Moment tensor inversion of very long period seismic signals from Strombolian eruptions of Erebus Volcano

Journal of Volcanology and Geothermal Research 177 (2008) 635–647

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Journal of Volcanology and Geothermal Research j o u r n a l h o m e p a g e : w w w. e l s e v i e r. c o m / l o c a t e / j v o l g e o r e s

Moment tensor inversion of very long period seismic signals from Strombolian eruptions of Erebus Volcano R. Aster ⁎, D. Zandomeneghi, S. Mah, S. McNamara, D.B. Henderson, H. Knox, K. Jones Department of Earth and Environmental Science and Geophysical Research Center, New Mexico Institute of Mining and Technology, Socorro, New Mexico, United States

a r t i c l e

i n f o

Article history: Received 8 February 2007 Accepted 22 August 2008 Available online 13 September 2008 Keywords: volcanism magma explosive eruptions seismology

a b s t r a c t Strombolian eruptions from the long-lived lava lake of Erebus volcano, Ross Island, Antarctica, generate repeating Very Long Period (VLP) signals, containing energy between approximately 30 and 5 s, that persist for several minutes and through the post-eruptive refilling of the lava lake. The initial approximately 10 s of this signal is moderately variable, particularly with respect to its initial polarity, while the following VLP coda has been observed to be stable since the earliest VLP observations were made (1996). To estimate forces and force couples consistent with the Erebus VLP signature, we perform moment tensor inversions for point sources using high signal-to-noise data stacks from the six-station, 18-component broadband seismographic network and Green's function forward calculations that incorporate topography. We infer a shallow (approximate depth of less than 400 m below the lava lake surface) source centroid that underlies the center to the northwestern rim of the main crater, east and north of the lava lake. Integrated Mii functions over the predominant (180 s) signal duration of VLP events show that the net scalar moments for these events are on the order of 4 × 1013 N m (corresponding to a moment magnitude mw ≈3) for typical sized VLP events. Moment rate tensors which characterize force couple components are dominated (85–97% of variance) by dilatational components. Approximately 25% of the data variance is attributable to single forces that are attributable to oscillatory reaction forces caused by fluid transport, however, the relative contributions of vertical forces and couples with this sparse network is poorly resolved for these shallow sources. The generally high degree of repeatability in the VLP signal across thousands of eruptions over the past decade indicates that the response of the conduit system to gas slug ascent and subsequent gravitational disequilibrium is stable, consistent with the generally unchanging surface manifestation of the convecting lava lake system, and arguing for a thermally and dynamically stable conduit system beneath the lava lake. © 2008 Elsevier B.V. All rights reserved.

1. Introduction and background Erebus Volcano has exhibited persistent Strombolian activity from its phonolitic lava lake for decades (e.g., Giggenbach et al.,1973; Kaminuma, 1994; Kaminuma et al., 1985; Dibble et al., 2008-this issue). The exposed Erebus magmatic system facilitates repeated close-range (to within several hundred meters; Fig. 1) study of diverse vent activity, with the most common eruptive activity by far consisting of characteristic eruptions from the lava lake. These eruptions are the explosive decomposition of large, generally single gas slugs that can reach 10s of m in diameter at the lava lake surface (Dibble, 1988, 1994; Aster et al., 2003; Johnson et al., 2003; Aster et al., 2004a; Jones et al., 2008-this issue). Erebus shows a distinct lack of internal earthquakes or volcanotectonic events (Rowe et al., 2000), consistent with a long-term open magmatic system that does not readily accumulate internal deviatoric stress or pressurization. Very long period (VLP) seismic signals at Erebus (Rowe et al., 1998; Aster et al., 2003) belong to a class of signals recorded ⁎ Corresponding author. E-mail address: [email protected] (R. Aster). 0377-0273/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.jvolgeores.2008.08.013

at a number of active volcanoes (Sassa, 1935; Chouet et al., 1999; Arciniega-Ceballos et al., 1999; Aster et al., 2000; Kawakatsu et al., 2000; Nishimura et al., 2000; Almendros et al., 2002; Chouet et al., 2003; Auger et al., 2006; Waite et al., 2008, Arciniega-Ceballos et al., 2008) that provide unique information on eruption- or transport-induced forces within volcanoes at periods ranging from seconds to hundreds of seconds. For sufficiently close seismographs (i.e. on the order of a seismic wavelength), these signals are observed as near-field elastic displacements that must be interpreted using near-field theory (as opposed to the more familiar far-field theory for P, S, and surface waves). Erebus Very Long Period (VLP) signals to date are uniquely associated with impulsive Strombolian eruptions from the lava lake system. Volcano instrumentation (Fig. 1) currently consists of a network of long-operational short-period seismic stations combined with infrasound, tiltmeters, geodetic GPS, gas, infrared, environmental, and stateof-health sensors installed since 2001 (Aster et al., 2004a). Most recently, seismic recording on Erebus has been greatly expanded with a 23station supplemental network of temporary IRIS PASSCAL (Aster et al., 2005) stations installed in 2007 scheduled to operate through early 2009, coupled with a tomographic shot program (Chaput et al., 2008;

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Fig. 1. Map of Erebus shaded summit topography, seismic, and video station locations showing locations of seismic stations and the video site (VID). Station RAY was destroyed by an eruption in 2007. Crater rim morphology taken from Csatho et al., 2008-this issue.

