Luigi Mereu
email: luigi.mereu@ingv.it

Simona Scollo
email: simona.scollo@ingv.it

Costanza Bonadonna
email: costanza.bonadonna@unige.ch

Franck Donnadieu
email: franck.donnadieu@uca.fr

Valentin Freret-Lorgeril
email: valentin.freretlorgeril@unige.ch

Fellow, IEEE Frank S Marzano
email: frank.marzano@uniroma1.it

Costanza Bonadonn

Ground-Based Remote Sensing and Uncertainty Analysis of the Mass Eruption Rate Associated With the 3-5 December 2015 Paroxysms of Mt. Etna

Keywords: Explosive eruptions, gas-tephra mixture density, lava fountains, mass eruption rate (MER), radar and thermal data, retrieval techniques, total erupted mass (TEM), uncertainties

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

up to tens of centimeters, are injected into the atmosphere [START_REF] Sparks | Volcanic Plumes, 1st ed[END_REF], [START_REF] Poret | Modeling eruption source parameters by integrating field, ground-based, and satellite-based measurements: The case of the 23[END_REF]. Volcanic particles subsequently settle under the action of gravity, plume/cloud, and atmospheric dynamics [START_REF] Bonadonna | Tephra Dispersal and Sedimentation[END_REF]. Atmospheric transport of tephra, depending on particle characteristics (size, density, shape) and altitude of release, can last from minutes up to several weeks. The area affected by tephra fallout can extend thousands of square kilometers around the eruptive vent.

Recent volcanic crises, such as those associated with Eyjafjallajökull (Iceland) in 2010 and Cordón-Caulle (Chile) in 2011, have demonstrated the need for a better real-time assessment of the eruption source parameters (ESPs), namely column height, mass eruption rate (MER), total erupted mass (TEM), and total grain-size distribution. Those are in fact the main input parameters of the dispersion models, widely used to forecast the tephra dispersal and fallout during explosive eruptions. Various geophysical strategies can be applied, such as the integration among different sensors, direct observations, and field measurement analysis, in order to increase the reliability of the ESPs assessment and mitigate volcanic risks [START_REF] Poret | Reconstructing volcanic plume evolution integrating satellite and ground-based data: Application to the 23rd November 2013 Etna eruption[END_REF], [START_REF] Bonadonna | Tephra sedimentation during the 2010 eyjafjallajökull eruption (Iceland) from deposit radar and satellite observations[END_REF]- [START_REF] Corradini | A multi-sensor approach for volcanic ash cloud retrieval and eruption characterization: The 23 November 2013 Etna Lava fountain[END_REF], [START_REF] Freret-Lorgeril | Examples of multi-sensor determination of eruptive source parameters of explosive events at Mount Etna[END_REF]. In particular, the MER which is a measure of the temporal rate of magmatic flow through the volcano vent [START_REF] Mereu | Retrieval of tephra size spectra and mass flow rate from C-band radar during the 2010 Eyjafjallajökull eruption Iceland[END_REF].

Mt. Etna, in Italy, is one of the most active volcanoes in the world, whose explosive eruptions represent a serious threat to the communities located around it, producing various dangerous effects on inhabitants, properties, and communication routes, and at the same time represents an ideal laboratory to improve volcano monitoring and forecasting of eruptive events [START_REF] Scollo | Volcanic ash concentration during the 12 August 2011 Etna eruption[END_REF]. The summit of Etna is made up of three main craters: Voragine Crater (VOR), NorthEast Crater (NEC), and SouthEast Crater (SEC), as shown in Fig. 1(a). From 2009 to 2013, the explosive activity was located on the east flank of SEC, which formed a new pyroclastic cone named New SouthEast Crater (NSEC) [START_REF] Behncke | The 2011-2012 summit activity of Mount Etna: Birth, growth and products of the new SE crater[END_REF]. Fig. 1(b) also shows the map of the network of sensors used in this work. Each sensor is marked with colored dots. In particular, the observations are obtained by the following.

(1) The X-band weather radar (XWR), at about 32 km from the VOR, which is a dual-polarization scanning radar of the Italian weather radar network, already successfully used to retrieve ESPs [START_REF] Carbone | Balancing bulk gas accumulation and gas output before and during lava fountaining episodes at Mt. Etna nature 2015[END_REF], [START_REF] Vulpiani | Mass discharge rate retrieval combining weather radar and thermal camera observations[END_REF], [START_REF] Poret | Reconstructing volcanic plume evolution integrating satellite and ground-based data: Application to the 23 November 2013 Etna eruption[END_REF]. This radar operates at 9.6 GHz, with transmitted peak power of 50 kW, half-power beam width of 1.3 degree, and considering a permittivity factor of tephra particles equal to 0.39 with respect to 0.93 for water particles (b) Map of the sensor network used in this work is shown. Each sensor and crater are geolocated in terms of latitude and longitude and highlighted with colored dots, specifying the elevation above sea level (asl), as listed in the legend. The colored dashed lines show the distance between each sensor and the Etna summit, as clarified for each one in km. [START_REF] Marzano | Synthetic signatures of volcanic ash cloud particles from X-band dual-polarization radar[END_REF]. The XWR performs a 3-D scan of the surrounding scene as a function of range, azimuth, and elevation with five azimuthal scans per minute.

(2) The fixed-pointing L-band Doppler radar VOLDORAD-2B (VDR, wavelength of 23.5 cm) for the near-crater detection of erupted material during Etna's explosive events. This Doppler radar measures both the radial velocity v r and the received backscattered power that characterizes the amount of detected tephra at high-time resolution (i.e., 0.2 s; cf. open-access data base used in this study [START_REF] Donnadieu | The Volcano Doppler Radar Data Base of Etna (VOLDORAD 2B)[END_REF]). From the observation geometry, it is possible to convert v r into exit velocity v ex (i.e., v ex = 3.89 v r ) ( [START_REF] Donnadieu | Volcanological applications of doppler radars: A review and examples from a transportable pulse radar in L-band[END_REF], [START_REF] Freret-Lorgeril | Mass eruption rates of tephra plumes during the 2011-2015 lava fountain paroxysms at Mt. Etna from Doppler radar retrievals[END_REF], [START_REF] Donnadieu | Near-source Doppler radar monitoring of tephra plumes at Etna[END_REF]), whereas from the specifications of the L-band radar and the radar constant, the backscattered power can be transformed into the L-band reflectivity factor Z hh [START_REF] Mereu | Maximum-likelihood retrieval of volcanic ash concentration and particle size from ground-based scanning lidar[END_REF].

(3) The thermal infrared camera (TIC), which is located at 15 km, in Nicolosi [START_REF] Bonaccorso | Eruptive processes leading to the most explosive lava fountain at Etna volcano: The 23 November 2013 episode[END_REF], on the south flank of Etna and belongs to the video-monitoring network system of the INGV-OE. TIC provides a time series of 640×480 pixels images with a spatial resolution of a few meters and a thermal sensitivity of 80 mK at 25°C. The images are displayed with a fixed color scale with a range of -10°C to 70°C [START_REF] Bonaccorso | Eruptive processes leading to the most explosive lava fountain at Etna volcano: The 23 November 2013 episode[END_REF].

