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Volcanic Plume Height Measured by Seismic Waves Based on a Mechanical Model 2

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Stephanie G. Prejean1 and Emily E. Brodsky2 4

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1. USGS Alaska Volcano Observatory 6

4210 University Ave., Anchorage AK 99508 7

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2. University of California, Santa Cruz 9

Department of Earth and Planetary Sciences 10

Santa Cruz, CA 95064 11

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Submitted to Journal of Geophysical Research, 4-4-10 23

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Abstract 24

In August 2008 an unmonitored, largely unstudied Aleutian volcano, Kasatochi, 25

erupted catastrophically. Here we use seismic data to infer the height of large eruptive 26

columns, like those of Kasatochi, based on a combination of existing fluid and solid 27

mechanical models. In so doing, we propose a connection between a common observable, 28

short-period seismic wave amplitude and the physics of an eruptive column. To construct 29

a combined model we estimate the mass ejection rate of material from the vent based on 30

the plume height, assuming that the height is controlled by thermal buoyancy for a 31

continuous plume. Using the calculated mass ejection rate, we then derive the equivalent 32

vertical force on the Earth through a momentum balance. Finally, we calculate the far-33

field surface waves resulting from the vertical force. Physically, this single force reflects 34

the counter force of the eruption as material is discharged into the atmosphere. In contrast 35

to previous work, we explore the applicability to relatively high frequency seismic waves 36

recorded at ~ 1 s. The model performs well for the 2008 eruption of Kasatochi volcano 37

and the 2006 eruption of Augustine volcano. The consistency between the seismically 38

inferred and measured plume heights indicates that in these cases the far-field 1 s seismic 39

energy radiated by fluctuating flow in the volcanic jet during eruption is a useful 40

indicator of overall mass ejection rates. Use of the model holds promise for 41

characterizing eruptions and evaluating ash hazards to aircraft in real time based on far-42

field short-period seismic data. 43

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1. Introduction and Motivation 47

Empirical studies have suggested that the amplitude of high frequency or 48

broadband seismic waves radiated during large volcanic eruptions generally scales with 49

the height of an eruption column [McNutt 1994]. However, a direct calculation of the 50

expected seismic wave amplitude based on physical models has not yet been successful. 51

Connecting commonly observable data, like seismic wave amplitudes, to a model of the 52

flow in the eruptive jet would provide a new tool to test and improve our understanding 53

of eruptive physics. For instance, small-scale turbulence is thought to play a major role in 54

entrainment of hot particles and gases and hence buoyancy of eruptive columns, yet there 55

are few measurements of the strength or distribution of small-scale features in real 56

eruptive columns [Andrews and Gardner, 2009]. Using seismic data to provide 57

observational constraints on eruption column flow processes is particularly attractive as 58

seismic data is often available even in remote settings. Thus use of the seismic database 59

would greatly increase the number and type of eruptions amenable to study. 60

Pragmatically, a physical model connecting plume height and seismic data would 61

allow the use of seismology as a remote sensing technology to determine volcanic plume 62

height. It would be a particularly useful tool for exploring eruption dynamics in remote 63

environments where direct observation is not possible, such as the volcanoes of the 64

northern Pacific Ocean. Although many volcano observatories world-wide are gradually 65

replacing short-period seismometers with broadband seismometers, we still largely rely 66

on short-period instruments for forecasting, monitoring and analyzing eruptions. These 67

realities motivate the development of the model described below. 68

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In this study we build on previous work to develop a physical model for the 69

expected amplitude of seismic waves from an eruption that generates a plume of a given 70

height. As a cautionary note we describe situations where the model is not applicable, 71

such as small eruptions or eruptions where most mass ejected is not entrained in the 72

plume. After reviewing previous work on co-eruptive seismology and the characteristics 73

of the Kasatochi and Augustine eruptions, we use the connection between plume height 74

and thermal ejection rate to predict the momentum ejection rate that generates the seismic 75

waves and then compare the model to the data for the Kasatochi eruption as well as the 76

2006 Augustine eruption. We show that for a reasonable range of parameters the model 77

performs well. 78

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2. Background 80

2.1 Co-eruptive Seismology 81

Volcanic activity produces seismic waves. These waves, which include brittle 82

failure earthquakes, long period and very long period events, and volcanic tremor, 83

generally result from movement of fluid in the Earth’s subsurface, directly and indirectly 84

[see Chouet 1996, McNutt, 2005, for reviews]. Seismology also directly senses magma 85

leaving the Earth in the form of both eruption tremor and discrete eruption earthquakes 86

[eg. Nishimura and Hamaguchi, 1993; Nishimura, 1995]. Such co-eruptive seismicity has 87

been documented in scientific literature since Omori’s early studies of Usu-san and 88

Asama volcanoes [1911-1913] and is the subject of this study. 89

By analyzing long period surface waves radiated by the May 18, 1980 eruption of 90

Mount St. Helens, Kanamori and Given [1982] show that the co-eruptive seismic source 91

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can be represented kinematically by a near-vertical downward single force. This single 92

force represents the counter force of eruption [Kanamori et al., 1984]. This model has 93

been used to investigate eruption source properties of volcanic eruption earthquakes at 94

other volcanoes including Mount Tokachi and Mount Asama, Japan [Nishimura, 1995; 95