Zandomeneghi et al., 2008). In the first study of Erebus VLP signals, accomplished with IRIS PASSCAL seismographs during 1996–1997, Rowe et al. (1998) noted the persistent association of VLP signals with Strombolian eruptions with vertical polarities uniformly directed downward, indicating a deflationary and/or downward-directed force operating on the elastic volcano volume during the VLP onset. Subsequent observations from 1999 onwards showed that some later events had upward initial polarities, corresponding to an inflationary, and/or upward-directed force. Mah (2003) performed an examination and classification of VLP signals using data from the prolifically active periods of 1999, 2001, and 2002, a period of elevated activity preceding an 18-month quietus in eruptive activity that ended in 2004. Mah has noted that, when the initial polarity and other characteristics could be easily discerned, the seismograms could be classified into 3 groups by initial polarity (assessed by crosscorrelation and confirmed by visual inspection), time function shape, and spectral content (Fig. 2). Group 1 eruptions show positive initial vertical motions (Fig. 2A). Group 2 eruptions show negative initial vertical motions (Fig. 2B) that tend to have lower frequency content. Group 3 eruptions are very rare (only 4 observed examples) and exhibit a relatively simple pulse-like shape (Fig. 2C). Although Group 3 events show clear high-frequency seismoacoustic arrivals consistent with Strombolian eruptions, eruptive and vent details are unknown due to their rarity and lack of accompanying video or other corroboration for the handful of observed cases. Many lava lake eruption signals are too noisy and/or emergent for to discern a definite character for the initial polarity and were classified by Mah as “indeterminate”.

To greatly increase signal-to-noise and thus facilitate study of the extended VLP source signal (e.g., Dreier et al., 1994; Aster et al., 2003), Mah (2003) and Aster et al. (2003) stacked events from each of the identified group populations (131 and 113 events from Groups 1 and 2, respectively). Fig. 3 shows normalized stacked displacement seismograms and corresponding power spectra. The characteristic VLP spectral peaks centered at T0 = 20.7, T1 = 11.3, and T2 = 7.8 s are nearly identical between the Group 1 and Group 2 events. Group 3 events show only a single very broad spectral peak near 25 s. Despite initial polarity differences, and timing relative to the short-period seismoacoustic signal created by the eruption, Groups 1 and 2 events show nearly identical VLP codas (Fig. 4), indicating that the post-eruptive VLP source mechanism for Group 1 and Group 2 events (which spans the lava lake refill period following the eruptive removal of the uppermost few 10's of meters of conduit material) is highly similar. Video taken approximately 350 m from the lava lake (Fig. 1) reveals key eruptive differences between Group 1 and Group 2 events. Fig. 5 shows a characteristic and well-observed Group 1 VLP verticalcomponent displacement signal at station E1S with corresponding video frames in 1 s intervals. The first frame shows the undisturbed lava lake just prior to the eruption. The second and third frames show a vertical jet-like eruption. Remaining frames show the immediate evisceration of the lava lake following eruption and a thermal ash and vapor plume. Fig. 6 similarly depicts a characteristic Group 2 eruption, where the video record reveals substantially different eruption characteristics from the Group 1 eruption shown in Fig. 5. The first

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Fig. 2. Representative self-scaled vertical-component displacement seismograms for each eruption family, showing characteristic differences, especially Group 1: positive initial VLP polarity), Group 2: negative initial VLP polarity, and Group 3: pulse shape. All data recorded at station E1S.

frame shows an undisturbed pre-eruptive lava lake surface. The second frame shows the lava lake surface start to noticeably inflate, a feature not observed in the Group 1 eruption, and is followed by somewhat

asymmetric westward (left in the video field of view) ejecta. Precursory inflation is associated with a prolonged (up to 10 s) initial downward VLP signal characteristic in Group 2 events. The onset of the explosive

Fig. 3. Stacked vertical-component displacement seismograms and corresponding power spectra (normalized power units) for the event types of Fig. 2.

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Fig. 4. Stacked vertical-component displacement seismograms for the event types of Fig. 2, taken from Fig. 3 (vertical gray line marks the approximate source origin time of the explosive seismoacoustic signal) showing pre-eruptive differences in polarity and timing, and the strong post-eruptive similarity in the post-eruptive VLP signal.

eruption in both cases (time zero) is revealed by the beginning of the much weaker seismoacoustic signal generated by the explosion. Analysis of a number of well-observed events by Mah (2003) appears to confirm the consistency of these eruptive differences. Mah (2003) further noted that Group 1 and Group 2 events are interspersed in time and overlap in size (with Group 2 events tending to be systematically larger), but display evolving proportions from season to season. It is important to note in the context of this paper that Group 1 and Group 2 events differ essentially solely in their first approximately 5–10 s of VLP signal, and their VLP signals are highly similar thereafter (Fig. 4).

within a conduit system acting upon the surrounding volcano, usually assumed to be elastic at these long periods and low strain rates (typically on the order of nanostrain or less). The seismograms are a convolution between forcing source time functions and an elastic (Green's function) impulse response. Therefore, responsible moment rate functions and force histories can be estimated as a linear inverse (deconvolution) problem. Increasingly, such results can be interpreted in the context of improved experimental and numerical modeling under increasingly realistic physical conditions. To investigate underlying forces and/or moment couples and the source location responsible for VLP signals at Erebus, we performed inversions (e.g., Chouet et al., 1999; Legrand et al., 2000; McNamara, 2004) utilizing data from events recorded by six three-component long-term broadband (Guralp 40-T, 30 s corner period) seismometers deployed at sites E1S, CON, HUT, NKB, RAY, and HOO (Fig. 1; Aster et al., 2004a). Our method of solution is essentially that of Ohminato et al. (1998) with Green's functions calculated using a 50-m resolution topographic model of the volcano using the method of using the TOPO finite-difference code (Ohminato and Chouet, 1997) generously provided to us with documentation by P. Dawson of the U.S. Geological Survey. Uniform elastodynamic parameters used in the model (Vp = 2.2 km/s, Vs = 1.27 km/s, and ρ = 2400 kg/m3) were based on the results of near-summit refraction experiments by Dibble et al. (1994). Moment rate and single force source functions were parameterized using 50% overlapping 0.5 s-wide triangular basis functions, b(t), to solve for K 200 s source time functions (M = 200/0.5 − 2 = 398 basis functions) using 200 s of 40 sample/s displacement seismogram data (N = 8000 points) for each seismic component. The general forward model is K