The MER is fundamental to accurately forecast the eruptive column evolution and the ash cloud dispersal. During a volcanic crisis, direct remote sensing measurements can provide information on the ongoing activity and allow evaluating the impact of an explosive event [START_REF] Spanu | Reconstructing eruptive source parameters from tephra deposit: A numerical approach for medium-sized explosive eruptions[END_REF]. However, the uncertainty related to the ESPs is difficult to assess due to the variability of instrument features and parameters used in the methodology. In general, the reliability in the uncertainty estimation depends on the degrees of freedom, or on the independent quantities that contribute to the MER estimates.

In this work, following and extending previous studies [START_REF] Marzano | Near-real-time detection of tephra eruption onset and mass flow rate using microwave weather radar and infrasonic arrays[END_REF], [START_REF] Mereu | Retrieval of tephra size spectra and mass flow rate from C-band radar during the 2010 Eyjafjallajökull eruption Iceland[END_REF], [START_REF] Marzano | Tephra mass eruption rate from ground-based X-band and L-band microwave radars during the November 23, 2013 Etna paroxysm[END_REF], [START_REF] Freret-Lorgeril | Mass eruption rates of tephra plumes during the 2011-2015 lava fountain paroxysms at Mt. Etna from Doppler radar retrievals[END_REF], we estimate the MER derived from the XWR, VDR, and TIC, and analyze in detail the various methodologies applied to the four Etna lava fountains occurred in 3-5 December 2015.

We select this case study mainly because the observations of the three different sensors, described previously, are available at the same time. Starting from previous studies ( [START_REF] Mereu | Retrieval of tephra size spectra and mass flow rate from C-band radar during the 2010 Eyjafjallajökull eruption Iceland[END_REF], [START_REF] Marzano | Tephra mass eruption rate from ground-based X-band and L-band microwave radars during the November 23, 2013 Etna paroxysm[END_REF], [START_REF] Mereu | Multisensor characterization of the incandescent jet region of lava fountain-fed tephra plumes[END_REF], [START_REF] Freret-Lorgeril | Examples of multi-sensor determination of eruptive source parameters of explosive events at Mount Etna[END_REF]), we aim at the following:

1) describing a new MER approach (Near Source Approach -NSA), which completes the set of the proposed methods; 2) comparing multisensor strategies for MER determination; 3) identifying the limits, advantages, and uncertainties correlated to each methodology; 4) exploring the possibility to integrate different approaches to estimate the MER; 5) calculating the gas-tephra mixture density near the eruptive vent and the relative uncertainty; and 6) identifying the best integrated strategy to reduce errors in MER retrievals. The rest of this article is organized as follows. Section II illustrates the Etna 2015 paroxysm observations by XWR, VDR, and TIC sensors. Section III describes the different MER methodologies used and relative uncertainties. Section IV is devoted to the MER retrievals and relative uncertainties, while Section V summarizes the MER results. Finally, Section VI concludes this article.

II. ETNA 2015 LAVA FOUNTAINS OBSERVATIONS

Between the early morning of 3 December and the afternoon of 5 December 2015, four powerful paroxysmal events were produced from the VOR crater. These events generated tall lava fountains above the crater and tephra plumes up to 12-16 km (all heights are asl) [START_REF] Corsaro | Monitoring the December 2015 summit eruptions of Mt. Etna (Italy): Implications on eruptive dynamics[END_REF], observed by the XWR, VDR, and TIC, at the same time.

A. Thermal Infrared Observations

In the first third of the volcanic plumes, we clearly identify a jet, called the incandescent jet region (IJR), which is mainly composed of lapilli and bombs mixed with hot gas [START_REF] Sparks | Volcanic Plumes, 1st ed[END_REF], [START_REF] Sparks | The dimensions and dynamics of volcanic eruption columns[END_REF]. Most procedures, used to identify the IJR, are based on setting a suitable threshold to the vertical spatial gradients and/or to edge-contour detection filters on thermal images [START_REF] Mereu | Multisensor characterization of the incandescent jet region of lava fountain-fed tephra plumes[END_REF], [START_REF] Tournigand | The initial development of transient volcanic plumes as a function of source conditions[END_REF]. By selecting the TIC frames at time intervals of 1 min, it is possible to derive the IJR top height in each image, as described in [START_REF] Borisova | Processes controlling the 2010 Eyjafjallajökull explosive eruption[END_REF], [START_REF] Dubosclard | Doppler radar sounding of volcanic eruption dynamics at Mount Etna[END_REF], [START_REF] Wilson | Ascent and eruption of basaltic magma on the Earth and Moon[END_REF], [START_REF] Donnadieu | Volcanological applications of doppler radars: A review and examples from a transportable pulse radar in L-band[END_REF], [START_REF] Ripepe | Ash-plume dynamics and eruption source parameters by infrasound and thermal imagery: The 2010 eyjafjallajökull eruption[END_REF], and [START_REF] Mereu | Multisensor characterization of the incandescent jet region of lava fountain-fed tephra plumes[END_REF].

In Fig. 2 we show the main frames displaying the lava fountains associated with the eruptive sequence occurring on: 1) 3 December 2015 at 03:00 UTC; 2) 4 December 2015 at 09:30 UTC; 3) 4 December 2015 at 21:00 UTC; and 4) 5 December 2015 at 15:30 UTC.

B. Microwave Observations

The comparative analysis of the XWR measurements provides the first interesting information on the vertical structure of the tephra plume. Focusing on the eruptive observations, Fig. 3 shows the vertical and horizontal maps of the four eruptive events analyzed in this work. We show the Plan Position Indicator (PPI) of XWR reflectivity factor Z hh (dBZ) at 6°elevation radar (a1), (a2), (a3), and (a4); vertical cut of the copolar reflectivity factor Z hh (dBZ) along the line connecting the radar site and the maximum horizontal ash plume extension (b1), (b2), (b3), and (b4). Regarding the PPI, the wind action in altitude is clearly perceivable, which can project the space-time evolution of the eruptive cloud in different directions of the horizon. In the panels (b1-b4) the top plume altitude H TP is clearly identifiable between 12-16 km and the plume is displaced with respect to the vertical line centered on the eruptive vent.

Fig. 4 shows the overlapping between the maximum values Z hh detected both by the XWR (red line) about 33 km far from the VOR crater, i.e., at the fifth elevation angle, and detected by the VDR (about 4 km from VOR), each 30 s (green line) and averaged over 10 min (blue lines). For the three events on 4 and 5 December, there is a qualitative match between both radar Z hh trends with maximum values slightly exceeding 60 dBZ [see Fig. 4(b)-(d)]. For the most powerful event on 3 December, the VDR trend of Z hh exceeds the XWR estimates with values of about 70 against 60 dBZ (see Fig. 4(a)).

