Volcanic eruptions can loft large amounts of ash into the atmosphere, endangering proximal populations and aircraft. Additionally, these events can produce long-lasting effects on both climate and ecosystems. Successful mitigation of these hazards requires a concerted effort by volcano observatories, governments, and civil protection agencies. However, anticipating plume behavior remains subordinate to our ability to reliably extract the conditions at the volcanic source. Probing the interior of proximal volcanic atmospheric flows in situ presents obvious technical and monetary challenges. Thus, much effort has been devoted to develop tools that can efficiently detect and interrogate the insides of volcanic plumes remotely and in real time.
Recently, a number of authors have shown that the presence of ash can be detected unambiguously using L-band Global Positioning System (GPS) networks deployed around active volcanoes (Aranzulla et al., 2013;Larson, 2013;Larson et al., 2017). Specifically, studies at Redoubt, Etna, and Okmok show that ash clouds can produce measurable attenuations in GPS signal-to-noise data. Using GPS to detect plumes is attractive for a number of reasons. First, many active volcanoes are already instrumented with high-quality GPS receivers, enabling novel science with low overhead. Second, measurements with GPS require a single link between a receiver and a satellite rather than observations with multiple satellites. Lastly, given that GPS must operate in a broad range of atmospheric conditions, the employed L-band signals have low sensitivity to water and, thus, any signal modification would result from the presence of suspende d silicates only (Larson et al., 2017). The attenuation of GPS signals propagating through volcanic clouds can be significant. Larson et al. (2017), for instance, detected attenuations of up to 15 dB in GPS signals traversing plumes during eruptions at Redoubt and Etna. Using a simple model, however, these authors showed that (for the parameter set used) the total path attenuation caused by the presence of silicates and ice would only be on the order of 1 dB. Larson et al. (2017) suggest that this discrepancy may be resolved by including larger particle sizes into their model. Here, we show that large scatterers are not required to attenuate L -band radiation if a plume comprises micron-to-millimeter-sized grains that are electrostatically charged.
Volcanic jets can produce a wide range of electrical phenomena (Aizawa et al., 2016;Behnke & Eric, 2015;Gilbert et al., 1991;Hatakeyama, 1947;Hatakeyama & Uchikawa,1951;Houlié et al., 2005;James et al., 1998James et al., , 2002;;Kikuchi & Endoh, 1982;McNutt & Williams, 2010;Miura et al., 1996Miura et al., , 2002;;Thomas et al., 2007). Large eruptions are often accompanied by brilliant displays of lightning, testifying to the high electrical potentials that can be generated by a diverse set of electrification mechanisms (including fractocharging, triboelectrification, and electrification processes similar to those found in thunderstorms; Cimarelli et al., 2014;Gilbert et al., 1991;Méndez Harper et al., 2015;James et al., 2000;Kikuchi & Endoh, 1982;Méndez Harper & Dufek, 2016). Beyond signals at optical wavelengths, volcanic lightning generates a wide array of electromagnetic (EM) signals that can be recorded from afar. Instruments like the Lightning Mapping Array (LMA) can provide extremely accurate four-dimensional maps of discharges in plumes. Processed LMA data have been used to show that charging and discharge processes reflect the kinematics of the associated volcanic atmospheric flow and can be used to obtain information that would otherwise be opaque to observation (Aizawa et al., 2016;Behnke et al., 2013;Behnke & Eric, 2015;Cimarelli et al., 2016;Méndez Harper et al., 2018;Thomas et al., 2007).
While electrification processes have been recognized in plumes since the time of Pliny the Younger (Walsh, 2006), the effect of particle charge on the propagation of EM waves has not been addressed in the context of volcanic systems. Standard Mie-Lorenz-Debye theory (Mie, 1908), which describes how EM waves interact with a suspended particle, implicitly assumes that said particle carries no excess surface charge. However, Bohren and Hunt (1977) and, more recently, Klacˇka and Kocifaj (2007) have proposed modifications to the theory to account for trapped surface charge. These authors have shown that electrostatic charge may influence the extinction properties of grains if such particles have radii R significantly smaller than the wavelength 𝜆 of the incident radiation. Specifically, charged particles for which 4𝜋R∕𝜆 is smaller than ≈0.01 may attenuate EM radiation by more than an order of magnitude more efficiently than their neutral counterparts (Klacˇka & Kocifaj, 2007). Using this framework, Kocifaj et al. (2015) have hypothesized that lightning hazards could be quantified in storms by illuminating thunderclouds with EM radiation and detecting anomalous backscatter. Similarly, Dou and Xie (2017) found that charging affects the propagation of EM waves through dust and sand storms.
