Extracting source characteristics and dynamics of the August 2010 Mount Meager landslide from broadband seismograms

Authors

  • Kate Allstadt

    Corresponding author
    1. Department of Earth and Space Sciences, University of Washington, Seattle, Washington, USA
    • Corresponding author: K. Allstadt, Department of Earth and Space Sciences, University of Washington, Box 351310, 4000 15th Ave. NE, Seattle, WA 98195-1310, USA. (allstadt@uw.edu)

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Abstract

[1] Seismic methods can substantially improve the characterization of the dynamics of large and rapid landslides. Such landslides often generate strong long-period seismic waves due to the large-scale acceleration of the entire landslide mass, which, according to theory, can be approximated as a single-force mechanism at long wavelengths. I apply this theory and invert the long-period seismic waves generated by the 48.5 Mm3 August 2010 Mount Meager rockslide-debris flow in British Columbia. Using data from five broadband seismic stations 70 to 276 km from the source, I obtain a time series of forces the landslide exerted on the Earth, with peak forces of 1.0 × 1011 N. The direction and amplitude of the forces can be used to determine the timing and occurrence of events and subevents. Using this result, in combination with other field and geospatial evidence, I calculate an average horizontal acceleration of the rockslide of 0.39 m/s2 and an average apparent coefficient of basal friction of 0.38 ± 0.02, which suggests elevated basal fluid pressures. The direction and timing of the strongest forces are consistent with the centripetal acceleration of the debris flow around corners in its path. I use this correlation to estimate speeds, which peak at 92 m/s. This study demonstrates that the time series recording of forces exerted by a large and rapid landslide derived remotely from seismic records can be used to tie post-slide evidence to what actually occurred during the event and can serve to validate numerical models and theoretical methods.

1 Introduction

[2] Direct time-dependent observations of natural landslides are critical to improving our understanding of landslide dynamics and hazard. However, such observations can be hard to come by due to the destructive nature of landsliding events, uncertainty about when and where they will occur, and their sometimes remote locations. Seismology is a potential tool to span this observational gap. Assuming the landslide under investigation radiates enough seismic energy to be recorded at existing seismic stations and the seismograms can be correctly interpreted, seismic data can provide a time series recording of landsliding events that can be used to extract information about landslide dynamics and source characteristics. This approach is comparable to how seismologists have been using seismograms to study earthquakes for over a century, but landslide seismology has the added benefits of being able to access the source area and of knowing the driving stress—gravity. When combined with field investigation, theoretical methods, and numerical landslide modeling, a much clearer interpretation of the event being investigated can emerge [e.g., Favreau et al., 2010, Moretti et al., 2012, Guthrie et al., 2012].

[3] In order to correctly interpret seismic signals of landslides, one must first understand how landslides radiate seismic energy. Energy is radiated on two scales: coherent long-period waves at periods of tens to hundreds of seconds generated by the acceleration and deceleration of the failure mass as a whole [Kanamori and Given, 1982; Eissler and Kanamori, 1987], and a more stochastic higher-frequency signal at periods from a few seconds to frequencies of tens of hertz generated by momentum exchanges on smaller scales such as flow over smaller-scale topographic features, frictional processes [e.g., Schneider et al., 2010], and impacts of individual blocks [e.g., Huang et al., 2007].

[4] Large and rapid landslides, in particular, are effective at generating strong long-period seismic waves, particularly surface waves, which attenuate slowly and can be detected at seismic stations for hundreds to thousands of kilometers. These long-period seismic waves are not sensitive to heterogeneities in the crust on much shorter scales than their wavelengths, so they can be studied using simplified Earth velocity models and are thus easier to work with than higher-frequency seismic energy. The long-period seismograms generated by such large landslides are often recorded at great distances and have been used for decades to study landslides.

[5] Some authors have directly interpreted the timing of pulses and variations in amplitude in long-period landslide seismograms to determine the occurrence, duration, speed, and timing of events [Berrocal et al., 1978; Weichert et al., 1994; McSaveney and Downes, 2002; Guthrie et al., 2012].

[6] Others have taken analysis further and used the long-period seismic waves to study the source process directly. In contrast to the double-couple mechanism of earthquakes, the equivalent force mechanism of a landslide is a single force applied to the surface of the Earth proportional to the acceleration and mass of the moving material. This is what generates the long-period seismic waves [Kanamori and Given, 1982; Eissler and Kanamori, 1987; Hasegawa and Kanamori, 1987; Kawakatsu, 1989; Fukao, 1995; Julian et al., 1998]. Many authors have used forward modeling to determine the amplitude and duration of the forces exerted on the Earth that could generate the observed seismic waves and used the result to estimate the mass or the acceleration of the landslide and to interpret the sequence of events [e.g., Kanamori and Given, 1982; LaRocca et al., 2004]. These methods have even been used to argue that what was thought to be an earthquake was actually a landslide [e.g., Eissler and Kanamori, 1987; Hasegawa and Kanamori, 1987]. Others have used this approach to constrain rheological characteristics. For example, Brodsky et al. [2003] estimated the coefficient of friction beneath three large volcanic landslides based on the forces they exerted on the Earth. Favreau et al. [2010] used long-period seismic observations of the 2004 Thurweiser landslide in Italy to determine the rheological parameters to use in a numerical landslide model. Moretti et al. [2012] used the long-period seismic signals generated by the 2005 Mount Steller landslide in Alaska to invert for the forces it exerted on the Earth and used both to constrain details about the flow dynamics in a numerical model. Most recently, Ekström and Stark [2013] inverted seismic waves generated by 29 large and rapid landslides recorded by the Global Seismographic Network. They used their catalog of landslide force inversions combined with field data to build empirical relations between maximum forces and mass, momentum, potential energy loss, and surface wave magnitude. These relations allow for rapid order of magnitude estimates of landslide size without having to wait for other evidence.

[7] However, the long-period seismic signals generated by the acceleration of the landslide as a whole that can be approximated as a single-force mechanism are not always observed. If the mass is too small and/or the average acceleration of the landslide too slow, the forces the landslide exerts on the Earth will be smaller [Kanamori and Given, 1982] and less likely to generate a long-period signal above the noise level on nearby seismometers. Furthermore, the period of the waves generated depends on the duration of the forcing [Kanamori and Given, 1982], so a slow landslide with an extended duration may not emit waves at seismic frequencies. The higher-frequency signal generated by smaller-scale processes, on the other hand, has been often observed for a wide range of landslide sizes and many authors have used this type of signal to study landslides as well. Often referred to as emergent, cigar shaped, or spindle shaped, the higher-frequency seismic energy typically builds up gradually, emerging from the noise without a clear onset or obvious phase arrivals, and then tapers back into the noise afterward [e.g., Norris, 1994; Dammeier et al., 2011; Deparis et al., 2008; La Rocca et al., 2004; Schneider et al., 2010; Suriñach et al., 2005].

[8] Such signals are useful for determining the occurrence, duration, and timing of landslides [e.g., Norris, 1994; Helmstetter and Garambois, 2010]. It is more challenging to use these signals to obtain quantitative landslide characteristics such as failure volume, fall height, or runout distance because only a small percentage of the energy is transmitted seismically for all landslides [Berrocal et al., 1978; Deparis et al., 2008; Hibert et al., 2011], and higher-frequency waves attenuate rapidly and are much more affected by smaller-scale heterogeneities in the crust. Despite this, several authors have been successful in estimating landslide characteristics, such as volume and runout length, within an order of magnitude or better, particularly in the presence of dense nearby seismic networks [Norris, 1994; Deparis et al., 2008; Hibert et al., 2011; Helmstetter and Garambois, 2010; Dammeier et al., 2011]. Schneider et al. [2010] investigated the physical basis for variations in amplitude in the higher-frequency seismic signal and found that increases in the relative amplitude could be attributed to increases in the loss of power due to frictional processes—the frictional work rate. The frictional work rate can be elevated, for example, after a sudden increase in speed after passing a step in the path, or when the sliding material hits a flatter area at high speeds and begins to decelerate as frictional resistance increases [Schneider et al., 2010]. Thus, the relative amplitude of the high-frequency seismic signature can be used to tie the timing of the seismic signal to the passage of material over particular sections of the sliding path.