2. Moment rate tensor and force inversions VLP signals generated by ascending gas slugs in Strombolian systems arise due to the integrated influence of inertial and pressurization forces

unj ðt Þ ¼ ∑ si ðt Þ⁎Gijn ðt Þ

ð1Þ

i¼1

where un is the nth component of displacement at seismic station j, si(t) is the time function corresponding to solution function i, ⁎ denotes

Fig. 5. Vertical-component displacement seismogram and corresponding video frames for a Group 1 event, taken at the camera position (VID) shown in Fig. 1. The field of view spans approximately 250 m of the inner crater from east to west going from left to right. The erupting vent at the left of the video frames is the lava lake. The seismograms have been shifted by a P-wave (2.1 km/s) estimated propagation delay. Note the predominantly sub-vertical ejecta.

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Fig. 6. Vertical-component displacement seismogram and video frames for a Group 2 event, shown as in Fig. 5. The seismograms have been time-shifted by an estimated P-wave (2.1 km/s) propagation delay. Such events show a characteristic asymmetry in ejecta towards the right (west), as can be seen by the large sheet of magma ejected to the right.

convolution, and Gijn(t) is the appropriate Green's function for solution function i, station j, and component n, including the Guralp 40T (30 s) seismometer response. Parameterization to solve for the amplitudes, Bil of the triangular basis functions, b(t −τl ), where i again denotes the index of the moment tensor source component and τl is the center time of the lth basis function, yields M K
 unj ðt Þ ¼ ∑ ∑ Bil bðt−τl Þ⁎Gijn ðt Þ

ð2Þ

l¼1 i¼1

The system of equations d ¼ Ax

ð3Þ

is formulated in terms of a composite data vector, d, of length D = NS, where S is the total number of single-component seismograms in the data set (18, in this case), and a very sparse system matrix of size D by KM. The rows of A are the given by time reversed responses of the system to appropriate unit height basis functions (which are easily generated using convolution and time shift operations). The resulting system is iteratively solved for the weighting functions Bil (assembled into the composite vector x) using the conjugate gradient least-squares (CGLS) algorithm (e.g., Aster et al., 2004b). The K time functions corresponding to the moment rate and forcing functions are finally reconstructed from basis function weights as M

si ðt Þ ¼ ∑ Bil bðt−τl Þ

ð4Þ

l¼1

Upon convergence (typically approximately 1500 iterations for six couples and three force functions), CGLS produces a least-squares model that maximizes the variance reduction V¼

 2 ∑Di¼1 d2i −∑Di¼1 di −dpred;i  100k ∑Di¼1 d2i

ð5Þ

and dpred is the composite data vector (the concatenation of all 27 individual seismograms) predicted by the solution.

Individual events at Erebus are typically too noisy for robust moment rate tensor inversion, because of the persistent and high microseismic noise of coastal Antarctica in the VLP band (Aster et al., 2008). Noise levels are particularly high during the summer months when the entire broadband network of six stations was operational due to favorable solar power and field support conditions. To optimize VLP signal-to-noise to allow for examination of the entire (minutes-long) VLP signal, including its relatively low-amplitude, minutes-long coda, we here perform inversions using a stacked data set from 293 similar Group 2 VLP lava lake eruptions that occurred between January 30, 2005 and April 18, 2006. The composite stack is representative of the larger downward initial motion events typical of this stacking period, and the VLP coda after about 30 s is highly representative of the VLP process as observed since 1996 (Fig. 7). This time period corresponds to a strong resurgence in eruptive activity that followed a lava lake eruption quietus between approximately November, 2002 through June, 2004 (Jones et al., 2008this issue). Each set of eruption seismograms, in native seismometer components, was aligned using station E1S vertical seismogram best lags. Seismograms were normalized by maximum amplitude, stacked, and then rotated into a (vertical, radial, tangential) right-handed lavalake-centric (radial component outwards), right-handed coordinate system. In cases where individual seismogram components were not available due to station downtime, stacking was performed on a subset of the data collected during operational periods. The correlation method, which relies on the agglomerative assembly of a consistent stack by selecting events that correlate with the total stack, initially with a relatively low correlation threshold (0.6) and culling those that do not, unbiasedly selects an ensemble of high signal-to-noise events for stacking. The maximum vertical displacement amplitude at station E1S for the stacked data set was nominally scaled to 10 µm, which is comparable to the signal amplitude experienced during medium to large single events that generate infrasonic overpressures at E1S of approximately 40 Pa (Johnson et al., 2003, 2004; Jones et al., 2008-this issue). We note that the largest recently observed eruptions (January 2005–January 2007) were approximately five times this size as measured by infrasonic