We posit that similar effects could be observed within charged volcanic plumes. GPS signals have wavelengths of tens of centimeters, implying that charge on particles may produce measurable signal attenuations if particles have diameters of a few hundred microns or less-typical sizes for particles in plumes (see Figure 1) . Using a broad set of plume parameters, we investigate the degree to which charge can impact the propagation of GPS signals or other L-band radiation. Our work suggests that the extinction efficiency of a group of charged, micron-sized particles is more than one order of magnitude larger than that associated with an equivalent neutral parcel. We show that this modified Mie theory can explain the relatively large attenuations of GPS signals propagating through plumes observed at Redoubt and Etna. Thus, any future work involving GPS to monitor plumes must take into account the electrostatic charging of pyroclasts. Finally, our results open the possibility of using GPS to explore charging in volcanic systems. We propose that GPS may be particularly suited to studying plume electrification if coupled with millimeter-wave RADAR.
Just over a century ago, Mie (1908) described the interaction of an EM plane wave with an isotropic particle by means of an infinite series of spherical multipole partial waves. The solution has been used to characterize the scattering and absorption of light by particles across a wide range of fields, from astrophysics to plasmonics (Eaton, 1984;Lal et al., 2007;Rowe et al., 2008). However, the standard Mie formulation assumes that the illuminated particle is electrostatically neutral-that is, the particle carries no excess charge on its surface-through the magnetic field strength boundary condition at the interface between the particle and the medium:
as well as through the electric displacement field boundary condition: In equations ( 1) and (2), n12 is the normal vector from medium 1 (the fluid surrounding the particle) to medium 2 (the particle), H is the magnetic field intensity, and D is the electric displacement field vector. Recently, Bohren and Hunt (1977) and Klacˇka and Kocifaj (2007) have relaxed the assumption of charge neutrality through modified boundary conditions:
In equation ( 3), E is the electric field vector and K is the surface current density equal to 𝜎sE1,T where 𝜎s is the phenomenological surface conductivity resulting from the surface charge and E1,T is the tangential component of the electric field at the boundary. We refer the reader to those works for the complete solution derivation. The degree to which EM radiation will be extinguished (i.e., scattered and/or absorbed) can be expressed in terms of cross sections, Cext (extinction), Csca (scattering), Cabs (absorption):
In equations ( 5) and ( 6), an and ab are scattering coefficients and k is the wave number. The cross sections are effective areas which quantify the likelihood that an incident beam will be extinguished when impinging on a particle. Generally, these cross sections are different from the particle's geometrical cross section 𝜋R 2 . Integrating the modified boundary conditions from equations ( 3) and ( 4), the Mie scattering coefficients can be written as follows (see Bohren &Hunt, 1977, andKlacˇka &Kocifaj, 2007, for detailed derivation):
| |
where
Above, 𝜇1 and 𝜇0 are the relative permeabilities of the particle and the medium, respectively (here we assume that 𝜇1 = 𝜇0); m ̃ is the complex refractive index; 𝜓 n(𝜌) is the Riccatti-Bessel of the first kind and 𝜉n(𝜌) is the Riccatti-Bessel of the third kind; R is the particle radius; the nondimensional size parameter x is defined as the ratio of particle size to the wavelength 4𝜋R∕𝜆; me and e are the mass and charge of an electron, respectively; kB is the Boltzmann constant; ℏ is the reduced Plank constant; and T is temperature in Kelvin (we assume a constant temperature of 273 K). Klacˇka and Kocifaj (2010) propose that 𝛾f = 0.1. The square of the plasma frequency (equation ( 15)) incorporates the effects of electrification through the potential 𝜙:
where 𝜂 is the surface charge density and kC is Coulomb's constant. Setting 𝜂 = 0 reduces equations ( 8) and ( 10) to the classical Mie scattering coefficients.
Using the model described above, we investigate the conditions under which charged volcanic particles impact the propagation of GPS signals. The following computations are performed for two wavelengths, 1,575.42 and 1,227.60 MHz, corresponding to GPS bands L1 and L2, respectively. Furthermore, these calculations were done for particles with diameters in the range of 1e -6 to 1e -2 m, for compositions ranging from basalts (low-silica content) to rhyolites (high-silica content), and for a range of charges observed in both the field and laboratory experiments.