[9] In this study, I inverted the long-period seismic signals generated by the August 2010 Mount Meager rockslide and debris flow in British Columbia to solve for the source process that generated them—the forces the landslide exerted on the Earth over time. I built on the initial characterization of the landslide by Guthrie et al. [2012] and show that the characterization of the dynamics and source process of the landslide can be substantially improved by these methods. The time series recording of forces exerted on the Earth during the landslide can be used to significantly reduce the level of interpretation required to tie post-slide observations to what actually happened during the event. Though this study is based on the same seismic records Guthrie et al. [2012] used as part of their characterization, they used just the raw seismograms of this event to qualitatively interpret the timing of events from peaks in amplitude of the signal. By inverting the seismic signals, I take a much more direct and quantitative approach to obtain information about the source process by determining what forces actually generated the observed seismic waves at the source location. This is comparable to the inversion of seismic signals generated by earthquakes to obtain information about the source history of an earthquake. Just as this type of analysis has advanced our understanding of earthquake physics, such an analysis of the seismic signals generated by landslides can contribute to a greater understanding of the landslide physics.

[10] In the following sections I first detail the known characteristics of the Mount Meager landslide and the seismic data available. Then I describe the inversion methods used, test these methods with synthetic data, and invert the long-period (T = 30 to 150 s) seismic signals to determine the forces exerted on the Earth by the landslide with time. I compare this result to the envelope of the higher-frequency portion of the signal and piece together the sequence and timing of events. I also calculate the speed of the landslide with higher certainty than was possible using just the raw seismic data. This provides a validation for other less direct landslide speed estimation methods. I use this result to discern the direction of failure of subevents, extract the coefficient of dynamic friction during the rockslide, and observe changes in the behavior of the debris flow over time—characteristics that are otherwise difficult to determine in the absence of seismic analysis. This case study illustrates the benefits of including a seismic source analysis in landslide investigations and its potential to improve numerical models, an option that is becoming more readily available as seismic networks become denser and high-quality digital data more accessible.

2 Data

[11] On 6 August 2010 at about 10:27 UTC, 3:27 A.M. local time, the secondary peak (gendarme) and southern flank of Mount Meager, part of the Mount Meager Volcanic Complex in British Columbia, collapsed in a massive rockslide that quickly mobilized into a debris flow [Guthrie et al., 2012]. A rockslide is a failure of bedrock where sliding occurs dominantly on a single failure surface [Cruden and Varnes, 1996], while a debris flow is a poorly sorted, internally disrupted, and saturated flowing mass controlled by both solid and fluid forces [Iverson et al., 1997]. According to the interpretation of the event by Guthrie et al. [2012], once the rockslide converted to a debris flow, it traveled down Capricorn Creek valley, turning two corners, sloshing up the sides of the valley, and plowing down swathes of trees. When it reached the end of the 7.8 km long valley, the debris flow burst out into the adjacent Meager Creek valley and ran 270 vertical meters up the opposing valley wall. It then split and flowed upstream and downstream 3.7 and 4.9 km, respectively, where it finally stopped, leaving vast fields of deposits and temporarily blocking the Lilloet River and its tributary, Meager Creek. This sequence of events is illustrated on Figure 1 and an image of the headscarp and source area is provided in Figure 2. Field evidence showed that some deposition began almost immediately below the initiation zone, though most material was deposited after the convergence with Meager Creek. Very little material was entrained along the path, though later activity incised into the primary deposits [Guthrie et al., 2012].

Figure 1.

Overview of the sliding path overlain on a post-landslide satellite image with approximate boundaries outlined. Refer to inset map on Figure 3 for regional location of Mount Meager.

Figure 2.

Image looking northward at the source area of the landslide. The top of the headscarp is indicated with a dashed line; note the helicopter circled at upper right for scale. Image courtesy of Dave Steers.

[12] The source material was an estimated 48.5 million m3 of highly fractured and hydrothermally altered rhyodacite breccias, tuffs, and flows, with a porphrytic dacite plug in the steeper areas [Guthrie et al., 2012]. Additionally, the source mass was highly saturated, evidenced by the rapid mobilization of the rockslide to a debris flow, requiring the availability of a lot of water, as well as surface seepage and large springs observed along the failure surface [Guthrie et al., 2012]. Assuming a density range of 2000–2500 kg/m3, representative of typical values for these types of rocks, the total mass was 1.0–1.2 × 1011 kg.

2.1 Seismic Data

[13] This highly energetic event generated strong seismic signals that were visible above the noise level at over 25 three-component broadband seismometers throughout Canada, Washington State, and Alaska. A record section of the seismograms of this event recorded across British Columbia and Washington State by the Canadian National Seismograph Network (CNSN) and the Pacific Northwest Seismic Network (PNSN) (Figure 3) shows that the entire seismic signal lasted about 5 min before fading into the noise. The onset of the seismic signal is dominated by long-period pulses, which are then overtaken by a more chaotic short-period signal. The two distinct frequency bands, corresponding to the two types of signals radiated by landslides described earlier, are apparent in the spectrum of the signal, and their distinct characters are made more apparent by high- and low-pass filtering the same signal around 0.2 Hz (Figure 4). Note that it is impossible to pick out the P and S wave arrivals separately because the amplitudes of the body waves in the higher-frequency signal are below the noise at the start of the event (Figure 4b).

Figure 3.

Record section of vertical component broadband velocity seismograms generated by the Mount Meager rockslide and debris flow. Seismograms are corrected for station response and are all plotted on the same vertical scale, which is shown under the inset map. The inset map shows station locations relative to Mount Meager, ring-shaped symbols indicate stations used in the inversion of long period seismograms.

Figure 4.

The vertical component velocity seismogram recorded at WSLR (a) low-pass filtered below 0.2 Hz to isolate the long-period pulses that start first at point 1 and (b) high-pass filtered above 0.2 Hz to isolate the shorter period signal that emerges from the noise about 20 s later at point 2. (c) The two distinct frequency bands are apparent as two separate broad peaks on the velocity spectrum of the entire signal. By point 3, the long-period signal shifts to shorter periods. By point 4 the seismic signal is largely over but does not fade completely back into the noise until point 5, nearly 5 min after the start of the signal.

[14] The consistency in the signal between distant stations (Figure 3) demonstrates that both the low- and high-frequency portions of the signal largely reflect source effects and not path effects or site effects at individual stations. However, one interesting difference is that the amplitudes of the higher-frequency ground motion are significantly stronger at station LLLB than station SHB, though it is only 3 km farther from the source area. This could be due to lower attenuation rates east of the source area, site amplification at the location of LLLB, or both. Another contributing factor could be that the landslide moved toward station LLLB and away from SHB. This change in location would affect the amplitudes of the shorter-period waves more than the longer periods because anelastic attenuation is strongly frequency dependent. Therefore, the movement of the source 10% closer to LLLB could result in a noticeable reduction in the attenuation of higher-frequency waves over the course of the event. Suriñach et al. [2005] observed a similar effect for smaller landslides. In this study, landslide speeds are almost 2 orders of magnitude slower than the seismic wave velocities, so this is probably not true directivity as observed during earthquakes, which is related to the Doppler effect [Douglas et al., 1988].

[15] From the seismic data available, I selected those that had the best signal quality in the frequency band of interest by visual inspection. Five of the seismic stations had significantly better signal quality than the rest. Rather than using more data with questionable signal quality, I used just the data from these five stations because test inversions (detailed in section 3) showed that high-quality seismic signals from just a few seismic stations should be more than sufficient to recover the force-time function. The seismic stations used for the inversion are shown in relation to the location of the landslide on the map in Figure 3 and are detailed in Table 1. At least one component of each station had a signal to noise ratio (SNR) above 8 in the frequency band used in the inversion (T = 30–150 s). The highest SNR was 57. The noise characteristics of each component of each station used are displayed in Table 2.

Table 1. Broadband Seismic Stations Used in the Inversion
 LatitudeLongitudeDistance (km)Source to Station Azimuth (Clockwise From N)
WSLR50.1265°−122.9212°70142°
LLLB50.6090°−121.8815°11590°
SHB49.5930°−123.8805°118192°
PASS48.9983°−122.0852°208150°
MRBL48.5183°−121.4845°276149°
Table 2. Noise Characteristics and Solution Misfit at Each Station
Station NameComponentaSignal to Noise Ratio (Amplitude)% noiseWeight (1/RMS Noise × 100)RMS Noise (nm)RMS Misfit Solution (nm)
  1. a

    Z, R, T refers to vertical, radial, and transverse components, respectively.

