Relationships between frequency and intensity of volcanic eruptions are actively sought by geophysicists for both monitoring and research purposes. By analyzing surveillance videos of persistent volcanic activity at Stromboli (Italy), we derived the frequency and jet height of >4000 explosions that occurred in 72 h-long time windows sampled yearly from 2005 to 2009. We found a positive relationship linking explosion frequency and jet height (linked to eruption intensity) when averaging the two parameters over time intervals from hours to days, with a stronger correlation for longer intervals. We interpret this behavior as the response of the magmatic system to variable influx of magma and gas at depth, increased flux at depth causing more frequent and stronger explosions at the surface. This relationship entails concurrent control of source processes over explosion frequency and intensity, directly impacting modeling of explosion sources at persistently active volcanoes in general and hazard assessment at Stromboli in particular.
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 Discrete, jet-like Strombolian explosions characterize persistent, long-lived activity at several mafic volcanoes, including Stromboli (Italy), Erebus (Antarctica), Yasur (Vanuatu), Kīlauea (USA), and Nyiragongo (Democratic Republic of the Congo). These explosions are commonly related to the recurrent formation, ascent, and bursting of conduit-filling gas pockets (slugs) [e.g., Chouet et al., 1974; Blackburn et al., 1976; James et al., 2009; Chouet et al., 2010], originated either by bubble accumulation at geometrical discontinuities of the plumbing system [Jaupart and Vergniolle, 1988] or by bubble capture via differential rise velocity in the conduit [Parfitt and Wilson, 1995]. The persistent and relatively mild nature of this kind of activity offers unique chances to investigate the long-term evolution and the control of deep plumbing system processes on their surface manifestations.
 Here we use infrared image time series of volcanic activity at Stromboli to assess the link between frequency and intensity of Strombolian explosions, with implications for their source processes and hazard mitigation. The current activity at Stromboli, persisting since at least 1.3 ka [Rosi et al., 2000], is characterized by continuous degassing and Strombolian explosions from multiple vents on a crater terrace at 800 m above sea level. Typically, every few minutes, explosions eject pyroclasts 100–200 m above the different vents over seconds to tens of seconds [e.g., Ripepe et al., 2009; Taddeucci et al., 2012]. Single vents show highly variable activity behavior (e.g., spatter fountaining, ash, and gas emission) over relatively short time scales (hours to weeks), also reflecting changing vent geometry, size, opening/obstruction, and location within the crater terrace. Fluctuations at yearly time scale appear in time series of (1) Strombolian explosion frequency [Andronico et al., 2008], (2) deep gas influx [Carapezza et al., 2004], (3) seismic activity [Falsaperla et al., 1998], and (4) thermal emission [Harris and Stevenson, 1997]. These fluctuations are accompanied by variations in the magma level in the conduit, occasional crater overflows, lava flank eruptions, and paroxysmal explosions [e.g., Spampinato et al., 2011].
 Despite this large observational body, the relationships of eruption frequency and intensity at Stromboli and their long-term variations have been scarcely investigated beyond their seismic manifestations [e.g., Bottiglieri et al., 2005]. Thus, key factors controlling the size and frequency of slug formation and related explosion parameters are still poorly defined. Here, by analyzing surveillance infrared videos sampled from several years of activity at Stromboli, we found a direct correlation of height versus frequency of explosion jets over different time scales. This correlation put constraints on the formation of gas slugs and the eruptive behavior and may provide a basis for rapid assessment of the overall state of the volcanic activity and impending hazards.
 We analyzed videos from one of the continuously operating surveillance cameras of the Istituto Nazionale di Geofisica e Vulcanologia (INGV; Osservatorio Etneo) network at Stromboli. In particular, we used images collected every 2 s by an infrared (8–14 µm band) camera (OPGAL EYE-M320B, resolution 320 × 240 pixel), located at Pizzo sopra La Fossa, about 100 m above and 240 m SE of the crater terrace, resulting in a pixel size of 0.87 × 0.87 m (Figure 1).