A standard way to quantify how well a single particle extinguishes radiation that strikes its surface is to compute efficiency factors. These factors are the cross sections defined above in equations ( 5) through (7) normalized by the particle's geometrical cross section:
,
Then, to determine how charge affects the extinction of EM radiation for the same single particle system, we calculate the ratios of the efficiency factors of a charged grain Q(c) to the efficiency factors of a geometrically equivalent, but neutral pyroclast Q.
From extinction cross sections (equations ( 5) through ( 7)), the attenuation coefficient 𝛼 for a monodisperse group of particles (with radii R) can also be computed: where NT(R) is the total number of particles with radius R per unit volume. This expression has units of Nepers per meter. For long path-lengths, the above quantity is more conveniently expressed in dB per kilometer by multiplying equation ( 21) by 4.343 × 10 3 :
For a more realistic cloud containing particles of different diameters, the attenuation coefficient can be expressed as (Dou & Xie, 2017;Hendrik Christoffel & van de Hulst, 1957)
min Above, Rmin and Rmax are the minimum and maximum radii in the particle size distribution, respectively, and N(R)dR is the total number of particles with radii R to R + dR per unit volume. However, we restrict our analysis to clouds of monodisperse particles using equation ( 22). In general, as we will show further on, charge will have a significantly greater effect on ash clouds with abundant submillimeter grains than in plumes with particles in the millimeter-to-centimeter range. The total attenuation experienced by a signal can be then obtained by multiplying the attenuation coefficient by the path-length through the obscuring medium (in this case, a cloud of particles).
Implementation of the extinction model was accomplished by modifying the freely available MatScat code (Schäfer, 2011) to include the effects of surface charge. All computations were performed in GNU Octave. Inputs to the model included the dielectric constants of ash (in order to compute the relative indices of refraction in equations ( 8) through ( 14)) and estimates of the charge densities expected on particles in plumes.
The following subsections describe these input parameters in detail.
The dielectric properties of products from several volcanoes at microwaves frequencies have been obtained by Adams et al. (1996) and Oguchi et al. (2009). Between 4 and 19 GHz, these authors showed that the relative permittivity of ash are virtually independent of frequency. We assume that this invariance holds down to the L-band frequencies of interest. Adams et al. (1996) did find that the real and imaginary components of the relative permittivity decrease with increasing silica content. From the relative permittivity, one can compute the complex refraction index as m ̃ 2 = 𝜖̃ = 𝜖 ′ + 𝑗𝜖 ′′ (as in Bohren & Donald, 2008, the convention used here r r utilizes a nonnegative imaginary component). The real and imaginary components of index of refraction for ash from six volcanoes are listed in Table 1 (In an attempt to help preserve Native American heritage, we emphasize the use the volcanoes' pre-Columbian names in the present work as defined in Anderson, 2013;Concepción, 1991;Coombs et al., 2006;Maxwell & Hill, 2006).
Multiple experimental and field efforts have attempted to characterize the charge densities acquired by volcanic particles through processes thought to operate in volcanic plumes. Gilbert et al. (1991) measured the charge on individual grains falling out of a Sakurajima plume using an electrostatic separator. These investigators found that eruptions could produce particles with maximum charge densities approaching the theoretical limit of 2.7 × 10 -5 C/m 2 . Similar values were obtained by Miura et al. (1996) at the same volcano. Experimentally, James et al. (2000) reported that charge densities on the order of 10 -8 -10 -6.5 C/m 2 were generated during the reduction of pumice samples through either fracture or abrasion. More recently, Méndez Harper et al. (2017) reported charge densities ranging between 10 -8 and 10 -6 C/m 2 on Popocatépetl and Atéxcac grains charged frictionally in a controlled experimental environment. We note that most of these experiments used andesitic to rhyolitic compositions. Little work has been conducted to determine whether low-silica compositions effectively charge in the volcanic context. However, experiments using JSC-1 Mars regolith simulant (weathered volcanic ash from Pu'u Nene) under near-surface Martian conditions suggest that low-silica materials do indeed charge triboelectrically (Forward et al., 2009b;Sternovsky et al., 2002). Thus, we assume that the charge that can be collected by grains in volcanic granular flows lies between 10 -8 and 10 -5 C/m 2 , regardless of composition.
