A Brownian walk model for slow earthquakes

GEOPHYSICAL RESEARCH LETTERS, VOL. 35, L17301, doi:10.1029/2008gl034821, 2008 A Brownian walk model for slow earthquakes Satoshi Ide 1 Received 29 May 2008; revised 10 July 2008; accepted 16 July 2008; published 3 September 2008. [1] Along some subduction plate boundaries, slow deformation is observable as seismically detected deep low-frequency tremor and geodetically detected slow slip events. These phenomena are considered as different manifestations of slow earthquakes characterized by fairly constant seismic moment rate. This paper presents a simple model of slow earthquakes that can explain wide variety of observed features including the steady moment rate and scaled energy, characteristics of tremor signals both in time and frequency domains, and the migration of the source location. In this model, slow earthquakes are represented as shear slip on circular faults whose radius is a random variable that is governed by a Langevin equation and three parameters, a diffusion coefficient, a damping coefficient, and a slip rate coefficient. This model expands on a previous scaling law for the slow earthquakes by providing a specific image of kinematics. Allowing for spatial variations of the parameters could potentially explain differences in behavior of slow slip events worldwide. Citation: Ide, S. (2008), A Brownian walk model for slow earthquakes, Geophys. Res. Lett., 35, L17301, doi:10.1029/ 2008GL034821. 1. Introduction [2] Recently, a variety of slow deformation processes, such as non-volcanic low-frequency tremor [Obara, 2002; Rogers and Dragert, 2003] and slow slip events [Dragert et al., 2001; Hirose and Obara, 2005; Schwartz and Rokosky, 2007], have been discovered adjacent to megathrust earthquake source areas in some subduction zones. Despite successive findings of interesting features that differentiate them from ordinary earthquakes, their physical mechanism has not been clearly identified. One of the most prominent features observed is the proportionality between seismic moment and duration. Ide et al. [2007a] proposed this scaling law and interpreted various phenomena as different manifestations of a single slow earthquake process. Seismologically, slow earthquakes are observed as continuous tremor dominant in 2-8 Hz, or isolated pulse-like low frequency earthquakes (LFE) of about moment magnitude (M w ) 1.5 [Shelly et al., 2006; Ide et al., 2007b]. Broadband seismometers have detected slow earthquakes as very lowfrequency earthquakes (VLF) of M w 3-3.5 with 20 s duration [Ito et al., 2007], and longer events up to M w 4 and 200 s [Ide et al., 2008]. Geodetically, slow earthquakes are modeled by slow slip events (SSE) of M w 6 7 lasting several days to months [Hirose and Obara, 2005]. 1 Department of Earth and Planetary Science, University of Tokyo, Tokyo, Japan. [3] The facts discovered for slow earthquakes in western Japan are summarized as follows. [4] F1: The seismic moment is proportional to the duration with a constant of 10 12 13 Nm/s [Ide et al., 2007a]. [5] F2: The seismic energy in 2 8 Hz is in direct proportion to the seismic moment with a constant, scaled energy [Kanamori and Rivera, 2006], of about 10 10 [Ide et al., 2008]. [6] F3: Tremor source regions migrate about 10 km and 100 km within 10 min and 10 days, respectively [Shelly et al., 2007a, 2007b; Ito et al., 2007]. [7] F4: The amplitude of tremor observed at surface seismometers is less than 1 mm/s and sometimes rapidly decreases after large peaks. [8] F5: VLFs are always observed during tremor, but tremor sometimes occurs without visible VLF [Ito et al., 2007]. [9] F6: The velocity spectra of tremor and VLFs are almost constant, independent of frequency [Ide et al., 2007a]. [10] F7: The relation between tremor amplitude and cumulative duration is described by an exponential law rather than a power law [Watanabe et al., 2007]. [11] Tremor waveforms look like random noise, suggesting the existence of some stochastic processes behind them. Therefore, we try to find a stochastic model consistent with the above facts. 