - Long, S.M.; Grosfils, E.B. Modeling the effect of layered volcanic material on magma reservoir failure and associated deformation, with application to Long Valley caldera, California. J. Volcanol. Geotherm. Res.
**2009**, 186, 349–360. [Google Scholar] [CrossRef] - Castaldo, R.; Gola, G.; Santilano, A.; De Novellis, V.; Pepe, S.; Manzo, M.; Manzella, A.; Tizzani, P. The role of thermo-rheological properties of the crust beneath Ischia island (Southern Italy) in the modulation of the ground deformation pattern. J. Volcanol. Geotherm. Res.
**2017**, 344, 154–173. [Google Scholar] [CrossRef] - Trasatti, E.; Giunchi, C.; Agostinetti, N.P. Numerical inversion of deformation caused by pressure sources: Application to Mount Etna (Italy). Geophys. J. Int.
**2008**, 172, 873–884. [Google Scholar] [CrossRef] - Battaglia, M.; Gottsmann, J.; Carbone, D.; Fernández, J. 4D volcano gravimetry. Geophysics
**2008**, 73, WA3–WA18. [Google Scholar] [CrossRef] [Green Version] - Carbone, D.; Poland, M.P.; Diament, M.; Grecco, F. The added value of 4D microgravimetry to the understanding of how volcanoes work. Earth Sci. Rev.
**2017**, 169, 146–179. [Google Scholar] [CrossRef] - Charco, M.; Fernández, J.; Luzón, F.; Rundle, J.B. On the relative importance of selfgravitation and elasticity in modeling volcanic ground deformation and gravity changes. J. Geophys. Res.
**2006**, 111, B03404. [Google Scholar] [CrossRef] - Johnson, D.J.; Eggers, A.A.; Bagnardi, M.; Battaglia, M.; Poland, M.P.; Miklius, A. Shallow magma accumulation at Kīlauea Volcano, Hawaii, revealed by microgravity surveys. Geology
**2010**, 38, 1139–1142. [Google Scholar] [CrossRef] - Bagnardi, M.; Poland, M.P.; Carbone, D.; Baker, S.; Battaglia, M.; Amelung, F. Gravity changes and deformation at Kīlauea Volcano, Hawaii, associated with summit eruptive activity, 2009–2012. J. Geophys. Res.
**2014**, 119, 7288–7305. [Google Scholar] [CrossRef] - Currenti, G. Numerical evidence enabling reconciliation of gravity and height changes in volcanic areas. Geophys. J. Int.
**2014**, 197, 164–173. [Google Scholar] [CrossRef] - Charco, M.; Camacho, A.G.; Tiampo, K.F.; Fernández, J. Spatiotemporal gravity changes on volcanoes: Assessing the importance of topography. Geophys. Res. Lett.
**2009**, 36, L08306. [Google Scholar] [CrossRef] - Fokker, P.A.; Wassing, B.B.T.; van Leijen, F.J.; Hanssen, R.F.; Nieuwland, D.A. Application of an ensemble smoother with multiple data assimilation to the Bergermeer gas field, using PS-InSAR. Geomech. Energy Environ.
**2016**, 5, 16–28. [Google Scholar] [CrossRef] - Vasco, D.W.; Wicks, C.; Karasaki, K.; Marques, O. Geodetic imaging: Reservoir monitoring using satellite interferometry. Geophys. J. Int.
**2002**, 149, 555–571. [Google Scholar] [CrossRef] - Tiampo, K.F.; Ouegnin, F.-A.; Valluri, S.; Samsonov, S.; Fernández, J.; Kapp, G. An elliptical model for deformation due to groundwater fluctuations. Pure Appl. Geophys.
**2012**, 169, 1443–1456. [Google Scholar] [CrossRef] - Geertsma, J. Land subsidence above compacting oil and gas reservoirs. J. Pet. Technol.
**1973**, 25, 734–744. [Google Scholar] [CrossRef] - Geertsma, J.; Van Opstal, G. A numerical technique for predicting subsidence above compacting reservoirs based on the nucleus of strain concept. Verh. Kon. Ned. Geol. Mijnbouwk
**1973**, 28, 63–78. [Google Scholar] - Segall, P. Stress and subsidence resulting from subsurface fluid withdrawal in the epicentral region of the 1983 Coalinga Earthquake. J. Geophys. Res.
**1985**, 90, 6801–6816. [Google Scholar] [CrossRef] - Vasco, D.W.; Karasaki, K.; Doughty, C. Using surface deformation to image reservoir dynamics. Geophysics
**2000**, 65, 132–147. [Google Scholar] [CrossRef] - Walsh, J.B. Subsidence above a planar reservoir. J. Geophys. Res. Solid Earth
**2002**, 107, 2202. [Google Scholar] [CrossRef] - Brown, N.J.; Woods, A.W.; Neufeld, J.A.; Richardson, C. Constraining Surface Deformation Predictions Resulting From Coal Seam Gas Extraction; Geoscience Australia: Symonston, Australia, 2014. [Google Scholar] [CrossRef]
- González, P.J.; Tiampo, K.F.; Palano, M.; Cannavó, F.; Fernández, J. The 2011 Lorca earthquake slip distribution controlled by groundwater crustal unloading. Nat. Geosci.