Nishimura and Hamaguchi, 1993]. Brodsky et al. [1999] calculated vertical mass 96

discharge rates for the May 18, 1980 eruption of Mount St. Helens based on this model. 97

Our work follows these studies. 98

Other models for the source of co-eruption seismicity exist. For example, Uhira 99

and Takeo [1994] show that in the case of 1986 eruptions of Sakurajima volcano, Japan, 100

a single force model cannot explain long period (several seconds) seismic radiation 101

patterns as well as a model containing volumetric components of expansion and 102

contraction of rock around the source. Chouet et al. [in press] and Chouet et al. [in prep] 103

suggest that in the case of Kilauea volcano, explosive degassing bursts associated with 104

Strombolian eruptions are best modeled by a single force at the lowest frequencies (VLP 105

band) followed by higher frequency oscillations in the conduit and short-period energy 106

from the slug burst itself. 107

Overall the source of low-frequency energy radiated co-eruptively has been more 108

thoroughly studied than high-frequency energy radiated co-eruptively. McNutt and 109

Nishimura [2008] model co-eruptive tremor as resulting from radial oscillations of 110

conduit walls, but do not connect the amplitudes to the plume dynamics. Here we take a 111

different approach and model the seismic wave generation as resulting directly from the 112

mass discharge of the erupting column. 113

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2.2 The 2008 Eruption of Kasatochi Volcano 115

Kasatochi volcano is a 3 km wide island with a large crater lake in the Andreanof 116

Islands in the Aleutian Arc (figure 1). Though the volcano’s summit is 314 m in 117

elevation, the volcanic vent is near sea level. Prior to its 2008 eruption, little was known 118

about this volcano. Kasatochi volcano is not seismically or geodetically monitored. The 119

nearest seismometers are a short-period network operated by the Alaska Volcano 120

Observatory (AVO) at Great Sitkin volcano, 40 km west of Kasatochi (figure 1). The 121

nearest broadband station functioning at the time of the eruption, ATKA operated by the 122

Alaska Earthquake information Center (AEIC), is 80 km southeast of Kasatochi on Atka 123

Island. 124

After a brief but powerful earthquake swarm with earthquakes as large as M 5.8 125

[Ruppert et al., submitted] Kasatochi volcano had three large ash producing eruptions 126

with satellite-detectable plumes to 18 km above sea level (22:01 UTC 7 August 2008, 127

01:50 and 04:35 UTC 8 August 2008) [Carn et al., submitted; Waythomas et al., 128

submitted; Webley et al., submitted]. Co-eruptive tremor associated with each of these 129

three explosions was readily apparent in real-time seismic data for at least 20 minutes 130

(Figure 2). Infrasound and seismic data indicate that two additional smaller explosions 131

occurred in the following 5 hours during the episode of continuous ash emissions 132

[Arnoult et al., submitted]. Satellite data indicate that the third large explosion was 133

significantly more ash-rich and had a higher plume than the first two [Waythomas et al., 134

submitted, Fee et al., submitted]. Seismic data support this observation as the far field 135

amplitude of the third explosion was more energetic than the previous explosions. 136

Following the third explosion, ash vented continuously for roughly 16 hours. Satellite 137

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data indicate that the continuous ash emission phase had a decreasing density of ash 138

emission with time [Waythomas et al., submitted]. This observation is consistent with 139

seismic data, as the prolonged continuous ash emission phase radiated less seismic 140

energy than the initial 10 minutes of the third explosion . 141

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2.3 The 2006 Eruption of Augustine Volcano 143

The 2006 eruption of Augustine Volcano was far better monitored than the 2008 144

eruption of Kasatochi, in part because the volcano erupted previously in 1971, 1976, and 145

1986. Augustine, a volcano located in the lower Cook Inlet with summit at 1260 m above 146

sea level (figure 1), is instrumented with both broadband and short-period seismometers. 147

Increasing seismicity and deformation was noted months before the eruption onset 148

[Power and Lalla, in press]. The Augustine pre-eruptive seismic swarm was notably less 149

intense than that of Kasatochi. The largest earthquake associated with the eruption was M 150

2.1. The explosive phase of the eruption began on 11 January and continued through 28 151

January [see Power and Lalla, in press]. The ash producing eruptions of Augustine, 152

though numerous, were shorter in duration and lower in altitude that the Kasatochi 153

eruption. Augustine produced 13 ash plumes ranging in altitude from 9 to 14 km above 154

sea level. The co-eruptive tremor durations for these explosions, as determined from far-155

field seismic data, ranged from 3 – 7 minutes. The co-eruptive seismicity associated with 156

the Augustine eruption plumes was observed at a maximum distance of ~ 200 km, 157

whereas Kasatochi co-eruptive seismicity was visible to over 350 km distance. 158

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3. Relating Plume Height to Seismic Wave Amplitudes 161

Here we develop a composite model that combines existing fluid and solid 162

mechanical models to connect observable seismic wave amplitudes to volcanic plume 163

height. We test the model on the 2008 eruption of Kasatochi volcano and the 2006 164

eruption of Augustine volcano. The model consists of three distinct steps as illustrated in 165