WSLRZ232%2.540124
R411%0.25301002
T49%0.3316434
LLLBZ84%2.051131
R75%1.286168
T572%4.72166
SHBZ88%0.6157170
R818%0.4282657
T312%0.5206416
PASSZ316%1.189105
R410%0.9110266
T716%0.4239232
MRBLZ233%3.23288
R826%0.5220306
T77%1.193191

[16] I prepared the seismic data by deconvolving the instrument response, integrating the seismograms from ground velocity to ground displacement, and rotating the horizontal components to the radial and transverse direction for each station. The radial component refers to motion directly toward or away from the source and transverse is perpendicular to radial. I then band-pass filtered the data between periods of 30 and 150 s using a second-order minimum-phase (causal) Butterworth band-pass filter. Using shorter-period waves in the inversion would require a detailed velocity model of the region because shorter-period seismic waves are more sensitive to smaller-scale heterogeneities and topography. Periods longer than 150 s could not be used because such long periods are beyond the falloff in the response curves of all of the seismic stations and including them amplified the noise at these longer periods and overwhelmed the signal. The data were weighted for the inversion by the inverse of the root-mean-square (RMS) value of the noise before the signal at the periods used. These weights are reported in Table 2.

3 Methods

[17] For a landslide simplified as a block of constant mass m sliding down a slope, the magnitude of the slope parallel force F comes from the driving force of gravity opposed by the frictional force Ff:

display math(1)

where g is the magnitude of the acceleration due to gravity, θ is the slope angle, and bold font indicates a vector quantity. The magnitude of the frictional force on the block is equal to

display math(2)

where μ′ is the apparent dynamic coefficient of friction, which accounts for both friction and the basal pore fluid pressures, and Fn is the magnitude of the normal force. The sum of forces in the direction perpendicular to the slope is zero because the landslide does not accelerate into or out of the slope, so the magnitude of the normal force is equal to

display math(3)

[18] This also means that the magnitude of the net force Fnet is equal to the magnitude of the slope parallel force F and equation (1) can be rewritten as

display math(4)

[19] If the frictional force and gravitational driving force are unbalanced, as they are in the case of a mobilizing landslide, the block will begin to accelerate. According to Newton's second law, the net force acting on an object is equal to its mass times its acceleration, a, so equation (4) becomes

display math(5)

[20] According to Newton's third law, the forces of two bodies on each other are equal and opposite. Therefore, at the long wavelength limit, as the sliding block feels a force due to its gravitational acceleration the Earth feels an equal point force, Fe, in the opposite direction. Fe is a three-component vector that can be written more explicitly as a time-dependent phenomenon:

display math(6)

where bolding indicates a vector. This is the equivalent force system that is responsible for generating the observed long-period energy generated by large rapid landslides [Kanamori and Given, 1982; Eissler and Kanamori, 1987; Hasegawa and Kanamori, 1987; Kawakatsu, 1989; Fukao, 1995; Julian et al., 1998]. Fe(t) is comparable to the source-time function of an earthquake so I refer to it as the force-time function in this study. Friction and slope angle control the acceleration of the sliding mass (equation (5)), which in turn determines the forces exerted on the Earth (equation (6)). Equation (6) dictates that the force felt by the Earth will be in the opposite direction to the landslide acceleration. So as the source mass accelerates downslope, the Earth feels a single force applied in the upslope direction. Then, as the mass decelerates (accelerates upslope), the Earth feels a single force in the same direction as the sliding mass, i.e., downslope. When the landslide banks a curve, the acceleration is toward the center of the curve (centripetal acceleration), and thus the equivalent force points away from the center of the curve. Julian et al. [1998] also describe the torque due to lateral displacement of the landslide mass as a potential source of seismic radiation. However Brodsky et al. [2003] did not find it to be a significant contributor of seismic waves in their analysis of similarly large and rapid landslides. For this study, the inclusion of torque as a seismic source was not required to fit the data and was not incorporated into the analysis.

[21] In this study, I inverted the seismic data to determine Fe(t). Once obtained, the force-time function can be used with equations (5) and (6) combined with the field evidence, imagery, and geospatial calculations compiled by Guthrie et al. [2012] to extract information about the source characteristics and dynamics of the landslide. However, there are a few caveats to using equations (5) and (6) directly for interpretation: The rigid block model of a landslide is of course a simplification of reality and these equations do not account for all spatiotemporal dependencies. First, the area of application of the force will migrate with the landslide mass and will change in total area over time. In spite of this, the landslide can still be treated as a stationary single force point source for long-period (and thus long wavelength) seismic waves even though, in this case, the sliding mass moved more than 12 km. This is because the shift in arrival times between waves generated at the start and the end of the sliding path would be less than 2 s at all stations for the slowest waves in the frequency band used: Rayleigh waves of a 30 s period. Two seconds is a negligible fraction of the wavelength. The difference in arrival times would be even smaller for the faster longer period waves.

[22] Second, the mass may vary with time. Mass may be added due to erosion and entrainment and removed due to deposition. Significant changes in mass over time are important for inferring acceleration from force (equation (6)) and must be taken into account in the interpretation. Furthermore, the failure mass spreads out in space and becomes internally disturbed and agitated when it transforms from a rockslide to a debris flow. To be a pure single force, the dislocation of the sliding mass needs to be spatially uniform; otherwise higher order force components can contribute seismic radiation [Fukao, 1995]. The complexities of debris flow motion and its elongation over its sliding path can hinder the straightforward interpretation of the force-time function. This is particularly true in the case of Mount Meager where there are several sharp bends in the sliding path and segments of the debris flow may be accelerating in different directions simultaneously resulting in opposing forces.

[23] In order to invert the seismograms to obtain Fe(t), I first set up the forward problem relating how the source process translates to the observed seismograms. The seismogram recorded at each station (after the station response has been removed) represents the effects of the source itself as well as its path through the Earth. To a good approximation, at long periods, the Earth is a linear system that can be characterized by the seismogram that would be recorded at the seismometer location from an impulse force applied at the source location. The set of impulse responses between each source and station pair are known as the Green's functions and they account for all types of seismic waves as well as attenuation along the wave path. The seismograms for a realistic source can then be obtained by convolving the Green's functions with a source-time function that describes the evolution of the source process over time [e.g., Stein and Wysession, 2003]. In the case of a landslide, this is the force-time function Fe(t).

[24] Green's functions can be calculated if the velocity structure of the material the seismic waves are passing through is known. The periods used in this study (T = 30–150 s) have wavelengths on the order of hundreds of kilometers and a low sensitivity to smaller-scale heterogeneities in the regional velocity structure or topography. For this reason, a generalized Earth model was sufficient to use in the calculation of the Green's functions. In this study I used the 1-D ak135Q Earth velocity and anelastic attenuation model [Kennett et al., 1995]. I calculated the Green's functions between each station and the landslide location using the wave number integration method [Bouchon, 1981] as implemented in Computer Programs in Seismology (CPS) [Hermann, 2002]. Only the source to station distance and the velocity and anelastic attenuation model were required for this step of the calculation. The radiation patterns of the seismic waves were accounted for in the inversion equations based on the source to station azimuth as explained in Appendix A. The source to station distance did not change over time in the inversion because the location of the single force applied to the Earth by the landslide is assumed to be a stationary point source for reasons explained earlier in this section. Once the forward problem was set up, I inverted for the force-time function using damped least squares [e.g., Aster et al., 2005]. Complete details of the inversion process are located in Appendix A.

3.1 Validation

[25] To validate the robustness of this inversion method, I first tested it with synthetic data to see if it was capable of recovering an input force-time function and to test the noise tolerance. I started by using the forward model (equation (A1)) to generate three-component synthetic seismograms for an arbitrary force-time function. I then progressively increased the noise in the synthetic seismograms by adding Gaussian noise with a standard deviation equal to a percentage of the maximum absolute peak of the noise-free synthetic data (Figure 5). I then used these data to invert for the force-time function using the process described in Appendix A to see how well it returned the original signal after the addition of noise. The results of this inversion for a range of noise levels first using a single three-component station (WSLR) and then three three-component stations (WSLR, LLLB, SHB) are shown in Figures 6 and 7, respectively.

Figure 5.

The vertical component of one station (WSLR) of the synthetic seismograms generated for an arbitrary force-time function with 0 to 40% Gaussian noise added to illustrate the range in signal quality used in the test inversions.

Figure 6.

The three-component force-time function recovered by the test inversion of synthetic seismograms for a single three-component station (WSLR) with 0 to 40% noise added. The original force-time function that was used to generate the synthetic seismograms (bottom line) is shown for comparison. All force-time functions are scaled identically. The relative root-mean-square errors (in arbitrary units) between the force-time functions obtained by the inversions and the original are indicated at the right of each signal.

Figure 7.

Same as Figure 6 but for the inversion of synthetic data for 3 three-component seismic stations (WSLR, LLLB, and SHB) with varying amounts of noise added to the signal.