The scattering coefficients, a n and b n in equations ( 8) and ( 9), depend on the mass of the charged species deposited on a particle's surface. Specifically, equations ( 10) through ( 16) indicate that the effects of charge on any impinging EM radiation are more pronounced when grains are electrified with less massive charged species. Indeed, one can show that excess electrons on a dielectric surface may result in elevated extinction, whereas excess ions have relatively insignificant effects. Thus, according to the above model, any attenuation of GPS signals through charged plumes would have to be the result of excess electrons, not ions, on particles. While the movement of electrons is often invoked to explain electrostatic build -up in dusty media (refer, e.g., to the trapped-electron model; Forward et al., 2009a;2009b;Lacks & Levandovsky, 2007;Lacks & Mohan Sankaran, 2011;Lowell & Truscott, 1986;Pahtz et al., 2010), strong evidence exists to support an ionic role in granular electrification. For instance, the release of positive and negative ions (in addition to electrons) has been observed during fragmentation processes (e.g., Dickinson et al., 1981;James et al., 2000). Furthermore, recent modeling and experimental work indicates that ions, specifically water ions, may also be responsible for triboelectric phenomena (Gu et al., 2013;Lee et al., 2018;Xie et al., 2016). However, the relative contributions of electrons and ions to the total electrification in a mobilized granular flow remain unknown. Until these questions are answered satisfactorily, we operate under the assumption that granular electrification is driven, at least to an important degree, by the exchange and concentration of electrons.
The effect of surface charge on the spectral characteristics of an ash particle was assessed by computing the ratios of the efficiency factors of a charged particle Q(c) to those associated with a geometrically equivalent but neutral particle Q. Efficiency ratios Q(c)∕Q for absorption Q(c)abs∕Qabs, scattering Q(c)sca∕Qsca, and total extinction Q(c) ext ∕Q ext as functions of size parameter are rendered in Figures 234, respectively. Q(c)∕Q ratios for the L1 channel are displayed in Figures 2a-4a, whereas those for the L2 channel are shown in Figures 2b-4b. For this calculation, we have assumed that the illuminated particle carries a charge density equal to the theoretical maximum value of 2.7 × 10 -5 C/m 2 for standard atmospheric conditions (air, 1 bar, 273 K). For all ash compositions, charge increases absorption when the size parameter ( x = 4𝜋R∕𝜆) is smaller than 10 -1 (Figure 2) There is also an increase in the scattering efficiency for size parameters smaller than 10 -3 (Figure 3). However, comparison of Figure 2a with Figure 4a and Figure 2b with Figure 4b reveals that the increase in total extinction, (Qext = Qabs + Qsca), is overwhelmingly controlled by an increase in absorption, not scattering. Note, also, that a mafic (low-silica content magma) charged particle extinguishes L-band radiation up to three times more effectively than its rhyolitic (high-silica content magma) counterpart. Such compositional dependence arises from the contrast between the complex dielectric constant of the particle and that of the medium (assumed to be free space, or very close to that of air, in the present work). Overall, Figures 2 through 4 show that charge significantly modifies extinction (an increase of at least 10%) when the size parameter is less than ≈0.05. In other words, for the L-band wavelengths (on the order of centimeters) considered here, excess surface charge has an impact on the spectral characteristics of volcanic particles with radii smaller than ≈1,000 𝜇m. For both L1 and L2, the increase in extinction is largest for an interaction with a 10-𝜇m particle, or when the size parameter is ≈0.0001. As particles become larger (>1,000 𝜇m) relative to the wavelength, the effects of charge become negligible.