The main features for each event of the 2015 Etna eruption are listed in Table I, as well as the temporal duration, the top plume altitude H TP (asl) and the VARR-estimated tephra classes. The maximum H TP value is detected both from ground-based XWR data and from satellite spectroradiometers [START_REF] Corradini | Proximal monitoring of the 2011-2015 Etna lava fountains using MSG-SEVIRI data[END_REF]. During the 4 and 5 December cases, the satellite-based H TP is higher than December 2015. Each curve is related to the bin volume near the summit crater, that is the fifth elevation angle for the XWR and eighth and ninth range bins for the VDR, considering that the latter points directly near to the volcanic source of VOR. The reflectivity factor Z hh (dBZ) derived from the XWR is sampled every 10 min (red line) and from the VDR directly every 30 s (green line) or linearly interpolated every 10 min (doted blue line).

TABLE I ETNA ERUPTION FEATURES ON 3-5 DECEMBER 2015, DERIVED FROM THE XWR ELABORATION APPLYING THE VARR METHODOLOGY AND FROM LITERATURE

the XWR one (about 17.6 km versus 13.1 km and 14.1 km versus 13.1 km, respectively, for the satellite [START_REF] Corradini | Proximal monitoring of the 2011-2015 Etna lava fountains using MSG-SEVIRI data[END_REF] and XWR [START_REF] Mereu | Retrieval of tephra size spectra and mass flow rate from C-band radar during the 2010 Eyjafjallajökull eruption Iceland[END_REF], [START_REF] Vulpiani | Mass discharge rate retrieval combining weather radar and thermal camera observations[END_REF]). This trend is reversed on 3 December, where we find 12.5 and 14.6 km for the satellite and XWR, respectively, probably due to the effect of larger tephra particles. The four eruptive columns measured by XWR are generally composed of more than 60% of coarse ash (CA). For the first two events, proportions of 10-30% and less than 2% are found for fine Lapilli (FL) and large Lapilli (LL), respectively [START_REF] Marzano | Synthetic signatures of volcanic ash cloud particles from X-band dual-polarization radar[END_REF], [START_REF] Mereu | Multisensor characterization of the incandescent jet region of lava fountain-fed tephra plumes[END_REF].

III. MASS ERUPTION RATE METHODOLOGIES AND UNCERTAINTIES

The block diagram of the extended VARR algorithm to implement the four MER (hereafter indicated with Q M ) methods and derive the input parameters used in each approach is shown in Fig. 5. Some MER retrieval methods, already described in previous works [START_REF] Marzano | Near-real-time detection of tephra eruption onset and mass flow rate using microwave weather radar and infrasonic arrays[END_REF], [START_REF] Mereu | Retrieval of tephra size spectra and mass flow rate from C-band radar during the 2010 Eyjafjallajökull eruption Iceland[END_REF], [START_REF] Marzano | Synthetic signatures of volcanic ash cloud particles from X-band dual-polarization radar[END_REF], [START_REF] Mereu | Multisensor characterization of the incandescent jet region of lava fountain-fed tephra plumes[END_REF], are based on the analysis of observations collected by XWR, VDR, and TIC sensors. The four approaches to estimate MER are as follows: 1) Mass continuity approach (MCA), related to time-space variation of mass concentration of the whole plume detected by the XWR [START_REF] Mereu | Retrieval of tephra size spectra and mass flow rate from C-band radar during the 2010 Eyjafjallajökull eruption Iceland[END_REF], [START_REF] Marzano | Tephra mass eruption rate from ground-based X-band and L-band microwave radars during the November 23, 2013 Etna paroxysm[END_REF], [START_REF] Mereu | Multisensor characterization of the incandescent jet region of lava fountain-fed tephra plumes[END_REF]. 2) NSA, correlated to tephra concentration variation in the beam lower surface nearest to the summit crater, obtained by XWR. This Q M estimation method is new and based entirely on the XWR measurements, adapted from [START_REF] Vulpiani | Mass discharge rate retrieval combining weather radar and thermal camera observations[END_REF] and [START_REF] Freret-Lorgeril | Examples of multi-sensor determination of eruptive source parameters of explosive events at Mount Etna[END_REF]. 3) Top Plume Approach (TPA), using semiempirical parametric models integrating the top plume altitude derived from XWR [START_REF] Marzano | Tephra mass eruption rate from ground-based X-band and L-band microwave radars during the November 23, 2013 Etna paroxysm[END_REF], [START_REF] Mereu | Retrieval of tephra size spectra and mass flow rate from C-band radar during the 2010 Eyjafjallajökull eruption Iceland[END_REF], [START_REF] Degruyter | Improving on mass flow rate estimates of volcanic eruptions[END_REF]. 4) Surface Flux Approach (SFA), that links the MER to the exit velocity v ex , derived from XWR, VDR, or TIC [START_REF] Marzano | Tephra mass eruption rate from ground-based X-band and L-band microwave radars during the November 23, 2013 Etna paroxysm[END_REF], [START_REF] Mereu | Multisensor characterization of the incandescent jet region of lava fountain-fed tephra plumes[END_REF]. The gas-tephra mixture density is estimated using the XWR technique. For each method, we show the percentage uncertainty ε Q = 100(δQ M /Q M ), evaluated according to the first-order error propagation theory for each independent variable [START_REF] Marzano | Tephra mass eruption rate from ground-based X-band and L-band microwave radars during the November 23, 2013 Etna paroxysm[END_REF].

A. Mass Continuity Approach (MCA)

The general approach starts from the conservation of mass, as described in [START_REF] Mereu | Retrieval of tephra size spectra and mass flow rate from C-band radar during the 2010 Eyjafjallajökull eruption Iceland[END_REF], [START_REF] Marzano | Tephra mass eruption rate from ground-based X-band and L-band microwave radars during the November 23, 2013 Etna paroxysm[END_REF], [START_REF] Mereu | Multisensor characterization of the incandescent jet region of lava fountain-fed tephra plumes[END_REF]. The a priori hypotheses used to compute the MER are: (i) no source-sink term; (ii) negligible advection; and (iii) time sampling of radar. Assuming a spacetime discretization related to the XWR operating mode, the MER can be expressed as follows:

Q (MCA) M (t) ∼ = Q M dif (t) + Q M adv (t) (1)
where Q Mdif (t) is correlated to the variation in each radar volume bin between two consecutive sampling times, and the Q Madv (t) is linked to the thrust flow outside the plume volume [START_REF] Mereu | Retrieval of tephra size spectra and mass flow rate from C-band radar during the 2010 Eyjafjallajökull eruption Iceland[END_REF]. The latter term is mainly related to the XWR horizontal velocity deduced from the cross-correlation between two consecutive XWR vertical-horizontal section of the eruptive event. To optimize the Q M estimates, we identify the maximum volume involved in the eruptive plume, as detected by the XWR. We search for the base area of the volume, centered over the Etna crater (i.e., VOR in our test), which optimizes the Q M estimates. The overall uncertainty ε Q (MCA) (7) as reported in [START_REF] Marzano | Tephra mass eruption rate from ground-based X-band and L-band microwave radars during the November 23, 2013 Etna paroxysm[END_REF] and shown in Table IV is estimated using the error propagation theory for independent errors, and is equal to 22.3% in relative percentage, assuming the variations δQ M dif = 0.20Q M and δQ M adv = 0.10Q M adv (see Table V). The advantage of this technique is the use of the direct XWR observables, making this approach applicable in near-real time.