[26] The success of the inversion in retrieving the input test signal confirms that a force-time function very close to the original can be retrieved using data from just a few high-quality three-component stations if the noise levels are low and random. Even with just one three-component station and a significant amount of noise, this inversion method recovered a force-time function close to the actual input model. The ability of the inversion to recover the signal starts to break down when noise levels reach 30–40%, but the noise levels for the real seismic signals of the landslide in the time immediately before the earthquake were mostly below 10% (Table 2).

4 Results

[27] Using the methods described above and in Appendix A, I inverted the seismic data to solve for the force-time function of the Mount Meager landslide. The result is shown in Figures 8a–8c. This is compared to the raw seismogram from the closest station (Figure 8d) and the azimuth of the force vector at each point in time (Figure 8e). To quantify the fit of the resulting force-time function, I generated synthetic seismograms from the inversion solution by plugging the force-time function back into the forward model. The synthetic seismograms fit the real data remarkably well (Figure 9) with a variance reduction of 80%. The model can even closely reproduce data that were not used in the inversion (EDB, bottom of Figure 9). The worst misfits were for the stations and components that had strong long-period noise in the signal, so it was encouraging that the solution did not fit that noise when it was present before and after the landslide signal. The root-mean-square (RMS) errors between the original and synthetic seismograms are comparable to the RMS of the noise prior to the signal (Table 2), though the RMS errors are nearly all higher than the RMS of the noise. Because the forward model is approximate, I am fitting 15 channels of noisy data simultaneously, and the noise may not be entirely random as it was in the test inversions.

Figure 8.

(a–c) The three-component force time function for both the broadband five-station inversion and the solely long period inversion. The vertical force is positive up and the zero of the time scale corresponds to the start of the landslide. (d) The vertical component of the original unfiltered broadband seismic signal at the closest station (WSLR) and the envelope of the energy above 0.2 Hz in the signal, shifted to line up in time with the force-time function. Intervals 1-5 correspond with events described in text. Points a–e indicate the peak in the force-time function within each interval that was used to tie the arrival of the center of mass of the landslide with arrival at points of peak forcing along the path. The location and horizontal direction of the force at each of these points are plotted in Figure 11. (e) The azimuth of the force vector over time.

Figure 9.

Comparison of observed displacement seismograms (dashed line) with the synthetic seismograms generated by the force-time function obtained in the broadband inversion (T = 30–150 s) when plugged back into the forward model (solid line). The number in the right corner of each box indicates the relative weighting of the original data used in the inversion. Data from EDB (shaded), 270 km from the source, are shown to demonstrate how well the model reproduces data from a station not used to develop the model. Vertical units are centimeters.

[28] The force-time function of the Mount Meager landslide starts at 10:26:55 UTC and has a duration of about 215 s and peak force amplitudes on the order of 1011 N. This is of the same order of magnitude as the forces generated by the similarly sized Mount Steller landslide [Moretti et al., 2012], but 1 to 2 orders of magnitude smaller than some much larger landslides in other volcanic areas [Kanamori and Given, 1982; Brodsky et al., 2003], and 4 orders of magnitude smaller than some huge submarine landslides [Hasegawa and Kanamori, 1987; Eissler and Kanamori, 1987]. This is to be expected because force scales linearly with mass (equation (6)).

[29] The forces are primarily horizontal, consistent with the findings of other studies of landslide single-forces [e.g., Kanamori and Given, 1982; Brodsky et al., 2003]. The amplitudes of the vertical forces are much lower and more prone to noise in the solution. In particular, the dip in the amplitudes prior to the start of the landslide, an artifact that only occurs on the vertical component, suggests the overall amplitudes may be reduced by an unknown amount. As a result, I focused on the horizontal components in the quantitative interpretation requiring absolute amplitudes. I use the vertical component of force only for the relative direction (up or down) of the horizontal forces and timing of events along the path.

[30] With knowledge about the sliding path, the timing and changes in direction of the force vector can be attributed to events along the path. The initiating rockslide is well approximated as a sliding block allowing us to make first-order calculations about landslide dynamics using equations (5) and (6) directly. The subsequent debris flow is not well approximated by a sliding block, but we can also make some inferences about the debris flow behavior and estimate speeds based on the timing of changes in direction of the forces relative to the debris flow path. Numerical landslide modeling might be necessary to fully interpret the features of the force-time function, but that is beyond the scope of this study.

4.1 Rockslide Initiation

[31] The Mount Meager rockslide failed toward the south, and according to theory, the direction of the initiating single force should be in the opposite direction: northward and upslope. That is, in fact, what occurs first in the force-time function, but instead of one wide pulse, there are two pulses superimposed on each other with about 20 s between their peaks (Figure 8, interval 1). The acceleration direction of the first pulse had an azimuth of 191 ± 3°, followed by another pulse of failure in a more southwesterly direction (217 ± 3°). The errors in azimuth are estimated as the standard deviation of the slope angle in interval 1 in Figure 8e. This sequence suggests a progressive mobilization of the flank of the mountain. Based on the shape of the source volume (Figures 1 and 2), the landslide may have started with the release of material lower down on the flank of the slope toward the south, generating the first pulse. As this started to mobilize, it may have destabilized the material above, the bulk of which is to the northeast of the lower flank and may have failed in a more southwesterly direction, as the azimuth of the second pulse suggests. This two-part failure is consistent with the report of two loud cracks heard at the start of the landslide by campers nearby [Guthrie et al., 2012].

[32] Though this indicates the flank mobilized in two pulses, they occur close enough in time to act as one bulk movement in the generation of the longest-period seismic waves, which are not sensitive to shorter timescale subevents. This is apparent in the force-time function obtained by inverting only the longer period waves (T = 75–150 s, Figures 8a–8c). In this result, the overall mobilization of the rockslide now appears as a longer-period single pulse with an overall acceleration toward 213 ± 5° (Figure 8).

[33] After these initiating pulses, at t = 40 s, the force vector starts to point downward and toward the south, possibly due to the rockslide starting to decelerate, but this is interrupted by a sharp upward and then downward force suggesting a rapid vertical collapse and impact (start of interval 2, Figure 8). This could signify the collapse of part of the secondary peak (gendarme) of Mount Meager or other steep material from the headwall that was left unsupported as the flank below mobilized. It is probably not the entire secondary peak, however, because its total volume was estimated as 8 to 10 Mm3 (N. Roberts, personal communication, 2013), which would generate a vertical force of about 2 × 1011 N if it collapsed vertically (using equation (6) and assuming a density of 2300 kg/m3), much higher than that observed. The vertical force observed has an amplitude of 4–6 × 1010 N, depending on what point is taken as its starting point, so its volume would be more on the order of 1 Mm3 if the collapse was nearly vertical.

[34] According to Varnes [1978] classification, the sliding surface of a rockslide is along one or a few shear surfaces within a narrow zone, and though the source material is disintegrating, it is moving en masse and is not yet elongated in space and flowing. This type of behavior can be approximated as a sliding block, which allows for a few simple calculations. First, equation (6) can be used to determine the acceleration of the block and determine the trajectory of the mobilizing flank as a whole, assuming the mass is relatively constant. Using the longer-period version of the force-time function (T = 75–150 s) to represent the whole-scale mobilization of the rockslide, I calculated the horizontal acceleration of the mass at each second in time. Unfortunately, it is not possible to calculate the acceleration of the subevents separately because their respective masses are unknown. I integrated the acceleration twice to obtain the displacement of the center of mass of the landslide at each moment in time and fit the displacement curve with the equation of motion: d(t) = do + vot + 0.5at2 to obtain a best estimate of the average horizontal acceleration of the rockslide (a), where d(t) is the horizontal displacement with time t. The initial horizontal displacement do and initial velocity vo were set to zero. The best fit was an average horizontal acceleration of 0.39 m/s2 (Figure 10). The median slope angle of the source area from the postslide digital elevation model (15 m resolution) resolved in the azimuth of slope failure (231 ± 5°) was θ = 23 ± 1° so the vertical acceleration corresponding to the horizontal acceleration calculated above should have been 0.17 m/s2 and the total slope-parallel acceleration, 0.43 m/s2. By this calculation, the rockslide as a whole mobilized slowly, taking 36 s to accelerate to a speed of 15 m/s, traveling about 250 m horizontally, and dropping about 110 m in that time.

Figure 10.

The horizontal trajectory of the rockslide calculated from the long period (T = 75–150 s) force-time function assuming a mass of 48.5 million m3. The solid line shows a quadratic fit to the trajectory to the form of the equation of motion relating displacement, acceleration, and time. The best fitting average acceleration was 0.39 m/s2.

