PUBLICATIONS. Journal of Geophysical Research: Solid Earth

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1 PUBLICATIONS Journal of Geophysical Research: Solid Earth RESEARCH ARTICLE Key Points: Volcanic tremor may result from cataclastic deformation along conduit wall Tremor source completely masked by conduit resonance/waveguide phenomena Correspondence to: R. P. Denlinger, Citation: Denlinger, R. P., and S. C. Moran (2014), Volcanic tremor masks its seismogenic source: Results from a study of noneruptive tremor recorded at Mount St. Helens, Washington, J. Geophys. Res. Solid Earth, 119, , doi:. Received 19 SEP 2013 Accepted 30 DEC 2013 Accepted article online 4 JAN 2014 Published online 31 MAR 2014 Volcanic tremor masks its seismogenic source: Results from a study of noneruptive tremor recorded at Mount St. Helens, Washington Roger P. Denlinger 1 and Seth C. Moran 2 1 Cascades Volcano Observatory, Vancouver, Washington, USA, 2 U.S. Geological Survey, Vancouver, Washington, USA Abstract On 2 October 2004, a significant noneruptive tremor episode occurred during the buildup to the eruption of Mount St. Helens (Washington). This episode was remarkable both because no explosion followed, and because seismicity abruptly stopped following the episode. This sequence motivated us to consider a model for volcanic tremor that does not involve energetic gas release from magma but does involve movement of conduit magma through extension on its way toward the surface. We found that the tremor signal was composed entirely of Love and Rayleigh waves and that its spectral bandwidth increased and decreased with signal amplitude, with broader bandwidth signals containing both higher and lower frequencies. Our modeling results demonstrate that the forces giving rise to this tremor were largely normal to conduit walls, generating hybrid head waves along conduit walls that are coupled to internally reflected waves. Together these form a crucial part of conduit resonance, giving tremor wavefields that are largely a function of waveguide geometry and velocity. We find that the mechanism of tremor generation fundamentally masks the nature of the seismogenic source giving rise to resonance. Thus multiple models can be invoked to explain volcanic tremor, requiring that information from other sources (such as visual observations, geodesy, geology, and gas geochemistry) be used to constrain source models. With concurrent GPS and field data supporting rapid rise of magma, we infer that tremor resulted from drag of nearly solid magma along rough conduit walls as magma was forced toward the surface. This is an open access article under the terms of the Creative Commons Attribution-NonCommercial-NoDerivs License, which permits use and distribution in any medium, provided the original work is properly cited, the use is non-commercial and no modifications or adaptations are made. 1. Introduction The eruption of Mount St. Helens was dominated by the extrusion of a series of spines of solidified dacite lava [Schilling et al., 2008; Vallance et al., 2008], producing a lava dome complex ~ m 3 in volume (S. Schilling, U.S. Geological Survey (USGS), written communication, 2013). The eruption was accompanied by millions of earthquakes, with individual earthquakes dominantly low frequency in nature and highly repetitive both in waveform and in the time interval between events [Iverson et al., 2006; Matoza et al., 2009; Moran et al., 2008b; Thelen et al., 2008; Waite and Moran, 2009]. The vent-clearing phase of 23 September to 5 October 2004 was particularly seismogenic, with M > 2 earthquakes occurring every minute during the peak [Moran et al., 2008b]. The intense seismicity associated with the vent-clearing phase provides a relatively unique opportunity to investigate near-surface mechanisms of conduit formation at a dacitic stratovolcano. Two notable features of the vent-clearing seismicity motivate the present study. The first was a rapid decrease in seismic energy release following each explosion and/or tremor episode from 1 to 5 October 2004 [Moran et al., 2008a, 2008b]. The two most notable decreases occurred following a phreatic explosion on 1 October (the first explosion of the eruption), which was followed by ~6 h of quiescence, and a 50 min long energetic tremor episode on 2 October, which was followed by a ~2 h long decline. The second notable feature was the 2 October tremor itself, which was recorded at stations over 240 km away and was energetic enough to cause the USGS Cascades Volcano Observatory to raise the level-of-concern color code to red [Scott et al., 2008]. However, no explosion or anomalous steaming or gas discharge occurred in association with the 2 October tremor [Moran et al., 2008a], providing a rare documented instance of energetic nonexplosion tremor. Although there was no explosion or gas/steam emission associated with the 2 October tremor, there is ample evidence that the tremor occurred in association with rise of magma within the conduit. A bulging of the glacier-covered crater floor south of the lava dome (Figure 1) was first observed on DENLINGER AND MORAN The Authors. 2230

2 crater bulge Figure 1. The deforming crater floor at Mount St. Helens as it appeared on 14 October 2004 (photo by W. E. Scott, USGS). A crater formed by an explosion the day before and lies adjacent to the steaming area, and a bulge is rapidly growing to the right. 26 September [Scottetal., 2008] and became progressively larger with time, reaching ~ m 3 in volume by 4 October [Vallance et al., 2008]. This bulge heralded the emergence of a spectacular recumbent spine (Figure 2). As the bulge was growing, concomitant up and northward motion of a continuous GPS station located on the lava dome just north of the bulge and down and southward motion at Johnston Ridge Observatory (JRO), ~9 km north of the bulge, collectively indicated upward displacement of magma within the conduit at depths to 9 km (Figure 3) [Lisowski et al., 2008]. After the dome GPS station was destroyed by an explosion on 1 October, the motion of GPS station JRO continued down and toward the volcano throughout the 4 year long eruption [Lisowski et al., 2008]. The rapid decline of seismicity immediately following both the 1 October explosion and the 2 October tremor implies that pressure Figure 2. The spectacular grooved, gouge-covered spine emerged from the vent (at the base of the smooth spine) in late October and is still extruding in this 21 February 2005 photograph (looking southeast; photo taken by S. Schilling, USGS). The cross section of the spine is approximately 100 m by 200 m, and the smooth portion of the spine is about 400 m long. DENLINGER AND MORAN The Authors. 2231

