**1. Introduction**

Approximately 10% of the Earth's population live under the direct threat of one of 1508 active volcanoes [1,2]. Understanding the nature and impact of volcanic hazards and developing monitoring systems to detect and track eruptions in (near) real time, is central to effective early warning and successful risk mitigation. The type and severity of hazards posed by volcanic activity depend on the style of eruption. Effusive eruptions often affect settlements and infrastructure such as during the recent 2018 rift eruption of Kilauea (Hawaii, USA) [3]. At the other end of the spectrum of volcanism, explosive and sustained activity can result in loss of life, such as during the 2018 eruption of Volcan de Fuego (Guatemala), and have a severe impact on local communities [4]. In general, long-lived eruptions can have significant, negative, repercussions on the economy at the national and international scale. During volcanic explosions fragments of pulverised rock, referred to as volcanic ash, are ejected from one or more active vents at high speed, driven into the atmosphere by hot gasses. It is well-known that airborne volcanic ash presents a direct threat to aviation; it can cause disruption to flight operations such as during the 2010 eruption of Eyafjallajokull (Iceland) and damage to infrastructure, with consequent loss of revenue [5].

During volcanic crises nine international Volcanic Ash Advisory Centers (VAAC), part of the International Civil Aviation Organization's (ICAO) Air Navigation Commission, are deputed to dispatch Volcanic Ash Advisories (i.e., alerts on the presence of airborne volcanic ash and estimates of its movement) to aviation authorities [6]. Numerical models are routinely used by VAACs to forecast atmospheric dispersal of ash clouds in order to evaluate the potential impacts of eruptions on aviation. The volume of erupted material, the rate at which mass is ejected from volcanic vents and the initial height of eruption plumes, collectively referred to as eruption source parameters [5], are key inputs into these models. Assessment of these parameters relies on empirical laws, which were derived from measurements of well-documented eruptions [7–10]. These empirical relationships are integrated in routine protocols to estimate mass eruption rates (MER) from measurements of ash plume height [6], despite large uncertainties when results are compared with values obtained from direct measurements of eruption deposits and durations [5]. Mastin et al. [5] discussed whether MER could be alternatively estimated by means of numerical modelling: based on the results of 1D numerical simulations, they argued that empirical equations still perform better than numerical models in reproducing field observations.

In recent times, acoustic infrasound has emerged as an increasingly popular tool for volcano remote sensing, and its potential to assess volcanic emissions in (near) real time has been extensively investigated. The term infrasound identifies atmospheric acoustic waves with frequencies typically <20 Hz, below the audible range of humans. Volcanoes are prolific radiators of infrasound, generated by large-scale eruptive processes, which inject gas and pyroclasts into the atmosphere causing its rapid acceleration [11]; these low-frequency acoustic waves can travel distances of up to several thousands of kilometres [1] lending themselves to volcano monitoring at different scales, from local to global. The use of acoustic infrasound in regional and local eruption monitoring, with applications in volcano early warning has become increasingly popular [12–16]. A substantial body of literature demonstrate the use of infrasound to detect, locate and track explosive volcanic eruptions, and its potential to provide estimates of eruption source parameters to inform volcanic plume rise and ash dispersal modelling [17–23].

In this manuscript, we provide a review of past work in the field of volcano infrasound, and describe recent developments of its use to assess eruption source parameters. Here, we focus on applications of acoustic infrasound at local distances (i.e., within 10–15 km from eruptive vents) and their potential to provide robust estimates of the amount of material ejected into the atmosphere during eruptions. We present a summary of the theory of linear acoustics and discuss equivalent models for the representation of volcano-acoustic sources. Methods derived from this theory that exploit volcano-acoustic time series to retrieve eruptive jet velocities, and invert infrasound waveforms to obtain estimates of mass and volume flow rates (of the eruptive mixture of gas and particles) are reviewed. We discuss the recent integration of numerical modelling of acoustic wavefield propagation within these workflows. We show examples from the published literature, as well as applications to new case studies. Finally, we offer a perspective on the potential for use of acoustic infrasound in real-time volcano monitoring.