Figure 3: 1) Relate maximum ash plume height to mass ejection rate of material from the 166

volcanic vent. 2) Relate mass ejection rate to a single downward force acting on the solid 167

Earth. 3) Calculate far-field seismic wave amplitudes resulting from the single force. 168

Each step is described in detail below. 169

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3.1 Step 1: Relate Plume Height to Mass Ejection Rate 171

We begin by assuming that plume height is controlled by thermal buoyancy for a 172

continuous ash-rich plume with dense rock equivalent volumetric ejection rate Q. This 173

fundamental physics for column height has its origins in studies of forest fire plumes 174

[Morton et al., 1956] and has been verified using eruption data [Carey and Sigurdsson, 175

1989; Sparks et al., 1997]. Slightly different relationships between Q and plume height 176

exist in the literature depending on the datasets used for calibration. We use the widely 177

cited Sparks et al. [1997], equation 5.1 178

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H=1.67 Q0.259 (1) 180

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where H is the column height in km above the vent and Q is measured in m3/s. Equation 182

(1) is an empirical regression with its form is motivated by analytic models of thermally 183

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buoyant continuous plumes in a stratified atmosphere [equation 4.15 of Sparks et 184

al.,1997; Wilson et al, 1978]. Observations used to constrain Equation 1 include 185

uncertainties both in plume height and ejecta volumes as discussed in Mastin et al. [2009] 186

and Sparks et al. [1997]. We use the term continuous to describe a plume where the 187

volcano is still activity venting at the time the plume reaches its maximum height. This is 188

often referred to as an attached plume. 189

An alternative model is a rapid ejection eruption where all of the mass is released 190

so quickly that the bottom of the plume has left the vent before the top of the plume 191

reaches its maximum height. Though this may have been the case for the first two 192

Kasatochi explosions, satellite data show that this was not likely the case for the third 193

explosion, as that explosion was the beginning of a 13 hour phase of continuous ash 194

emission [Waythomas et al., submitted]. Nonetheless, it is useful to realize that even in 195

this extreme case, the scaling in equation (1) is preserved and the only major 196

modification is the inclusion of the ejection duration. Morton et al. [1956] also developed 197

a relationship for a quickly released plume and found, 198

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H=1.87 x 10-3 E1/4 (2) 200

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where H is height in kilometers and E is thermal energy in Joules. The thermal energy is 202

primarily carried by the solid particles and E=cp ρ ΔT Q τ where cp is the heat capacity, ρ 203

is the density of the solid rock, ΔT is the temperature difference between the plume and 204

the ambient air and τ is the duration of the ejection phase. Substituting in reasonable 205

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values of the parameters (cp = 1 KJ/kg/K, ρ = 3000 kg/m3, ΔT = 103 K, τ=100 s), yields 207

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H=1.4 Q1/4 (3) 209

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which is nearly identical to the protracted source result (equation 1). Thus, the model is 211

insensitive to the assumed relative duration of the source. 212

We next relate the thermal ejection rate to the mass ejection rate, as the later 213

quantity is most directly associated with the momentum flux which ultimately generates 214

seismic waves. If dense rock equivalent density is ρ, then 215

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ρQM =•

(4) 217

where •

M is the mass ejection rate. 218

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3.2 Step 2: Relate Mass Ejection Rate to Single Force 220

We assume that the force system acting on the Earth during eruption can be 221

represented as a single force following the work of Kanamori and Given [1982], 222

Kanamori et al. [1984], and Brodsky et al. [1999]. Due to a lack of quality broadband 223

seismic data and poor coverage of the focal sphere typical of Aleutian eruptions, we are 224

not able to invert the seismic radiation field of co-eruptive Kasatochi for an equivalent 225

force system. Thus we cannot directly demonstrate that a single force is the most 226

appropriate source model for Kasatochi. Instead we start with this simple seismic source 227

model and test its applicability. 228

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Using the mass ejection rate from (4), we can derive the equivalent vertical force 229

F on the Earth using a momentum balance following Brodsky et al. [1999]. 230

FvMdtMomentumd ==•

)( (5) 231

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where v is the velocity of material exiting the vent. This is sometimes termed the rocket 233

equation [Thompson, 1972]. In the case of a rocket the force propelling a rocket upward 234

is equal to the momentum discharge rate of the fuel behind it. In the case of a volcano the 235

rocket is inverted, as it discharges momentum upward. 236

To use equation 5, we need to estimate v. For large-scale eruptions, the extreme 237

overpressures at the vent require supersonic flow. In steady-state the flow is choked at the 238

vent with the vent velocity equal to the sound speed of the erupting material [Kieffer, 239

1981; Woods, 1995; Mastin, 2007]. The mixture of ash and gas in a plume has a greatly 240

depressed sound velocity relative to either the solid or gas end member [Rudinger, 1980]. 241

For instance, the May 18, 1980 blast of Mount St. Helens is thought to have a sound 242

speed of ~100-250 m/s based on the temperature and relative mass fraction of gas and 243

solid phases [Kieffer, 1981; Brodsky et al., 1999]. Kieffer and Sturtevant [1984] argue 244

that the low sound velocity of particulate-laden flows allows vent velocities as low as 245

meters per second. Alternative derivations of the vent velocity based on conduit flow 246

models also arrive at the range of 100-250 m/s [Papale and Longo, 2008]. McNutt and 247