**2012**, 5, 821–825. [Google Scholar] [CrossRef] [Green Version] - Charco, M.; Galán del Sastre, P. Efficient inversion of 3D finite element models of volcano deformation. Geophys. J. Int.
**2014**, 196, 1441–1454. [Google Scholar] [CrossRef] - Samsonov, S.V.; Tiampo, K.F.; Camacho, A.G.; Fernández, J.; González, P.J. Spatiotemporal analysis and interpretation of 1993–2013 ground deformation at Campi Flegrei, Italy, observed by advanced DInSAR. Geophys. Res. Lett.
**2014**, 41, 6101–6108. [Google Scholar] [CrossRef] - Camacho, A.G.; Fernández, J.; Cannavò, F. PAF: A software tool to estimate free-geometry extended bodies of anomalous pressure from surface deformation data. Comput. Geosci.
**2018**, 111, 235–243. [Google Scholar] [CrossRef] - Camacho, A.G.; Montesinos, F.G.; Vieira, R. A 3-D gravity inversion by means of growing bodies. Geophysics
**2000**, 65, 95–191. [Google Scholar] [CrossRef] - Davis, R.O.; Selvadurai, A.P.S. Elasticity and Geomechanics; Cambridge University Press: Cambridge, UK, 1996; p. 201. [Google Scholar]
- Saleh, B. Underground deformation measurements using new quarts instruments. In Proceedings of the 95th Annual CIG Geomatics Conference, Can. Inst. of Geomatics, Ottawa, ON, Canada, 8–23 July 2002. [Google Scholar]
- Rundle, J.B. Gravity changes and the Palmdale uplift. Geophys. Res. Lett.
**1978**, 5, 41–44. [Google Scholar] [CrossRef] - Walsh, J.B.; Rice, J.R. Local changes in gravity resulting from deformation. J. Geophys. Res.
**1979**, 84, 165–170. [Google Scholar] [CrossRef] [Green Version] - Currenti, G.; Del Negro, C.; Ganci, G. Modelling of ground deformation and gravity fields using finite element method: An application to Etna volcano. Geophys. J. Int.
**2007**, 169, 775–786. [Google Scholar] [CrossRef] - Williams, C.A.; Wadge, G. The effects of topography on magma chamber deformation models: Application to Mt. Etna and radar interferometry. Geophys. Res. Lett.
**1998**, 25, 1549–1552. [Google Scholar] [CrossRef] - Battaglia, M.; Hill, D.P. Analytical modeling of gravity changes and crustal deformation at volcanoes: The Long Valley caldera, California, case study. Tectonophysics
**2009**, 471, 45–57. [Google Scholar] [CrossRef] - Biot, M.A. General theory of three-dimensional consolidation. J. Appl. Phys.
**1941**, 12, 155–164. [Google Scholar] [CrossRef] - Tiede, C.; Tiampo, K.F.; Fernández, J.; Gerstenecker, C. Deeper understanding of non-linear geodetic data inversion using a quantitative sensitivity analysis. Nonlinear Processes Geophys.
**2005**, 12, 373–379. [Google Scholar] [CrossRef] [Green Version] - Farquharson, C.G.; Oldenbourg, D.W. Non-linear inversion using general measures of data misfit and model structure. Geophys. J. Int.
**1998**, 134, 213–227. [Google Scholar] [CrossRef] [Green Version] - Bertete-Aguirre, H.; Cherkaev, E.; Oristaglio, M. Non-smooth gravity problem with total variation penalization functional. Geophys. J. Int.
**2002**, 149, 499–507. [Google Scholar] [CrossRef] [Green Version] - Camacho, A.G.; Nunes, J.C.; Ortiz, E.; França, Z.; Vieira, R. Gravimetric determination of an intrusive complex under the Island of Faial (Azores): Some methodological improvements. Geophys. J. Int.
**2007**, 171, 478–494. [Google Scholar] [CrossRef] - Tarantola, A. Inverse Problem Theory; Elsevier: Amsterdam, The Netherlands, 1987; p. 613. [Google Scholar]
- Camacho, A.G.; Montesinos, F.G.; Vieira, R. A 3-D gravity inversion tool based on exploration of model possibilities. Comput. Geosci.