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Fig. 7. E1S vertical component displacement seismograms (293 similar events between January 30, 2005 and April 18, 2006, selected based on high mutual crosscorrelation and aligned to minimize all 2-event crosscorrelation lags; Rowe et al., 2002) plotted in greyscale (white = up). Stacked E1S vertical component seismogram used in moment tensor inversion shown at right. The stacked data is representative of a Group 2 (first motion down) event, as can be seen in the stack trace and characteristic dilatations (black, downward) signature. Data are low-pass filtered at 5 s. This data period was selected to encompass the optimal operational period for the six-station broadband network (Aster et al., 2004a).

overpressure. To remove short-period energy generated by the distinct surface explosion process that is not due to the VLP seismic source (Aster et al., 2003) seismogram stacks were low-pass filtered at 0.2 Hz using a zero-phase 4-pole bidirectional Butterworth frequency response. We calculated moment rate tensors for force couples and forces using a 600 m by 600 m, 150 m-spacing grid of source epicentroid locations and source depths across a search volume (175 sources). The lateral boundaries of the search region were constrained by the leastsquares azimuthal disagreement between the observed VLP signal and the (geometric) station-source radius (Fig. 8). We justify this procedure because we expect the VLP source to be dominated by pressurization

and vertical force terms that will tend to produce radial VLP particle motions, an assumption that is bolstered by the predominantly radial particle motions observed in the data and shown in Fig. 8). The azimuthal fitness function applied in this determination F¼

1 L  b b 2 ∑ lr d e L i¼1 i i 1;i

ð6Þ

has a theoretical maximum value of 1 and is based on the eigenvalue– eigenvector decomposition of the variance tensor of horizontal displacement particle motions at all L stations, where the linearity, l

Fig. 8. Contoured particle motion azimuthal fit function, F (Eq. 6; maximum = 0.802) used to laterally bound a search region for the VLP centroid source. The applied grid of trial source epicentroids across the maximum region of F is shown. The maximum of F occurs approximately 330 m WNW of the lava lake, in the vicinity of the northwestern-central main crater (Fig.1), but the function is nearly flat in a triangular region bounded by the locations of stations RAY, E1S and NKB. Locations of the lava lake (Fig.1), minimum F, and the epicentroid inferred from moment tensor inversion are indicated. Epicentral source search region is shown by the black box. Extent of the Erebus crater and lava lake location are those shown in (Fig. 1).

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Fig. 9. Variance reduction (5) for six couple, three force solutions as a function of source position and depth, with best-fit locations indicated by stars. The lateral location of search region corresponds to the black box in Fig. 8. The location of the main crater rim from Fig. 1 is shown for reference.

(e.g., Aster and Shearer, 1990) is defined by the eigenvalues, λi as l = (λ1 − λ2)/(λ1 + λ2), br is the station-source unit vector, and ê1 is the unit eigenvector corresponding to the largest eigenvalue, λ1 (and is the unit least-squares best-fit vector to the horizontal particle motion). Each of the 25 epicentroid location shown in Fig. 8 was indexed to the nearest (50-m spacing) node in the finite-difference model of the volcano. Greens functions were calculated using the TOPO code for source elevations of 3300, 3200, 3100, 1050, 2800, 2500, and 2200 m. For comparison, the tip of the magmatic conduit as indicated by the mean lava lake surface is at an elevation of approximately 3350 m (Csatho et al., 2008-this issue). Initial inversions were performed (Aster et al., 2006) utilizing the AIC metric, and approximate half-space Green's functions (Johnson, 1974). In this work, moment tensor inversion was explored for ascending degrees of source complexity, using moment and force components appropriate for a fluid pressure and/or mass transport system (Aster et al., 2006). Models tested included 1) A singlecomponent isotropic Mogi (Mogi, 1958) source (K = 1); 2) A 3component dilatational source (K = 3); 3) A 3-component dilational source plus a vertical single force (K = 4); and 4) A 3-component dilatational source plus three orthogonal single forces (K = 6), and a full moment tensor of six couples plus three forces. Solution appropriateness (number of source components versus goodness of fit) was evaluated by finding the minimum of the Akaike information criterion (Akaike, 1974) under normally distributed error assumptions AIC ¼ 2KMσ 2 þ R

ð7Þ

where R is the square of the residual 2-norm, σ is the noise standard deviation (estimated from the pre-event noise in the VLP data), and the product KM is the total number of parameters in this inverse problem.

Aster et al. (2006) suggested a best solution for K = 6 rate functions and a source hypocentroid 150 m north of the minimum azimuthal error epicentroid and approximately 330 m west–northwest (283° E of N) of the center of the lava lake at an elevation of approximately 3100 m. However the inversions discussed here, using significantly more complex topographically corrected Green's functions calculated using the TOPO code, clearly produce best-fit solutions for a full suite of K = 9 forces and couples. These inversion results are shown in terms of their color-contoured variance reduction in Fig. 9 as a function of depth, with corresponding solution locations indicated by stars. Corresponding maximum variance reduction solutions for each depth are shown in Figs. 10 and 11. Representative data fits (for the maximum variance solution at 3300 m elevation) are shown in Fig. 12. We summarize the data fit and some useful solution metrics in Tables 1 and 2. 3. Discussion The best-fitting solution is found for an extremely shallow source (3300 m; V = 0.89), it is clear from the spatial pattern of variance reduction (Fig. 9) that there are a range of almost equally suitable solutions from the standpoint of fitting the data. However, all of the best-fitting solutions show common features that suggest some robustly resolved features of the VLP source. One robust feature is that the distribution of best-fitting solutions clearly indicates a very shallow VLP source; data fit falls off dramatically below 3050 m. This source region is consistent with earlier work (Aster et al., 2006), and indicates that the VLP source centroid lies in the upper extent of the volcano, at a depth that we can state with a high degree of uncertainty is less approximately 400 m beneath the lava lake (2900 m). If the VLP centroid represents a significant feature within the conduit system, this bodes well for