While charge densities approaching the breakdown limit of 2.7 × 10 -5 C/m 2 have been measured on both pyroclasts falling out of plumes and in laboratory experiments using volcanic ash and analog materials (Gilbert et al., 1991;James et al., 2000;Méndez Harper et al., 2015), the same measurements indicate that a large fraction of material carries much smaller surface charge densities. Thus, we also explored how lesser degrees of charging impact the propagation of L-band signals in dusty, volcanic media. The percentage increase in total extinction as a function of charge and particle diameter (1 𝜇m to 1 cm)-that is, (Q(c) -Q)∕Q × 100-appears in Figures 5a and 5b for L1 and L2, respectively. This computation was done for the charge density ranges described in section 2 (𝜂 = 10 -8 -10 -5 C/m 2 ; Gilbert et al., 1991;James et al., 2000;Méndez Harper & Dufek, 2016;Méndez Harper et al., 2017). As demonstrated by Figure 5, important increases in attenuation (larger than 10 %) occur when the charge density is above 𝜂 = 10 -7 C/m 2 and the particle size is smaller than 50 𝜇m. Up until this point, we have considered the interaction of EM radiation with a single particle. Traversing a plume, however, an EM wave undergoes extinction from the interaction with innumerable grains. Using equation ( 21), we calculated the attenuation coefficient 𝛼 for a monodisperse collection of particles. For brevity, these computations only used the basaltic (Chi Q'aq') and rhyolitic (Atitlan) end-members in Table 1. We assumed four different mass loadings: 1e -4 , 0.001, 0.01, and 0.1 kg/m 3 , which correspond to volume fractions of 10 -8 , 10 -7 , 10 -6 , and 10 -5 , respectively. Particles are given charge densities in the range of 0 to 27 𝜇C/m 2 and diameters between 1 𝜇m and 10 cm (note this range of particle diameters is slightly larger than that used in previous calculations). Attenuation coefficients for L1 are shown in Figures 6 and 7 (basalt and rhyolite, respectively), while those associated with L2 are rendered in Figures 8 and 9 (basalt and rhyolite, respectively). There, each panel displays equation ( 21) as a function of particle size and charge density (curve color and thickness) for the four different loadings. Several behaviors are worth highlighting. First, as mentioned previously, charge exacerbates attenuation by at least 10% when grains are smaller than 1,000 𝜇m in diameter, but may increase losses by more than an order of magnitude for certain particle sizes. Second, the particle size under which attenuation is maximum increases with increasing charge density. Third, the effect of charge on the attenuation of L-band signals is more prominent for basaltic compositions than for silica-rich materials. Finally, as to be expected, the attenuation increases with mass loading.
With GPS? Larson et al. (2017) performed first order calculations using a Rayleigh approximation to determine whether suspended particles could produce the attenuations of GPS signals observed during eruptions of the Redoubt and Etna volcanoes. These investigators concluded that hydrometeors would produce negligible attenuation. They also computed the effect of pyroclasts, showing that grains with diameters ranging between 6 and 13 mm would generate path attenuations in the vicinity 0.014-0.109 dB/km from absorption. Because at those diameters, losses due to absorption equal those associated with scattering, the effective path attenuations would double to 0.028-0.218 dB/km. These path attenuations would produce a maximum attenuation of 1 dB for an arbitrarily defined 5 km path-length (Larson et al., 2017). Yet, Aranzulla et al. (2013), Larson (2013) The modified Mie model presented here suggests that the excess charge on particles may have been responsible for the attenuations observed at Redoubt and Etna. In agreement with Larson et al. (2017), our model consistently predicts that uncharged ash clouds will produce attenuation coefficients below or at the lower limit of those observed in the field for ash-sized pyroclasts. Large attenuations do occur for higher mass loadings, but require an abundance of clasts with diameters of several tens of centimeters (large peaks on the left in Figures 6789). Conversely, if the material is charged, micron-to submillimeter-sized pyroclasts can effectively generate losses congruent or exceeding those reported in the field studies. Note, however, that even with highly charged particles, large attenuations are more readily observed for the densest of flows (i.e., those with mass loading in the range of 10-100 g/m 3 ), suggesting that L-band signals are most affected by particle jets which are either very dense or proximal. Such inference conforms well with observations at Etna and Redoubt (Larson et al., 2017). At Redoubt, signals that traversed small, dilute plumes did not display important attenuations. Similarly, at Etna, only the satellite tracks (i.e., the signal paths linking transmitter and receiver) that fell within 3 km of the vent were associated with signal reductions. Additionally, studies at Augustine, Redoubt, and Sakurajima employing LMA sensors and high-speed cameras indicate that flows exiting the vent may have volumetric charge densities significantly larger than those found in conven- tional thunderstorms (Aizawa et al., 2016;Behnke et al., 2018;Cimarelli et al., 2016). This combination of high mass loading and elevated charge densities in proximity to the vent likely contributed significantly to the anomalous GPS attenuations observed at Redoubt, Etna, and other volcanoes.
The calculations presented above suggest that the propagation of GPS signals through plumes is strongly influenced by particle surface charge. Consequently, any efforts to use GPS to remotely characterize atmospheric volcanic flows must take into consideration electrostatic effects. By the same token, our work implies that GPS may offer novel opportunities to study electrification itself in volcanic ash clouds. A clear advantage of GPS-based sensor network for analyzing electrification in plumes is abundance of GPS receivers around active volcanic systems. This ubiquity implies that data collection campaigns similar to those undertaken by Larson et al. (2017) could be performed with little investment by many groups. However, correctly interpreting modified GPS waveforms requires additional measurements. Indeed, Figures 6 through 9 demonstrate that, in addition to charging, L-band radiation is affected by mass loading and, to a lesser degree, magmatic composition. Furthermore, while the propagation of L-band radiation is insensitive to grain size (for clasts smaller than few centimeters) in the case of an uncharged collection of particles, important size dependences emerge when charge is included into the system (see Figures 6 through 9).