**2002**, 28, 191–204. [Google Scholar] [CrossRef] - Gottsmann, J.; Camacho, A.G.; Marti, J.; Wooller, L.; Fernández, J.; Garcia, A.; Rymer, H. Shallow structure beneath the central volcanic complex of tenerife from new gravity data: Implications for its evolution and recent reactivation. Phys. Earth Planet. Inter.
**2008**, 168, 212–230. [Google Scholar] [CrossRef] - Corrado, G.; Guerra, I.; Lo Bascio, A.; Luongo, G.; Rampoldi, R. Inflation and microearthquake activity of Phlegraean Fields, Italy. Bull. Volcanol.
**1976**, 40, 1–20. [Google Scholar] [CrossRef] - Berrino, G.; Corrado, G.; Luongo, G.; Toro, B. Ground deformation and gravity changes accompanying the 1982 Pozzuoli Uplift. Bull. Volcanol.
**1984**, 47, 187–200. [Google Scholar] [CrossRef] - Orsi, G.; Civetta, L.; Del Gaudio, C.; De Vita, S.; Di Vito, M.A.; Isaia, R.; Petrazzuoli, S.M.; Ricciardi, G.P.; Ricco, C. Short-term ground deformation and seismicity in the resurgent Campi Flegrei caldera (Italy): An example of active block-resurgence in a densely populated area. J. Volcanol. Geotherm. Res.
**1999**, 91, 415–451. [Google Scholar] [CrossRef] - Gottsmann, J.; Berrino, G.; Rymer, H.; William-Jones, G. Hazard assessment during caldera unrest at the Campi Flegrei, Italy: A contribution from gravity-height gradients. Earth Planet. Sci. Lett.
**2003**, 211, 295–309. [Google Scholar] [CrossRef] - Berrino, G. Gravity changes induced by height-mass variations at the Campi Flegrei caldera. J. Volcanol. Geotherm. Res.
**1994**, 61, 293–309. [Google Scholar] [CrossRef] - Manconi, A.; Walter, T.R.; Manzo, M.; Zeni, G.; Tizzani, P.; Sansosti, E.; Lanari, R. On the effects of 3-D mechanical heterogeneities at Campi Flegrei caldera, southern Italy. J. Geophys. Res.
**2010**, 115, B08405. [Google Scholar] [CrossRef] - Lanari, R.; Berardino, P.; Borgström, S.; Del Gaudio, C.; De Martino, P.; Fornaro, G.; Guarino, S.; Ricciardi, G.P.; Sansosti, E.; Lundgren, P. The use of IFSAR and classical geodetic techniques for caldera unrest episodes: Application to the Campi Flegrei uplift event of 2000. J. Volcanol. Geotherm. Res.
**2004**, 133, 247–260. [Google Scholar] [CrossRef] - Casu, F.; Manzo, M.; Lanari, R. A quantitative assessment of the SBAS algorithm performance for surface deformation retrieval from DInSAR data. Remote Sens. Environ.
**2006**, 102, 195–210. [Google Scholar] [CrossRef] - Gottsmann, J.; Rymer, H.; Berrino, G. Unrest at the campi flegrei caldera (Italy): A critical evaluation of source parameters from geodetic data inversion. J. Volcanol. Geotherm. Res.
**2006**, 150, 132–145. [Google Scholar] [CrossRef] - Berrino, G.; Rymer, H.; Brown, G.C.; Corrado, G. Gravity height correlations for unrest calderas. J. Volcanol. Geotherm. Res.
**1992**, 53, 11–26. [Google Scholar] [CrossRef] - Talwani, P.; Acree, S. Pore pressure diffusion and the mechanism of reservoir-induced seismicity. Pure Appl. Geophys.
**1984**, 122, 947–965. [Google Scholar] [CrossRef] - Todesco, M.; Berrino, G. Modeling hydrothermal fluid circulation and gravity signals at the Phlegraean Fields caldera. Earth Planet. Sci. Lett.
**2005**, 240, 328–338. [Google Scholar] [CrossRef] - Berrino, G.; Corrado, G.; Riccardi, U. Sea gravity data in the Gulf of Naples: A contribution to delineating the Phlegraean Volcanic District. J. Volcanol. Geotherm. Res.
**2008**, 175, 241–252. [Google Scholar] [CrossRef] - Gottsmann, J.; Folch, A.; Rymer, H. Unrest at Campi Flegrei: A contribution to the magmatic versus hydrothermal debate from inverse and finite element modeling. J. Geophys. Res.
**2006**, 111, B07203. [Google Scholar] [CrossRef] - Battaglia, M.; Obrizzo, F.; Pingue, F.; De Natale, G. Evidence for fluid migration as the source of deformation at Campi Flegrei caldera (Italy). Geophys. Res. Lett.
**2006**, 33, L01307. [Google Scholar] [CrossRef] - Berardino, P.; Fornaro, G.; Lanari, R. A new algorithm for surface deformation monitoring based on small baseline differential SAR interferograms. IEEE Trans. Geosci. Rem. Sens.
**2002**, 40, 2375–2383. [Google Scholar] [CrossRef] [Green Version] - Usai, S. A least squares database approach for SAR interferometric data. IEEE Trans. Geosci. Rem. Sens.
**2003**, 41, 753–760. [Google Scholar] [CrossRef] [Green Version] - Samsonov, S.; van der Koij, M.; Tiampo, K. A simultaneous inversion for deformation rates and topographic errors of DInSAR data utilizing linear least square inversion technique. Comput. Geosci.
**2011**, 37, 1083–1091. [Google Scholar] [CrossRef] - Ferretti, A.; Prati, C.; Rocca, F. Permanent scatterers in SAR interferometry. IEEE Trans. Geosci. Rem. Sens.