3 Figure 3. Seismicity and deformation associated with the 2 October 2004, tremor episode. SHW is a seismic station, and RSAM stands for relative seismic amplitude measurement, a running average of the velocity transducer output. The tremor occurred during a period when seismic energy release was fluctuating and dropping rapidly. Seismic energy release reached a peak on 1 October just prior to the first large explosion, which was followed by 4 h of seismic quiescence. On 2 October rapid-fire M2-M3.5 earthquakes abruptly stopped when tremor started, and this tremor was followed by nearly an hour of seismic quiescence. A short burst of tremor also occurred the following day. Details of the seismicity are given in Moran et al. [2008a], and the deformation is described by Lisowski et al. [2008]. relief in the conduit magma following the explosion and tremor temporarily relieved stresses. This observed sequence of events motivates us to consider amodelforvolcanictremorthatdoesnotinvolve energetic gas release from magma but does involve movement or pressure relief of conduit magma through extension. In this paper we analyze the tremor observed around Mount St. Helens on 2 October 2004 (local seismic network in Figure 4), to look for clues as to its source. We propose that the conduit wavefield that generates the surface waves comprising these signals develops and propagates in essentially the same way as that proposed by Jousset et al. [2003] and Neuberg et al. [2000]. We elaborate on the two modes of conduit resonance mentioned by Jousset et al. [2003] for wide conduits and show how P and S head waves induced by any seismic source within a conduit filled with low-velocity magma or rock interferes with critically reflected waves trapped inside the conduit to produce resonance. As shown by numerical work [Jousset et al., 2003] and replicated here, this wavefield propagates along the conduit to the surface and forms surface waves. Drawing on detailed analyses of seismic waves along boreholes [Paillet and White, 1982; Paillet et al., 1987], we demonstrate that the observed tremor displays the characteristics of borehole tremor, or equivalently tremor in a waveguide. Interpreted in this way, the mechanism of tremor masks the true nature of its seismogenic source and instead reflects the width of the conduit relative to the seismic velocity of the magma filling it. The duration and spectral content of the 2 October tremor requires a sustained, nearly hourlong, broad bandwidth seismic source operating within the conduit. Although other source mechanisms may be invoked, we consider the most likely seismogenic source to be brittle failure induced by shearing of nearly solid magma along rough conduit walls. 2. Seismic Tremor on 2 October 2004 The 2 October tremor began at 1917 UTC, immediately following a ~1 km deep, low-frequency M2.7 earthquake with a normal faulting focal mechanism (as determined by the Pacific Northwest Seismic Network). DENLINGER AND MORAN The Authors. 2232

4 Figure 4. Map of Mount St. Helens showing distribution of seismic stations surrounding the volcano. The local seismic net within 9 km of the crater was used in this analysis. Station JRO is three component; all other stations are vertical component only. The Nyquist frequency for all stations is 50 Hz. Although the timing is suggestive of a direct relationship between the earthquake and tremor (Figure 5), we note that the earthquake had waveforms similar to earthquakes occurring over the previous 24 h, indicating that the earthquake was part of a long-lasting, quasi steady-state process that terminated with the onset of tremor. Tremor amplitudes progressively intensified, reaching peak reduced displacement amplitudes of cm 2 after 30 min (Figure 6) that were recorded at stations 240 km distant [Moran et al., 2008b]. At 1955 UTC amplitudes began to decay to background levels. The strong tremor then briefly flared for 4 5 min before finally declining to nearly ambient noise levels at 2008 UTC (1208 PST in Figure 6). This tremor was relatively broadband in nature, with dominant frequencies in the Hz range [Moran et al., 2008b] and weaker signals out to 5 Hz; the tremor envelope and duration were virtually identical regardless of direction or distance for several hundred kilometers (Figure 5). After tremor ceased there was a ~50 min long decline in earthquake rates and sizes, which was not as abrupt or as long-lived as the 6 h hiatus after the 1 October explosion [Moran et al., 2008b]. We first attempted to determine if the tremor was partially or mostly composed of rapid-fire repetitive low-frequency events, as has been found in multiple tectonic settings [Houston et al., 1998; Ide et al., 2007; DENLINGER AND MORAN The Authors. 2233