 From the videos, we extracted those frames capturing explosive activity at any vent, marking the UTC (Coordinated Universal Time) time of the first frame of each explosion. Then for each explosion, we selected a single representative frame, in which the erupted gas-pyroclast jet attains its maximum apparent height before getting dispersed (Figure 1). Each jet (represented by a continuous bright area) was isolated to remove other bright features (i.e., vent areas and hot pyroclasts from previous explosions). Each representative frame was analyzed through the ImageJ freeware software [Abramoff et al., 2004], which, after thresholding the image grey tones to a fixed value, fits the jet area with an ellipse with the long axis equal to the jet height (Figure 1). Measured height was then corrected for perspective by considering the angle between the camera viewing direction and the average jet direction, assumed as vertical, as confirmed by INGV surveillance videos shot perpendicularly to the viewing direction. Each explosion is thus characterized by two parameters: jet height (h) and time of occurrence, from which inter-explosion time (t) and explosion frequency (1/t) are derived.
 Balancing available data, the need for broad time coverage, and a representative number of observations, we sampled 72 consecutive hours of video with good crater visibility from five consecutive years from 2005 to 2009, totaling 4275 explosions (counting a single event with cumulated h in case of simultaneous explosions from multiple vents) as follows: 503 from 29 April to 1 May 2005, 1409 on 11–13 January 2006, 591 on 20–22 September 2007, 922 on 7–9 July 2008, and 850 on 19–21 July 2009.
 In the analyzed periods, explosive activity occurred simultaneously at several (four to six) active vents, t and h ranging <2–2650 s and 8–138 m, respectively, without any systematic difference of h for different vents. In search for a general relationship, our analysis below focuses on h and explosion frequency irrespective of the source vent.
 Both t and h distributions are best fitted by a Weibull distribution (R2 always >0.99):
with parameters λ and κ ranging 174–580 and 0.87–1.20 for t and 47.3–61.1 and 2.49–5.07 for h, respectively (Figure 2). As a whole, although no simple t-h relationship is observed on a one-to-one basis, higher h values seem to characterize periods with shorter inter-explosion times (i.e., with more frequent explosions; Figure 2). We test the hypothesis of a general t-h correlation by considering the mean and cumulative h values (hereafter hmean and hcum, respectively) of all the explosions occurred within 1, 24, and 72 consecutive hours (Figure 3). It appears that a positive relationship links the total number of explosions to their hmean and hcum (i.e., hmean and hcum increase with decreasing average t), with best fits provided by a linear and a power law regression for hmean and hcum, respectively. The correlation coefficient is higher for hcum, and for both hmean and hcum, it increases with the duration of the averaging period. An analogous but less marked relationship holds for explosions at individual vents and when considering explosion frequency versus the jet area or volume at all vents (the latter calculated assuming either a cylindrical or an ellipsoidal jet of height h and width equal to the maximum width of the fitting ellipse). Beyond the presence of multiple, even simultaneously active vents and the observed variability of h and t at all time scales, the above results clearly show that more frequent explosions produce, on average, higher jets.
4.1 Significance of the Parameter h
 The parameter h is not representative of the maximum height reached by pyroclasts in a single explosion; individual bombs often outrun the jet front to greater heights (Figure 1). In fact, h characterizes a gas-pyroclast mixture hot and dense enough to appear as a well-defined entity to the camera sensor. The eruptive mixture is accelerated out of the vent by expanding gas as a well-defined, coupled gas-pyroclast jet as long as gas thrust dominates over individual particle momentum and air entrainment. The maximum extent of the jet thrust phase is a function of the initial temperature, pressure, and mass (hence volume and density) of the mixture and of the concentration and grain size distribution of pyroclasts. Theoretical analysis of Strombolian explosions points out a positive, nonlinear correlation between volume and pressure of the gas driving the explosions [James et al., 2009; Del Bello et al., 2012]. On these grounds, we consider h as a first-order proxy for both relative intensity (related to the peak mass eruption rate, function of pressure) and magnitude (related to the total erupted mass) of explosions.
 Besides the intrinsic h variability that actually reflects changing erupting mixtures [Patrick et al., 2007] and vent/conduit geometries over time (at a single vent) and space (at different vents), the measured h values may also subordinately depend on the 2 s inter-frame time acquisition (e.g., the maximum jet height might be reached during the time lapse between two consecutive frames) and possible changes in environmental (weather, degassing) conditions that reflect in the jet-ambient contrast in the frames. Despite these uncertainties, it appears that when averaged over tens to hundreds of measurements, h is a robust parameter that highlights a strong link between explosion frequency and relative intensity. In particular, the plot of hcum versus number of explosions links two basic parameters representative of Strombolian activity, beyond any specific distribution function of h, and irrespective of the chosen cumulative interval.