**2001**, 39, 8–20. [Google Scholar] [CrossRef] - Hooper, A. A multi-temporal InSAR method incorporating both persistent scatterer and small baseline approaches. Geophys. Res. Lett.
**2008**, 35, L16302. [Google Scholar] [CrossRef] - Samsonov, S.; d’Oreye, N. Multidimensional time series analysis of ground deformation from multiple InSAR data sets applied to Virunga volcanic province. Geophys. J. Int.
**2012**, 191, 1095–1108. [Google Scholar] [CrossRef] - Samsonov, S.; d’Oreye, N.; Smets, B. Ground deformation associated with post-mining activity at the French-German border revealed by novel InSAR time series method. Int. J. Appl. Earth Obs. Geoinf.
**2013**, 23, 142–154. [Google Scholar] [CrossRef] - Samsonov, S.; d’Oreye, N.; González, P.; Tiampo, K.; Ertolahti, L.; Clague, J.J. Rapidly accelerating subsidence in the Greater Vancouver region from two decades of ERS-ENVISAT-RADARSAT-2 DInSAR measurements. Rem. Sens. Environ.
**2014**, 143, 180–191. [Google Scholar] [CrossRef] [Green Version] - Amoruso, A.; Crescentini, L.; Sabbetta, I. Paired deformation sources of the Campi Flegrei caldera (Italy) required by recent (1980–2010) deformation history. J. Geophys. Res. Solid Earth
**2014**, 119, 858–879. [Google Scholar] [CrossRef] - Chiodini, G.; Caliro, S.; De Martino, P.; Avino, R.; Gherardi, F. Early signals of new volcanic unrest at Campi Flegrei caldera? Insights from geochemical data and physical simulations. Geology
**2012**, 40, 943–946. [Google Scholar] [CrossRef] - Aloisi, M.; Bonaccorso, A.; Cannavò, F.; Gambino, S.; Mattia, M.; Puglisi, G.; Boschi, E. A new dyke intrusion style for the Mount Etna May 2008 eruption modelled through continuous tilt and GPS data. Terr. Nova
**2009**, 21, 316–321. [Google Scholar] [CrossRef] - Palano, M.; Rossi, M.; Cannavò, F.; Bruno, V.; Marco, A.; Daniele, P.; Mario, P.; Siligato, G.; Mattia, M. Etn@ref: A geodetic reference frame for Mt. Etna GPS networks. Ann. Geophys.
**2010**, 53, 49–57. [Google Scholar] - Bonaccorso, A.; Bonforte, A.; Currenti, G.; Del Negro, C.; Di Stefano, A.; Greco, F. Magma storage, eruptive activity and flank instability: Inferences from ground deformation and gravity changes during the 1993-2000 recharging of Mt. Etna volcano. J. Volcanol. Geotherm. Res.
**2011**, 200, 245–254. [Google Scholar] [CrossRef] - Maronna, R.A.; Martin, D.R.; Yohai, V.J. Robust Statistics: Theory and Methods; John Wiley & Sons Ltd.: Chichester, UK, 2006; p. 436. [Google Scholar]
- Patané, D.; De Gori, P.; Chiarabba, C.; Bonaccorso, A. Magma ascent and the pressurization of Mount Etna’s volcanic system. Science
**2003**, 290, 2061–2063. [Google Scholar] [CrossRef] - Aloisi, M.; Mattia, M.; Monaco, C.; Pulvirenti, F. Magma, faults, and gravitational loading at Mount Etna: The 2002–2003 eruptive period. J. Geophys. Res.
**2011**, 116, B05203. [Google Scholar] [CrossRef] - D’Auria, L.; Giudicepietro, F.; Martini, M.; Lanari, R. The 4D imaging of the source of ground deformation at Campi Flegrei caldera (southern Italy). J. Geophys. Res.
**2012**, 117, B08209. [Google Scholar] [CrossRef] - Fernández, J.; Rundle, J.B. Gravity changes and deformation due to a magmatic intrusion in a two-layered crustal model. J. Geophys. Res.
**1994**, 99, 2737–2746. [Google Scholar] [CrossRef] [Green Version] - Rundle, J.B. Viscoelastic-gravitational deformation by a rectangular thrust fault in a layered Earth. J. Geophys. Res.
**1982**, 87, 7787–7796. [Google Scholar] [CrossRef] - AEMET, Agencia Estatal de Meteorología. Available online: www.aemet.es/es/datos_abiertos/ AEMET_OpenData (accessed on 15 January 2018).
- Bourgois, J.; Mauffret, A.; Ammar, A.; Demnati, A. Multichannel seismic data imaging of inversion tectonics of the Alboran Ridge (western Mediterranean Sea). Geo Marine Lett.