B. Near Source Approach (NSA)

A further method, employing parameters derived by processing XWR measurements, can be introduced. The tephra concentration C t , derived from the VARR technique [START_REF] Marzano | Near-real-time detection of tephra eruption onset and mass flow rate using microwave weather radar and infrasonic arrays[END_REF], [START_REF] Mereu | Retrieval of tephra size spectra and mass flow rate from C-band radar during the 2010 Eyjafjallajökull eruption Iceland[END_REF], [START_REF] Mereu | Multisensor characterization of the incandescent jet region of lava fountain-fed tephra plumes[END_REF], [START_REF] Marzano | Tephra mass eruption rate from ground-based X-band and L-band microwave radars during the November 23, 2013 Etna paroxysm[END_REF], is the input parameter used in the near surface approach (NSA) as well as v ex and the area of the eruptive flow near the VOR crater as detected from the XWR scans [START_REF] Vulpiani | Mass discharge rate retrieval combining weather radar and thermal camera observations[END_REF].

We can define A NS as the near source area above the crater when C t is greater than a fixed threshold value. This area is related to the radar bins above the VOR, i.e., corresponding to the fourth and fifth elevation angle of the XWR in which the erupted mass flow is more intense. We calculate the Q M starting from the integration of C t flowing with v ex through the elementary area dA of A NS , as defined in the following relation:

Q (NSA) M (t) = A NS C t (t) v ex (t)dA (2)
where v ex is derived from H IJR as discussed previously [START_REF] Mereu | Multisensor characterization of the incandescent jet region of lava fountain-fed tephra plumes[END_REF]. The advantage of this method is that each parameter is derived independently from the XWR alone and, for this reason, it can be applied in real time.

In the same way as before, the uncertainty ε Q (NSA) is computed assuming for each independent parameter the following variations: δC t = 0.10C t , δH IJR = 0.20H IJR , and δA NS = 0.20A NS . Consequently, the relative percentage error ε Q is equal to 26% (see Table V).

C. Top Plume Approach (TPA)

Different nonlinear parametric relations between Q M and top plume H TP are known in literature [START_REF] Sparks | Volcanic Plumes, 1st ed[END_REF], [START_REF] Carazzo | On the rise of turbulent plumes: Quantitative effects of variable entrainment for submarine hydrothermal vents, terrestrial and extra-terrestrial explosive volcanism[END_REF], [START_REF] Bringi | Polarimetric Doppler Weather Radar[END_REF], [START_REF] Mastin | A multidisciplinary effort to assign realistic source parameters to models of volcanic ash-cloud transport anddispersion during eruptions[END_REF], [START_REF] Morton | Turbulent gravitational convection from maintained and instantaneous sources[END_REF], [START_REF] Degruyter | Improving on mass flow rate estimates of volcanic eruptions[END_REF], [START_REF] Wilson | Explosive volcanic eruptions-IV. The control of magma properties and conduit geometry on eruption column behaviour[END_REF]. In this work, we consider the model of Degruyter and Bonadonna, 2012, (DB12) [START_REF] Degruyter | Improving on mass flow rate estimates of volcanic eruptions[END_REF], that includes both the wind and buoyancy contributions at a given instant [START_REF] Mereu | Retrieval of tephra size spectra and mass flow rate from C-band radar during the 2010 Eyjafjallajökull eruption Iceland[END_REF], using the following relation:

Q M (TPA) (t) = αH T P 4 + βH T P 3 (3)
where α, β are two variables that include a series of environmental parameters. For the considered case studies, a wind velocity of 7 m/s is obtained, averaging the wind velocity derived from the radio-sounding (Trapani observations) along the vertical altitude from the VOR and the maximum altitude reached by the volcanic plume.

The uncertainty ε Q (TPA) in this case is equal to 30%, assuming for each independent variable the following variations: δα = 0.20α, δβ= 0.20β, and δH TP = 0.20H TP (see Table V).

D. Surface Flow Approach (SFA)

The divergence theorem allows us to transform a volume integral of the vector divergence into an integral area over the surface that defines the volume of the detected plume from the XWR [START_REF] Mereu | Retrieval of tephra size spectra and mass flow rate from C-band radar during the 2010 Eyjafjallajökull eruption Iceland[END_REF], [START_REF] Vulpiani | Mass discharge rate retrieval combining weather radar and thermal camera observations[END_REF]. We select the nearest area to the volcanic vent where v ex is normal to the surface S v = (π • r 2 v ) and under a spatial uniform flow assumption, we estimate Q M as follows:

Q (SFA) M (t) ∼ = ρ x v ex (t) S v ( 4
)
where ρ x is the density (kg/m 3 ) of the erupted mixture that can be assumed (i.e., from literature) or retrieved from the XWR and r v is the vent radius (m) assuming a circular vent shape.

We can estimate the vent radius r v implementing the regressive power law for circular vent shape, as in the case of Etna, derived from [START_REF] Wilson | Ascent and eruption of basaltic magma on the Earth and Moon[END_REF], correlating this quantity with the MCA Q M . In the relation r v = cQ d M , the variables c (0.01709, 0.007967, 0.01149) and d (0.4331, 0.4902, 0.4819) are the regressive coefficients, varying as a function of the exsolved water content w t (0.75%, 0.20%, and 0.06%, respectively) [START_REF] Wilson | Ascent and eruption of basaltic magma on the Earth and Moon[END_REF] (see Fig. 6). The regressive correlation is computed also considering a fissure with length L and the same exsolved water content. Taking from the MCA, a Q M of 10 6 kg/s, the r v can be assumed to be around 10 m [START_REF] Sparks | Volcanic Plumes, 1st ed[END_REF], [START_REF] Wilson | Ascent and eruption of basaltic magma on the Earth and Moon[END_REF]. This radius estimate is confirmed by analyzing the thermal-infrared images measuring the average size of the detected vertical column [START_REF] Marzano | Tephra mass eruption rate from ground-based X-band and L-band microwave radars during the November 23, 2013 Etna paroxysm[END_REF]. The uncertainty ε Q

(SFA)
is equal to 31.6% as previously described in (5) in [START_REF] Marzano | Tephra mass eruption rate from ground-based X-band and L-band microwave radars during the November 23, 2013 Etna paroxysm[END_REF] and shown in Table V, while the assumptions relating to the mixture density are reported in the next paragraph. It is worth noting that, once r v is estimated, the gas-tephra mixture density ρ x can be retrieved starting from the Q M (MCA) independent retrievals derived from the MCA and the estimations of v ex . The mixture density ρ x is much greater than the ambient air when the gas-mass fraction is small, but decreases rapidly during entrainment, as the gas-mass fraction increases. At high temperatures, the gas is heated by the solids, expands, and has a very low density. As a result, even with dense solid particles, the mixture density becomes substantially less than the environment. As air is added, the mixture density eventually attains a minimum.