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Fig. 10. Moment rate and force rate functions corresponding to the maximum variance reduction solutions for elevations of 3300, 3200, and 3100 m (Fig. 9). Coordinate convention is (xˆ1, xˆ2, xˆ3) = (East, North, Up).

ongoing tomographic efforts to examine the velocity structure of the upper mountain above the 2000 m contour (Chaput et al., 2008; Zandomeneghi et al., 2008). An additional common best solution location feature is that they all lie from 150 m to 300 m east and from 0 to 150 m north of the lava lake. Depending on depth, this places the VLP centroid clearly to the west of the lava lake and beneath the central to northeastern quadrant of the main crater, consistent with our aforementioned particle motion analysis (Fig. 8). The implication is that the magmatic conduit system at shallow depths resides more towards the center to northwest of the crater complex, rather than beneath the significantly off-center lava lake (Fig. 1; Csatho et al., 2008-this issue). The presence of a substantial underlying magmatic system to the west of the lava lake is also supported by the intermittent presence of a second small lava lake near the western edge of the inner crater that has produced a few very small Strombolian eruptions since 2005. Absolute source size can be estimated from moment rate tensor functions using a scalar moment rate function (e.g., Stein and Wysession, 2003) Mr0 ðt Þ ¼

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ∑ Mii ðt Þ2

ð8Þ

i

which can be integrated to obtain a net scalar moment M0 ¼ ∫0∞ Mr0 ðt Þdt:

ð9Þ

For these results, we find a range of net scalar moments ranging between 3.2 × 1013 and 5.3 × 1013 N m. For comparison to general

seismic sources, this scalar moment can be converted into a moment magnitude (e.g., Hanks and Kanamori, 1979), using mw ¼ 2=3ðlog10 M0 −9:1Þ:

ð10Þ

The corresponding mw of a typical Erebus eruption VLP signal is thus approximately 3–3.1. This is substantially larger than the lowerbound moment magnitude equivalent estimated from the first pulse amplitude using a Mogi source approximation, mw ≈ 1.9, by Aster et al. (2003) for comparably sized VLP signals. This apparent discrepancy arises because the scalar moment calculated here is for the integrated contribution across 200 s of oscillatory source activity. Because the VLP process is obviously causally linked to forces arising from the transport of a gas slug to the vent and subsequent recovery of the conduit system after eruptive removal of ejecta (James et al. (2006), one might expect the moment rate functions to be dominated by dilatational components, as the pressurization of a near-summit magmatic system would propagate at P wave speeds and, for a system within the uppermost extent of the volcano, such pressurization would occur rapidly relative to VLP periods. An examination of the shear couples in the solutions show that the shear couple moment rate functions are, as expected, relatively small, but not entirely negligible. The ratio of the dilatational to total net scalar moments ranges between 85 and 97%. VLP source non-couple forces might similarly be expected to include reaction forces generated by the acceleration of conduit fluids. All solutions show single force terms that are predominantly in the (x 1,x 3), or (east, vertical) plane, where the initial force is downward and eastward. This is consistent with oscillatory upward and westward mass advection

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Fig. 11. Moment rate and force rate functions corresponding to the maximum variance reduction solutions for elevations of 3050 and 2800 m (Fig. 9). Coordinate convention is (ˆx1, xˆ2, xˆ3) = (East, North, Up).

and with the western trend of ejecta in Group 2 eruptions (Fig. 6). However, the eruption time scale, as observed by infrasound and shortperiod seismic radiation, is much shorter than the VLP time scale, and the single force components tend to show period content near 7–8 s (see further discussion on this point below). We note that the maximum amplitudes of the Mii and Fi components for characteristic Erebus eruptions are of a similar size to those reported for Stromboli by Chouet et al. (2003); Mii,max ≈ 2 × 1012 N m/s and Fi,max ≈ 5 × 108 N here compared to Mii,max ≈ 2 × 1012 N m/s and Fi,max ≈ 1 × 108 N at Stromboli, with a maximum force-to-maximum moment rate ratio (Table 1) on the order of 10− 4 m− 1 in both cases. However, the Erebus VLP signal has a significantly more sustained duration than that of Stromboli (several minutes at Erebus versus approximately 20 s at Stromboli). The above quantification and localization of the VLP source is consistent across the best suite of VLP inversions shown in Figs. 10 and 11. We next discuss the differences between the solutions. A largest progressive difference as the test hypocentroid depth increases is a transition in the dilatational terms between a solution that is dominated by a vertical (M33) force couple (e.g., 3300 m), such as might be produced by a pressurized subhorizontal crack or sill, to one that is dominated by M11 and M22 couples (e.g., 2030 m), as might be generated by a prolate, vertical conduit or chamber. Mii ratios, decomposed as eigenvectors and eigenvalues of the moment rate tensor, can provide key information on the orientation and aspect ratios of the VLP source (Davis, 1986; Yang et al., 1988; Chouet et al., 2003). A consistency check is thus to examine the correlation between the Mii functions. To be consistent with the pressurization of a crack or