One possible avenue to reduce the uncertainties associated with GPS measurements involves complimenting them with millimeter-wave (30 to 300 GHz) RADAR measurements (Bryan et al., 2017;Speirs, 2014). While RADAR has been used for some time in the context of volcanic eruptions, most studies have relied on C-band weather RADAR (4-8 GHz; Harris & Rose, 1983;Marzano et al., 2006;Rose et al., 1995) or L-band doppler RADAR (Donnadieu et al., 2005;Dubosclard et al., 2004;Valade & Donnadieu, 2011). Additionally, some works have used systems operating in the S (2-4 GHz; Marzano et al., 2013) and X bands (8-12 GHz;Vulpiani et al., 2011). However, like GPS, these longer wavelengths cannot directly measure pyroclast sizes unless the particles are very large (centimeter size) and concentrated. Recent numerical modeling by Bryan et al. (2017) indicates that ground-based K-band (18-26 GHz) and millimeter-waves (180-235 GHz) RADARs may be better suited to study airborne granular flows rich in micron-to millimeter-sized particles and volatiles (namely, water). Indeed, those authors suggest that a system operating at 220 GHz would offer over six orders of magnitude more sensitivity to fine ash than conventional 5 GHz weather RADAR. While K-band and millimeter-wave radiation, with wavelengths approaching the size of suspended pyroclasts ( x ≈ 1), can directly measure the size and distribution of ash, they are insensitive to any charge on the surfaces
Foundation Graduate Research Fellowship Program as well as the Blue Waters Graduate Fellowship Program for providing support for J.M.H. (2) and grant EAR 1645057 for J.D. All data for this paper are contained in the main text. The MatScat code can be downloaded online (https://www.mathworks.com/ matlabcentral/fileexchange/36831-matscat).
of particles. This analysis is supported by experiments conducted by Méndez Harper (2017), who used a Fabry-Perot resonator to show that extinction of millimeter-waves was not affected by particle charge for micron-sized grains. Thus, in principle, such wavelengths could be used to characterize the physical properties of a plume-information which could then be employed to determine the electrostatic state of a plume from GPS or other L-band signals. A hypothetical combined GPS/millimeter-wave radar systems is shown schematically in Figure 10. Millimeter-wave systems are becoming ubiquitous in many fields and it is likely that similar trends will be seen in the geosciences. Indeed, low-power millimeter-wave systems can be much more portable than their low-frequency counterparts and can be constructed from low-cost off-the-shelf components (Bryan et al., 2017). Such technology may be able to compliment GPS measurements without intractable overhead expenses for the purpose of studying electrification in plumes.
In this work, we have numerically shown that charging affects the propagation of GPS signals or other L-band radiation. Specifically, we used the modified Mie model described by Klacˇka and Kocifaj (2007) to show that charging in volcanic flows could influence the propagation of L-band radiation or other EM waves with wavelengths at least two orders of magnitude larger than the particle size through increased absorption.
Our findings can be summarized as follows:
1. For particles charged to the breakdown limit (2.7 × 10 -5 C/m 2 ), there is increased extinction of L-band radiation when the size parameter is smaller than 0.05. This size parameter corresponds to grain sizes smaller than 1,000 𝜇m. Maximum increase in extinction occurs when the grain size is close to 10 𝜇m. 2. A charged volcanic particle may attenuate GPS signals by almost two orders of magnitude more effectively that an geometrically equivalent neutral particle. 3. Extinction increases with decreasing silica content. 4. For particles undersaturated in charge (i.e., when the charge density is below the breakdown limit), increases in extinction larger than 10% are predicted for grains smaller than 50 𝜇m and charge densities larger than 10 -7 C/m 2 . 5. Our modeling demonstrates that enhanced absorption due to charged particles may explain the anomalous attenuations in GPS signals observed by various investigators at a number of volcanoes-decreases that cannot be accounted for solely by the presence of suspended particles.
Because numerous active volcanoes are instrumented with high-quality receivers, finding ways to use GPS (or GLONASS and GALILEO) to study volcanic plumes remotely is an attractive proposition. In this work we show that L-band signal attenuation is a function of particle size, charge, and number density-quantities which have substantial uncertainties. Thus, beyond binary detection of plumes, we propose that GPS may be an effective tool to study electrostatic processes in plumes if coupled with millimeter-wave RADAR measurements.