**1992**, 12, 117–122. [Google Scholar] [CrossRef] - Martínez-Díaz, J.J. Stress field variation related to fault interaction in a reverse oblique-slip fault: The Alhama de Murcia fault, Betic Cordillera, Spain. Tectonophysics
**2002**, 356, 291–305. [Google Scholar] [CrossRef] - Palano, M.; González, P.J.; Fernández, J. Strain and stress fields along the Gibraltar Orogenic Arc: Constraints on active geodynamics. Gondwana Res.
**2013**, 23, 1071–1088. [Google Scholar] [CrossRef] - Cerón, J.C.; Pulido-Bosch, A. Groundwater problems resulting from CO 2 pollution and overexploitation in Alto Guadalentín aquifer (Murcia, Spain). Environ. Geol.
**1996**, 28, 223–228. [Google Scholar] [CrossRef] - Confederación Hidrográfica del Segura. Plan especial de actuación en situaciones de alerta y eventual Sequía. Tech. Rep.
**2006**, 298. [Google Scholar] - IGN. Instituto Geográfico Nacional. Available online: http://www.ign.es/ (accessed on 15 January 2018).
- Tomás, R.; Herrera, G.; Lopez-Sanchez, J.; Vicente-Guijalba, F.; Cuenca, A.; Mallorqui, J.J. Study of the land subsidence in Orihuela City (SE Spain) using PSI data: Distribution, evolution and correlation with conditioning and triggering factors. Eng. Geol.
**2010**, 115, 105–121. [Google Scholar] [CrossRef] - Herrera, G.; Tomás, R.; Monells, D.; Centolanza, G.; Mallorquí, J.J.; Vicente, F.; Navarro, V.D.; Lopez-Sanchez, J.M.; Sanabria, M.; Cano, M.; et al. Analysis of subsidence using TerraSAR-X data: Murcia case study. Eng. Geol.
**2010**, 116, 284–295. [Google Scholar] [CrossRef] - Zhu, L.; Gong, H.; Li, X.; Wang, R.; Chen, B.; Dai, Z.; Teatini, P. Land subsidence due to groundwater withdrawal in the northern Beijing plain, China. Eng. Geol.
**2015**, 193, 243–255. [Google Scholar] [CrossRef] - Pacheco-Martínez, J.; Cabral-Cano, E.; Wdowinski, S.; Hernández-Marín, M.; Ortiz-Lozano, J.A.; Zermeño-de-León, M.E. Application of InSAR and gravimetry for land subsidence hazard zoning in Aguascalientes, Mexico. Remote Sens.
**2015**, 7, 17035–17050. [Google Scholar] [CrossRef] - Solari, L.; Ciampalini, A.; Raspini, F.; Bianchini, S.; Moretti, S. PSInSAR analysis in the Pisa urban area (Italy): A case study of subsidence related to stratigraphical factors and urbanization. Remote Sens.
**2016**, 8, 120. [Google Scholar] [CrossRef] - Chaussard, E.; Milillo, P.; Bürgmann, R.; Perissin, D.; Fielding, E.J.; Baker, B. Remote sensing of ground deformation for monitoring groundwater management practices: Application to the Santa Clara Valley during the 2012–2015 California drought. J. Geophys. Res. Solid Earth
**2017**, 122, 8566–8582. [Google Scholar] [CrossRef] - Chen, G.; Zhang, Y.; Zeng, R.; Yang, Z.; Chen, X.; Zhao, F.; Meng, X. Detection of land subsidence associated with land creation and rapid urbanization in the Chinese Loess Plateau using time series InSAR: A case study of Lanzhou New district. Remote Sens.
**2018**, 10, 270. [Google Scholar] [CrossRef] - Hanssen, R.F. Radar Interferometry: Data Interpretation and Error Analysis; Kluwer Academic Publishers: Dordrecht, The Netherlands, 2001; p. 308. [Google Scholar] [CrossRef]
- Samieie-Esfahany, S.; Hanssen, R.F.; van Thienen-Visser, K.; Muntendam-Bos, A.; Systems, S. On the effect of horizontal deformation on InSAR subsidence estimates. In Proceedings of the Fringe Workshop, Frascati, Italy, 30 November–4 December 2009. [Google Scholar]
- Mora, O.; Ordoqui, P.; Iglesias, R.; Blanco, P. Earthquake rapid mapping using ascending and descending Sentinel-1 TOPSAR interferograms. Procedia Comput. Sci.
**2016**, 100, 1135–1140. [Google Scholar] [CrossRef] - Blanco-Sánchez, P.; Mallorquí, J.J.; Duque, S.; Monells, D. The coherent pixels technique (CPT): An advanced DInSAR technique for nonlinear deformation monitoring. Pure Appl. Geophys.
**2008**, 165, 1167–1193. [Google Scholar] [CrossRef] - Ezquerro, P.; Guardiola-Albert, C.; Herrera, G.; Fernández-Merodo, J.A.; Béjar-Pizarro, M.; Bonì, R. Groundwater and subsidence modeling combining geological and multi-satellite SAR data over the alto guadalentín aquifer (SE Spain). Geofluids
**2017**, 1–17. [Google Scholar] [CrossRef] - Pascal, K.; Neuberg, J.; Rivalta, I. On precisely modelling surface deformation due to interacting magma chambers and dykes. Geophys. J. Int.
**2014**, 196, 253–278. [Google Scholar] [CrossRef] - European Plate Observing System. Available online: https://www.epos-ip.org/ (accessed on 25 July 2019).
- Okada, Y. Surface deformation due to shear and tensile faults in a halfspace. Bull. Seismol. Soc. Amer.
**1985**, 75, 1135–1154. [Google Scholar]