As more air is added, the density tends toward the original density of the surrounding atmosphere, since the mass fraction of ash present is so small that its effect becomes negligible [START_REF] Sparks | Volcanic Plumes, 1st ed[END_REF]. Generally, the density of a well-mixed parcel of hot tephra, as magmatic gas and air, varies according to the temperature and particle fraction of pyroclasts in the mixture [START_REF] Corradini | A multi-sensor approach for volcanic ash cloud retrieval and eruption characterization: The 23 November 2013 Etna Lava fountain[END_REF], [START_REF] Dubosclard | Doppler radar sounding of volcanic eruption dynamics at Mount Etna[END_REF], [START_REF] Morton | Turbulent gravitational convection from maintained and instantaneous sources[END_REF]. As reported in [START_REF] Marzano | Tephra mass eruption rate from ground-based X-band and L-band microwave radars during the November 23, 2013 Etna paroxysm[END_REF], lava fountains are characterized by a range of gas content between 2% and 3% [START_REF] Carbone | Balancing bulk gas accumulation and gas output before and during lava fountaining episodes at Mt. Etna nature 2015[END_REF], [START_REF] Borisova | Processes controlling the 2010 Eyjafjallajökull explosive eruption[END_REF], [START_REF] Dürig | Earth, mass eruption rates in pulsating eruptions estimated from video analysis of the gas thrust-buoyancy transition-A case study of the 2010 eruption of Eyjafjallajökull, Iceland[END_REF], a gas density of 0.15 kg/m 3 , and a magma density of 2700 kg/m 3 [START_REF] Bonadonna | Tephra sedimentation during the 2010 eyjafjallajökull eruption (Iceland) from deposit radar and satellite observations[END_REF], resulting in a gas-tephra mixture density ρ x in the rising part of the conduit between 5-15 kg/m 3 [START_REF] Parfitt | A discussion of the mechanisms of explosive basaltic eruptions[END_REF], [START_REF] Ripepe | Ash-plume dynamics and eruption source parameters by infrasound and thermal imagery: The 2010 eyjafjallajökull eruption[END_REF]. An estimate of ρ x for the events considered herein is about 7.5 kg/m 3 . Indeed, we can retrieve ρ x inverting (4), knowing

Q M (MCA) , r v, and v ex ρx = Q (MCA) M 1 πr v 2 v ex . ( 5
)
The tephra exit velocity v ex in ( 5) is sonic and nearly independent of the crater pressure, but the crater pressure and hence the mixture density is closely coupled to the mass eruption rate and thus to the vent radius [START_REF] Sparks | Volcanic Plumes, 1st ed[END_REF]. In this study, we consider v ex derived from each sensor. 1) SFA based on the XWR v ex retrievals. Applying the methodology discussed in [START_REF] Mereu | Multisensor characterization of the incandescent jet region of lava fountain-fed tephra plumes[END_REF], which combines the opposite trend between radar reflectivity and correlation coefficient, we derive the H IJR . The tephra exit velocity v ex is then derived inverting the Bernoulli relation [START_REF] Marzano | Tephra mass eruption rate from ground-based X-band and L-band microwave radars during the November 23, 2013 Etna paroxysm[END_REF], [START_REF] Vulpiani | Mass discharge rate retrieval combining weather radar and thermal camera observations[END_REF]. 2) SFA based on VDR v ex retrievals. The VDR provides the maximum positive Doppler velocity V + max (t), that is geometrically correlated to the ejection velocities v ex assuming vertical jets [START_REF] Scharff | Pulsed Vulcanian explosions: A characterization of eruption dynamics using Doppler radar[END_REF], [START_REF] Freret-Lorgeril | Mass eruption rates of tephra plumes during the 2011-2015 lava fountain paroxysms at Mt. Etna from Doppler radar retrievals[END_REF], [START_REF] Dubosclard | First testing of a volcano Doppler radar (Voldorad) at Mount Etna, Italy[END_REF], [START_REF] Donnadieu | Volcanological applications of doppler radars: A review and examples from a transportable pulse radar in L-band[END_REF], [START_REF] Donnadieu | Near-source Doppler radar monitoring of tephra plumes at Etna[END_REF]. 3) SFA based on TIC v ex retrievals. Applying a temperature gradient algorithm on TIC images, as discussed in [START_REF] Marzano | Tephra mass eruption rate from ground-based X-band and L-band microwave radars during the November 23, 2013 Etna paroxysm[END_REF] and [START_REF] Mereu | Multisensor characterization of the incandescent jet region of lava fountain-fed tephra plumes[END_REF], we identify the H IJR used to derive the v ex applying the Bernoulli equation. A possible limit could be due only to the reduced TIC visibility due to the superimposition of meteorological clouds on the scene under observation. The uncertainty correlated to gas-tephra mixture density is given by:

ε ρ = δQ M dif Q M dif 2 + δQ M adv Q M adv 2 + 1 2
δr v r v 2 + δv ex v ex 2 (6)
where we assume the same values for each independent parameter and δv ex = 0.20v ex , the relative percentage error ε ρ is equal to 24%.

E. Time-Integrated Eruption Mass and Volume

We can introduce a quantity complementary to Q M , named volume eruption rate and indicated by Q V , due to pyroclastic material moving through surface per unit time, that can be retrieved if H TP derived by XWR is known, applying the Mastin et al. relation [START_REF] Mastin | A multidisciplinary effort to assign realistic source parameters to models of volcanic ash-cloud transport anddispersion during eruptions[END_REF]

Q V = aH TP
b with the coefficients a and b equal to 0.4786 and 3.512, respectively. It is possible to convert the estimates of

Q M into Q V through the relation Q V = Q M /ρ m , assuming a magma density ρ m of 2700 kg/m 3 .
Integrating Q M and Q V in the whole eruption time range Δt, we obtain the time-integrated eruption mass (TIEM) and the time-integrated eruption volume TIEV. The last value of TIEM and TIEV at the end of the eruption event represents the total erupted mass (TEM) and total erupted volume (TEV) of the eruptive event.

IV. MASS ERUPTION RATE RETRIEVALS

In this section we present and analyze the Q M estimates, derived by processing the XWR, VDR, and TIC measurements.

To compare retrieval methodologies and evaluate their differences for the eruptive events on 3-5 December 2015, we plot Q M and Q V in Fig. 7. In panels (a1)-(a4), we show the Q M and Q V estimates as a function of time t, derived from: (i) TPA methods, i.e., using the top-plume altitude H TP , retrieved from the XWR within the DB12 model; ii) MCA method, and iii) NSA method, both derived from the XWR measurements. The retrievals, obtained using only the SFA method with ρ x = 7.5 kg/m 3 , are shown in panels (b1)-(b4). They use the v ex retrieved from the TIC and XWR (inverting the Torricelli relation) and v ex directly derived from the VDR. Each value is sampled every 10 min, which is the minimum common sampling time imposed by the XWR operational schedule. The Q M and Q V retrievals are plotted with colored solid lines, whereas the uncertainties are expressed as colored areas around each method and limited by dashed lines in the same color, following the relation

Q M ± (ε Q Q M )
, where ε Q is expressed as decimal value between 0 and 1.