2. Brownian Walk Model [12] Slow earthquakes occur in a belt-like zone that follows the 35 km iso-depth curves of the top of the Philippine Sea Plate. The accurate locations of LFEs [Shelly et al., 2006] and the focal mechanisms of LFEs [Ide et al., 2007b], VLFs [Ito et al., 2007], and SSEs [Hirose and Obara, 2005] strongly suggest that they are shear slip on the planar interface between two tectonic plates. Therefore, we consider a simple circular fault with a constant slip rate v 0 on the plate interface (Figure 1). v 0 must be quite slow, as demonstrated later. Because the observed locations of LFEs are clustered in small area of about 10 km [e.g., Shelly et al., 2006], we regard each circular fault as one of these clusters. Inspired by repeated migration of LFEs in each cluster (F3) [Shelly et al., 2007b], we assume that the randomness of tremor signal arises from random expansion and contraction of the fault area. Namely, the radius behaves as a Brownian walk. However, without a stopping mechanism, a Brownian walk will go to infinity, which is unrealistic for an earthquake model. Hence we also assume that there is a damping term proportional to the size of the source. Based on these assumptions, we obtain a stochastic differential equation [e.g., Øksendal, 1998] for the radius r t, Copyright 2008 by the American Geophysical Union. 0094-8276/08/2008GL034821 dr t ¼ ar t dt þ sdb t ð1þ L17301 1of5

Figure 1. Schematic illustration of a Brownian walk model of slow earthquakes. We assume circular patches of slow earthquakes on the subducting plate interface, between locked area and deeper stable sliding zone. The radius of a patch, r t, increases or decreases randomly. where db t is Gaussian white noise with zero mean and variance 1. a is the coefficient of damping, and a 1 is the characteristic damping time. s is the diffusion coefficient. This is the Langevin equation describing Brownian motion. In other words, this is an Ornstein-Uhlenbeck process [Uhlenbeck and Ornstein, 1930] studied for analysis in various fields such as interest rates in finance engineering and spike trains in biological science. [13] The mean and variance of r t approach constant values, 0 and s 2 /2a after a time longer than a 1. The seismic moment rate is given as _M o = pmv 0 r t 2, where m is the rigidity. Note that a random variable is introduced only in the squared form, r t 2, which is the essential feature of this model. Therefore, r t can be negative in equation (1), which looks unphysical, but it would not be a practical problem. The expectation of seismic moment rate also approaches a constant value pmv 0 s 2 /2a, which explains the scaling law F1. The stochastic differential equation for _M o is calculated using Ito s lemma [e.g., Øksendal, 1998], as d _M o ¼ pmv 0 s 2 2art 2 dt þ 2pmv0 r t sdb t : ð2þ The first term in the right side is small compared to the second term when we consider a small increment of time. In fact, the proposed process is unrealistic for a very small scale as discussed later. Nevertheless we can assume a realistic value of dt that is small enough to neglect the first term. Then equation (2) is approximated to be d _M o 2pmv 0 r t sdb t : The seismic energy rate is calculated for a point source, neglecting a small contribution from P-wave and using (db t ) 2 = dt [e.g., Øksendal, 1998], as _E s 1 2 2pmv M 2 0 s2 rt 2 10pmv 3 o s 5v 3 s dt ¼ 2v 0s 2 5v 3 s dt _M o where v s is the S wave velocity. Equation (4) explains the direct proportionality between seismic moment and seismic energy functions, e.g., fact F2. The proportionality constant containing dt means that the ratio depends on the minimum time interval in which this process works. Thus this simple model is consistent with two strong constraints for slow earthquake (F1 and F2). ð3þ ð4þ [14] Next, we estimate the values of parameters, a, s, and v 0. Fact F3, the migration of slow earthquakes at a slower velocity for a longer distance, suggests a diffusional process, which is a built-in process of our model where s is the diffusion coefficient. From the observation of migration velocity, 10 km within 10 minutes p[shelly ffiffiffiffiffiffiffi et al., 2007a, 2007b], s is estimated as 10000 / 600 400 m/s 1/2. Actually this is a little large to explain the observed longrange migration [Ito et al., 2007], for which additional mechanisms may be required. For s = 400 m/s 1/2,aslow earthquake area expands 40 m within 0.01 s at the average velocity of 4000 m/s, which is equal to the S-wave velocity, v s, in the source region. Therefore, because a crack cannot plausibly be expected to expand faster than S-wave velocity, governing physical process must be different below this size, which means that dt = 0.01 s is the minimum duration to be considered. 