**Figure 1.**(

**a**) Location of survey benchmarks for repeated gravity and leveling in Campi Flegrei. (

**b**) Temporal changes for gravity (red) and elevation (blue) at Serapeo from 1980 to 2000. The vertical dimension of the symbols is representative of the errors. A high correlation is observed between both data types, but the elevation values show a more continuous pattern. Modified from [1].

**Figure 2.**(

**a**) LOS deformation velocity computed from ascending passes for the period 1993–2000 and (

**b**) LOS deformation velocity computed from descending passes for the period 1992–2000. Modified from [1].

**Figure 3.**Three cross sections of the 3-D model for depressurization: Horizontal (depth 1500 m) and NNE–SSW and WNW–ESE vertical sections of the under-pressure model across a central position resulting from the simultaneous inversion of the gravity changes, leveling changes, and DInSAR data in Campi Flegrei for 1992–2000 assuming an elastic half-space [1].

**Figure 4.**Some additional views of the LOS ascending data fit: (

**a**) residual map and (

**b**) WE central profile for observed-modeled comparison [1].

**Figure 5.**MSBAS results, 1993–2013, for the images detailed in Table 1. (

**a**) Vertical cumulative component of deformation in centimeters, 1993–2013; (

**b**) east-west cumulative component of deformation in centimeters, 1993–2013; (

**c**) time series of vertical and east-west components shown in panels (

**a**) and (

**b**) at location of maximum subsidence, identified with the green dot. The reference location for MSBAS processing is located at 4532380, 4335351 (14.23°N, 40.94°E). Modified from [53].

**Figure 6.**Source location, depth and shape for deflation period of 1993–1999. (

**a**) 3D perspective; (

**b**) map view of source below Campi Flegrei caldera; (

**c**) EW vertical profile; (

**d**) NS vertical profile [53].

**Figure 7.**(

**a**) Observed vertical displacement rate (left), cm/yr, for the subsidence period, 1993–1999; modelled vertical displacement rate from inversion (center); and residual of observed and modelled displacements (right). (

**b**) Observed EW displacement rate (left), cm/yr, 1993–1999; modelled EW displacement from inversion (center); and residual of observed and modelled displacements (right). Images were saturated at the corresponding scales in order to improve comparison and highlight differences among panels [53].

**Figure 8.**Source location, depth and shape for deflation period of 1999–2000. (

**a**) 3D perspective; (

**b**) map view of source below Campi Flegrei caldera; (

**c**) EW vertical profile; (

**d**) NS vertical profile [53].

**Figure 9.**(

**a**) Observed vertical displacement rate (left), cm/yr, for the subsidence period, 1999–2000; modelled vertical displacement rate from inversion (center); and residual of observed and modelled displacements (right). (

**b**) Observed EW displacement rate (left), cm/yr, 1999–2000; modelled EW displacement from inversion (center); and residual of observed and modelled displacements (right). Images were saturated at the corresponding scales in order to improve comparison and highlight differences among panels [53].

**Figure 10.**Source location, depth and shape for deflation period of 2000–2005. (

**a**) 3D perspective; (

**b**) map view of source below Campi Flegrei caldera; (

**c**) EW vertical profile; (

**d**) NS vertical profile [53].

**Figure 11.**(

**a**) Observed vertical displacement rate (left), cm/yr, for the subsidence period, 2000–2005; modelled vertical displacement rate from inversion (center); and residual of observed and modelled displacements (right). (

**b**) Observed EW displacement rate (left), cm/yr, 2000–2005; modelled EW displacement from inversion (center); and residual of observed and modelled displacements (right). Images were saturated at the corresponding scales in order to improve comparison and highlight differences among panels [53].

**Figure 12.**Source location, depth and shape for deflation period of 2005–2007. (

**a**) 3D perspective; (

**b**) map view of source below Campi Flegrei caldera; (

**c**) EW vertical profile; (

**d**) NS vertical profile [53].

**Figure 13.**(

**a**) Observed vertical displacement rate (left), cm/yr, for the subsidence period, 2005–2007; modelled vertical displacement rate from inversion (center); and residual of observed and modelled displacements (right). (

**b**) Observed EW displacement rate (left), cm/yr, 2005–2007; modelled EW displacement from inversion (center); and residual of observed and modelled displacements (right). Images were saturated at the corresponding scales in order to improve comparison and highlight differences among panels [53].

**Figure 14.**Source location, depth and shape for deflation period of 2007–2013. (

**a**) 3D perspective; (

**b**) map view of source below Campi Flegrei caldera; (

**c**) EW vertical profile; (

**d**) NS vertical profile [53].

**Figure 15.**(

**a**) Observed vertical displacement rate (left), cm/yr, for the subsidence period, 2007–2013; modelled vertical displacement rate from inversion (center); and residual of observed and modelled displacements (right). (

**b**) Observed EW displacement rate (left), cm/yr, 2007–2013; modelled EW displacement from inversion (center); and residual of observed and modelled displacements (right). Images were saturated at the corresponding scales in order to improve comparison and highlight differences among panels [53].

**Figure 16.**Planar view (

**a**) and EW vertical cut view (

**b**) of the main elements from the real time modelling process covering the inflation period and the eruption on 13 May 2008, and possible connections between the source bodies and the location of the earthquakes (blue circles) for 12–14 May 2008. Contours correspond to surface topography. Gray triangles indicate the location of the GPS stations. Shaded gray area outlines the position of the High Velocity Body (HVB) [100]. In green, the cumulated sources during the yearlong recharging phase and in orange the sources active during the day of the eruption. Cannavò et al. [17] suggested fast magma ascent from the reservoir level (2–3 km depth bsl) along paths indicated by the source geometries and the earthquake locations (dashed lines in the figures) [17].