more general cavity, we would expect the dilatational terms to be in phase so that the correlation of the respective time functions are highly positive. These metrics are reported in Table 2. The correlation between M11 and M22 is moderately high for all best solutions, ranging from 0.71 (3200 m) to 0.89 (3050 m). The correlation between M11 and M33 (Table 2; Figs. 10 and 11), however, is highly variable, showing relatively high correlations for the shallow and deep (3300 m and 2800 m) solution end members of 0.79 and 0.66, respectively, but very low correlations between these depths. The corresponding nigh degree of anticorrelation between M33 and F3 for the shallow depths is highly suggestive of poor resolution in the inversion. The tradeoff between these two source terms is expected for shallow sources in a least-squares inversion because of similarities in the Green's functions for M33 and −f3 (Uhira and Takeo, 1994; Chouet et al., 2003). We thus contend that the low correlation between the horizontal and vertical moment rate terms is due to this effect. Mii ratios between the vertical and horizontal components are therefore highly affected by this tradeoff. M11 to M22 amplitude ratios vary between 2.85 (3300 m) and 1.06 (3100 m), showing that the azimuthal aspect of the source is also not particularly well constrained, although the trial source locations below 3300 m have a relatively consistent ratio between 1.38 and 1.06, suggestive of an equant azimuthal source geometry. The modeling here incorporates two significant approximations. First, the source is modeled as a superposition of couples and single forces applied at a common point within the volcano, and is therefore a highly idealized representation of a spatially distributed source. However, the point approximation is a good one when scaled to the

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R. Aster et al. / Journal of Volcanology and Geothermal Research 177 (2008) 635–647 Table 2 Solution summary metrics, continued. H is elevation, c values are correlation coefficients between indicated rate functions, and RM is the proportion of data variance due to the moment tensor terms alone in the forward calculation H (m)

c(M11,M22)

c(f3,M33)

c(M11,M33)

|M11|/|M22|

|M11|/|M33|

RM (%)

3300 3200 3100 3050 2800

0.80 0.71 0.84 0.89 0.86

−0.89 −0.86 −0.97 −0.92 −0.26

0.79 0.02 −0.29 0.09 0.66

2.85 1.33 1.06 1.15 1.38

0.45 0.74 1.83 2.93 1.86

75 74 78 79 76

stations, and this suspicion is borne out in the relatively large misfit for seismograms at this station (Fig. 12. A probably smaller source of modeling error is the use of a uniform elasticity and density structure, which was constrained using the near-summit refraction estimates of Dibble et al. (1994). A simple test of the veracity of the Dibble et al. velocity model is to examine the crosscorrelation lags between observed and forward modeled waveforms for the various VLP pulses across the network. We find this fit to be generally consistent relative to the long periods that characterize the VLP data. The largest such discrepancy, as measured by the entire VLP signal crosscorrelation lags between the complete observed and predicted three-component signals of Fig.12, is a negative residual of approximately 2 s seen at HOO, located 5.5 km from the source epicentroid. This indicates, not surprisingly that the elastic moduli of the volcanic edifice far outside of the central conduit region (Fig. 1) are appreciably higher than the velocity model. Conversely, the near-summit stations LEH and NKB show positive residuals of 0.1 and 0.38 s, respectively, suggesting that the northern sector of the nearsummit region out to ranges of approximately 2 km is more compliant than specified by the half-space parameters. The best lag for the net crosscorrelation between observed and predicted data across all 18 components is zero to sample resolution. The contribution of the single force components to total variance reduction in this inversion is consistently approximately 25% for all solutions (Table 2). Single forces also appear as a primarily posteruptive component of the source. Single forces are expected to be generated by magmatic momentum change within the conduit. For mass-conserving systems like the Erebus conduit during its posteruptive magmatic refill phase, the generated internal forces are simply the spatial integral of the magma density multiplied by its corresponding acceleration distribution Fig. 12. Data, forward modeling, and residual corresponding to the 3300 m maximum variance reduction solution of Fig. 10.

f ¼ ∫ ρðV ÞaðV ÞdV:

ð11Þ

V

wavelength of seismic waves at these periods (10 km or longer) and the total size of the array (on the order of an S wave wavelength), similar inversions on other volcanoes and synthetic tests have shown that this issue may not be highly significant except for very sparse networks (P. Dawson, pers. commun.). Second, we incorporate the effects of topography using a fairly crude approximation with a 50 m finite-difference node spacing. Topographic effects should be especially notable for station RAY, which was nearly perched on the crater rim overlooking the lava lake (Fig. 1). We suspect that Green's functions for RAY are significantly less accurate than for the other Table 1 Solution summary metrics for the five best (and shallowest) maximum variance reduction solutions, with corresponding time functions shown in Figs. 10 and 11. H is elevation, V is variance reduction (5), M0 is the integrated seismic scalar moment, MΔ is the integrated seismic moment for the dilatational (Mii) components, F0 is the integrated total force H (m) 3300 3200 3100 3050 2800

V (%) 89 88 88 87 83

M0 (N m) 13

5.5 × 10 3.2 × 1013 4.1 × 1013 4.8 × 1013 5.4 × 1013

MΔ (N m) 13

5.3 × 10 3.0 × 1013 3.4 × 1013 4.1 × 1013 4.9 × 1013

%Δ 97 94 85 85 90

F0 / M0 (m- 1)