For the event of 3 December 2015, the lower Q M and Q V climax values of about 2.010 6 kg/s and 0.810 3 m 3 /s, which are the values relating the transition to stable fountains [START_REF] Freret-Lorgeril | Mass eruption rates of tephra plumes during the 2011-2015 lava fountain paroxysms at Mt. Etna from Doppler radar retrievals[END_REF], are estimated with the NSA. Because v ex is similar among the three sensors, the temporal trends of the 3 SFA estimates are similar between 02:00 UTC and 03:30 UTC (2-1.810 6 kg/s during climax). The Q M (Q V ) retrievals derived from the TPA and MCA methods during the climax vary between 2.5 and 4.210 6 kg/s (1.0 -1.610 3 m 3 /s), in agreement with the independent climax Q M estimate of 2.710 6 kg/s obtained by combining VDR echo power and velocities [START_REF] Freret-Lorgeril | Mass eruption rates of tephra plumes during the 2011-2015 lava fountain paroxysms at Mt. Etna from Doppler radar retrievals[END_REF]. The uncertainties, plotted as colored areas in Fig. 7, are minimal among the SFA methods (about 0.510 6 kg/s). Instead, the uncertainties of methods completely based on the XWR show some differences, about ± 1.010 6 kg/s for the MCA and ± 0.510 6 kg/s for the NSA; in particular, the NSA method tends to underestimate Q M and associated uncertainty due to the uncertainty in the estimated area A NS .

For the paroxysm of 4 December 2015, during the climax (09:10 -10:00 UTC), the trends of Q M and Q V range between 0.5 and 1.910 6 kg/s (0.3-0.710 3 m 3 /s) for the MCA and TPA methods and come closer near the peak in Q M at 09:20 (1.510 6 kg/s versus 1.910 6 kg/s, respectively). The NSA shows a time shift with a higher peak at 09:30 UTC of 2.310 6 kg/s [see Fig. 7(a2)]. The SFA approaches vary from 0.7 to 1.210 6 kg/s (0.2 -0.510 3 m 3 /s). In this case, the estimate using the velocity derived from VDR seems lower than others, probably due to a reduced v ex retrieval for a pointing that does not perfectly intercept the region above the eruptive crater.

The eruptive event of the evening of 4 December 2015 starts from 20:10 to 21:40 UTC. For this event, the Q M and Q V peak retrievals range between 1.210 6 kg/s and 2.210 6 kg/s (0.410 3 and 0.810 3 m 3 /s) for the MCA, TPA, and NSA methods. Once again, the NSA estimate is delayed with a peak value at 21:00 UTC, 20 min after the MCA and TPA. Regarding the SFA methods, the Q M and Q V values range between 0.610 6 and 1.810 6 kg/s (0.210 3 m 3 /s -0.610 3 m 3 /s). Regarding TIC, the presence of meteorological clouds at the summit between 20:20 and 20:50 UTC prevented the continuous determination of Q M and Q V with only one peak value at 20:40 (1.210 6 kg/s and 0.510 3 Fig. 7. Evolution of Q M (kg/s) and Q V (m 3 /s), axis on the right and on the left of each panel, respectively, and the relative uncertainties identified by the colored area bounded by dashed lines around each estimate (solid line) on (a1, b1) 3 December 2015, (a2, b2) 4 December 2015, (a3, b3) 4 December 2015, and (a4, b4) 5 December 2015. The left panels indicate the Q M and Q V retrievals, sampled every 10 min (and then linearly interpolated), as a function of time t, derived from: TPA methods, i.e., using the top-plume altitude H TP retrieved from the XWR within the parametric relationships DB12 (setting a wind velocity of 7 m/s); MCA and NSA methods, both derived from the XWR elaboration. On the right panels, the retrievals are deduced from SFA methods using the exit velocity obtained from TIC, XWR, and VDR, respectively. m 3 /s, respectively). The SFA-VDR highlights a trend with lower values than TIC and XWR between 20:10 and 21:40 UTC.

Finally, for the event of 5 December 2015, the paroxysm lasts from 14:20 to 17:00 UTC. In this case, the TIC estimates are restricted to 14:20-15:10 UTC. During the climax, the Q M and Q V range between 0.7 and 1.310 6 kg/s (0.210 3 -0.510 3 m 3 /s) for the SFA approaches. For the SFA-XWR and SFA-VDR, the time range of Q M and Q V begins around 14:10 UTC and ends at 16:40 UTC. For the MCA we observe a lower Q M (Q V ) estimation with a maximum of 0.710 6 kg/s (about 0.210 3 m 3 /s) at 15:00 UTC. The TPA method shows greater Q M and Q V values with respect to the MCA and NSA, with a peak at 15:00 UTC. The NSA peak (1.210 6 kg/s, i.e., 0.410 3 m 3 /s) is again delayed by about 20 min. In comparison with previous cases, we observe a lower agreement among the different trends. In particular, the SFA-TIC shows a trend that differs more from the others due to the limited ability of the sensor to observe the same eruptive event under cloud cover conditions. The SFA-XWR shows a greater trend in terms both of estimated values and of time interval compared to the other methods. This is related to the specific sensitivity of the sensor in observing the eruptive event in its temporal evolution.

In all cases previously described, the maximum Q M and Q V estimates are derived from the TPA (DB12) for 3 and 5 December, whereas for both events on 4 December the maximum is obtained with the NSA. The TPA method depends on both meteorological data (pressure, wind velocity, etc.) and mainly H TP estimates derived from the XWR. Generally, when the visibility of each sensor is not degraded, a good agreement of the respective MER is noted. The Q M peak values are generally in the same time instants for three employed sensors, except for the NSA, which shows a time delay in estimating the peak of the event related to the variation in time of the surface area crossed by the mass flow near the crater. The different trend of Q M and Q V retrievals, observed on 5 December 2015, is confirmed by the analysis of the thermodynamic variables derived from the radio sounding in Trapani, where a decrease of both temperature and atmospheric pressure is observed at an altitude of about 4 km asl event, again due to the flow area.

To analyze the correlation among Q M estimates, we show in Fig. 8 the dispersion of the Q M retrievals for the four eruptive events of Etna: Q M from the NSA, TPA, and SFA (TIC, VDR, and XWR) methods are reported as a function of the Q M from the MCA method. The NSA method shows a smaller dynamic, whereas the TPA method covers a larger set of Q M values, although with wider scattering for low Q M . Although equally distributed around the 1:1 line, the SFA methods display a more asymmetrical distribution with respect to the MCA. This is since v ex does not vary much among paroxysms; in fact, we have a larger dynamic in the parameters that are related to the quantity of detected tephra (echo power, radar reflectivity factor, etc.) hence, unlike the MCA, SFA values are less varying among all paroxysms [START_REF] Freret-Lorgeril | Examples of multi-sensor determination of eruptive source parameters of explosive events at Mount Etna[END_REF].