**Figure 17.**(

**a**) Best windowing. In blue the mean (for all the stations and components) standard deviations, for different window sizes, of the estimated displacement time series from GPS solutions. (

**b**) Mean autocorrelation function with shaded error bar for all the 1-Hz GPS time series. It is possible to see the steep decay of autocorrelation after lags of a few hundreds of seconds. Modified from [17].

**Figure 18.**Model for the inflation phase. Cumulated magmatic body (in green) modelled for the period 1 January 2007 to 12 May 2008 and time evolution (in red) for the sub-period that corresponds to the months of June and July 2007, when the system was subject to a recharging episode, which appears as an ascending high-pressure body. Parallelepiped sources represent active point pressure sources within the same volume. Blue circles correspond to location of the earthquakes for 12–14 May 2008 [17].

**Figure 19.**Time sequence of pressurized source models (described by aggregation of parallelepiped cells which map active point pressure sources) corresponding to key instants from the real time inversions during the 13 May 2008, eruption of Mount Etna volcano at different UTC times: (

**a**) at 7:00, (

**b**) at 8:30, (

**c**) at 11:30, and (

**d**) at 15:00. Red volumes are positive-pressure sources while blue volumes are negative pressure sources. Structures marked in orange correspond to the cumulated sources describing the main source structures across the sequence. Contours correspond to surface topography. Gray triangles indicate the location of the GPS stations. Blue circles correspond to location of the earthquakes for 12–14 May 2008 [17].

**Figure 20.**Graphic summary of the sources modeled in the studied 2008 Mt. Etna eruption.

**HVB**: High velocity body,

**F**: the dyke as modelled by the present study and previous works [102],

**M**: main source body for the inflation phase preceding the eruption. UTM coordinates and depth are expressed in meters. The azimuth view is 40° [17].

**Figure 21.**3D uncertainty maps for the considered GPS network of 10 stations, representing, respectively, (

**a**) the pressure changes, (

**b**), the depth changes, and (

**c**) the horizontal deviations required to produce a surface deformation with quadratic mean value 1 mm. Map (

**d**) represents the minimum horizontal deviation for an isolated body of 1 km

^{3}with a pressure change given in panel (

**a**) to produce a deformation at the GPS stations with rms magnitude of 1 mm [17].

**Figure 22.**(

**a**) Location of the Alto Guadalentín Basin, the Bajo Guadalentín Basin and the Guadalentín River that formed the two basins. Black lines depict main faults in the area. The locations and names of the main cities in the area are shown. The topography has been obtained from MDT05 2015 CC-BY 4.0 digital elevation model [111]. (

**b**) Subsidence area detected in previous studies31 by means of DInSAR techniques along the Alto Guadalentín Basin. Subsidence rates have a maximum of 16 cm/yr for the period 2006–2011 located ~4 km south-west the city of Lorca. The black stars are damage locations due to the M = 5.1 May 2008 Lorca earthquake. Red lines are main faults (AMF, Alhama de Murcia Fault). The contour lines indicate 2 cm/yr DInSAR subsidence due to groundwater pumping. (

**c**) Location of the monitoring GNSS control stations deployed in the area of Alto Guadalentín. The network consists of 33 monitoring stations (blue circles show their location) and covers an area of about 70 km

^{2}. The network is designed to allow high accuracy GNSS surveys and also includes two existing continuous GNSS stations. Main population centers are depicted with white stars. Modified from [5].

**Figure 23.**Displacement rates determined from GNSS observations. Results corresponding to the period November 2015–February 2017. (

**a**) Annual vertical displacement rates, subsidence, measured with standard confidence bars. (

**b**) Average annual horizontal displacements with standard confidence regions. Additional results are shown in the Supplementary Information from [5]. (

**c**) and (

**d**) show the results obtained from the A-DInSAR processing using CPT technique. Both geometries, ascending and descending, have been processed using a multilook window of 3 × 13 pixels (azimuth × range) which generates a square pixel of about 60 × 60 meters in ground resolution. Coherence method has been used for pixel selection coherence method. Results are shown for the period November 2015–February 2017. (

**c**) Line of Sight (LOS) velocity values obtained for the ascending orbit. (

**d**) LOS velocity values for the descending orbit. Black dots locate the GNSS stations. Modified from [5].

**Figure 24.**East-West and Vertical displacements obtained by A-DInSAR. (

**a**) Horizontal (East-West) and (

**b**) vertical (Up-Down) displacement rates estimations obtained by decomposition of the LOS detected velocity using ascending and descending orbits. GNSS displacements are also plotted with arrows to compare. Results are shown for the period November 2015–February 2017. Modified from [5].

**Figure 25.**Representation of the inversion results obtained for the 1D, 2D, 3D and 2D+3D considered data sets. (

**a**) Obtained source for Case A; (

**b**) for Case B; (

**c**) for Case C; (

**d**) Results for Case D; (

**e**) for Case E; (

**f**) for Case F; (

**g**) for Case G; (

**i**) for Case I and (

**j**) for Case J. Blue color indicates negative pressure value cells, produced by water extraction. White color indicates positive pressure change cells. These positive pressure sources adjust the errors and the effects of other deformation sources, different from water extraction (e.g., of tectonic origin) [5].

**Figure 26.**Schematic flow diagram of the described inversion methodology [1,5,17,53,54] where we start from the data set, the medium characteristics and its 3D gridding, to get (using the direct model equations and complementary conditions) a 3D source model of the anomalous sources via a growth process.