#### **2. The Acoustic Fingerprint of Volcanoes**

Volcanic activity encompasses a broad spectrum of eruptive styles ranging from effusive discharge of lava onto the Earth's surface, to energetic explosions that inject large amounts of gas-laden pyroclasts in the atmosphere. Acoustic waves are produced when the atmosphere is perturbed from its background state by a moving source causing unsteady fluid motion, and pressure disturbances propagate through the air. Multiple processes occur in volcanic environments making them ideal sources of acoustic waves; these include gas-driven oscillations of the surface of lava lakes or magma columns [24], surface mass movements and impacts (rockfalls, pyroclastic flows and lahars [25]) explosive fragmentation of magma at different scales (from Strombolian to Plinian [17,20]), and vigorous ash-and-gas jetting during sustained eruptions [18]. Volcanoes produce acoustic waves that deliver energy over a broad spectrum of frequencies; however, because of the comparatively large spatial scale of source mechanisms, the majority of sound is radiated in the infrasonic band, approximately between 0.01 and 20 Hz [11].

**Figure 1.** Infrasound waveforms associated with explosions at volcanoes that exhibit regular activity. Excess pressures are reduced to a distance of 1 km from the vent for comparison across volcanoes: (**a**) small degassing explosions recorded at Etna, Italy; (**b**) small degassing explosions recorded at Santiaguito, Guatemala; (**c**,**d**) ash-rich impulsive explosions from Tunghurahua, Ecuador, and Fuego, Guatemala; (**e**,**f**) large amplitude signals exhibiting blast wave characteristics recorded at Fuego, Guatemala, and Sakurajima, Japan; (**g**,**h**) short-duration signals featuring multiple pulses from Fuego, Guatemala; (**i**) monochromatic signal recorded at Mt. Etna associated with degassing activity from one of the summit vents; abd (**j**,**k**) explosion signals with impulsive onsets followed by an extended coda from Fuego, Guatemala, and Sabancaya, Peru, associated with observations of sustained ash plumes.

Volcano infrasound waveforms range from impulsive transients (Figure 1) to tremor-like sustained signals, with a number of other intermediate types [26]. Waveforms with short duration (5–15 s) and sharp onsets (Figure 1a–f) are frequently generated by impulsive explosion-like sources. These signals may feature compression onsets followed by rarefaction with a nearly symmetrical shape (Figure 1a); this symmetry is indicative of a flow rate source time function symmetrically distributed, in time, around a peak value [27]. In many instances, waveforms are less symmetrical, exhibiting rapid compression onsets (e.g., Figure 1e) followed by rarefaction phases with reduced amplitudes (Figure 1b–f); waveform asymmetry may represent either a non-symmetrical flow rate source function, or reflect a shock-type source mechanism similar to blast waves produced by chemical explosions [27,28]. Typically, blast waves can be separated from other explosion mechanisms due to their characteristic appearance and much larger peak amplitudes, of the order of several hundreds of Pa within a few hundred metres from the source. Diversity in the characteristics of infrasound signals reflects variety in source mechanisms, and may additionally provide clues on whether explosions are gas- or tephra-rich [29]. Observational evidence suggests that waveforms featuring impulsive onsets followed by several additional pulses (Figure 1g–h), or by a prolonged coda (Figure 1j–k), are frequently associated to tephra-rich plumes with durations up to a few minutes. Open-vent volcanic systems, where active degassing is persistent, often produce nearly monochromatic waveforms [24] due to resonance taking place within the shallow conduit and crater system (Figure 1i).

Longer duration (several minutes to hours) infrasound is typically associated with sustained open-vent degassing, or continuous eruptive activity. Figure 2 illustrates three examples of extended duration waveforms associated with persistent pulse-like degassing, a sequence of intermittent small Strombolian explosions, and an episode of sustained lava fountaining. A wider variety of tremor-like signals have been reported in the literature; a comprehensive account of these signals is, however, beyond the scope of this manuscript. For additional details, the reader is referred to a number of insightful reviews published in recent years on the nature and characteristics of volcano infrasound [11,26,30,31].