4.2 Implications for Explosion Source and Hazard Assessment
 Given the lack of a systematic difference in h at different vents, we infer the intensity and frequency of Strombolian explosions to reflect, rather than shallow conduit conditions, the size and release frequency of pressurized gas pockets (slugs) at depth. Our findings (Figure 3) imply that at Stromboli, when slugs form more frequently, they also tend to incorporate greater masses of gas, thus feeding stronger explosions. Gas slugs in low-viscosity magmas result by the coalescence of smaller bubbles due to (1) bubble accumulation either at geometrical discontinuities in the plumbing system [Jaupart and Vergniolle, 1988] or below cooled magma plugs at the conduit top and (2) size-related differential velocities of bubbles rising through an almost stagnant magma column, leading to a transition from bubbly to slug flow [e.g., Parfitt and Wilson, 1995]. In this regard, James et al.  showed that inclined conduits, with respect to vertical ones, promote a shift from bubbly to slug flow, where the increase in the size of gas pockets occurs at the expense of their release frequency. Other factors that may control the size and frequency of gas slugs include the capture of bubbles released from the turbulent tail of just passed-by slugs [Llewellin et al., 2011], and conduit-scale magma convection and gas percolation [Burton et al., 2007].
 In spite of experimental and theoretical investigations of the above processes, a general, quantitative relationship linking the size and release frequency of slugs has not yet been developed. We note that if the slug release frequency is controlled by a specific threshold for, e.g., a minimum gas volume accumulation or critical gas pressure, then our frequency-intensity relationship would imply that the same threshold holds for the slug size. If slugs form by bubble capture during buoyant rise, then greater initial bubble size and higher number density at a given depth would result in increasing size and release frequency of slugs [Krepper et al., 2005], consistent with our findings. The specific distributions of h and t and their correlation provide the first field-based benchmarks against which to compare the results of any future model of slug formation at Stromboli. In particular, the values of the parameter κ of the Weibull distributions for h and t imply a strong clustering of slug sizes and a quasi-Poissonian, time-independent distribution of slug release frequencies, this last finding in agreement with previous infrared and seismic evidence [Ripepe et al., 2009, De Martino et al., 2011]. The relatively low exponent (~1.2) for the power law regression of hcum versus number of explosions (Figure 3) quantitatively expresses the dominant role of slug release frequency over slug size in modulating the total amount of gas and magma released by the volcano. In addition, our analysis shows that the process that simultaneously controls the frequency and intensity of explosions at Stromboli characterized the behavior of the volcanic system at least on a 5 year time scale and does not seem to have been affected by the intervening eruptive crisis with lava flank effusion, collapse of the crater terrace, and paroxysmal explosion that occurred in February–April 2007 [Barberi et al., 2009]. We relate this process to the input of magma and gas from the deep part of the plumbing system, with higher input rates leading to the combined formation of larger and more frequent slugs.
 At Stromboli, an increase in the input of deep magma and gas has been considered the cause for a concomitant increase of seismic tremor, amplitude of the infrasonic pressure signals, and number of explosions/day before the onset of the 2007 eruptive crisis, leading to the use of these geophysical signals as the critical parameters in setting hazard level thresholds [Ripepe et al., 2009]. We have shown that, as for the associated geophysical parameters, explosion frequency and intensity are strictly related. Thus, the observed frequency-intensity relationship sets the basis for combining different monitored quantities into a unified measure of the input of deep magma and consequent hazard level at Stromboli. Provided that the travel range of ballistic volcanic bombs can be deduced from h, automated, continuous monitoring of h and t could be used to simultaneously assess the overall state of the volcano, with respect to the likeliness of eruptive crisis occurrence, and the local hazard from ballistic bombs impending on the Pizzo sopra La Fossa area, daily hosting tens of visitors during tourist season.
 We thank E. Biale, E. Pecora, and D. Reitano for their support on the INGV-CT camera network. This work was partially funded by DPC-INGV Project V2 “Paroxysm.” G. Saccorotti, I. Lokmer, and reviewers (M. James and anonymous) provided additional insights and suggestions.
 The Editor thanks Mike James and an anonymous reviewer for their assistance in evaluating this paper.