5 Figure 5. The tremor observed up to 128 km away from the crater of MSH, with time given in UTC, followed an earthquake visible here at about 100 s. All signals shown here were filtered with a band-pass filter from 1 to 3 Hz. Tremor recorded up to 240 km away from the volcano consists of a packet of surface waves with a nearly constant amplitude envelope, indicating strong near-surface velocity gradients in this region that produced an Airy phase that contained the tremor. Shelly, 2010]. We selected a 100 min long window starting at 1900 UTC that included ~20 min of time before and after the tremor episode. We only used data from station JRO, where a broadband sensor was installed following the 1 October explosion (see McChesney et al. [2008] for details on the Mount Saint Helens (MSH) seismic network), as other stations with on-scale recordings for the entire tremor episode were > 30 km distant. We followed a modified version of the methodology of Shelly [2010] and broke the 2 October tremor into a set of s long template events with 1.9 s overlap between bins and crosscorrelated each resultant template event against the entire continuous waveform. The templates with most significant correlations corresponded to time windows before or after the tremor, mostly when no earthquakes or other discrete events were occurring. The tremor itself mostly had very poor correlations with time windows before and after the tremor as well as with most windows within the tremor, with the exception of a few intratremor template windows that showed significant correlation (R = max) with other tremor windows. Inspection of waveforms in these windows found nothing that looked like a discrete event; rather, the waveforms were dominated by sinusoidal 1 3 Hz waves that were more reminiscent of surface waves. We found similar results using different window lengths and overlaps. The lack of correlation of windows, along with the distance independence of the tremor envelope and duration, led us to hypothesize that the tremor consisted largely of surface waves. Figure 6. Observed signal at JRO (9 km from the crater) and spectrogram of the 2 October 2004, tremor and earthquakes [Moran et al., 2008a]. The instrumentcorrected, reduced displacement of this signal is cm 2. The tremor is 49 min long and has a wider frequency band and contains much higher frequencies than the earthquake spectra. Detailed analysis of the tremor spectrum recorded by our network supports an interpretation of these signals as surface waves. Particle motions at three-component seismic stations JRO (Figure 4) and LON (at Mount Rainier) show elliptical motions consistent with Rayleigh and Love waves, and the decline in tremor envelope amplitude with distance (Figure 5) varies DENLINGER AND MORAN The Authors. 2234

6 Figure 7. The tremor recorded at all stations around the volcano that remained unclipped is shown here scaled to the signal at JRO and time shifted slightly so that all station signals can be compared on the same plot. Each signal consists of the earthquake immediately preceding the tremor followed by 49 min of tremor. These signals are divided into ten bins (0 9), with 0 being the earthquake. Bins 1 9 are each 5.4 min long. The signals within each bin are analyzed for changes in signal character with frequency and with time. inversely with the distance from the crater. All stations local to MSH that were operating on 2 October (Figure 4) had vertical-component short-period seismometers except for the JRO broadband sensor. We used data from JRO (Figure 5) as well as from the next-nearest broadband station LON, located ~68 km to the north near Mount Rainier. After deconvolving the instrument response of each station, a nearly constant shape for the amplitude envelope of the tremor observed across a large part of the network (Figure 6) allows us to scale and plot the signals on a single graph (Figure 7) and look for ways to deconstruct the signal into its components. Attenuation and scattering reduce amplitude of nearby stations, such as HSR perchedhighonthesideofthehollow cone, so that their amplitude is comparable to the amplitude at JRO (which is in a vault). We break the tremor signals into nine 5.4 min bins numbered 1 through 9 (Figure 7), noting that there is a trade-off between the size of each bin and the resolution with which we can determine the frequency spectrum of each bin. We then examine the data in the tremor bins to look for variations in time. First, we determine the cross-spectral correlation of the three components from the two closest broadband stations at JRO and LON (Figure 8), in which the cross correlation of two components x and y in the time domain, with lag h, or γ xy ðhþ ¼ 1 n Xn h ðx i h i ¼ 1 is transformed into the frequency domain to obtain the cross-spectrum f xy or Þðy i Þ (1) f xy ðþ υ ¼ Xh ¼ h ¼ γ xy ðhþe 2πiυh (2) The cross spectrum of components at JRO (9 km away, azimuth = 345 ) contains the same peaks as that at LON (68 km away, azimuth = 30 ) as shown in Figure 8. We interpret the frequencies unique to the correlation of north and vertical components to be composed of Rayleigh waves and the frequencies unique to correlation of east and north components to be composed of Love waves. The largest amplitudes are associated with Love waves as shown by the relative magnitudes of peaks in Figure 8. The decrease in amplitude in the relatively high-frequency peak L8 is interpreted as the disappearance of the surface layer carrying this phase between JRO and LON. The difference in relative amplitudes of peaks L3 and L6 results from the change in azimuth. The sharp spectral peaks forming the cross spectrum between the north and vertical components for JRO also are found in the normalized sum of the amplitude spectra for all vertical-component-only stations surrounding the volcano (Figure 4). Using a multitapered Fourier transform [Prieto et al., 2009], with five tapers and a time-bandwidth product of 4, we found that the number and bandwidth of these common spectral peaks systematically varied with signal amplitude for each time bin in Figure 7, as shown in Figures 9a and 9b. These plots clearly show that the buildup and decline of the spectrum with time around bin 5, and each major spectral peak is also found in the correlation of vertical and north components at JRO, supporting the interpretation of these signals as Rayleigh waves. DENLINGER AND MORAN The Authors. 2235