**Figure 2.** Extended duration infrasound waveforms: (**a**) Twenty minutes of very regular, pulse-like, degassing recorded at Fuego, Guatemala. This type of activity is often referred to as "chugging" and accompanied by audible degassing with a noise similar to that of a steam train. (**b**) Sequence of small Strombolian explosions recorded at Mt. Etna. This activity recorded in the summer of 2008, persisted for days. (**c**) Infrasound signature of a lava fountain recorded at Etna, Italy in May 2008. This episode lasted for several hours with relatively constant infrasonic amplitudes. (**d**–**f**) Enlargements of selected portions (shaded gray) of signals shown in (**a**–**c**).

#### **3. Linear Acoustic Theory and Multipole Acoustic Sources**

Fluid flow associated with volcanic explosions causes acceleration of the atmosphere making them efficient radiators of acoustic infrasound. The propagation of density (and thus, pressure) perturbations in a fluid medium at rest can be investigated with the Navier–Stokes equations, by solving an associated wave equation [32]. Early, seminal, work by Lighthill [33] and Woulff and McGetchin [32] set out an elegant formulation for the solution of the acoustic wave equation in free- and half-space, to describe the radiation of sound from monopole, dipole, and quadrupole sources. A monopole is the simplest source of acoustic waves, linked to unsteady fluid flow (i.e., injection of mass) such as during a volcanic explosion, and it is the most efficient form of acoustic radiation. Dipole radiation results from the application of a force field at the source location, and it is obtained as the linear superposition of two adjacent monopoles oscillating out of phase by 180◦. Dipole sources, unlike monopoles, do not introduce net mass at the source; they impart momentum to the fluid causing its movement, and thus, to radiate sound. Woulff and McGetchin [32] suggested that dipole radiation may result from the interaction of fluid flow from volcanic vents with the crater walls. Quadrupole radiation represents turbulence within the gas jet itself, and it results from the combination of two opposite dipoles. Woulff and McGetchin [32], building on previous work by Lighthill [33], and using dimensional analysis, presented a suite of scaling laws that link the acoustic power radiated by monopole (Π*m*), dipole (Π*d*) and quadrupole (Π*q*) sources to gas velocity during fluid flow from volcanic vents:

$$\begin{array}{rcl} \Pi\_m &=& K\_m \frac{\rho\_p A\_v}{c} v\_v^4 \end{array} \tag{1}$$

$$
\Pi\_d \quad = \quad \mathcal{K}\_d \frac{\rho\_p A\_v}{c^3} v\_c^6 \tag{2}
$$

$$\Pi\_q \quad = \quad \mathcal{K}\_q \frac{\rho\_p A\_v}{c^5} v\_e^8 \tag{3}$$

where *Km*, *Kd* and *Kq* are dimensionless constants, *ρ<sup>p</sup>* is the density of the volcanic jet, *c* is the speed of sound in the atmosphere, *ve* is the gas exit velocity at the vent, and *Av* is the cross-sectional area of the vent.

Recently, Kim et al. [34] presented a detailed review of the derivation of the formal integral solution to the wave equation in a half-space for monopole and dipole sources, using the Green's Function (GF) solution to the inhomogeneous Helmholtz wave equation [35,36]. Under the assumptions of compact acoustic sources (i.e., their characteristic dimensions are much smaller than the acoustic wavelength) and that far-field volcano infrasound propagates in half-space (i.e., an atmosphere bounded by a flat solid topography), the transient solutions to the wave equation for monopole, and horizontal and vertical dipole radiation at distance *r* from the source can be written as:

$$p\_m(\mathbf{r}, t) \quad = \underbrace{1}\_{2\pi r} \dot{S} \left(t - \frac{r}{c}\right) \tag{4}$$

