Quantifying the time function of nonvolcanic tremor based on a stochastic model

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1 JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 115,, doi: /2009jb000829, 2010 Quantifying the time function of nonvolcanic tremor based on a stochastic model Satoshi Ide 1 Received 24 December 2009; revised 22 April 2010; accepted 3 May 2010; published 27 August [1] Nonvolcanic tremor is the seismically observable component of slow earthquakes, which have been recently discovered in subduction zones around the world, including major study areas at the Nankai and Cascadia subduction zones. Although tremor appears to occur randomly in both areas, Cascadia tremor has a longer duration than Nankai tremor. In the present study, this difference in tremor duration is quantified using a Brownian slow earthquake model that explains several features of tremor and slow earthquakes. A previous Brownian model is improved and applied to explain the cumulative distribution function of tremor amplitude, which is approximated by a c 2 distribution. The model also shows that the power spectrum of tremor amplitude has a simple analytic formula, including a characteristic time. An inversion method is developed to measure the characteristic time from the tremor spectrum in the presence of non Gaussian background noise. The method is applied to several tremor sequences in the Nankai and Cascadia subduction zones to quantitatively confirm the apparent differences in tremor behavior between the two areas. The constants for Cascadia and Nankai tremor are s and s, respectively, with temporal increases in these values observed over the course of 1 day records of activity. The difference in characteristic time between the two areas may reflect geometric constraints such as the width of the tremor region. Citation: Ide, S. (2010), Quantifying the time function of nonvolcanic tremor based on a stochastic model, J. Geophys. Res., 115,, doi: /2009jb Introduction [2] An important topic in solid earth science is the discovery of a new class of earthquake like phenomena that includes nonvolcanic tremor [Obara, 2002; Kao et al., 2005], low frequency earthquakes (LFEs) [Katsumata and Kamaya, 2003; Shelly et al., 2006], and slow slip events (SSEs) [Dragert et al., 2001; Hirose and Obara, 2005; Schwartz and Rokosky, 2007]. Because these phenomena occur simultaneously [Rogers and Dragert, 2003; Obara and Hirose, 2006], they are commonly referred to as episodic tremor and slip (ETS). The close proximity of ETS events to megathrust earthquake source areas suggests that a detailed understanding of slow processes may help to characterize and predict catastrophic fast events. The discovery of similar ETS phenomena worldwide, mainly in subduction zones, indicates that they reflect essential processes that govern the slow deformation near the earth s surface. [3] ETS is considered to occur at the plate interface. It may be natural that a large SSE of moment magnitude (M w )6 7 occurs as part of long term relative plate motion. Much 1 Department of Earth and Planetary Science, University of Tokyo, Tokyo, Japan. Copyright 2010 by the American Geophysical Union /10/2009JB smaller LFEs of M w 1, which represent a constituent element of tremor [Shelly et al., 2007a], occur very close to the plate interface within the Nankai subduction zone [Shelly et al., 2006], as are tremor in the Cascadia subduction zone [La Rocca et al., 2009]. Moreover, in a study of the Nankai, Cascadia, and Costa Rica subduction zones, Brown et al. [2009] showed that LFEs occur between the subducting oceanic crust and the continental mantle located beneath the Moho discontinuity in the upper plate. There exists an overlap in the areas in which tremor and SSEs occur in the Nankai and Cascadia subduction zones, suggesting a strong relation between the two phenomena. However, it should be noted that long term SSEs in the Nankai subduction zone [e.g., Ozawa et al., 2002; Hirose and Obara, 2005] may be unrelated to tremor, and that an analysis of Mexican ETS reveals that tremor and SSEs occur in distinct areas [Kostoglodov et al., 2003; Payero et al., 2008]. Given that SSEs without accompanying tremor have been documented worldwide [Schwartz and Rokosky, 2007; Ide et al., 2007b], it is likely that the occurrence of tremor is not a necessary condition for the occurrence of SSEs. [4] In addition to a possible relationship regarding the location of occurrence, SSEs and tremor involve similar movements: shear slip consistent with the regional plate motion. The orientation of SSE faults and the direction of slow slip are generally consistent with plate motion. Data regarding the mechanism of LFEs in the Nankai subduction zone [Ide et al., 2007a] and the particle motion of tremor in 1of13