7 Figure 8. Cross-spectral correlation (equations (1) and (2)) between the components of velocity at JRO (Figure 4) and LON (68 km north (N) at an azimuth of ~30 o from MSH crater). Both JRO and LON are broadband, three-component seismographs. The R symbol indicates Rayleigh motion (vertical ellipse with retrograde motion) and the L symbol indicates Love waves (oscillations in a horizontal plane, expected in correlation between N and east (E) in plot). The Rayleigh wave signal also reveals symmetry in the spectrum during buildup and decay of this tremor. Spectral cross correlation of station HSR with the vertical component of station JRO (Figure 10) shows a symmetric pattern around the peak of observed tremor in bin 5 of Figure 7, with the same peaks recurring in bins 1-9, 2-8, 3-7, and 4-6. The broadest and most unique spectrum was observed in bin 5, and this time increment also had the highest amplitude in Figure 7. At all stations, the spectral bandwidth increases and decreases with tremor amplitude, as can be seen by comparing Figure 7 with Figures 9a and 9b; each bin had spectral peaks in common across the singlecomponent network. These common frequencies also are found in the correlation of the vertical and horizontal components of three-component stations JRO and LON. Using a band-pass filter to isolate the band 0.8 to 1.5 Hz that contains most of this spectral content, we found that the largest principal components in the velocity field [Konstantinou and Schlindwein, 2002] are horizontal with directions shown in Figure 11. The vertical components are a small, fluctuating fraction of the horizontal components in each time bin, consistent with the spectral cross-correlation results in Figure 8. Based on these results and observations, we have interpreted the spectral peaks defined by correlation of horizontal and vertical DENLINGER AND MORAN The Authors. 2236

8 Figure 9. (a) Sum of the amplitude spectra for time bins 1 5 in Figure 7, for all stations in Figure 4 (only vertical component of JRO used; all other stations single-component vertical). A few prominent peaks corresponding to Rayleigh waves interpreted in Figure 8 are labeled here and in Figure 9b. The increase in the bandwidth of the spectrum from bin 1 to bin 5 is clearly shown. (b) Sum of the amplitude spectra for time bins 5 9 in Figure 7, for all stations in Figure 4 (only vertical component of JRO used; all other stations single-component vertical). A few prominent peaks corresponding to Rayleigh waves interpreted in Figure 8 are labeled here and in Figure 9a. There is a more gradual decline in the bandwidth of the spectrum from bin 5 to bin 9 than in the buildup visible in Figure 9a. components as produced by Rayleigh waves and those peaks defined by correlation of horizontal components alone as produced by Love waves and labeled them as R and L, respectively, in Figures 8 and 9 and in Table Summary of Tremor Data From the above analyses the primary observations are (1) that the spectral content and shape of the envelope varies little with azimuth or distance from the vent (Figures 1, 4, 6, and 7), (2) that the tremor bandwidth is much larger than earthquakes in the volcano (Figure 5), and (3) that the tremor bandwidth grew and declined with signal amplitude (Figures 7 9). The azimuthal variation of the envelope amplitude (not shown here) DENLINGER AND MORAN The Authors. 2237

9 Figure 9. (continued) indicates that the vent is the source of the surface waves carrying the tremor. Though both Rayleigh and Love waves are produced, the largest amplitudes are horizontal, and the spectral contributions from Rayleigh and Love waves alternate in the frequency domain (Figure 8). All of these observations are consistent with a seismic wavefield radiated from a resonating waveguide similar to those produced by seismic sources in fluid-filled boreholes, as described by Paillet and Cheng [1991]. We follow this lead and test the observed conduit resonance with a waveguide model to see if it makes sense given what is known of the Mount St. Helens magma system. 4. Tremor in a Wide Conduit Produced by a Broadband Source The observed radiation pattern for the 2 October tremor constrains the tremor surface wavefield to emanate from the top of the conduit where a spine emerged in late October (Figure 2) and close to the area where a vent-clearing explosion produced a small crater on 1 October (Figure 1). The principal components of these DENLINGER AND MORAN The Authors. 2238

10 Figure 10. Normalized cross-spectral density (equations (1) and (2)) between the vertical component of velocity at JRO and the vertical single-component station HSR (Figure 4), for each time bin in Figure 7. This plot shows that the tremor begins and ends with the same frequencies, is broadest in bin 5, and that there is symmetry in the frequencies positioned symmetrically in time on either side of bin 5 (1-9, 2-8, etc) as shown here. Comparison with the paired tremor is made in red. This indicates that as the tremor built up and then wound down, it approximately replicated a previous spectrum in each bin, creating a symmetric pattern. Compare these peaks to Figure 8. data (Figure 11) show that the largest forces were delivered at a slight angle to horizontal, consistent with our interpretation of strong Love waves observed at JRO and LON (Figure 8). Based on these results and the results of previous studies of conduit tremor [Jousset et al., 2003; Neuberg et al., 2000], we consider a model geometry in which a conduit filled with low-velocity magma or rock terminates near the surface. We base the following analysis on these studies and the physics of waveguides [Paillet and Cheng, 1991]. If the seismic velocities of the magma or rock filling the conduit are less than the velocities of the host rock, then the only necessary condition for conduit resonance is that seismic wavelengths within the conduit are comparable to or less than the conduit diameter. Lower frequencies simply radiate outward as body waves, with the wavefront modified slightly by the presence of the conduit. However, a broad-spectrum seismogenic source will produce a few wavefronts for which constructive interference with internally reflected waves is possible, and those waves are preferentially propagated along the conduit margins as hybrid P or S head waves. Each hybrid interface wave is a mixture of Stoneley, head, and internally reflected waves, as described by Paillet and Cheng [1991]. If conduit velocities are less than the shear wave velocity of the host rock, then most of the resonant energy will be pushed into these hybrid shear waves, though compressional hybrid waves will still be DENLINGER AND MORAN The Authors. 2239