$$p\_h(\mathbf{r}, t) \quad = \underset{\mathbf{r} \, \, \mathbf{r} \, \mathbf{r} \, \mathbf{c}^2}{\text{Tr} \, \mathbf{r} \mathbf{c}^2} \left[ \frac{\mathbf{x}}{r} D\_x(t - r/c) + \frac{y}{r} D\_y^\dagger(t - r/c) \right] \tag{5}$$

$$p\_v(\mathbf{r}, t) \quad = \frac{1}{2\pi r c^2} \left[ \frac{z}{r} \frac{z\_0 \partial^3 D\_z(t - r/c)}{\partial t^3} \cos\theta \right] \tag{6}$$

where *pm*, *ph*, and *pv* are monopole, horizontal and vertical dipole pressure fields, respectively; *S*˙ is the time history of mass acceleration at the source; (*x*, *y*, *z*) are coordinates in a Cartesian system centred at the source location; *Dx*,*y*,*<sup>z</sup>* are dipole momenta (in units of kg m s<sup>−</sup>1) in the (*x*, *y*, *z*) directions; *θ* is the angle between the axis of the dipole and the vertical direction; *z*<sup>0</sup> is source elevation; and *c* is the sound speed. Any infrasound time series can be written as the radiation from a multipole source, that is the linear superposition of a monopole, with horizontal and vertical dipoles (and terms of higher order):

$$p(\mathbf{r},t) = p\_{\rm{fl}}(\mathbf{r},t) + p\_{\rm{h}}(\mathbf{r},t) + p\_{\rm{v}}(\mathbf{r},t) \tag{7}$$

The formulation of Equations (4)–(7) represents the foundation for acoustic waveform inversion that is discussed below.

#### **4. Eruption Source Parameters and Their Potential Use in Ash Plume Modelling**

In this section, we discuss the use of infrasound for the evaluation of eruption source parameters, and their integration into models of ash plume rise. We review some case studies and expand on past work to include recent developments in numerical modelling of volcano acoustic wavefields, and their integration into infrasound waveform inversion schemes for the assessment of volume and mass flow rates during explosive eruptions.

#### *4.1. Fluid Flow Velocity from Acoustic Infrasound*

Early work from Woulff and McGetchin [32] demonstrated the application of Equations (1)–(3) to assess volcanic jet dynamics—in particular fluid flow velocity—from records of the sound emitted by explosions at Stromboli (Italy) and fumaroles at Acatenango (Guatemala). An experiment at Acatenango where acoustic power, the cross-sectional area of the vent, and flow velocity could be directly measured in the field, allowed calibration of the dimensionless constants *Km*, *Kd*, and *Kd* in Equations (1)–(3), and to infer that the sound radiated in either case had, predominantly, dipolar nature. Several authors [17–20] built on the original work of Woulff and McGetchin [32] on translating time series of acoustic pressure into gas flow velocities. Acoustic power (Π) can be estimated directly from the recorded infrasound [20] as:

$$
\Pi = \frac{\pi r^2}{\rho\_{atm} c \mathcal{T}} \int\_0^\tau \Delta p^2(t) dt \tag{8}
$$

where Δ*p*(*t*) is a time series of atmospheric pressure, *r* is the source-receiver distance, *ρatm* is atmospheric air density, *c* is the atmospheric sound speed and *τ* is the source duration. Once acoustic power is calculated by solving the definite integral in Equation (8), the flow velocity at the source can be easily retrieved by inversion of Equations (1)–(3). Gas velocity is an important eruption parameter easily translated into volume flow rate (for a known cross-sectional area of the vent), which is used as input into one of many empirical models to evaluate the initial height of eruption plumes. The most used of models is the one from Sparks et al. [8]:

$$Q\_{\text{magma}} = \left[\frac{H\_{\text{plumec}}}{1.67}\right]^{\frac{1}{0.259}}\tag{9}$$