[35] Using a rigid sliding block approach, it is also possible to estimate the areally averaged apparent dynamic friction at the base of the rockslide given the angle of the sliding plane. Using a rearrangement of equation (5)

display math(7)

yields a best estimate of the apparent friction coefficient of μ′ = 0.38 ± 0.02 assuming a slope-parallel acceleration of 0.43 m/s2 and a slope angle of θ = 23 ± 1°. The apparent friction coefficient accounts for the effects of both friction and basal fluid pressure, and this value suggests high basal fluid pressures. To quantify this, the apparent coefficient of friction can be related to the true coefficient of friction, μ that would be felt in the absence of pore fluids by

display math(8)

where P is the mean basal pore pressure and σn is the mean normal stress at the base of the sliding mass. When rearranged, equation (8) relates the difference between the apparent and true coefficient of friction to the ratio of basal pore pressure over basal normal stress:

display math(9)

[36] While we do not know what the actual dynamic coefficient of friction in the absence of basal fluids was for the rocks composing Mount Meager, we know from lab experiments that μ is nearly always greater than 0.6 for dry rocks of a wide variety of lithologies [Byerlee, 1978], so equation (9) indicates that basal fluid pressures were at least 22% of the basal normal stress during the rockslide for μ′ = 0.38. This is corroborated by the large springs and surface seepages found throughout the source area after the landslide [Guthrie et al., 2012], suggesting that groundwater was instrumental in triggering this slide and its rapid mobilization to a debris flow. The value obtained here is within the bounds on the coefficients of friction of μ′ = 0.2 to 0.6 that Brodsky et al. [2003] found for three large landslides also in volcanic environments, but significantly higher than the value used by Guthrie et al. [2012] to numerically model this part of the event, μ′ = 0.06. The reason for this discrepancy is addressed in the discussion section.

4.2 Debris Flow

[37] After the initiating pulses, two longer-period, horizontal oscillations dominate the force-time function (Figures 8a–8c, intervals 2 and 3). Based on the direction of these vectors (Figure 8e), they are most likely the manifestation of centripetal accelerations of the debris flow material turning the two major corners in its sliding path. Unfortunately, it is not possible to take the same approach as above and use equation (6) to estimate the trajectory of the debris flow directly because a debris flow is poorly approximated as a sliding block. The flow becomes elongated in space, which could result in differing flow directions amongst segments of the failure volume, and the material is flowing and agitated, and can have complicated flow patterns [Iverson et al., 1997; Zanuttigh and Lamberti, 2007]. However, under the assumption that the timing of peaks in the force-time function corresponds to the times when the center of mass reached points of maximum forcing, the timing of peaks can be tied to points along the sliding path to estimate the debris flow speeds. I defined the starting point as the center of mass of the landslide, calculated from the depleted thickness map from Guthrie et al. [2012] (marked with a cross in Figure 8) and placed the location of the first peak, point a, 75 m downslope from there, corresponding with the distance traveled by the rockslide by the time it reached its peak at t = 21 s (Figure 10).

[38] For intervals 2 and 3 in Figure 8, I assumed that the peak in the force-time function corresponded to the time that the center of mass of the debris flow passed the portion of the path with the highest curvature (i.e., highest centripetal acceleration). To find these points quantitatively, I fit a polynomial to the horizontal sliding path that was delineated by Guthrie et al. [2012] and calculated the curvature of this polynomial analytically. I then placed the peak force vectors for each curve at these peak points of curvature (Figure 11, points b and c). If the assumption that the peak force corresponds to the center of mass arriving at the location of peak forcing is valid, the direction of the acceleration should point toward the center of a circle tangent to the point of maximum curvature. To test this, I placed circles with radii equal to the radius of curvature tangent to the point of maximum curvature, and the peak acceleration vectors do point to within a few degrees of the center of these circles, as expected for a centripetal acceleration (Figure 11).

Figure 11.

Outline of sliding path with the azimuth of the major peaks in the force-time function from Figure 9 placed at locations of maximum forcing (black arrows). The acceleration is in the opposite direction (gray arrows). The path trace (dashed line) is from Guthrie et al. [2012], except from segment d to e, which was estimated from satellite imagery. The cross marks the approximate location of the start of the center of mass. Peaks in the force-time function correspond to the arrival of the center of mass of the landslide at the designated locations and were used to estimate the average speed of the landslide between these intervals, as summarized on Table 3 and labeled along the path.

Table 3. Distances and Speeds Between Points Along Path and Corresponding Azimuth of the Force at Each Point
PointaDistance From Previous Point (m)Time From Previous Point (s)Average Speed, Best Estimate (m/s)Possible Range (m/s)Azimuth of Force
  1. a

    Refer to Figure 8 for the timing and Figure 11 for the location of points a–e.

a7521 ± 243-433° ± 5
b1850 ± 40048 ± 23932-45205° ± 5
c3360 ± 40044 ± 27664-907° ± 4
d2400 ± 40026 ± 29271-117114° ± 29
e700 ± 35052 ± 2136-22290° ± 3

[39] The next interval (interval 4) is characterized by an eastward and upward force that I interpreted as the debris flow decelerating rapidly upon reaching the confluence with Meager creek and running into the opposing valley wall (point d). At this point, the debris flow split and flowed both upstream and downstream and also left significant deposits at the confluence of the two valleys. This complicated the interpretation of the force-time function in interval 5 because there were opposing accelerations and a significant decrease in moving mass. However, the distinct pulse of force toward the northwest (point e) is consistent with another centripetal acceleration of some of the debris as it sloshed up the side of a steep hill before making its final turn to the northern depositional area (Figures 8 and 11, point e). I placed the location of this peak force vector at the estimated point of maximum curvature of this turn and used it to estimate the speed for this last segment.

[40] To conservatively account for uncertainties in the determination of distances between points, I assumed a possible distance interval range of ±400 m. I assumed a timing error range of ±2 s. The location assigned to each major peak labeled on Figure 8 is shown on Figure 11. The direction of the force vector at each of these peak times is also plotted along with the corresponding acceleration vector pointing in the opposite direction. I estimated the errors in the azimuth of the peak force as the standard deviation of the angle within the interval containing the peak force (Figure 8e). All error ranges are within a few degrees except for point d, probably due to the opposing flow directions of the material at this confluence (Table 3).

[41] The average speed of the center of mass of the landslide between each of these points, calculated by simply dividing the distance over the time, is reported in Table 3 and plotted as the solid black line in Figure 12. By this analysis, the average speed from the starting point to the first curve was 39 m/s, increasing to 76 m/s going into the second curve and to 92 m/s as the debris flow traversed the third segment and burst into the adjacent valley. After reaching the opposing valley wall at point d, the average speed decreased significantly and the portion of debris that continued downstream slowed to an estimated 13 m/s before turning the final curve to main depositional area.

Figure 12.

The average speed of the center of mass along path distance from this study (black dots with error bars). The average speed estimates are plotted halfway between the distance markers defining each interval. The solid line shows a linear interpolation to these points, shading shows the error range. The zero point corresponds to the location of the center of mass of the source area and the average speed estimates were plotted halfway between the distance markers used. This result is compared to speed estimates by the three methods used by Guthrie et al. [2012].

[42] Schneider et al. [2010] showed that the relative amplitude of the envelope of the high-frequency seismic signal correlates well with the total frictional work rate. Frictional forces will be higher than the gravitational forces when the landslide reaches a slope shallower than the arc tangent of the coefficient of friction [Schneider et al., 2010]. For the apparent coefficient of friction determined in this study, μ′ = 0.38, that angle would be 21°. The slope of the sliding path is consistently less than 21° just past the toe of the source area so there should have been an increase in the amplitude of the higher-frequency signal almost immediately after the rockslide left the source area. This is, in fact, what occurred: The envelope of the higher-frequency (>0.2 Hz) energy started to build up about 20 s after the start of the landslide (Figure 8d). As the mass started to disintegrate, flow, and reach high speeds, the random kinetic energy should have increased as well. This, in turn, should have increased the frictional resistance due to higher shearing rates and thus higher seismic amplitudes. This is observed: The envelope of the higher-frequency energy rose and reached a plateau after turning the first curve in the path (intervals 2–3). This is the part of the sliding path where the debris flow reached high speeds (~76 m/s) over a relatively straight interval and random kinetic energy should have been high. After passing the second curve (Figure 8, point c), the amplitude of the envelope reached an even higher plateau, this is where the estimated speeds reached their highest point but were followed by a rapid deceleration as the debris hit the wall, was deflected, and slowed down as it spread over the depositional area. This section (interval 4) has the highest amplitudes of high-frequency shaking, most likely due to high frictional resistance forces as the mass rapidly decelerated.