11 Figure 11. The principal components for velocity at JRO were found for the dominant frequency range in the cross-spectral density common to both JRO and LON, 0.8 to 1.5 Hz, using the polarization analysis described in Konstantinou and Schlindwein [2002]. The amplitudes of the vertical velocity component at JRO are much smaller than the amplitudes of the horizontal components, so the nearly horizontal vectors are plotted in map view here. The vectors at LON are nearly the same for each bin as those at JRO. produced. Both types of hybrid waves result from interference with the seismic wavefield trapped by total internal reflection within the conduit and travel along the conduit wall as interface waves. This process of resonance is robust and occurs in tabular as well as cylindrical waveguides [Paillet and White, 1982]. In Figure 12 we show results from several full wavefield simulations we performed employing finite volume methods (see Appendix A for references and description). The first simulation in Figure 12a shows strong reverberations fed by S head waves that are typical of waveguide resonance in an idealized, low-velocity conduit. The simulations in Figure 12b compare the resonance of two different sources: (1) an explosion initiated in the center of the conduit (Figure 12a) and (2) a downgoing P wave (the first column vector in equation (A9) in Appendix A) across the conduit. The nearly identical reverberation patterns are indicative of the resonance process within the waveguide, illustrating that such waves are sensitive primarily to conduit properties rather than to the nature of the source. The influence of conduit resonance on tremor wavefields is noted by other authors. Ferrazzini and Aki [1987] described two groups of interface waves that propagate along conduit margins. Neuberg et al. [2006], in discussing numerical simulations of conduit resonance resulting from an explosion in the conduit, stated that While most of the seismic energy is trapped in the conduit, there is, at t = 1 s, an initial, weak P wave visible, followed immediately by high-amplitude interface waves. As shown in this work and in Jousset et al. [2003], these interface waves then interact with Earth s Table 1. Frequencies Attributed to Rayleigh (R) andlovewaves(l) at JRO Based Upon Cross Power Spectral Density of the Components at JRO and LON, as Well as the Cross Power Spectral Density Between JRO Vertical Component and the Vertical Component Stations in Figure 4 a Rayleigh Waves Hertz Love Waves Hertz R L R L R L R L R L R L R L R L R L R L R L a These frequencies are labeled in the plots in Figures 8 and 9. surface to generate surface waves that are recorded in the far field by seismic stations as tremor. Though only one of the source mechanisms and bandwidths in the simulation shown in Figure 12 is equivalent to the point source used by [Jousset et al., 2003] and Neuberg et al., their results and ours display the same features of conduit resonance: P and S head waves, a composite interface wave train composed of Stoneley and head waves interacting with internal reflections [Paillet and Cheng, 1991; Roever et al., 1959], and conduit wall fluctuations associated with tube waves. This pattern of waveguide resonance in a conduit relies only on source bandwidth, conduit velocity, and geometry. One significant result from this DENLINGER AND MORAN The Authors. 2240

12 Figure 12. (a) Synthetic wavefield created in an infinite waveguide consisting of low-velocity fluid sandwiched between two planar elastic solids. The source is an explosion near the center of the conduit. Here the frequency content (0 30 Hz) is high enough to generate both P and S head waves that travel along the solid side of both walls as well as at least three modes of internal reflections (equation (4)) that feed the high-intensity interface waves, called hybrid waves here. These are a mix of internal reflections, S head waves, and Stoneley waves, and with the tube wave generate the largest surfaces as this field reflects from the Earth s surface[jousset et al., 2003]. The mechanics of this conduit resonance are described in Paillet and Cheng [1991]. (b) Comparison of the wavefield in a tabular waveguide in response to two different sources: (1) an explosive source in the center of the conduit, on the left, and (2) a downgoing P wave source across the conduit (from first column vector, equation (A9), Appendix A), on the right. Both sources are given by a Ricker wavelet in velocity, with a maximum frequency of 30 Hz. The similarity in the pattern of resonance induced by these disparate sources shows how waveguideresonancemasksthetruenatureoftheseismicsource.the hybrid waves are a mix of head waves, Stoneley waves, and are amplified by energy from internal reflections [Paillet et al., 1987; Roever et al., 1959]. DENLINGER AND MORAN The Authors. 2241