Note the particularly good agreement of the SFA-VDR with the MCA-XWR for Q M above 610 4 kg/s.

As mentioned in Section III-C, the knowledge of MCA Q M allows us to compute the eruptive crater radius r v , through a power regressive law, and the gas-tephra mixture density ρ x . Fig. 9 shows the time trend of ρ x as a function of time.

The red area bounded by the red dashed line identifies the value computed combining ( 5) and ( 6), that is ρ x ± (ε ρ ρ x ) , where ε ρ is expressed as decimal value between 0 and 1. The dashed red lines define the variability of ρ x derived from the MCA density (green dashed line). Important variations, up to 25 kg/m 3 during purely volcanic signals, occur as a function of eruption time both for the TPA-derived and MCA-derived ρ x , with larger averaged values for the former. The constant mixture density value, computed as the average of the mixture density derived by the MCA and TPA, is used in the SFA methods in Figs. 7 and8. Across all four paroxysms, this averaged value ranges between 4 and 12 kg/m 3 , as a function of particle fraction f N previously fixed [START_REF] Marzano | Tephra mass eruption rate from ground-based X-band and L-band microwave radars during the November 23, 2013 Etna paroxysm[END_REF], [START_REF] Trolese | The footprint of column collapse regimes on pyroclastic flow temperatures and plume heights[END_REF], [START_REF] Ongaro | Modelling pyroclastic density currents from a subplinian eruption at la soufrière de Guadeloupe (West Indies, France)[END_REF]. We observe higher values of ρ x during the last event when meteorological clouds covered Etna's summit area (especially after 15:30 UTC), as confirmed by the radio sounding and TIC images. In this case ρ x represents the combined contributions of volcanic particles and hydrometeors.

V. SUMMARY AND COMPARISON OF

Q M AND Q V RESULTS
We summarize the results derived from each retrieval approach, comparing them with data available in literature, to highlight congruencies and identify the potential of the various approaches.

Tables II andIII show the time-averaged value of Q M (kg/s) and Q V (m 3 /s) and the TEM (kg) and TEV (m 3 ), respectively, for Etna's paroxysms on 3-5 December 2015.

In the table, row a shows the values derived from the TPA model (DB12), whereas rows b-d refer to SFA using the video TIC, XWR, and VDR, respectively. Rows e and f show the results for the MCA and NSA retrievals, whereas the rows g, h, and i show the values derived from literature [START_REF] Vulpiani | Mass discharge rate retrieval combining weather radar and thermal camera observations[END_REF], [START_REF] Freret-Lorgeril | Mass eruption rates of tephra plumes during the 2011-2015 lava fountain paroxysms at Mt. Etna from Doppler radar retrievals[END_REF].

For 3 December 2015, Q M and Q V time-averaged values in [START_REF] Vulpiani | Mass discharge rate retrieval combining weather radar and thermal camera observations[END_REF] and [START_REF] Freret-Lorgeril | Mass eruption rates of tephra plumes during the 2011-2015 lava fountain paroxysms at Mt. Etna from Doppler radar retrievals[END_REF] and MCA-XWR values are larger than other estimates in this study, up to a factor 2 to 3 with respect to SFA methods (extrema: 4.510 6 kg/s for SFA-VDR, 14.510 6 kg/s in [START_REF] Freret-Lorgeril | Mass eruption rates of tephra plumes during the 2011-2015 lava fountain paroxysms at Mt. Etna from Doppler radar retrievals[END_REF]). For both eruptions of the 4 December, the highest estimates are obtained with the TPA. For the 5 December, the approach that shows the largest time-averaged Q M value is the SFA-XWR. Generally, the TPA model, when it has both reliable environmental data and measurements of the wind speed in function of the altitudes, shows time-averaged Q M values that concur with the other estimates. In between, the SFA-XWR estimates retrieved from real-time XWR data show generally medium-high values between 3.910 5 and 610 5 kg/s. The MCA method, if the advection term is neglected, tends to show lower time-averaged Q M and Q V . The NSA method provides estimates of time-averaged Q M between MCA and TPA ones.

It is noteworthy that the differences with the time-averaged estimates in [START_REF] Vulpiani | Mass discharge rate retrieval combining weather radar and thermal camera observations[END_REF] remain unchanged in each case study analyzed independently from the considered approach. This is also the case for the TEM and TEV shown in Table III. On 3 December the largest values are noted with the TPA, SFA-TIC, and SFA-XWR techniques; on 4 December morning and evening with the TPA and SFA-XWR. Again, the TPA model displays the largest values with respect to other approaches, whereas on 5 December the larger value is with SFA-XWR.

In Table IV we listed the four Q M approaches with their respective equations and extrapolated uncertainties. The MCA method is the one with the lowest overall uncertainty of about 22%, followed by the NSA method (26%), TPA (30%), and finally SFA (32%). The MCA method is more self-consistent from the radar point of view, as it is less dependent on parameters derived from other sensors and/or geophysical hypotheses formulated a priori. Mainly for this reason, the uncertainty associated with the MCA Q M estimation is lower with respect to other Q M estimates. The correlation matrix related to the Q M retrievals is shown in Table V, where the coefficient values are always positive and the correlation between pairs of Q M retrievals is highlighted by colored boxes. The highest correlation is observed between the SFA-VDR and both the TPA (0.94) and MCA (0.92), the lowest values are for the NSA and SFA-TIC (∼0.4). Among the SFA approaches, the SFA-VDR shows the best correlation with the MCA and TPA methods, both trained by input parameters derived from the XWR, whereas the NSA has the lowest correlation. The TPA method is more correlated to the MCA and NSA methods, as well as the SFA-XWR and SFA-VDR.

VI. CONCLUSION

In this work, we analyzed the Etna paroxysms on 3-5 December 2015 to investigate and optimize the different strategies for Q M and Q V estimation and the related TEM and TEV as well as their uncertainties. Starting from the three methodologies already known in the literature (MCA, TPA, and SFA), a new NSA approach was introduced and tested. For each method, advantages and limits were also discussed. A comparison between the various retrieval approaches, in terms of Q M , Q V , TEM, and TEV, and corresponding data available in the literature was carried out to highlight similarities and differences.

Using ground in situ measurements, the analysis of this event showed that the MCA method has the lowest uncertainty of 22.3%, whereas other approaches exhibited a higher uncertainty (i.e., 26.4% for NSA, 30% for TPA, and 31.6% for SFA). If weather radar observations are not available, the other approaches are valuable to derive a first reliable MER. The TPA method generally showed the largest MER retrieved values because it is particularly sensitive to the H TP estimates that are influenced by the cross wind variability and consequent plume bending. The SFA-based results were consistent with each other, but significantly dependent on the estimate of the tephra exit velocity. Regarding the NSA method, entirely based on XWR, a more accurate estimate of the tephra flow crossing area just above the crater should improve its overall accuracy in the MER retrieval.