4.3 Aftershock

[43] About 2 min after the end of the main landslide event, there was an “aftershock,” a smaller landslide that occurred after the main event. This landslide also generated a long-period seismic pulse. I inverted the long-period seismograms of this event to obtain its force-time function (Figure 13). I had to use shorter-period seismic waves (T = 20–50 s) in the inversion to capture this smaller event. The shortest-period waves used are too short to be accurately represented by the Green's functions so this is a more approximate inversion than the main event. I also only used the closest three stations in the inversion because their mean signal to noise ratio (SNR) was 4, while the two more distant stations had a mean SNR of 2, with the lowest component having an SNR less than 1. The force-time function for this event indicates a primarily vertical collapse toward the south-southeast, probably off the now over-steepened headscarp. The collapse quickly took on a more horizontal trajectory in the southeast direction. The absence of a strong subsequent downward force suggests that there was no impulsive vertical impact so the material may have disintegrated along the slope as it fell. The force of the vertical collapse is an order of magnitude smaller than the main event, peaking around 6 × 109 N. The second vertical pulse, occurring about 30 s after the first, could have been a second collapse or could be forces generated along the sliding path of the first collapse. Since there is no way of estimating the volume of this secondary landslide from satellite data because it occurred just minutes after the main event, I used the result of this inversion to estimate the volume solely from the seismic data. To do this, I assumed the same frictional value found for the main event, μ′ = 0.38, and took the arc tangent of the vertical over the horizontal forces at the peak of the vertical collapse to estimate a slope of θ = 72°. This results in an acceleration of 8.2 m/s2 by equation (7). Since the magnitude of the forces at this point was 5.8 × 109 N, using equation (6), this yielded an estimated mass of 7 × 108 kg, or a volume of 0.3 million m3 assuming a density of ~2300 kg/m3. This volume of material is 2 orders of magnitude smaller than the main event, yet the forces are only 1 order of magnitude smaller, probably because accelerations were higher due to the near-vertical failure direction.

Figure 13.

(a) A second smaller landslide, an “aftershock,” occurred a few minutes after the end of the main landslide, as shown in the seismic data. (b–d) Three components of force of the aftershock obtained by inverting seismic data at periods of 20 to 50 s from the three closest three-component stations (WSLR, LLLB, and SHB). The time scale is consistent between all four plots.

5 Discussion

5.1 Improvements to Landslide Characterization

[44] The time series recording of the forces exerted by the landslide allowed for significant improvements in the characterization of the dynamics of this event, particularly when combined with the extensive field evidence and geospatial data compiled by Guthrie et al. [2012]. In particular, the availability of the force-time function reduces the qualitative interpretation typically required to tie postslide evidence to what happened during the event. It also eliminates guesswork required to interpret the source of pulses in the raw seismograms directly by using the seismograms to find what the source was quantitatively.

[45] As mentioned, the force-time function showed that the rockslide initiation occurred in two pulses, which I interpreted as a progressive failure of the flank of the mountain. These events were followed soon after by a nearly vertical collapse that was smaller in volume. This sequence of events is in contrast with the interpretation of Guthrie et al. [2012]. They proposed that the near-vertical collapse of the steep gendarme occurred first and its impact onto the shallower slopes below caused the flank to mobilize due to undrained loading and rapidly turn into a debris flow. This is a reasonable interpretation from the evidence that was available at the time, but with the additional information provided by the force-time function, it is clear that the failure of the flank of the mountain started first and was likely the cause of the vertical collapse—though the collapse of this material onto the already-mobilizing flank still could have been responsible for its rapid disintegration and mobilization to a debris flow. This is a reasonable argument given that the higher frequency portion of the signal attributable mainly to the debris flow rises quickly after the vertical collapse.

[46] The results of this study also improve on the interpretation of Ekström and Stark [2013], who included the 2010 Mount Meager landslide in their inversion of long-period seismograms from 29 catastrophic landslides worldwide recorded on the Global Seismographic Network. Though they do not include the entire force-time function in their results, they reported peak forces of 1.48 × 1011 N for this event, comparable but higher than the peak magnitude of the force found in this study of 1.0 × 1011 N. However, the start time they reported for the start of the event was two minutes later than the start time found in this study and reported by Guthrie et al. [2012]. Their start time corresponds instead to the time of the peak at point d (Figures 8 and 11). This suggests they interpreted what this study found to be a centripetal acceleration of the debris flow around the second curve in the path as the initiation of the rockslide. This discrepancy may explain why their estimated runout of 4.6 km was much shorter than the actual runout (>12 km) and highlights the care that must be taken when interpreting the force-time functions of landsliding events.

[47] Though the methods used to estimate the speed of the landslide from the force-time function in this study still required some interpretation, they are the closest to a direct measurement of the options available. Guthrie et al. [2012] used two common methods of estimating landslide speeds: a theoretical method called superelevation [Chow, 1959] and a numerical landslide model, DAN-W [Hungr, 1995; Hungr and McDougall, 2009]. I validated these methods against the more physically based measurement available from the results of this study, since the opportunity is not often available for natural landslides. As mentioned, Guthrie et al. [2012] also used features of the raw seismograms to estimate speeds, and I include that result in the comparison as well. The four methods are plotted for comparison in Figure 12 and were all adjusted to the same path starting point used in this study.

[48] The speeds predicted by the numerical model presented by Guthrie et al. [2012] are much higher at the start of the debris flow than that determined in this study. The initial acceleration in the first 20 s was about 4.5 m/s2, which should have generated a force an order of magnitude higher than that observed in the force-time function, so the speeds cannot have been that high at the onset. The exceptionally low coefficient of friction they used in this simulation (μ′ = 0.06, compared to 0.38 determined in this study) may partially explain the difference. Guthrie et al. [2012] chose this value to best fit the runout distance and velocities of the landslide interpreted from the raw seismograms and did not fit the superelevation estimates well. This low value for μ′ is probably because they were trying to match initial speed estimates that were too high. This mismatch highlights the potential of using the force-time functions and other information derived from seismic waves to validate and calibrate numerical models and tie them to the physical world, as Moretti et al. [2012] did in their study. The limitation is that for complicated events—such as this one, with subevents of unknown relative masses and a complex path causing opposing forces at times—numerical models may be needed to fully interpret the force-time function. So in reality, the comparison between numerical models and landslides may be more of an iterative process that may not necessarily have a unique solution.

[49] The landslide speeds estimated roughly from the raw seismograms in Guthrie et al. [2012] also had a much higher estimate of the initial speed (Figure 12). This is due to the aforementioned difference in their interpretation of mobilization sequence. They assumed that the majority of the landslide mass—the flank of the mountain—did not mobilize until the secondary peak collapsed on it about 45 s after the start of the landslide. With this interpretation, there is very little time between the mobilization of the flank of the mountain and its arrival at the first corner, thus resulting in the extremely high initial speeds they report. As explained above, the force-time function shows the reverse sequence of events: the flanks of the mountain mobilized first, followed by a vertical collapse. If they had this information in the initial interpretation of the seismograms, the initial speed estimate would have been much closer to the speeds obtained by this study. Though the uncertainties are higher, the best estimates of the speeds from the raw seismograms are close to those estimated in this study: between 10 and 30% lower but with error bars that overlap. The main benefit of using the results of the seismic inversion to estimate speeds is the elimination of most of the guesswork involved in correlating pulses in the seismic record to exact locations of events along the path.

[50] The speeds determined in this study compare most favorably with the values determined using the superelevation method [Chow, 1959], a theoretical method of determining speed by how much higher the debris flow reaches on the outside corners of turns than the inside corners. The three superelevation measurement points are between 5 and 11% different from the projection of the speeds calculated in this study and are well within the error bars (Figure 12), providing a validation of this theoretical method with real data. The main discrepancy is the average speed for the path segment between points c and d coming out of the last corner in the valley, where the average speed from this study exceeds the projection line between superelevation estimates by 35%. However, it is not possible to conclude whether this discrepancy is real or not because the superelevation measurements are effectively point measurements at the points before and there is no information in between.