Nishimura [2008] summarize vent velocities for a large number of eruptions globally and 248

arrive at a range of 28-420 m/s with the caveat that the lowest end of the range is likely 249

an underestimate due to the neglect of the coupled gas flow. We consider vent velocities 250

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within either the observational or theoretical ranges, i.e., 28-420 m/s, reasonable and 251

consistent with prior work. 252

The simplest application of equation 5 would be using data with periods of 253

minutes, consistent with the time it takes to evacuate the magma chamber, following 254

Kanamori et al. [1984]. We lack data at those frequencies, so we explore the possibility 255

that the higher frequency seismic waves radiated by turbulence in the volcanic jet scale 256

with energy radiated at low frequencies. This requires some discussion of the role of 257

turbulence in the volcanic jet and its effect on seismic waves. 258

As the generation of seismic waves is a linear process, the observed seismic 259

spectrum is a product of the propagation spectrum in the elastic medium (see equation 6 260

below) and the spectrum of momentum ejection from the vent (source spectrum). The 261

spectrum at the source can reflect the spectrum of vent velocities, which in turn are 262

controlled by properties of the flow. In a turbulent flow fluid velocity generally fluctuates 263

with well-defined statistical properties. For instance, for homogeneous, isotropic 264

turbulence in an incompressible flow, the energy density spectrum, S, of the flow is 265

related to frequency S ~ f -5/3, and the integral of the energy spectrum over all 266

wavenumbers is proportional to the autocorrelation of the velocity field [Kolmogorov, 267

1954; Matthieu and Scott, 2000]. A volcanic eruption involves transient, compressible 268

flow that has multiple origins of velocity fluctuations including the turbulent field, 269

internally generated periodic collapses and evolving vent shape [Ogden et al., 2009]. For 270

simplicity, in this initial study we assume that the vent velocity spectrum is flat and the 271

spectral amplitude is constant for all frequencies. We also do not consider here the 272

influence of density fluctuations on the source spectrum. These assumptions are difficult 273

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to evaluate in the absence of direct observations of the turbulent structure of plumes and 274

thus can only be justified by testing the consistency of the seismic data with the model. 275

We use the spectral simplifications as a starting place to explore the link between plume 276

height and seismic waves. For future studies infrasound data from large eruptions could 277

potentially be used to provide constraints on the true turbulence spectrum of volcanic jets 278

[eg. Matoza et al., 2009; Fee et al., submitted]. 279

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3.3 Step 3: Relate Single Force to Far-Field Seismic Waves 281

Finally, we calculate the far-field surface waves resulting from a single force. For 282

a vertical single force F, the displacement amplitude of a fundamental mode Rayleigh 283

wave with wavenumber k traveling in a half-space is 284

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u =0.07F

μk

r (6) 286

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where μ is the shear modulus and r is the distance from the source [Aki and Richards, 288

1980]. Equivalently, equation 6 describes the propagation spectrum as a function of 289

wavenumber k. 290

Equation 6 is applicable for a single mode. For real data, we apply a narrow 291

bandpass that captures multiple modes. Since the modal spectrum is dense near 1 s, it is 292

difficult to analytically add all of the relevant modes together to construct the amplitude. 293

Instead, we generate synthetic seismograms using a wavenumber-frequency summation 294

method and bandpassed the synthetics to obtain a correction for the finite band-pass 295

(Figure 4) [Herrmann, 1987]. Empirically, we find that an impulse source for a given F 296

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bandpassed at 0.5-1.5 Hz at the regional distances studied here produces an amplitude 2.4 297

times greater than that predicted by equation 6 for a half-space homogenous velocity 298

model. This correction factor also holds for a more realistic velocity model like AK135 299

[Kennett et al., 1995; Montaigner and Kennett, 1995]. Therefore, we apply this factor of 300

2.4 correction to our predicted amplitude found using equation 6. A potential 301

complication to this seismic model is the dominance of scattering at high-frequencies. 302

For this initial study, we confine the calculation to the direct energy. 303

By assuming values for μ, v, and ρ, we can calculate ground displacement as a 304

function of column height and observed wavenumber k. Combining equations 1 with 4-6 305

with the finite band-pass correction yields 306

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u =0.17ρv

μk

rH / 1.67( )1/0.259

(7) 308

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Equation 7 predicts the amplitude of the seismic waves resulting from a certain column 310

height based entirely on known physics. The column height is an observed quantity and 311

the wavenumber and epicentral distance r are generally well-resolved for seismic data as 312

is the dense rock density ρ and the shear modulus μ. Therefore, utilizing Equation 7 313

requires estimating only one relatively poorly constrained eruptive quantity, the vent 314

velocity v. 315

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3.4 Model Applicability 317

We now discuss the range of conditions and eruption styles for which this model 318

is appropriate. Because multiple physical processes produce seismic waves at a variety of 319