3. Numerical Simulation [15] To find reasonable estimates for a and v 0, we numerically solve equation (1), fixing s = 400 m/s 1/2 and dt = 0.01 s. We calculate seismic waves assuming that the velocity wavefield consists of only intermediate-field and far-field S-waves and taking the rms average of corresponding radiation patterns of the equation (4.33) of Aki and Richards [2002], and write as, _uðþ¼ t 2:4 4pmr 2 M _ o ðt r=v s Þþ 0:63 M 4pmv s r o ðt r=v s Þ; ð5þ where r is the distance from the source to the station. The rigidity m is 40 GPa and r = 40 km is adopted to compare the observation of slow earthquakes at the station KIS, maintained by National Research Institute for Earth Science and Disaster Prevention. While v 0 affects only the amplitude of the seismic waves, a changes the appearance of the waveforms considerably (Figure 2a). When a =0.02s 1, the calculated seismic waves explain overall characteristics of observed waves both in high-and low-frequency bands. If we choose v 0 =5mm/s (Figure 2), the amplitude of waves is consistent with the observation. Rapid decrease after a large amplitude peak (fact F4) is frequently observable. This is a consequence of the size-dependent damping term in equation (1). Not all high-frequency peaks correspond to low-frequency peaks (fact F5). Note that the calculated waves are too clean, without background noise and coda waves, which accounts for the apparent difference from the observation. While smaller a may also explain these features, the synthetics with a large a are quite different from the observation. [16] Similar to the observation, the velocity spectrum of the synthetics is flat for higher frequencies (fact F6), which is not surprising as a nature of Brownian walks (Figure 2b). The combination of parameters, a =0.02s 1 and v 0 =5mm/s, gives a constant moment rate of 5 10 12 Nm/s and the scaled energy 5 10 10. Both numbers are in good agreement with the observations (facts F1 and F2). If we compare the seismic energy rate function calculated using equation (4) and seismic moment rate function, the two functions are almost identical (Figure 3). The relation between tremor amplitude and the cumulative duration 2of5

Figure 2. (a) Observed and calculated waveforms for three different values of a. For each setting, high-frequency (2 8 Hz) and low-frequency (0.003 0.02 Hz) waveforms are shown on the top and bottom, respectively. (b) Power spectrum of the waveforms. In Figure 2b (left), observed spectrum (black) is compared with typical noise at the station (dark gray) and USGS low noise model [Peterson, 1993] (light gray). In Figure 2b (right), calculated spectrums for three different values of a are shown. (c) Comparison between amplitude and cumulative duration. Observation (black) and calculation for a = 0.02 (blue) are shown using semilog (top: exp) and log-log (bottom: power) plots. Figure 3. Moment rate functions and seismic energy rate functions. (a) Observed seismic moment rate function (gray area) and seismic energy rate function (bold black line) for the broadband seismograms at KIS, on September 29, 1999 [Ide et al., 2008]. (b) Calculated moment rate function and seismic energy rate function using a =0.02s 1, for which waveforms are shown in Figure 2a. The moment rate function is shown after decimation at 4 samples per second (sps). The energy rate function is calculated from the moment acceleration function using equation (4), smoothed with a 10 s boxcar window, and decimated at 4 sps. (c) Zoom up of the initial part of Figure 3b (dotted square). Only the moment rate function without decimation is shown. Vertical gray lines show the separation of events at every time when the amplitude is zero. Black thin lines show the realistic measurements of event duration with a threshold of 5 10 11 Nm/s (dashed line). 3of5