F0 (N) 10

1.8 × 10 1.6 × 1010 1.8 × 1010 1.8 × 1010 1.9 × 1010

3.3 × 10− 4 4.9 × 10− 4 4.5 × 10− 4 3.8 × 10− 4 3.5 × 10− 4

The initial abrupt ejection of materials from the Erebus lava lake generates a single force that undoubtedly contributes to the shortperiod signal (Henderson, 2007). However, this jet force has a period content that is peaked near 1 s, as observed in infrasound (Jones et al., 2008-this issue), which is much shorter that of the 5–30 s VLP band (Aster et al., 2003). Indeed, the eruption onset, as defined by the bubble burst, does not visibly appear as a feature on the VLP signals or moment rate functions when the data are low-passed below several seconds. In a rough calculation we can assume that the total force in the prolonged VLP signal is due to rapid shallow accelerations of magma in a constricted upper conduit region during lava lake refill. Using the eviscerated volume from the eruption (roughly cylindrical with a radius of 25 m and a depth of 20 m) as an approximation for the dimensions of the volume of magma in motion during subsequent refill, and using an approximate shallow Erebus magma density of 2000 km/m3 estimated by Dibble (1994), we obtain estimated total mass in motion of approximately 8 × 107 kg. Eq. (11) then implies that the inverted force magnitudes on the order of the observed 1 × 108 N could be generated by a mass of this magnitude undergoing a vector average acceleration with a magnitude on the order of 1 m/s2. An interesting consistent aspect of these moment tensor inversions is the asymmetric spectral partitioning of VLP spectral energy

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Fig. 13. Normalized power spectra for representative moment (M11) and force (F3) components of the rate functions of Fig. 10. VLP spectral components are denoted (Aster et al., 2003) as T0 ≈ 20.7 s, T1 ≈ 11.3 s, and T2 ≈ 7.8 s. The unequal partitioning of energy between the Mii moment components and the Fi single forces, with the single force accounting for the majority of the shortest period (T2) spectral component, is a general feature of these solutions.

peaks (Fig. 3) between the vertical and horizontal Fi and Mii rate functions (Fig. 13), particularly with respect to the T2 =7.8 s period peak being predominantly fit by the single force and/or M33 components. Single forces inferred from moment rate inversions are ideally attributable to magmatic recharge acceleration forces, but (as described above) show a high level of correlation in this study (Eq. (11)) that may be a resolution artifact of the inversion. This suggests that the T2 component of the VLP spectrum might be linked to a distinct process compared to the T0 and T1 spectral components. This partitioning is seen in all solutions, and is most apparent for the deepest solution (2800 m) shown in Fig. 11 (however this solution has significantly degraded variance reduction relative to shallower solutions). One hypothesis, that could be tested in the future with higher resolution data sets from an augmented broadband network is that the partitioning of the T2 component into the (predominantly vertical) force component system suggests that the resonance time for surging recharge into the eviscerated post-eruptive lava lake is distinct from the pressurization resonances of the system (T0 and T1). The stability of the VLP periods further implies that, despite thousands of Strombolian eruptions during the past decade, the repeating and self-reconstructing conditions of the lava lake and VLP system have been maintained. If the T2 component of the VLP spectrum were the fundamental “surging” excitation of the lava lake system to gravitational disequilibrium, then modeling this component of the VLP signal as a volcanic analogue of a hydraulic “slug test” in a highpermeability aquifer bounded by impermeable strata (e.g., Guenther and Mohamed,1986; Guenther et al.,1987) might be fruitful. In this analogue, a near-summit magma chamber/conduit system corresponds to the aquifer. The height of the magma column is only intermittently visible in video records during its highest amplitude excursions because of the camera view being obscured by ash and vapor. It may be possible in future modeling efforts to estimate this parameter, however, by doubly integrating the inferred acceleration derived from the F3 component of the moment rate tensor inversion, under the assumption of homogeneous oscillatory laminar flow of an appropriate conduit segment. We note that ascribing the T2 component of the VLP spectrum to oscillatory recharge is consistent with previously noted video evidence in that rare, exceptionally clear views of the immediate post-eruptive lava lake were observed to display surging behavior with a period of 8.8 ± 1.6 s. Additionally, source Q estimates of the principal VLP modes from spectral peak widths, showing that the decay of the T2 component is significantly more rapid than the T1 and

T0 components, with Q2 = 4, Q1 = 18, and Q0 = 11, again suggesting that it may be associated with distinct processes (Aster et al., 2003). Notable other studies of conduit-associated VLP signals at active volcanoes include Aso, Japan (Kawakatsu et al., 2000; Legrand et al., 2000), Stromboli (Chouet et al., 1999, 2003), and Popocatepetl (ArciniegaCeballos et al., 1999; Chouet et al., 2005). In the case of Aso volcano, a similar gravitationally-driven inertial mechanism to that proposed here for Erebus is suggested, with the important distinction that a mixture of fluid and rock interacting with a sub-crater hydrothermal reservoir (heated by deeper magma) is invoked at Aso to model the mechanism of oscillation. Preferred models for VLP moment tensor inversions for Stromboli and (vulcanian) Popocatepetl eruptions, however, invoke magmatic transport through constrictive, single- or multiple crack-like upper conduit structures embedded in compliant media (Chouet,1996). In this respect Erebus, with its highly oscillatory and exceptionally longduration VLP signals may constitute a near end-member “open” or “underdamped” example of such systems, where inertial forces generated by the relatively unimpeded transport and back-flow of recharging magma are dominant, as opposed to shorter duration VLP signals largely controlled by a more restrictive crack-like conduit geometry. A long-lived stable open conduit system is furthermore consistent with geochemical modeling of characteristic abundant anorthoclase feldspar phenocrysts erupted from the lava lake system, which require long-term (e.g., decades to millennia) convective circulation of magma at depth (between approximately 400 m and the surface) for their formation (Dunbar et al., 1994). The displacement of the VLP source centroid by approximately 330 m from the lava lake, and somewhat closer to the geometric center of the main crater and the consistent orientation of single forces in these moment rate inversions are consistent with a shallow kink in the magmatic system beneath the lava lake. 4. Conclusions Strombolian eruptions from the Erebus lava lake consistently produce repeating VLP signals that show high degrees of correlation across their several minute durations, but which show variable first motion polarity encompassing the first approximately 5–10 s. First motion differences are correlated with ejecta direction characteristics observed in video observations and are thus reflective gas slug delivery and eruption and the immediate conduit response. Aside from these early seismogram variations, time- and frequency-domain