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frequencies during large ash producing volcanic eruptions, it is sometimes difficult to 320

connect broadband seismic amplitudes with eruptive properties directly, particularly in 321

the near field and for small eruptions. For example, local processes including rock falls 322

and pyroclastic flows can dominate seismic recordings local to the volcano. 323

Because the calculated mass ejection rate only accounts for erupted material that 324

provides energy for buoyant ascent of ash-rich plumes, our model is not applicable for 325

phreatic eruptions or eruptive phenomenon apart from the plume such pyroclastic flows 326

and lava fountaining. In the case of eruptions where the majority of the mass ejected is 327

not entrained in the plume, the model may overestimate plume height for a given 328

amplitude of ground shaking. 329

The model is also only appropriate for plumes that rise to altitudes of 5 km or 330

higher for three reasons: 1) The empirical regression in Sparks et al. [1997] is not 331

constrained for plume heights below 5 km. 2) The assumption that the upward mobility 332

of the plume is thermally driven may not be correct at heights where material can travel 333

ballistically. The maximum ballistic height of a projectile with initial velocity v occurs 334

when the initial kinetic energy equals the gravitational potential energy, i.e., ½ M v2 = M 335

gh, where M is mass, g is the acceleration due to gravity, and h is ballistic height. Thus, 336

the maximum ballistic height is ½ v2 / g. Assuming v = 300 m/s or less, ballistics could 337

be ejected to 4.5 km. 3) The gas thrust region can be dominated by the dynamics of the 338

compressible flow rather than the thermal buoyancy [Ogden et al., 2009]. In the case of 339

small eruptions at Tungurahua and Santiaguito volcanoes Johnson et al [2004] and 340

Johnston et al. [2005] show that amplitudes of co-eruptive seismic waves scale poorly 341

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with eruption intensity. In both cases however, eruption plumes analyzed were less than 4 342

km in height. 343

This model is also not appropriate in cases where the source of co-eruptive 344

volcanic tremor is not the volcanic vent, but elsewhere in the system. For example during 345

the 2008 eruption of Okmok volcano, co-eruptive tremor amplitude correlated poorly 346

with volcanic plume height after the opening phase of the eruption. Haney [submitted] 347

estimates that the VLP tremor occurred along the length of a dike at 2 km depth. Thus in 348

the case of Okmok, Haney concludes that the primary tremor source likely reflected the 349

flux of magma from the shallow magma chamber into the dike rather than mass ejection 350

at the vent directly. 351

Similarly, we must avoid performing this analysis on time series where the 352

seismic data is dominated by earthquakes sourced from deeper in the system. In the case 353

of the Kasatochi eruption, for example, the far-field seismic energy radiated by pre-354

eruptive earthquakes was up to 4 times larger in amplitude than the co-eruptive tremor 355

itself at the distances we consider when seismic energy is averaged over 10 minute 356

windows. For this reason, we limit our analysis at Kasatochi to a time period free of large 357

earthquakes. 358

The model is only applicable when the volcanic vent is open and freely emitting 359

material. Therefore, caution must be used when considering the initial stages of eruption. 360

During this time frame a large portion of the seismic energy may be related to “throat 361

clearing” or removing the lid that caps the magma body. For example, the first large ash 362

producing explosion during the 2006 eruption of Augustine volcano produced stronger 363

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ground shaking than the subsequent 20 ash producing explosions, although the plume 364

height of the first explosion was similar to that of many later explosions. 365

Finally, atmospheric effects should be considered when applying our model. For 366

example, the model is most appropriate for large eruptions with strong plumes where 367

tropospheric wind shear and other atmospheric instabilities are minimal. In addition, 368

because the buoyancy of volcanic ash plumes can increase significantly if atmospheric 369

water vapor is entrained in the plume [Woods, 1993; Tupper et al., 2009], our model is 370

more applicable in dry sub-polar environments such as Alaska than in moist tropical 371

environments. However, in all environments atmospheric humidity does not have an 372

appreciable effect on large eruption columns greater than 15 km in height [Woods, 1993]. 373

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4. Data Analysis and Results 375

4.1 Kasatochi 376

To apply our model to the eruption of Kasatochi, we begin by measuring co-377

eruptive ground displacement after correcting for instrument response at short-period and 378

broad-band seismometers primarily located west of the volcano. Stations analyzed are 379

shown in Figure 1. The geometry of the Aleutian Islands provides us with a nearly linear 380

array. We are not able to use data recorded at Great Sitkin volcano because co-eruptive 381

seismic waves are clipped there (Figure 2). We also do not use data east of Atka Island 382

because seismic waves generated by Kasatochi volcano would likely be contaminated by 383

those generated by the simultaneous eruption of Okmok volcano, roughly 350 km east of 384

Kasatochi. 385

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This use of far field data is an advantage because far field data is not 386

contaminated by energy radiated from secondary surficial processes such as the huge 387

pyroclastic and block and ash flows which left the island inundated in up to 15 meters of 388

tephra. Secondary seismic sources such as rock falls and pyroclastic flows decay more 389

quickly with distance than a single force source associated with magma removal because 390

these secondary sources: 1) transport a small percentage of mass compared to the total 391

mass ejected, 2) are multi-pole sources distributed in time and space on the volcano’s 392

flank in contrast to the concentrated single force mono-pole associated with overall mass 393

ejection [Kanamori et al., 1984, Appendix A], and 3) are largely surficial sources. Jolly et 394

al. [2002] show that in the case of Soufriere Hills Volcano, Monstserrat, seismic waves 395

generated by pyroclastic flows are highly attenuated (Q=20) due to their surficial travel 396

paths. 397

We focus our study on the initial pulse of energy radiated by the third and most 398

ash rich explosion at Kasatochi (22:01 UTC, 7 Aug 2008) (Figure 5), as this explosion 399

began a 13 hour phase of continuous venting and is not contaminated by earthquakes. 400