2 the Cascadia subduction zone [Wech and Creager, 2007] reveal that the sense of movement during these small scale processes is consistent with that of large scale slow slip. Furthermore, in the Nankai subduction zone, intermediatefrequency tremor signals are interpreted to represent verylow frequency earthquakes (VLFs) of about M w with shear slip mechanisms consistent with the regional plate motion [Ito et al., 2007]. Given the occurrence of repeated small slip events (LFEs and VLFs), significant part of the total slip would be observed by geodetic instruments as a SSE. In this sense, tremor is a sufficient condition for the occurrence of a SSE. This proposal is supported by the strong temporal correlation observed between the size of tremor and SSEs in the Nankai subduction zone [Hiramatsu et al., 2008]. [5] There are several known characteristics of ETS. Ide et al. [2007b] found that the seismic moment is proportional to event duration for LFEs (tremor), VLFs, and SSEs, and proposed that these phenomena are different aspects of a single process summarized under the term slow earthquakes. A similar linear relationship between seismic moment and event duration has been identified for SSEs occurring worldwide with M w > 6 [Schwartz and Rokosky, 2007]. Several other important characteristics of slow earthquakes have been reported in the literature. Ide et al. [2008] noted that seismic moment is proportional to seismic energy for slow earthquakes of M w 3 4. The velocity spectra of seismic waves from slow earthquakes are almost flat in a broad frequency range from 0.01 to 10 Hz, similar to the spectrum of white noise. In addition, migration of the tremor source area is often observed. The tremor source has been observed to propagate a distance of about 10 km in 10 min, at a velocity of 10 m/s [Shelly et al., 2007b], whereas long distance migration (>100 km) requires several weeks at a much slower velocity of 0.1 m/s [Dragert et al., 2004; Ito et al., 2007]. The cumulative distribution function (CDF) of tremor amplitude resembles an exponential function [Watanabe et al., 2007]. [6] To explain the occurrence of slow earthquakes observable as tremor in seismic observations and as SSEs in geodetic observations, Ide [2008] proposed a stochastic model, termed a Brownian slow earthquake (BSE) model, which comprises a Langevin equation and a circular slow slip area. This model is able to explain the main characteristics of slow earthquakes: (1) proportionality between seismic moment and duration, (2) proportionality between seismic moment and seismic energy, (3) flat velocity spectra, (4) tremor migration at size dependent velocities, and (5) exponential CDF of tremor amplitude. This model is used in the present study, after improvements related to characteristics 2 and 5. [7] Following the discovery of ETS like phenomena in many locations worldwide, a comparison of the various phenomena is emerging as an important topic of study. Given that tremor occurrence appears to be random or, at the very least, complicated, appropriate tools are required to characterize this complexity. The CDF represents one approach in this regard, but it shows only minor variation among ETS phenomena, as demonstrated below. To quantify tremor activity, this paper presents a method of determining a time constant of tremor activity from spectrum analysis based on the BSE model. The target activity is tremor that occurs in the Nankai and Cascadia subduction zones, two of the most intensively studied areas worldwide in terms of tremor activity. [8] The remainder of the paper is organized as follows. In section 2, it is shown that the continuity of tremor activity varies for different types of activity in the Nankai and Cascadia subduction zones. Seven examples of tremor records are prepared for investigation. Section 3 introduces the BSE model and describes improvements made to the model as part of the current study. In addition, observed and modeled CDFs are compared. Section 4 contains an account of how to estimate the time constant in the BSE model using an inversion method. The method is applied to a tremor data set to obtain time constants and their temporal change. The significance of differences in the time constant and tectonic environment is considered in section 5, and the main conclusions are presented in section Records of Tremor at the Nankai and Cascadia Subduction Zones [9] Tremor in western Japan occurs in a belt like region along the 35 km depth contour of the Philippine Sea Plate, which is subducting toward the northwest at the Nankai Trough. Impulsive signals in tremor activity are detected by the Japan Meteorological Agency as LFEs [Katsumata and Kamaya, 2003] and the source locations are determined based on the arrival times of body waves (Figure 1). In most regions, LFEs occur over a distance of about 20 km in the subduction direction. These LFEs represent just a small fraction of the continuous tremor that occurs. The tremor source distribution [Obara, 2002] is wider than the LFE distribution, although this difference may be explained by the large error involved in calculating the locations of tremor sources. [10] Figure 1 shows examples of tremor records observed at four stations (NAA, KIS, OKW, and TSA) of F NET, the broadband seismometer network maintained by the National Research Institute for Earth Science and Disaster Prevention (NIED), Japan. These records are 1 day ground velocity records of the vertical component. Each record represents a portion of intermittent tremor activity recorded over several days. For example, the TSA record shows a 1 day period from within a 1 week period of activity from 15 April to 22 April 2006, which was studied in detail by Shelly et al. [2007b]. Tremor behavior is generally variable, making it difficult to define a typical record; consequently, relatively long lasting tremor waveforms were selected from many years of continuous seismogram records at each station. Nevertheless, the tremor recorded at station KIS is spiky compared with that recorded at TSA. This spiky feature is typical of tremor that occurs around KIS; there is no evidence of longer lasting tremor activity in the 13 year record at this station. [11] The above difference between KIS and TSA may be insignificant compared with the difference between the Nankai and Cascadia tremor. Figure 2 shows examples of tremor that occurred in May 2008, as observed at three stations: B001 and B926 operated by the Plate Boundary Observatory, and HDW maintained by Washington University. These stations are located immediately above the tremor sources determined by Wech and Creager [2008] 2of13