where *Qmagma* is the magma volume flow rate in units of m3/s and *Hplume* is the plume height in units of kilometres. Vergniolle and Caplan-Auerbach [20] used acoustic data recorded during the sub-Plinian phase of the 1999 eruption of Shishaldin volcano (USA) to infer gas velocity and total gas volume erupted; they assumed dipole radiation during the sub-Plinian phase of the eruption, and confirmed that the gas velocities obtained were realistic using independent plume height measurements and Equation (9). They noted that the assumption of dipole radiation would not be appropriate outside the period of sub-Plinian activity; for instance, discrete explosions are more effectively described as monopole sources. A dipole source is often the preferred choice for explosive events lasting up to few tens of seconds associated with plumes rising to their highest elevation within a few minutes. In these situations, the force field exerted by the conduit and crater walls on the volcanic flow, and the interactions of the gas phase with solid pyroclasts are well described by a dipole. During sustained eruptions, lasting up to several hours, infrasound is generated by internal turbulence within the plume, a source best represented as a quadrupole [17].

Caplan-Auerbach et al. [17] also favoured a dipole source to describe the infrasound recorded during the 2006 eruption of Augustine volcano (USA); they analysed nine eruptive events with durations of the order of several minutes that generated largely buoyant plumes reaching elevations of 8–15 km above the vent. Here (Figure 3), we have reproduced their results following the workflow presented in the original manuscript for two of the explosions (Events 1 and 4, in the original publication). Estimates of flow velocity at the vent were obtained, in the original manuscript and, here, by using Equation (8) to calculate acoustic power and then inverting Equation (2). They calculated volume flow rates for all nine explosions assuming a vent with circular cross-section, and used these values to predict plume heights. Modelling the plumes as buoyant thermals rather than continuously fed by eruption (an implicit assumption of Equation (9)), they showed that gas fluxes retrieved from analysis of acoustic data are a good indicator for the rise height of volcanic plumes. Ideally, the choice of a specific source mechanism should be cross-validated using other geophysical measurements, when available. For example, high-resolution visual and thermal infrared imagery can be used to assess eruption dynamics and to calculate the exit velocity of volcanic jets in the near-vent region, two crucial factors in controlling sound generation. Ripepe et al. [18] employed multi-parameter data to investigate the 2011 eruption of Eyjafjallajokull (Iceland); plume heights were retrieved from infrasound-derived fluid flow velocities assuming a dipole acoustic source, and cross-validating these measurements with continuous thermal IR measurements of the plume in the near-vent region.

**Figure 3.** Infrasound-derived time series of gas velocity for two explosions during the 2006 eruption of Augustine volcano, USA: (**a**,**c**) Raw infrasound of explosive events on 11 January 2006 and 13 January 2006, respectively. Signals recorded by a microphone at 3.2 km from the vent. (**b**,**d**) Five-second moving-window gas velocity time series obtained by inverting Equation (2), assuming a dipole acoustic source. The figure reproduces results originally presented in Caplan-Auerbach et al. [17].

Careful consideration, when using infrasound-derived flow velocities for the assessment of plume rise, should be given to the nature of the plume itself, and whether its initial development is affected by local atmospheric conditions. Empirical relationships such as Equation (9) are usually derived for strong plumes where internal drag is negligible and entrainment of atmospheric air is approximately proportional to the vertical velocity of the flow [37]. For weak plumes, where internal shear is near normal to the plume axis, turbulence may be important and some of the empirical relationships found in the literature may not be appropriate [38]. Furthermore, plumes that develop from low to intermediate velocity volcanic jets are more likely to be affected by local wind conditions in the lower troposphere, causing plumes to bend. These effects are included in more complex numerical models of ash plume rise. Woodhouse et al. [39] developed an integral model of volcanic plume rising in a windy atmosphere and investigated how wind restricted the rise height of volcanic plumes at Eyjafjallajokull. Lamb et al. [19] used the same plume rise model to investigate two events during the 2009 eruption of Mt. Redoubt, USA. Fluid flow velocities were estimated from acoustic infrasound for monopole, dipole and quadrupole sources, and used to produce models of plume rise, including local wind conditions (Figure 4). Comparison of plume modelling [39] with syn-eruptive Doppler radar measurements of the plumes [40] showed good agreement with the results obtained for infrasound dipole radiation.