The strongest phase of this energy packet is several minutes in duration and can be 401

clearly identified as it moves out appropriately with distance away from Kasatochi 402

(Figure 2). We measure the peak amplitude of ground displacement at 1 second period by 403

bandpass filtering data between 0.5 and 1.5 Hz (Figure 5). We choose 1 second period 404

because it is well within the response range of short-period instruments, it is below the 405

frequency range generally radiated by pyroclastic flows [eg. Zobin et al., 2009], yet it is 406

low enough to minimize site effects. At higher frequencies site effects cause significant 407

scatter in our measurements. Figure 6 shows that the relative amplitude of the observed 408

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seismic waves fall off with distance as r1 1/2, where r is radius from the source, as would 409

be expected for Rayleigh waves. This fall-off with distance is a helpful corroboration of 410

the direct Rayleigh wave interpretation of the co-eruptive seismicity in equation 6. To 411

evaluate fit of our model, we fit a line to our data points by forward modeling, varying v. 412

We assume a shear modulus of μ = 3 x1010 Pa, and specify cfk π2= , where the 413

frequency in the middle of the observed bandpass f = 1 s and c = 3x 103 m/s. Dense rock 414

equivalent, ρ, is 2700 kg/m3. 415

Figure 6 shows ground displacement measurements for the Kasatochi eruption 416

and predicted ground displacement with distance. In our preferred model (black line) v = 417

80 m/s in the case of an 18 km plume. This velocity is in the range of those predicted by 418

Kieffer and Sturevant [1984] and observational estimates for vent velocity of specific 419

eruptions as summarized in McNutt and Nishimura [2008]. However, this vent velocity is 420

at the lower end of the previously established ranges. The data therefore is consistent 421

with either the flat spectrum source model as proposed above, or may accommodate an 422

increase in spectral amplitude of the source with increasing period as would be expected 423

from turbulence models [Kolmogorov, 1941; Matthieu and Scott, 2000; Matoza et al., 424

2009]. 425

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4.2 Augustine 427

In order to investigate the robustness of the model to a different eruptive setting, 428

we test its applicability for the 2006 eruption of Augustine volcano. Although Augustine 429

is locally instrumented, for our analysis we use the short-period seismometer, OPT, 430

located 37 km from the erupting vent. This is required since our model uses far-field 431

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surface waves and because we do not want the tephra ejection signal contaminated by 432

more surficial local processes that dominate the short-period signals on the island, such as 433

rockfalls and pyroclastic flows. In the case of the 2006 eruption of Augustine McNutt 434

[submitted] shows that pyroclastic flows were only visible seismically on stations near 435

the flow travel path and were not visible on the volcano flanks opposite the flow path. 436

Figure 7 shows the 2 s – 1.5 Hz displacement amplitudes at OPT for explosions at 437

01:40 UTC 14 January and 16:57 UTC 17 January 2006, which had plume heights of 438

10.5 and 13 km above sea level respectively based on radar and pilot reports [Power and 439

Lalla, in press]. The far-field short-period amplitudes of these two explosions are similar 440

to other explosions in the sequence (with the exception of the initial 11 January 441

explosion). We select these explosions to analyze because they are representative of the 442

entire eruption. Because the initial explosions from the 2006 eruption of Augustine 443

volcano may have included energy from brittle failure vent clearing processes, we do not 444

analyze data from 11 January 2006. In the case of Augustine, the volcanic vent is well 445

above sea level at ~1260 m altitude. Thus, when applying our model we correct for the 446

difference between vent altitude and the reported plume heights, which are defined with 447

respect to sea level. 448

As in the Kasatochi case, we assume a shear modulus of μ = 3 x1010 Pa, and 449

specify cfk π2= , where the observed frequency is f= 1 s and c = 3x 103 m/s. Dense 450

rock equivalent, ρ, is 2700 kg/m3. In the case of the 13 km plume at Augustine on 17 451

January 2008, our model performs well, as it estimates the correct plume height for vent 452

velocities of 78 m/s based on seismic data. In the case of the 10.5 km plumes on 14 453

January 2008, our model estimates the correct plume height if the velocity at the vent is 454

21

95 m/s. These velocities are within the previously defined range of reasonable values (28-455

420 m/s), but as in the case of Kasatochi, they are at the lower end of the expected values. 456

Once again, there is ambiguity between the appropriateness of the flat spectrum and the 457

possibility of a decrease in spectral amplitude with increasing frequency. 458

459

5. Dynamical Implications 460

Because our model fits the data, the single force model appears to be an 461

appropriate force system in the case of the two eruptions we consider. The most 462

surprising aspect of the model’s success is the fact that we can relate plume height to 463

seismic wave amplitude using ~1 s radiated waves assuming a reasonable v of 78-95 m/s. 464