**Table 1.**Seven DInSAR data sets used in this study with continuous coverage from 1993 until 2013

^{a}[53].

DInSAR Data Set | Orbit | Coverage Period ^{b} | θ | Φ | N | M |
---|---|---|---|---|---|---|

ERS, track 129 | asc | 10/1/1993–17/9/2008 | 344.1 | 23.2 | 90 | 215 |

ERS, track 036 | dsc | 8/6/1992–25/12/2008 | 194.1 | 23.2 | 84 | 164 |

Envisat, track 129 | asc | 13/11/2002–16/12/2009 | 344.0 | 22.8 | 46 | 120 |

Envisat, track 036 | dsc | 5/6/2003–21/10/2010 | 195.9 | 22.8 | 60 | 158 |

R2, S3 | asc | 19/1/2009–2/8/2013 | 348.7 | 35.1 | 42 | 166 |

R2, S3 | dsc | 27/12/2008–3/8/2013 | 190.4 | 35.1 | 53 | 290 |

R2, F6 | asc | 29/12/2008–5/8/2013 | 351.0 | 48.3 | 50 | 158 |

^{a}Incidence angle φ (degrees), azimuth angle θ (degrees), number of available single-look complex SAR images in each set N, and number of computed highly coherent interferograms M. Combined coverage: 10 January 1993 to 3 August 2013. Total number of unique time steps (measured daily) = 385 (48 repeated by different sensors). Total M = 1271. Ascending (asc) and descending (dsc).

^{b}Dates are formatted as day/month/year.

**Table 2.**Six time periods used for modeling expansion/contraction sources below Campi Flegrei Caldera and associated location, pressure, volume, and depth to the center of mass

^{a}[53].

Time Period (Years) | X Location UTM (m) | Y Location UTM (m) | Depth (m bsl) | Pressure (MPa × 10 ^{9}/yr) | Volume (km ^{3}/yr) |
---|---|---|---|---|---|

(±50) | (±50) | (±60) | (±0.05) | (±0.7) | |

1993–1999 | 426573 | 4519313 | 1372 | −0.37 | 6.2 |

1999–2000 | 426419 | 4519781 | 1434 | +0.51 | 8.5 |

2000–2005 | 426469 | 4519501 | 1874 | −0.13 | 2.2 |

2005–2007 | 427388 | 4519326 | 2126 | +0.69 | 11.4 |

2007–2007 | 426184 | 4519238 | 1683 | −0.47 | 7.9 |

2007–2013 | 426669 | 4519315 | 1819 | +0.49 | 8.2 |

^{a}UTM = Universal Transverse Mercator.

CASE | Intensity MPa×Km ^{3} | Misfit (cm) | Mean Model Intensity(MPa × Km ^{3}) | Pres. (MPa) | Vol. (Km ^{3}) | Displacement Components Considered | Nº of Data Used |
---|---|---|---|---|---|---|---|

A | −41 | 0.36 | −42.7±3.8 (9%) [−40±1.4 (4%)] | −3 | 14.2 [13.3] | 1D | 1505 |

B | −39 | 0.31 | 1203 | ||||

C | −48 | 0.19 | 1572 | ||||

D | −32 | 0.30 | −33.9 ± 1.9 (6%) [−32.5 ± 1.1 (3%)] | −3 | 11.3 [10.8] | 2D | 1505 |

E | −31 | 0.28 | 1203 | ||||

F | −33 | 0.32 | 2708 | ||||

G | −37 | 0.45 | 3144 | ||||

H | −34 | 0.94 | 3D | 108 | |||

I | −34 | 0.43 | 2D + 3D | 2816 | |||

J | −36 | 0.63 | 3252 |

^{1}Data are grouped into two types: 1D data (Cases

**A**–

**C**), 2D and 2D +3D (Cases

**D**–

**J**). 2D data come from A-DInSAR results. 2D+3D data sets combine data obtained from the A-DInSAR study with those obtained from GNSS observation campaigns. For each of these two sets, which give rise to very similar results, mean values of intensity and values of volume variation in the aquifer, as a function of the pressure variation assumed, are given. Figures in square brackets denote average values determined not considering Cases

**C**, and Cases

**G**,

**H**and

**J**respectively. See text for more details.

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Camacho, A.G.; Fernández, J.
Modeling 3D Free-geometry Volumetric Sources Associated to Geological and Anthropogenic Hazards from Space and Terrestrial Geodetic Data. *Remote Sens.* **2019**, *11*, 2042.
https://doi.org/10.3390/rs11172042

**AMA Style**

Camacho AG, Fernández J.
Modeling 3D Free-geometry Volumetric Sources Associated to Geological and Anthropogenic Hazards from Space and Terrestrial Geodetic Data. *Remote Sensing*. 2019; 11(17):2042.
https://doi.org/10.3390/rs11172042

**Chicago/Turabian Style**

Camacho, Antonio G., and José Fernández.
2019. "Modeling 3D Free-geometry Volumetric Sources Associated to Geological and Anthropogenic Hazards from Space and Terrestrial Geodetic Data" *Remote Sensing* 11, no. 17: 2042.
https://doi.org/10.3390/rs11172042