[51] There is a possibility that the continuous increase in the debris flow speed until point d found in this study could be erroneous. However, the uncertainties incorporated in the calculation of the error bars of the speeds for this study were conservatively wide. The main source of error would have to be a misinterpretation of the force-time function in this study due to the complexity of flow at this junction. This is possible, but it is difficult to conceive where the pulse of eastward forces observed in interval 4 would come from, if not from the material running up against the adjacent primarily west-facing valley wall (point d), so I consider this unlikely.

[52] Assuming this late peak in speeds is real and the other methods are either erroneous or cannot resolve this peak, there are a few potential explanations. According to field evidence, there was very little entrainment of material along the path, in fact deposition started immediately below the initiation zone, but most material was deposited beyond the Capricorn creek valley [Guthrie et al., 2012] so the mass was relatively constant during the fastest intervals along the landslide path. A landslide of constant mass will continually increase in velocity if the angle of friction (tan−1 μ′) is shallower than the slope angle [Iverson, 2012]. The apparent coefficient of friction at the base of the rockslide is estimated in this study to be μ′ = ~0.38, which translates to an angle of friction of about 21°. The slope of the debris flow path is shallower than this, averaging around 10°, which means the coefficient of friction must have dropped below 0.18 (tan 10°) during the debris flow in order to still be accelerating along this path. A drop in the coefficient of friction of a debris flow can occur due to undrained loading of wet bedded sediments by the overriding debris flow that causes an increase in the pore pressures at the bed [Iverson et al., 2011]. This often also results in an increase in entrainment [Iverson et al., 2011], but entrainment was not observed in this case, possibly because of the exceptionally high speeds, which actually make entrainment less likely to occur [Iverson, 2012].

[53] Another contributing factor to the late high peak in speeds estimated in this study could be due to a common behavior of debris flows: the development of surging, also referred to as roll waves. As explained by Zanuttigh and Lamberti [2007], debris flow surges develop when smaller flow instabilities grow and form surface waves that overtake each other with growing wavelengths and amplitudes. As these instabilities progress downstream and continue to overtake each other, they can coalesce into bigger surges, often eventually forming one dominant first surge characterized by a concentration of boulders at the front, sometimes followed by subsequent smaller surges. Debris flow depths are often significantly higher in surge waves and the waves can travel up to three times faster and exert forces more than an order of magnitude higher than the rest of the regularly flowing mass.

[54] The development of one or more large coalesced surges along the flow path of the Mount Meager debris flow could have moved the speed of the center of mass of the debris flow forward faster than the average flow, but the forces exerted by the surges themselves were likely too short period and too low amplitude to be resolved in the force-time function. The timing of the speedup is consistent with the observation that larger coalesced surges tend to preferentially appear farther down the flow path because it takes time for the instabilities to grow large enough and overtake each other [Zanuttigh and Lamberti, 2007]. If the surge started to develop prior to reaching point c, the speed estimate between b and c could also be higher than the regular nonsurging flow. There is even a hint of what could be interpreted as two separate surges visible in the vertical component of the force-time function. The vertical component of the force at point d has a shorter duration than the eastward component and is followed by a second smaller upward pulse (Figure 8a). This could indicate that the bulk of the debris flow material did not run up vertically, but was primarily deflected horizontally, and it could have been the arrival of these two or three subsequent surges that were faster, deeper, and higher energy than the rest of the flow that was responsible for the high runup observed in the field. The surges do not appear as discrete events on the east component of the force-time function, however, though this could be because their masses are not large enough to contrast with the rest of the regular flow being deflected primarily horizontally.

5.2 Limitations

[55] One major limitation to the application of the methods used in this study to other events is that there is a whole spectrum of landslide types and behaviors [Varnes, 1978], but these methods can only be used to study the small percentage that generates the required long-period waves: exceptionally large and rapid landslides. Scaling these methods down to more common shorter duration, smaller rapid landslides would require a detailed characterization of the velocity and attenuation structure of the study area and better seismic coverage because they would generate lower amplitude and shorter-period seismic waves. Even if this can be done, it does not address the problem of potential overlap in the frequency domain between the coherent pulses from the bulk mobilization of the landslide mass and the signal generated by stochastic smaller scale processes. For larger landslides, the two sources of seismic radiation are manifested in distinct frequency bands (e.g., Figure 4c), but this may not be the case in a scaled down scenario and if they overlap, the two can no longer be isolated with simple filtering. On the other side of the spectrum, slower events that may have large masses but slow accelerations and long durations, such as large slumping events, may be too slow to generate waves of short enough period to be observed by seismic methods, though subevents or smaller-scale forcing may be resolvable.

[56] Another limitation is that the force-time function alone cannot be used directly to estimate useful information about the dynamics of the landslides. Other information is required. For example, even if the landslide under investigation can be approximated as a coherent sliding block, equation (6) dictates that the trajectory of the sliding block can only be calculated from the force-time function if the mass is known. Likewise, the mass can only be calculated if the trajectory is known. However, even if neither is known, order of magnitude estimates of mass can still be obtained due to physical limitations on acceleration.

[57] I showed that the methods used in this study can be used to determine the sequence of events and the occurrence of subevents. However, this can also be a limitation because subevents can complicate the force-time function and thus make straightforward interpretation and calculations from it challenging. In this study, I was able to get around the fact that the rockslide initiation occurred in two subsequent failures because the two subevents were close enough in time and space that they acted as a single failure mass at the long-period limit. Care must be taken to correctly assess when and where the rigid sliding block approximation is valid, and when it breaks down. For example, the approximation breaks down if subevents are too far apart in time or space to be considered one failure mass at long periods, or when the failure mass is not moving coherently enough, such as during the debris flow of this event, where the material became elongated in space and internal agitation and complex flow patterns dominated.

6 Conclusions

[58] In this study, I inverted the long-period seismic signals generated by the Mount Meager landslide to solve for the forces it exerted on the Earth as the failure mass accelerated, turned curves along its path, and decelerated. I used this result, the force-time function, to track landslide behavior over time. This analysis not only is useful for unraveling the sequence of events and facilitating a more direct interpretation than is typically possible between post-slide evidence and what actually happened during the event, but can also be used to make first-order calculations about landslide dynamics.

[59] Using the three component force-time function, I was able to identify the directions of the slope collapse and discern that the slope failure was progressive, with the massive flank of the mountain starting to mobilize first in two discrete but closely timed subfailures, followed by a much smaller nearly vertical collapse that might have been the collapse of part of the gendarme of the secondary peak of Mount Meager left unsupported as the flank mobilized. The addition of this new information clarified the sequence of events, suggesting that the initial interpretation made by Guthrie et al. [2012], where they proposed that the vertical collapse occurred first and caused the flank to mobilize, was actually reversed.

[60] Using the mass of the failure determined from satellite imagery by Guthrie et al. [2012], I was able to use the force-time function directly to estimate the trajectory of the center of mass of the rockslide, showing that it had an average horizontal acceleration of 0.39 m/s2, and estimated the apparent coefficient of friction at the base of the rockslide to be μ′ = 0.38, a low value that suggests basal fluid pressures were high—at least 22% of the normal stresses at the base of the slide, assuming a minimum coefficient of friction of 0.6 for dry rocks.

[61] Following the sequence of forces generated by the rockslide initiation, the direction and timing of the primarily horizontal forces were consistent with the debris flow turning two corners in its path and then running up the wall of an adjacent river valley. The horizontal forces from these centripetal accelerations were actually the highest forces overall. A debris flow is poorly approximated as a sliding block so I could not make calculations or estimate the trajectory directly from the force-time function, but instead tied the timing and direction of peak forces to points of peak forcing along the path to estimate the speed of the center of mass of the debris flow. I found that speeds increased continuously as the debris flow progressed down the first valley, peaking at 92 m/s before rapidly decelerating upon reaching the adjacent Meager Creek valley. The speeds I found by this method were very close to the speeds predicted by superelevation, providing a physically based validation to that theoretical method. The force-time function also provided a test of other speed estimation methods applied by Guthrie et al. [2012]. The initial speeds they estimated by numerical modeling were much higher than those estimated in this study, and would have required forces an order of magnitude higher than those observed in the force-time function. Their estimates of speed from the raw seismograms were close to those estimated in this study, though with higher uncertainties, but their speed estimates were also too high at the onset of the landslide because of their reverse interpretation of the sequence of initiating events. The differences show that the addition of the information provided by the force-time function can significantly clarify landslide characterization.

[62] Though the calculations I presented in this study are rough and large scale due to the long-period nature of the seismic waves used, and direct interpretation is difficult due to the complexities of debris flow behavior, it is remarkable that using data from five broadband stations 70 to 276 km from the source, it was possible to reconstruct details of this event that are unobtainable by other methods but nonetheless important for understanding landslide physics and improving numerical models. As there are currently networks of broadband seismometers worldwide for which data are freely available, the techniques I described could be applied to study other large landslides that occurred both in the past and to come.