This consistency implies that the high-frequency fluctuations of the plume are 465

comparable in magnitude to the long-period eddies. The fact that our v estimates are at 466

the low-end of the expected range allows that the source spectral amplitude may be 467

decreasing with frequency. This finding is not surprising as turbulent flows commonly 468

have stronger eddies at long-periods than short-periods [Kolmogorov, 1941; Matthieu 469

and Scott, 2000; Fee et al., submitted]. The spectrum can also be modified by additional 470

processes like supersonic flow [Smits and Dussauge, 2006]. The length scale of turbulent 471

eddies in a volcanic plume has only recently been measured in plumes [Andrews and 472

Gardner, 2009] and thus far there is little independent constraint on the relative strength 473

of various frequencies in the plumes. The seismic measurements presented in this paper 474

suggest that future work on the seismic source spectrum of plumes would be a fruitful 475

avenue into the fluid dynamics of erupting jets. 476

477

22

6. Practical Implications 478

For the Aleutian Islands and many other remote volcanoes in the North Pacific the 479

primary safety concern is the threat of ash clouds to aircraft. Airlines are most concerned 480

about ash clouds that reach aircraft cruising altitudes. Estimating volcanic plume heights 481

in remote environments in real-time (and even in retrospect) is always challenging, 482

particularly when clouds are obscuring activity and the timing of satellite passes is not 483

optimal. With further development this model could potentially be used to constrain 484

plume heights in near-real-time. In the case of the Kasatochi eruption, we would have 485

estimated plume heights of 13-19 km using our model based on seismic data alone, 486

potentially giving airlines notice that the eruption was large enough to be dangerous. To 487

narrow this range in height, we would need more accurate estimates of v and its spectrum 488

or a larger empirical database of v measurements from similar eruptions. These estimates 489

could possibly be obtained through other data sources such as infrasound or radar or by 490

studying a range of eruptions with our model and establishing a range of possible 491

velocities observationally. 492

To illustrate the potential utility of the approach, we examine our ability to 493

characterize future eruptions at Kasatochi Volcano using the nearest seismic stations, 494

incorporating some of the uncertainty in the free parameter, v. Figure 8 shows calculated 495

ground displacement at Great Sitkin stations estimated for eruptions of Kasatochi with a 496

variety of reasonable v values. We assume a detection threshold of 0.4 μm, as AVO 497

short-period seismometers in the Aleutian arc often have noise below this threshold for 498

raw unfiltered data. This is a conservative estimate for eruption detection provided the 499

recording environment is relatively quiet and the seismometers are functioning optimally. 500

23

Figure 8 shows that generally, we can expect to detect co-eruption seismicity for 501

explosive eruptions with plumes that reach 9 – 12 km height above sea level or higher at 502

Kasatochi volcano. 503

This same approach allows us to estimate detection capability for eruptions at 504

other unmonitored volcanoes. For instance, Bogoslof and Seguam volcanoes are small 505

unmonitored Aleutian Islands, like Kasatochi (Figure 1). Bogoslof most recently erupted 506

in 1992 and had 7 additional historic eruptions. Seguam most recently erupted in 1993 507

and had 5 additional historic eruptions. The nearest seismometers to Bogoslof and 508

Seguam are those of the Okmok and Korovin seismic networks respectively, ~50 km and 509

~100 km distant. Using figure 9, we estimate that large continuous ash producing 510

eruptions at Bogolof and Seguam would be visible in high frequency data at the nearest 511

seismic stations if the plumes reached roughly 12 km in altitude, in the conservative case 512

where v=200 m/s. Again, however, a better understanding of vent velocity is needed, as 513

reasonable variation in v (50 – 200 m/s) leads to scatter in predicted plume heights over a 514

4 km range. 515

With better constraints on v and its spectrum,, our model could provide a plume 516

height estimate independent of remote sensing techniques that could be considered along 517

with estimates from other sources. To do this more work would need to be done to 518

constrain errors, explore applicability to different eruptions, and test velocity model 519

assumptions. Another consideration is that several physical processes cause seismic 520

tremor in the Earth’s crust, so other data are needed to confirm that tremor observed is 521

co-eruptive tremor before applying our model. Uncertainties in tremor source process 522

24

would likely lead to far more false positives than false negatives. Thus our model could 523

be used as a conservative indicator of threats to aviation from volcanic ash. 524

525

7. Conclusions 526

Our model is generally successful in relating plume height to ground shaking in 527

the 2008 eruption of Kasatochi volcano and the 2006 eruption of Augustine volcano. This 528

suggests that using plume height to calculate the mass ejection rate and interpreting the 529

resulting force system as a single force are appropriate techniques. The single force 530

model appropriate at low frequencies also fits measurements reasonably well at 531

frequencies of ~ 1 s. This surprising observation suggests that force resulting from the 532

high frequency turbulence in volcanic jets can be a useful indicator of plume height. 533

The success and applicability of the model to other vulcanian and plinian 534

eruptions globally remains to be evaluated. However, the results from Kasatochi and 535