13 finding is that resonance will mask the seismogenic source. Stating this point more fully, any source that generates frequencies high enough to produce head waves (i.e., produce seismic wavelengths comparable to or less than conduit diameter, over a range of frequencies) will induce similar patterns of resonance through interaction of those head waves with critically reflected waves internal to the conduit. For this reason it is nearly impossible to deconvolve a signal measured on Earth s surface that is produced by such a conduit resonance to infer its seismogenic source at depth, for the same reason that it is nearly impossible to deconvolve the source of a borehole logging tool from the seismic signal it produces [Paillet and Cheng, 1991]. This resonant waveguide phenomenon involving head waves and internally reflected waves is not confined to cylindrical conduits, because two planar interfaces enclosing tabular waveguides will also interact and resonate in the same fashion. Analytical studies [Paillet and White, 1982; Scott et al., 2008] and numerical studies [Neuberg et al., 2000; Paillet and White, 1982] have reiterated the result, long established in the borehole geophysics literature [Paillet and White, 1982; Paillet and Cheng, 1991; Paillet et al., 1987], that both group- and phase-velocity dispersions for seismic wave fields in cylindrical and tabular waveguides are nearly identical. The reason is that the interaction of head waves along the margin of a cylindrical conduit with internally reflected waves trapped inside and the interaction of head waves along the margin of a tabular waveguide with internally reflected waves trapped inside is only weakly dependent upon the curvature of the interface. This analogy between cylindrical and plane geometries, exploited by Paillet et al. [1987] for seismic waves in boreholes and by Neuberg et al. [2000] and Jousset et al. [2003] for volcanic tremor, is also used here to illuminate the interaction between head waves and internally reflected waves that is the key process in this type of conduit resonance. Using the tabular/cylindrical waveguide analogy, we can increase our understanding of resonant modes by splitting the problem of a tabular waveguide in half and examining the full wavefield interaction on one of the opposing interfaces [Roever et al., 1959]. By measuring the interaction of a plane wave with a plane boundary for a wide range of angles of incidence, Roever et al. [1959] demonstrated the mechanisms of head wave formation and radiation, as well as the fusion of Stoneley and head waves when density was finite and comparable on both sides of the interface. His analytical and experimental results are consistent with many studies: his hybrid interface waves correspond to the interface waves of Biot [1952], to the crack waves of Chouet [1985], to the pseudo-rayleigh waves of Paillet and Cheng [1991], and to the high-energy interface waves of Neuberg et al. [2000] for conditions of a finite density fluid in contact with a solid interface. If the fluid density goes to zero, these hybrid waves become Rayleigh waves [Biot, 1952; Chouet, 1985; Paillet and Cheng, 1991]. We reproduce the results for an incident P wave onto a dense fluid-solid boundary in Figure 13, in which we use our numerical model (Appendix A) to determine how the amount of transmitted P to transmitted S energy varies with the angle of incidence. Here the ratio of transmitted kinetic energy to incident kinetic energy is calculated directly from the simulated wavefield using E ¼ 1 2 ρv2 MAX (3) where E is kinetic energy, ρ is density, and V MAX is the peak velocity amplitude. The energy radiated back into the low-velocity medium (the conduit) varies as a function of the incident angle. At angles much less than critical incidence, energy radiates away from the boundary. At the critical angle and beyond, energy is trapped by the boundary. It is the angle of incidence just less than the critical angle that dominates the leaky mode. In a tabular waveguide, two such interfaces enclosing low-velocity material will interact through radiation passed between them from refracted and reflected waves. In this way, conduit resonance is filtered by wave interactions as interface waves propagate along the conduit wall. For any source, seismic energy generated in the conduit radiates outward as body waves and is lost, except for those few nearly critically refracted wavefronts where constructive interference with internal waves is possible (as mentioned earlier). These hybrid waves preferentially propagate along the conduit walls, and it is these waves that form the high-energy interface waves noted by Neuberg et al. [2006] and which owe their hybrid nature to finite densities on both sides of the interface. When this energy reflects from the Earth s surface, it generates surface waves that are recorded and interpreted as volcanic tremor [Jousset et al., 2003; Neuberg et al., 2006]. The modes of resonance of internally reflected waves confined by conduit margins may be described using a Snell s law approximation that shows the filtering of source bandwidth. In particular, in order for the internally DENLINGER AND MORAN The Authors. 2242

14 Figure 13. The conduit resonance shown in Figure 12 is closely related to the problem of an incident P wave on a single boundary between a fluid and a solid, or between any low-velocity and high-velocity material excited with enough bandwidth. Here the wavefield obtained for conversion of a P wave to an S wave upon transmission across the boundary from low velocity to high velocity is shown for three different angles of incidence. There is an increase in energy (intensity of color) at the boundary in the transmitted shear wave as the incident angle approaches the critical angle for the interface. In hard rock most of the resonant energy along conduit waveguides is pushed into shear wave energy that travels as hybrid interface waves along the conduit walls as it continually interacts with internally reflected waves. In soft rock, where the compressional velocity of the conduit is greater than the shear velocity of the host medium, all of the energy along the conduit walls will consist of compressional interface waves. reflected wave to arrive back at the conduit margin at the same phase, the wavelength, conduit diameter, mode n, and angle of incidence are related by cosðϑþ ¼ D (4) 2nλ f where ϑ is the incident angle for critical refraction, D is the width or diameter of the conduit, nis the mode number of internal reflections before constructive interference occurs, and λ f is the wavelength of the compressional wave inside the conduit [Paillet and White, 1982; Paillet and Cheng, 1991; Paillet et al., 1987]. However, the precise nature of the interference between head waves and these internally reflected waves is not revealed by equation (4) and requires a full wavefield analysis [Paillet and Cheng, 1991; Paillet et al., 1987]. The results of full waveform simulations using the program in Appendix A show that there is a cutoff frequency above which an infinite number of higher-frequency modes exist, and below which resonance will not occur. For each frequency for which constructive interference occurs, there are two critical angles of incidence ϑ, one for a P head wave and one for an S head wave [Paillet et al., 1987], as shown in Figure 14. Since compressional wave velocity Vp is always greater than shear wave velocity Vs in the host rock, the critical angles for S head waves are always greater than critical angles for P head waves, and cutoff frequencies for S waves are always higher. As a consequence, waveguide resonance produces distinctive spectra in which spectral peaks from P and S head waves alternate in the frequency domain [Pailletetal., 1987]. Where a nearvertical conduit terminates at or within a fraction of a wavelength from the Earth s surface,s head waves produce Love waves and P head waves produce Rayleigh waves. The alternating spectral pattern of P and S peaks generated by conduit resonance will be reproduced as alternating peaks in surface wave spectra derived from DENLINGER AND MORAN The Authors. 2243