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characteristics of VLP signals are notably invariant across a decade of observation, despite widely varying eruptive frequency at the volcano, ranging from many events per day to months-long quiet periods. This degree of consistency in the VLP signature indicates a stable, selfreconstructing VLP source and stable upper conduit system, where the underlying plumbing system dynamically responds in a reproducible manner to departures in gravitational equilibrium induced by the gas slug-mediated removal of magma from its tip. Moment rate tensor inversion of a high signal-to-noise six-station, 18-component data constructed by stacking VLP signals produce solutions that constrain the source centroid to lie less than about 400 m below the lava lake and to be laterally displaced by up to several hundred meters to the west and north. Mii component ratios are variable with trial source depth, and tradeoffs between vertical forces and vertically-oriented moments are large due to the ill-posed nature of the inverse problem for this small network. These tradeoffs unfortunately make chamber and other geometric constraints on the summit magmatic system unfeasible with this limited network geometry. Net scalar moments range between 3.2 × 1013 and 5.3 × 1013 N m. Energy partitioning between spectral components of the VLP signal may indicate that the shortest-period predominant VLP mode (T2 ≈ 7.8 s) is preferentially associated with the single force system, and may thus reflect resonant recharge of magma into the post-eruptive lava lake system. Acknowledgments We thank UNAVCO and the IRIS PASSCAL Instrument Center at New Mexico Tech for facility support and field assistance on Mount Erebus. Donations of equipment from Extreme CCTV and VideoComm Technologies were essential for video observations. We thank the many Raytheon Polar Services Company individuals and groups at McMurdo who made this field effort possible. The development and testing of inverse methodologies was assisted by Brian Borchers and Christian Lucero (Lucero, 2007). The manuscript was significantly improved in revision following comments by P. Dawson and an anonymous reviewer. We additionally thank P. Dawson for providing us with the TOPO Green's function calculation code. This research was supported by NSF Awards OPP-9814291, OPP-0116577, OPP-0229305, and ANT-0538414 and by New Mexico Tech Research and Economic Development. References Akaike, H., 1974. A new look at the statistical model identification. IEEE Trans. Automat. Contr. AC-19 (6), 716–723. Almendros, J., Chouet, B., Dawson, P., Bond, T., 2002. Identifying elements of the plumbing system beneath Kilauea Volcano, Hawaii, from the source locations of very-long-period signals. Geophys. J. Int. 148, 303–312. Arciniega-Ceballos, A., Chouet, B., Dawson, P., 1999. Very long-period signals associated with vulcanian explosions at Popocatepetl volcano, Mexico. Geophys. Res. Lett. 26, 3013–3016. Arciniega-Ceballos, A., Chouet, B., Dawson, P., Asch, G., 2008. Broadband seismic measurements of degassing activity associated with lava effusion at Popocatepetl Volcano, Mexico. J. Volcano. Geotherm. Res. 170, 12–23. Aster, R., Shearer, P., 1990. Quantitative measurements of shear-wave polarizations at the Anza seismic network, southern California—implications for shear-wave splitting and earthquake prediction. J. Geophys. Res. 95, 12449–12473. Aster, R., Lees, J., Neuberg, J., 2000. Broadband seismic and acoustic observations of volcanic seismicity (editorial). J. Volcanol. Geotherm. Res. 101, vii–viii. Aster, R., Mah, S., Kyle, P., McIntosh, W., Dunbar, N., Johnson, J., 2003. Very long period oscillations of Mount Erebus volcano. J. Geophys. Res. 108, 2522. doi:10.1029/ 2002JB002101. Aster, R., McIntosh, W., Kyle, P., Esser, R., Bartel, B., Dunbar, N., Johns, B., Johnson, J., Karstens, R., Kurnik, C., McGowan, M., McNamara, S., Meertens, C., Pauly, B., Richmond, M., Ruiz, M., 2004a. New instrumentation delivers multidisciplinary real-time data from Mount Erebus, Antarctica. EOS trans. AGU. 85 (10) March 9. Aster, R., Borchers, B., Thurber, C., 2004b. Parameter Estimation and Inverse Problems. Elsevier Academic Press. 301 pp. Aster, R., Beaudoin, B., Hole, J., Fouch, M., Fowler, J., James, D., et al., 2005. IRIS PASSCAL program marks 20 years of scientific discovery. EOS. trans. AGU 86 April 26. Aster, R., Kyle, P., McIntosh, W., Lucero, C., Borchers, B., 2006. Very long period Strombolian eruption-associated seismic signals observed in the near field at Mount Erebus volcano. The Physics of Fluid Oscillations in Volcanic Systems Workshop, Lancaster, U.K., 7–8 September.

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