The MER retrieval methods, described and applied in this work, showed promising results that can be exploited to improve the tephra dispersal and fallout forecast at Etna in near real time. The potential of the remote sensors in operating both individually and in synergy with the other instruments should be further investigated using other case studies, where all sensors are contemporarily available. The high correlation of MER retrievals between XWR and VDR is particularly promising since it confirms the potential of XWR as scanning operational systems for volcanic eruption monitoring and near-source parameter estimation. Further work might be devoted to explore new techniques, using low-cost sensors for MER estimation and employing microwave radars as validation tools.

Fig. 1 .

 1 Fig. 1. (a) Map of the main craters of Etna volcano: Voragine Crater (VOR), North-East Crater (NEC), New Southeast Crater (NSEC), Southeast Crater (SEC). (b) Map of the sensor network used in this work is shown. Each sensor and crater are geolocated in terms of latitude and longitude and highlighted with colored dots, specifying the elevation above sea level (asl), as listed in the legend. The colored dashed lines show the distance between each sensor and the Etna summit, as clarified for each one in km.

Fig. 2 .

 2 Fig. 2. Frames of the thermal infrared camera in Nicolosi (named ENT) showing the lava fountains at the VOR on 3-5 December 2015, panels (a), (b), (c), and (d), respectively. Below each image, the date (dd-mm-yyyy) and UTC time (hh:mm:ss:00) of the Etna eruptive events. The color bar on the right side of each picture shows the uncalibrated temperature ranging between -10°C and 70°C.

Fig. 3 .

 3 Fig. 3. X-band radar data of Etna explosive events on (a1, b1) 3 December 2015 at 03:00 UTC; (a2, b2) 4 December 2015 at 09:30 UTC; (a3, b3) 4 December 2015 at 21:00 UTC; and (a4, b4) 5 December 2015 at 15:30 UTC. We can observe the PPI of copolar radar reflectivity factor Z hh (dBZ) at 6°elevation radar (panels a1-a4) and the RHI of the copolar radar reflectivity factor Z hh (dBZ) of XWR (panels b1-b4).

Fig. 4 .

 4 Fig. 4. Radar reflectivity factor time series of the four Etna explosive events on (a) 3 December 2015; (b) 4 December 2015; (c) 4 December 2015 and; (d) 5 December 2015. Each curve is related to the bin volume near the summit crater, that is the fifth elevation angle for the XWR and eighth and ninth range bins for the VDR, considering that the latter points directly near to the volcanic source of VOR. The reflectivity factor Z hh (dBZ) derived from the XWR is sampled every 10 min (red line) and from the VDR directly every 30 s (green line) or linearly interpolated every 10 min (doted blue line).

Fig. 5 .

 5 Fig. 5. Flow diagram of extended VARR algorithm, which starting from the sensors data, allows to implement the four methodologies for the Q M estimation (i.e., MER).

  TABLE II AVERAGED VALUES OF Q M (kg/s) AND Q V (m 3 /s) OVER THE ENTIRE DURATION OF ETNA'S PAROXYSMS ON 3-5 DECEMBER 2015Note: (a) Applying the TPA parametric models DB12; (b, c, and d) using the SFA method ingesting the XWR, VDR, and TIC data; (e) implementing the MCA, (f) NSA approaches, and (g, h, and i) derived from literature. The Case i) is related to climax for each event on 3-5 December 2015, that is 02:32-03:12 UTC, 09:07-09:30 UTC, 20:36-20:50 UTC, and 14:54-15:25 UTC.

Fig. 6 .

 6 Fig. 6. Variation of the circular vent radius r v with Q M and surface fissure width w f with Q M /L [mass eruption rate per unit length of fissure for an elongated vent applying the Wilson and Head (1981) relation as a function of the exsolved water content wt of 0.75%, 0.20%, and 0.06% (red, blue, and green; and cyan, magenta, and yellow solid and dashed lines, respectively].

Fig. 8 .

 8 Fig. 8. Plot showing on the x-axis Q M derived from the MCA method and on the y-axis Q M retrieved from the NSA, TPA, and the three SFA methods, identified by different colored dots for the four Etna lava fountain events.

Fig. 9 .

 9 Fig. 9. Time trend of gas and tephra mixture density: the red area identifies the mixture density uncertainty computed according to the relation ρ x ± ε ρ , whereas the dashed green and blue lines are related to ρ x using Q M derived from MCA and TPA methods, respectively. The fixed value, plotted with a dashed dark line is obtained by averaging both estimates. Each picture is related to (a) 3 December 2015, (b) 4 December 2015 (06:30-11:00 UTC), (c) 4 December 2015 (20:00-22:00 UTC), and (d) 5 December 2015.

TABLE III SAME AS

 IIIAS

  TABLE II BUT SHOWING THE TEM AND TEV, RESPECTIVELY, IN (kg) AND (m 3 )

TABLE IV SUMMARY

 IV TABLE OF QM METHODS WITH RELATIVE EQUATION AND ESTIMATED UNCERTAINTY TABLE V CORRELATION COEFFICIENTS MATRIX COMPUTED AMONG THE

ACKNOWLEDGMENT

The authors would like to thank for the operation of VOLDORAD-2B radar of OPGC on Etna in close collaboration with INGV-OE (M. Coltelli, M. Prestifilippo). VOLDORAD-2B processed data are made available by OPGC in the framework of EPOS with support of EUROVOLC. We also thank the Italian Department of Civil Protection (DPC, Italy) for the provision of the X-band radar data.

This work was supported in part by the "European Network of Observatories and Research Infrastructures for Volcanology" under European Projects EUROVOLC H2020-INFRAIA-02-2017 and 731070, and in part by the funding from the European Union's Horizon 2020 Research and Innovation Programme under Grant Agreement 731070 (EUROVOLC).

Luigi Mereu received the M.Sc. degree in telecommunication engineering and the Ph.D. degree in remote sensing from the Sapienza University of Rome, Rome, Italy, in 2012 and 2016, respectively.

In 2012, he joined the Department of Information Engineering, Sapienza University of Rome, Roma, Italy, and the Centre of Excellence CETEMPS, L'Aquila to cooperate on radar remote sensing of volcanic ash clouds within the ICT Ph.D. She was then awarded the position of Young Investigator at the University of Hawaii, and later appointed an Assistant Professor with the University of South Florida, Tampa, FL, USA. She is currently a Full Professor with the Department of Earth Sciences, University of Geneva, Geneva, Switzerland, where she is also the Vice-Dean of the Faculty of Sciences. Her research interests include modeling sedimentation from volcanic plumes, exploring new methodologies for the characterization of tephra-fallout deposits, and developing probabilistic analysis for the assessment of volcanic hazards and risks.

Prof. Bonadonna was a recipient of the President's Award of the Geological Society of London (2001), the IAVCEI Outstanding Recent Graduate (2004), the Outstanding Woman in Science Award of the Geological Society of America (2004), the USF Outstanding Faculty Research Achievement Award (2005), and the International Galileo Galilei Award of the Italian Rotary Clubs for Sciences (2020). In addition, she is currently the president-elect of the Volcanology, Geochemistry, and Petrology section of the American Geophysical Union and a member of the Academia Europaea.