15 Figure 14. The concentration in energy for interface waves near the critical angle of incidence results from the sharp variation in transmission of kinetic energy across a boundary near the incidence angle for complete refraction. These results are generated numerically with the finite volume model used to produce the results in Figure 13. As waves become refracted along the interface, they radiate back into the lowvelocity medium. It is the coupling of this returned radiation with trapped, internally reflected waves that produces the hybrid resonance shown Figure 12. Whereas this figure shows the transmission spectrum of a single interface, conduit resonance requires two opposing interfaces, separated by at least one wavelength, that then interact. Resonance occurs by interference of these interface waves with internally reflected waves regardless of whether the conduit is tabular or cylindrical [Paillet and White, 1982; Paillet and Cheng, 1991]. If the conduit is near-vertical and connects to the Earth s surface, then surface waves are generated as conduit wave packets are reflected [Jousset et al., 2003]. The contributions from individual P and S head waves along the conduit wall will produce spectral peaks that alternate in the frequency domain [Paillet and Cheng, 1991]. As these interface waves reflect and convert to surface waves at the Earth s surface, they impart the same frequency spectrum to the surface waves and produce alternating spectra similar to Figure 8. it. The alternating spectral peaks of the Love and Rayleigh waves in the recorded 2 October tremor is an indication that the tremor was filtered by such a waveguide path. These patterns of conduit resonance are visible in the previously cited studies of tremor as well. Since incident energy greater than each critical angle is trapped within the conduit, the intense interface waves visible in numerical simulations of conduit resonance [Jousset et al., 2003] and in the resonance shown in Figure 12 are produced by a narrow range of incident angles just less than the critical angle at which each P and S head wave is generated. If the width of the conduit supports the nth mode for internal reflection, and if the wavelength for that mode is radiated by the source, then the conduit resonance will contain that frequency. The resonance supported by a combination of P head waves, internally reflected waves, and S head waves will be excited by adjacent portions of a broadband source spectrum and is essentially equivalent to the mechanism proposed by Jousset et al. [2003]. The interface wave packets excited by a broadband conduit source and fed by conduit reverberations will propagate along the conduit margins, eventually reaching the free surface and forming surface waves. In their P-SV simulations of conduit resonance, Jousset et al. [2003] show the synthetic record obtained for a horizontal array of seismometers at the free surface in their Figure 7c, in which most of the transmitted energy is in two groups of Rayleigh waves. They show that these surface waves are produced by two distinct sets of interface waves which propagate along the conduit margins. The first set of interface waves correspond to reverberations associated with S head waves and the second set to hybrid waves, both of which are modeled in this study (Figure 12). Both phase and group velocities in the first set decrease as frequency increases, as these are pseudo-rayleigh waves, whereas both phase and group velocities increase with increasing frequency in the second set, as these waves are fed by internal reflections [Paillet and Cheng, 1991]. 5. Application to Conduit Resonance on 2 October 2004 If we interpret tremor as produced by a conduit resonating as a waveguide, then a simple model emerges for the 2 October tremor at Mount St. Helens. The duration of the signal, the broadening of the spectrum (Figure 9), the symmetry with which this occurred over time (Figure 10), and the growth of tremor bandwidth DENLINGER AND MORAN The Authors. 2244