Appendix A

[63] This section details the inversion process used to obtain the force-time function in this study. The forward problem to obtain the displacement seismograms generated by a single force applied at the source location is

display math(A1)

where the seismogram di(t) at each component of each station (i) is equal to the Green's functions Gij(t) for station component i for an impulse force applied at the source location in direction j convolved (∗) with the force-time function of the source, mj(t), where j is the directional component of the force (up, north, east). Note that mj(t) is equivalent to Fe(t)—the change in notation is for consistency with inverse theory conventions, where m refers to “model.” Equation (A1) assumes a stationary force vector applied to a single point on the surface of the Earth. Though the landslide is moving, it can be approximated as a stationary point force for the long-period wavelengths used in the inversion, as explained in the methods section.

[64] Green's functions are the response of the Earth to an impulse source. In this study I calculated the Green's functions relating an impulse force at the source location to the response at the distance and azimuth corresponding to each seismic station. I computed these using the wave number integration method [Bouchon, 1981] as implemented in Computer Programs in Seismology (CPS) [Hermann, 2002]. The velocity model used for these calculations was ak135Q [Kennett et al., 1995]. This is a one-dimensional radially stratified global Earth velocity model and does not account for regional or smaller-scale variations. However, it was sufficient for this study because the long-period waves (T = 30–150 s) used in the inversion are not sensitive to smaller-scale variability.

[65] Using CPS, I computed the Green's functions corresponding to each seismic station used in the inversion for a Dirac delta function impulse force applied to the surface of the Earth at the location of the landslide. Five Green's functions are required for each station used in order to calculate three-component synthetic time histories for a single-force mechanism applied in any direction at the source [Hermann, 2002]. The radial component of the seismogram is positive in the direction pointing directly away from the source. The transverse component is perpendicular to the radial direction, positive in the direction clockwise from north. Vertical is positive upward. The five Green's functions are abbreviated as follows:

ZVF

vertical component of the seismogram for a downward vertical force;

RVF

radial component for a downward vertical force;

ZHF

vertical component for a horizontal force in radial direction;

RHF

radial component for a horizontal force in radial direction;

THF

transverse component for a horizontal force in transverse direction.

[66] The first four Green's functions correspond to the P-SV system, while THF corresponds to the SH system. The Green's functions calculated for station SHB, 118 km from the source, are shown on Figure A1. These illustrate the role of the Earth's structure in the waveforms observed at station SHB due to an impulse force of 1 N at the source. The Rayleigh wave dwarfs the P and S arrivals in the top four Green's functions, but has not moved out and dispersed much yet because of SHB's proximity to the source. Note that due to the nature of causal filters, the relatively compact unfiltered Green's function becomes distorted when filtered; the energy is smeared out later in time. This is not a problem in the inversion because identical filters are applied to the data so it is distorted in the same way. This is confirmed in that a nearly identical force-time function is obtained when a zero-phase (acausal) filter is used. Also note that the duration of the source (force-time function) is much longer than the Green's functions, unlike for most regular earthquakes, which is why such a detailed force-time function can be derived from the seismograms.

Figure A1.

Green's functions calculated for this study for station SHB located 118 km from the source for a impulse force of 1 N. The functions are shown unfiltered (0.5 Hz sample rate) and band-pass filtered between periods of 30 to 150 s using the same causal (minimum phase) Butterworth filter applied to the data and Green's functions prior to inversion. The amplitude of the filtered Green's functions is amplified by a factor of 10 relative to the unfiltered version. See text for discussion.

[67] These Green's functions can be used to calculate synthetic ground displacement seismograms for any single force vector impulse or time series by

display math(A2)
display math(A3)
display math(A4)

[Hermann, 2002] where φ is the source to station azimuth measured clockwise from north. The single force f = (f1, f2, f3) is in a north (N), east (E), vertical (Z), respectively, Cartesian coordinate system, where Z is positive down (note, however, that Z was switched to positive up in the force-time function plots in this paper to be more intuitive). The ground displacement seismograms are in spherical coordinates local to each source-station pair, where vertical is positive up (uz), radial (ur) is positive in the direction away from the source and tangential (ut) is positive at a right angle clockwise from ur.

[68] In order to do the inversion, the forward problem (equation (A1)) was rewritten as matrix multiplication rather than a convolution. A convolution is equivalent to reversing one of the two signals being convolved in time and passing them by each other, multiplying all points and summing them up at each time interval. To convolve a Green's function with a force vector via matrix multiplication, the Green's functions were reversed in time and staggered by shifting the Green's function by one sample in each successive row to produce a convolution matrix. To illustrate this setup, Equation (A5) shows the convolution via matrix multiplication between a four-sample Green's function g and a three-sample force-time function m to obtain the seismogram d that is six samples long:

display math(A5)

[69] Equation (A5) illustrates the convolution with just one Green's function for one station and one component of the force-time function. The actual Green's function convolution matrix must be set up to include all the Green's functions corresponding to each component of each seismic station used and to incorporate the equations that relate the source to station azimuth to account for the radiation patterns of each type of wave (equations (A2)(A4)). This setup can be illustrated for 2 three-component stations by

display math(A6)

[70] Each dic in equation (A6) is a column vector that contains the seismogram for the station (i) and component (c) (z vertical, r radial, and t transverse). The left side of equation (A6) is a column vector consisting of all the data (displacement seismograms) concatenated end to end. Each entry in the G matrix is a Green's function convolution matrix (indicated by * superscript) for the Green's function specified by its abbreviation for each station (indicated by the superscript). Each mj is a column vector containing the force-time function for each component (j) of force. The sines and cosines come from equations (A2)(A4).

[71] For this study, as described in the text, I downweighted the noisy data based on the inverse of the root mean squared value of the noise before the signal at each component of each station. To include this in the inversion I constructed a weighting matrix, W. W was built by filling the weights in along the diagonal of W, organized so that the correct scalar weight value multiplied the entire seismic signal of the corresponding component of each station when W is multiplied with the data vector. For example, the weight for the vertical component of the first station, d1z, fills in the diagonal of W from W11 to Wii, where i is the length of d1z. W weights both the data vector d and the Green's functions matrix, G, (which, when weighted are denoted as dw and Gw, respectively) because the same processing must be done to both sides of the equation. For the same reason, the individual Green's functions and the seismograms were band-pass filtered identically as well prior to building the matrix G. In this study I used a minimum phase (causal) Butterworth filter, but a nearly identical result is obtained when using a zero-phase (acausal) filter because the same filtering is applied to both sides of the equation.

[72] The next step was to solve the damped least squares problem to invert for m:

display math(A7)

where superscript T indicates the transpose, I is the identity matrix, and α is the regularization parameter chosen as the trade-off between keeping the model small while still fitting the data well. The matrix setup of the forward problem using data from five seismic stations with three components of motion each becomes very large, but the inversion was still manageable: It took less than a minute to run in MATLAB on a desktop computer.

[73] An additional step I took in this study was to constrain that all components of the single force must add to zero in the end because the total momentum of the Earth must remain stable [Fukao, 1995]. This did not significantly change the solution, but is more physically correct. To constrain this in the solution, I added equations to the forward problem by concatenating matrix, A, a 3 × 3 N matrix, where N is the length of the data from one component of one station, to the bottom of the Green's function matrix, G. A was constructed so that the first third of the first row was ones and the rest was zeros so that it multiplied and added up just the z component of m, a corresponding zero is added to the bottom of the data vector d to constrain that these forces add to zero. The same was done for the next two rows to multiply and add up the north and east components of m to equal zero. The A matrix was scaled up to the same magnitude as the weighted Green's functions so that it influenced the final solution.

Acknowledgments

[74] Many thanks to Ken Creager who provided indispensable guidance on what started as a project for his Inverse Theory course, and to my advisor John Vidale. Also to Emily Brodsky, Dick Iverson, Rick Guthrie, John Clague, Anne Mangeney, and Hiroo Kanamori for helpful feedback, reviews, and discussions; Nicholas Roberts for helping with some geospatial calculations; the Pacific Northwest Seismic Network and Canadian National Seismograph Network for providing the seismic data; and Sharon Watson for tolerating frigid fast-flowing river crossings and quicksand to accompany me on a field visit of the landslide. Additionally, reviews by Jackie Caplan-Auerbach, Agnes Helmstetter and two anonymous reviewers helped improve the manuscript significantly.

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