Augustine volcanoes are sufficiently encouraging that more work in this field is 536

warranted. Future studies should focus on other eruptions to explore errors and the range 537

of applicability of the model, consider more complicated velocity structures and indirect 538

wave arrivals, consider atmospheric effects, and further investigate frequency variations 539

in energy radiated by the volcanic jet. Further development of this model may give us a 540

means to better understand the relationship of ground shaking to plume height and 541

explore eruption dynamics of the turbulent jet. In the best case scenario, this avenue of 542

research may lead us to constraining plume height based on seismic wave amplitudes in 543

near-real-time. 544

545

25

546

Acknowledgements 547

Review by Larry Mastin, Darcy Ogden, David Schneider, David Hill, and David Fee 548

significantly improved this manuscript. Seismic stations used in this study were originally 549

installed with funding from the Federal Aviation Administration. 550

551

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Figure Captions 732

733

Figure 1 – Map of central Aleutian arc showing locations of Augustine volcano, 734

Kasatochi Island and neighboring islands and volcanoes. Seismic stations used in this 735

study are shown with triangles. Red are short-period instruments. Green are broadband 736

instruments. Great Sitkin volcano had a functioning seismic network at the time of the 737

Kasatochi eruption. However, those data were clipped and not used in this study. 738

739

Figure 2 - Waveform of third ash producing explosion of Kasatochi as recorded on 740

short-period stations, GSMY, ETKA, KIKV, TASE, and GANE. Line shows moveout of 741

surface wave energy packets at 3.4 km/s. 742

743

Figure 3 – Cartoon depicting processes associated with the three calculation steps in our 744

model. Step 1 relates plume height to mass ejection rate. Step 2 relates mass ejection rate 745

to a downward acting sngle force. Step 3 calculates seismic waves radiated by the single 746

force. 747

748

34

Figure 4 - Synthetic seismograms from an impulse source of the same strength as 749

inferred for the Kasatochi eruption. The synthetics are bandpassed using the same filter as 750

the observed seismograms (0.5-1.5 Hz) to measure the amplitude in the narrow band-pass 751

corresponding to equation 7. The synthetics were produced using a regional f-k 752

integration and the ak135 model [Hermann, 1987; Kennett et al., 1995; Montaigner and 753

Kannett, 1995]. Synthetics are calculated for distances to stations GSMY, ETKA, KIKV, 754

TASE, and GANE. The resulting phase velocity matches the observations in Figure2. 755

756

757

Figure 5 – a) Raw waveform of third large ash producing explosion of Kasatochi, as 758

recorded at short-period station TASE at Tanaga Volcano, 177 km distant from 759

Kasatochi. b) First distinct burst of energy, which we analyze by bandpass filtering raw 760

waveform between 0.5 and 1.5 Hz and measure maximum displacement. c) Spectrogram 761

of energy recorded at TASE. 762

763

Figure 6 – Ground displacement at 1 s period of co-eruptive seismicity with distance 764

from Kasatochi volcano for the large ash producing explosion at 2008. Large dots are 765

broadband stations ATKA and AMKA. Small dots are short-period stations. Preferred 766

forward model fit shown with solid line (v=80m/s). 767

768

Figure 7- Ground displacement at 0.5 - 2 s period of co-eruptive seismicity with distance 769

from Augustine volcano for the large ash producing explosions on January 14 and 17, 770

2006. Model fits for velocities of 78 m/s and 95 m/s are shown with lines. 771

35

772

Figure 8 – Expected ground displacement at 1 s period due to a plume observed at Great 773

Sitkin, 40 km distant from Kasatochi, for three vent velocities, 50 m/s (dashed), 100 m/s 774

(solid), and 200 m/s (dotted). Vertical line indicates average noise level at Great Sitkin 775

seismic stations. 776

777

Figure 9 – Minimum detectable plume heights with distance, assuming a conservative 778

detection threshold of 0.4e-6 m at 1 second period and vent velocities of 50 m/s (dashed), 779

100 m/s (solid), and 200 m/s (dotted). 780

180 -178 -176

51

52

53

KasatochiGreat Sitkin

KanagaTanagaGareloi

Amchitka Island

Atka

Figure 1

0 400 kmAlaska

BogoslofSeguam Augustine

Moveout 3.4 km/s

100 200 300 400 500 600 700 time (s)

Velo

city

(nm

/s)

Figure 2

Figure 3

40 km70 km

118 km

177 km

227 km

3.4 km/s

Figure 4

20000 40000 60000 80000

x 10

-5000

0

5000

22000 26000 30000 34000�-5000

0

5000

Time (s)

Freq

uenc

y

0 10000 20000 30000 400000

0.5

1

a

b

c

Vel

ocity

(nm

/s)

Figure 5

10 2 10 310 -2

10�-1

10 0

10�1

10�2

Dis

plac

emen

t at 1

s pe

riod

(µm

)

Distance from Kasatochi (km)

Broadband seismometer

Short period seismometer

Figure 6

0 0.5 1 1.5 2 2.5 38

9

10

11

12

13

14

15

16

Plum

e H

eigh

t (km

)

Displacement at OPT, 37 km Distance (mm)

14 Jan. 2006

17 Jan. 2006

Figure 7

v = 78 m/s

v = 95 m/s

0 0.5 1 1.5 2 2.5 38

10

12

14

16

18

20

Plum

e H

eigh

t (km

)

Displacement (µm)

Figure 8

v = 50

v = 100

v = 200

10 10 10 10

6

8

10

12

14

16

18

Distance (km)

Plum

e H

eigh

t (km

)

v = 100 m/s

v = 50 m/s

v = 200 m/s

Figure 9

1 2 3 4