observed duration of LFEs. In noisier records, a larger threshold would limit detectable event periods shorter. Each red line shows the growth history, a moment-duration relation, of a slow earthquake sequence, in which we regard the whole sequence as one SSE. The moment increases with time, at a constant moment rate, after much longer time than a 1 =50s. Figure 4. A modified version of scaling law for slow earthquakes. Observed seismic moments and durations for LFE [Ide et al., 2007b], VLF [Ito et al., 2007] and longer events [Ide et al., 2008], SSE [Hirose and Obara, 2005] are shown by green symbols. Red circles are events defined in noise free environment ( noise free duration in Figure 3c), while gray circles are events separated using a threshold of 5 10 11 Nm/s, ( measurable duration in Figure 3c). Red lines show time history of seismic moment in different simulations. Dotted lines are previously proposed scaling law for slow earthquakes [Ide et al., 2007a]. A blue zone represents a typical relation between seismic moment and duration for regular earthquakes, in which M o is proportional to the cube of duration. during which amplitude is larger than the threshold is described by an exponential law rather than a power law (Figure 2c). Tremor amplitude is measured using the reduced displacement [Aki and Koyanagi, p 1981; Watanabe et al., 2007], defined by D = Ar /2 ffiffi 2, where r is the source-station distance and A is the root mean square displacement bandpass filtered between 2 8 Hz and averaged using a 6 s boxcar function. In summary, this very simple model of a circular slow slip can explain many reported features of slow earthquakes (F1 F7). [17] Although Ide et al. [2007a] proposed a diffusional earthquake model to explain the scaling law for slow earthquakes, the model presented in this paper is different from the previous one. Furthermore, the current model contains a slight modification of the scaling law. Figure 4 shows the relation between the size and duration of the event in our numerical simulation. The red circles are synthetic events separated by the time when r t becomes 0 (Figure 3c). These values have a trend, M o T 2, which is less steep than the predicted trend, but similar to the observation for slow earthquakes of about 100 s [Ide et al., 2008]. This trend cannot explain the observation of a LFE. However, the above measurement of event duration is only possible for noise free data. In the real situation, with the existence of noise, events are measured using a threshold, for example 5 10 11 Nm/s (Figure 3c), and detected as shown in gray circles in Figure 4. This explains the 4. Discussion and Conclusion [18] There are some reports of slow earthquakes without tremor, which may be explained by the variation of parameter values of Brownian motion. For example, the observed migration velocity is very slow for the long term SSE without tremor in the Tokai region that occurred in the shallower extension of tremor zone [Hirose and Obara, 2006; Miyazaki et al., 2006]. If s is smaller by one order, the evolution of these slow earthquakes is invisible in seismological frequency range, but the seismic moment rate is consistent for those observed for the Tokai long term SSE. The depth of slow earthquakes, about 30 35 km, is also close to the depth of Moho discontinuity, and the significant variation of materials along the plate interface may be responsible. [19] A feature that cannot be explained by this model is the intermittence of slow earthquake activity. It has been shown that tremor is controlled by tidal stress [Shelly et al., 2007b; Rubinstein et al., 2007a; Nakata et al., 2007] and long-period seismic waves from distant earthquakes [Miyazawa and Mori, 2005; Rubinstein et al., 2007b]. Such external effects may affect the activity of diffusional processes and be required for a complete simulation of slow earthquake activity. [20] The proposed model is sufficient to explain various features of slow earthquakes, from tremor to SSEs. This is a unique application of Brownian walk for fairly large scale phenomena. Even without any elementary physical mechanisms, this model connects the tremor migration velocity and slow slip amount and by capturing the properties of the entire moment-rate history. Moreover, it leads to a better estimation of stress build-up that is essential for the assessment of probability of future megathrust earthquake. Due to its simplicity, with just three free parameters, this model would be useful to characterize many similar phenomena in subduction zones and in other geophysical environments [e.g., Rogers and Dragert, 2003; Schwartz and Rokosky, 2007; Ide et al., 2007a]. [21] Acknowledgments. Comments from J. J. McGuire and H. 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