**Figure 4.** Infrasound-derived gas flow velocities, and plume models for an explosion at Mt. Redoubt, USA, on 23 March 2009: (**a**) raw infrasound signal recorded at 12 km from the vent; (**b**) five-second moving window gas velocity time series calculated for monopole, dipole, and quadrupole infrasound sources by inverting Equations (1)–(3); and (**c**,**d**) cross-section (in the downwind direction) and map view of modelled plumes using measurements of local atmospheric conditions at the time of eruption. Initial flow velocities in the plume models corresponds to peak gas velocities from infrasound for monopole, dipole and quadrupole sources (colour coding consistent across (**b**–**d**)). The figure reproduces and expands results originally presented in Lamb et al. [19].

#### *4.2. Acoustic Monopoles: Volume and Mass Flow Rate*

One of the advantages in the use of acoustic infrasound for assessment of volcanic emissions is that its source is directly linked to the unsteady injection of mass into the atmosphere and its acceleration, and to other forces acting in the near-source region. Equations (4)–(6), thus, provide a framework for retrieval of volume or mass flow rates of volcanic material ejected during eruptions. For a pure monopole, the relationship between volume or mass flow rate and the resulting acoustic pressure time series is given by Equation (4), as illustrated in Figure 5; for a source radiating as a monopole in a homogeneous and isotropic atmosphere, Equation (4) can be inverted to retrieve the rate at which atmospheric mass is displaced during an explosion by direct integration over time of the recorded pressure time series. Johnson [41], Johnson and Lees [42], and Johnson and Miller [43] demonstrated the application of the monopole source model to explosions recorded at Erebus (Antarctica), Santiaguito (Guatemala), and Sakurajima (Japan).

**Figure 5.** Modelling according to linear acoustic theory for a monopole source: (**a**) time-history of fluid injection into a uniform and homogeneous atmosphere (Gaussian); and (**b**) excess pressure time series at 1 km from the source, obtained as the time derivative of the source time function in (**a**).

Direct waveform integration according to Equation (4) rests on the assumption that infrasound amplitudes attenuate proportionally to the inverse of distance from the source (1/*r*, or geometrical spreading), and neglects the effect of topography on the acoustic wavefield. It is, however, well-known that acoustic wave amplitudes often do not attenuate proportionally to 1/*r* (e.g., Lacanna and Ripepe [44]), thus limiting the application of the monopole model to single-station recordings. Johnson and Miller [43] at Sakurajima proposed its use with data recorded at sites at proximal distance from the source and with unobstructed line-of-sight to the vent, conditions that provide a reasonable approximation to geometrical spreading and allow discarding major effects of topography. In Figure 6, we show an application of the monopole model to data collected by a temporary infrasound network at Fuego (Guatemala) during a moderate-size explosion in May 2018. This example illustrates well the effects of non-1/*r* attenuation and topography at different locations around the source. Acoustic waveform shape (Figure 6a–e) varies across the network, evidence of diffraction and scattering from topography; the source time functions of volume flow rate (Figure 6f,j) show variable shapes and inconsistent peak values, suggesting that Equation (4) cannot account for the true attenuation pattern, and for realistic scattering of the acoustic wavefield.

**Figure 6.** Volume flow source time functions derived from integration of infrasound waveforms at Fuego volcano, Guatemala: (**a**–**e**) explosion infrasound recorded at a range of distances from the source; and (**f**–**j**) time series of volume flow rate calculated from (**a**–**e**) according to Equation (10). The inconsistency of peak volume flow rate and variability in the shape of the volume flow rate source time functions suggest that: (i) the attenuation does not follow simple geometrical spreading corrects; and (ii) the effects of topography on acoustic wave propagation are present.