16 with tremor amplitude indicate a sustained source that energizes different widths of the conduit at different times. In particular, the increase in bandwidth with time (up to bin 5 in Figure 7) indicates an increase in width and length in the resonating portion of the conduit as tremor amplitude increased. Alternatively, the decrease in bandwidth following bin 5 is consistent with a decrease in length and width of the resonating conduit as tremor declined. If we consider the source bandwidth as constant then the filter producing the signal is the width of the conduit, which controls the number of head-wave frequencies that can be energized by a broadband seismic source. For this initial condition, the variation in observed spectrum reflects seismogenic excitation along a conduit of variable width. We consider the most likely source of broadband seismic energy, consistent with a duration of 49 min and reduced displacements of cm 2 to be that proposed by Tuffen and Dingwell [2005] and Neuberg et al. [2006]. In both models seismic energy is radiated during progressive comminution of nearly solid magma along the walls of a volcanic conduit as silicic magma is forced upward, shearing the wall rock and tearing the magma fabric along the wall. Field evidence for cataclastic structures within conduits is documented for silicic dikes in Iceland by Tuffen and Dingwell [2005], and their interpretation is consistent with laboratory evidence for viscoelastic rheology of silica-rich magmas [Webb and Dingwell, 1990]. At MSH, the evidence for cataclastic deformation of conduit margins is ubiquitous [Pallister et al., 2012]. The spines emplaced from 2004 to 2008 were covered in gouge at least a meter thick, and field exposures of dissected spines reveal cataclastic deformation that extends at least 3 m into the spine from gouge-coated margins [Kennedy et al., 2009]. Finally, GPS measurements show that the highest rates of deformation and inferred magma movement occurred during 2 October and the days preceding it [Lisowski et al., 2008]. The implication is that cataclastic deformation occurred along the conduit margins at MSH when strain rates were higher than could be accommodated by flow and that this deformation was seismogenic. For dacite magma, the strain threshold for cataclastic deformation is given approximately by η ε ¼ 10 7 Pa (5) where η is the magma + crystal viscosity and ε is the shear strain rate [Webb and Dingwell, 1990]. Using the known velocity of the JRO GPS station when the spines emerged at the surface to constrain the near-surface rate of volume intrusion on 2 October, we can model the upward velocity of magma in the conduit assuming that the conduit has the 100 m by 200 m cross-sectional area of the whaleback spine that eventually emerged [Schilling et al., 2008]. This gives an average velocity of magma at the time tremor occurred on 2 October of about 3 mm/s. If this velocity is accommodated across a 1 to 3 meter wide shear zone, then ε ranges from 0.001/s to 0.002/s. In the spine that emerged later in October, the average magma viscosity was high enough to support a whaleback shape with steep sides (Figure 2) that were sustained until they fractured and fell apart months later [Vallance et al., 2008]. Sustaining steep (80 ) walls for months requires a viscosity of at least Pa/s (it is likely that the temperature-dependent viscosity was much higher on the cooler margins of the spine, and where magma was ground into gouge). Thus, the field evidence supports the hypothesis that the threshold for brittle deformation of the magma was met during the eruption and that the thick gouge mantling the emerging spine in early November indicates that these conditions may have also been met at the time of the 2 October tremor. Given that the highest rates of magma movement in the conduit, as determined from the JRO GPS, occurred on 2 October, it is reasonable to expect that cataclasite-generating deformation was occurring along the conduit margins at that time. As invoked by Neuberg et al. [2006] for Montserrat, we suggest that cataclastic disruption of a nearly solid magma at Mount St. Helens, as the magma was forced to slide along a rough conduit wall, was the seismogenic source that sustained tremor for 49 min on 2 October. Can sliding of rough surfaces at a few millimeters/second, far less than shear wave velocities for the rock, generate and radiate seismic energy over our observed range of tremor frequencies? Numerous and detailed experiments using granular material [Behringer, 2013] suggest that this is possible. In controlled laboratory experiments, Behringer [2013] produced fluctuating forces with frequencies up to 10 Hz by shearing dry granular material at slow rates (millimeter/second). Though the theoretical framework for this phenomenon is not complete, theoretical analysis [Behringer, 2013] suggests that if the modulus and forces across individual grains are high enough, then the formation and breaking of linked chains of particles within a granular mixture (such as gouge) during slow shear is sufficient to produce and radiate high-frequency strain waves. The magnitudes of the oscillating forces generated are comparable to the magnitudes of normal forces DENLINGER AND MORAN The Authors. 2245

17 Figure 15. All interface waves and tube waves resulting from conduit resonance reflect from the Earth s surface and generate surface waves. Shown here is an idealized low-velocity conduit 200 m in width buried in a homogeneous space with velocities comparable to Mount St. Helens. The top surface is a free surface and all other sides are absorbing. Frictional slip along conduit margins generates a 3-D wavefield similar to that shown in Figure 12, causing interface waves and resonances that reflect from the free surface as shown in the top left plot. In the remaining surface plots, the Z coordinate is replaced with the velocity amplitude, with all plots at the same vertical exaggeration. These plots show the relative velocity amplitudes of the Love wave polarizations (shear in XY) and the two Rayleigh wave polarizations (shear in XZ and in YZ) generated by frictional slip (producing large normal forces) along the conduit margin. The Love wave amplitudes are 6 8 times stronger. In this snapshot, the reduced displacement [Aki and Koyanagi, 1981] of these waves ranges from 7 to 54 cm 2, proving that shear along conduit margins can generate relative surface wave amplitudes comparable to what we observe for the tremor of 2 October across the zone of shear. Thus, Behringer s experiments support the concept that shearing granular or pervasively fractured material between rough surfaces at differential velocities comparable to those inferred for magma movement on 2 October at MSH can, under the right conditions, produce and radiate seismic energy. If the fluctuating forces are large enough, then the radiation will have sufficient intensity and bandwidth to be observed over 100 km away. But how deep does slip need to be to generate sufficient force? We test our mechanism using an idealized medium with a P wave velocity of 1.8 km/s, an S wave velocity of 1.62 km/s, a 200 m wide conduit with a P wave velocity of 1.01 km/s and an S wave velocity of 0.31 km/s (given by effective shear modulus for 1 2 Hz oscillations of high-viscosity magma). We impose the following initial conditions along the conduit wall at a depth of 300 m for 0.1 s: σ n ¼ ρg depth τðþ t ¼ 0:6 σ n Rt ðþ (6) This is a Coulomb criterion, where σ n is the normal stress across the conduit wall and τ is a shear stress chosen so that the conduit material moves upward relative to the host material. The normal stress is held constant whereas the shear stress varies as a Ricker wavelet R(t) with a maximum frequency of 30 Hz and a duration of 0.1 s. This generates a wavefield that reflects off of the top free surface (all other sides are absorbing) and produces surface waves (Figure 15). These waves are plotted in three panels in Figure 15, each with a different velocity component. As expected, the component forming Love waves is much stronger than the components forming Rayleigh waves since the largest contributor to surface waves are intense hybrid interface waves and the tube waves that follow them (also noted by Jousset et al. [2003]). DENLINGER AND MORAN The Authors. 2246