3 Figure 1. Examples of tremor records from the Nankai subduction zone. The records are 24 h vertical velocity seismograms (low pass filtered at 2 Hz) obtained at four stations. The start time is shown in Universal Time (UT). Dots indicate LFE locations determined by the Japan Meteorological Agency. using cross correlation of the tremor envelope. The analysis revealed long range migration of the tremor source, from south to north, for about 1 month. In Figure 2, each section of the waveform shows the tremor with the largest amplitude at the station during the 1 month period of activity. Two or three periods of large amplitude waves are visible in every panel; these appear to be different from the waves in Figure 1. Another difference between the Cascadia and Nankai records is the occurrence of ordinary earthquakes, which are rare at Cascadia but common at Nankai. [12] Figure 3 shows the tremor envelope calculated from the seismograms in Figures 1 and 2. A band pass filter was applied to ground velocity records from 2 to 8 Hz, and the root mean square (RMS) was calculated for each 1 s window. Impulsive signals from ordinary earthquakes in each record were removed manually based on the JMA catalog for Nankai records and visual inspection for Cascadia records. The difference in large amplitude duration between Nankai and Cascadia is clearly shown in Figure 3. These functions are proportional to the square root of seismic energy functions, as the high frequency component is dominant in the integral for seismic energy calculation in the case of a flat velocity spectrum. Below, the duration of events is quantified based on the BSE model and a time constant is estimated from the envelope. 3. Characteristics of the Brownian Slow Earthquake Model and Tremor Data 3.1. Definition [13] Ide [2008] approximated a slow earthquake source using a circular patch in which shear slip occurs (Figure 4). The radius of the patch, r(t), is the only random parameter in the model and is controlled by the Langevin equation or a stochastic Ornstein Uhlenbeck process [e.g., Gardiner, 2004]: drðþ¼ rdt t þ dw ðþ; t where W(t) is a Wiener process, s is a controls parameter for the amplitude of r, and a is a parameter that controls the temporal behavior of the model via the damping term in equation (1). Although the equation (1) can make negative value of r(t), we consider only its absolute value in this ð1þ 3of13

4 Figure 2. Examples of tremor records from the Cascadia subduction zone. The records are 24 h velocity seismograms (band pass filtered between 2 and 8 Hz) obtained at three stations. The start time is shown in Universal Time (UT). Solid circles show tremor source locations determined by Wech and Creager [2008]. Figure 3. Examples of observed tremor envelopes, which is the absolute value of band passed velocity seismograms. Stations and periods are the same as those shown in Figures 1 and 2; a noise sequence on a quiet day is shown for comparison (TSA NOISE). Gray spikes are large noises or regional earthquake waves that were removed. 4of13

5 Figure 4. Conceptual illustration of the Brownian Slow Earthquake model. (left) Tremor is generated mainly at the periphery of slip areas with arbitrary shapes. Slip areas of various sizes may be activated simultaneously. (right) Circular slip area with a constant slip rate and time dependent radius, which is the only variable in this model. study. The inverse of a, a 1, is a characteristic time constant that is the focus of the present study. The value of r fluctuates around 0 with a variance of s 2 /2a and is approximated by a normal distribution N(0, s 2 /2a), with a much longer duration than a 1. [14] The moment rate function of the source is defined with a rigidity m and slip rate v slip, as follows: M o ¼ v slip r 2 : ð2þ Ide [2008] assumed a constant v slip to obtain the simplest model possible; however, as shown below, the assumption that v slip is proportional to the absolute value of the radius r provides a better explanation of the statistics of tremor amplitude. Since the variance of r is near constant over very long periods, r 2 and r 3 are also near constant. Therefore, these two assumptions predict that the expectation of M o is a constant, which corresponds to the proportionality coefficient between seismic moment and duration [Ide et al., 2007b]. [15] This model can be simulated by numerically solving equation (1), with Gaussian random noise as dw(t). Figure 5 shows examples of numerical simulations with different values of a 1 and the above two assumptions regarding v slip. Figures 5a and 5b correspond to the previous and new assumptions, respectively. Large a 1 results in a long lasting tremor with large amplitude. A major difference between observed and simulated tremor (Figure 5) is the existence of background noise. In the case that noise exists, only large peaks are recognized as events with a finite duration, which are sometimes termed VLFs [Ito et al., 2007]. The different assumptions regarding v slip affect the amplitudes and duration of peaks exceeding the noise level (Figure 5). The new assumptions appear to better explain the observations (as confirmed in section 3.2). Similar to the old assumption, the new assumption yields flat moment acceleration spectrum, which is proportional to the spectrum of far field velocity. [16] The assumptions of a circular source and only one random parameter r(t) are made to keep the model as simple as possible. A change in the radius of the source mimics size dependent propagation velocity, which is natural because the Langevin equation (1) is one representation of the diffusion equation, though this simple 1 D model cannot explain the long range unidirectional migration observed at Nankai [Ito et al., 2007] and Cascadia [Dragert et al., 2001]. Furthermore, this simple 1 D assumption yields a negative value of r(t), which appears physically unlikely, although this is not a problem because only the absolute value is required in calculating the moment rate function and spectrum. To undertake more realistic modeling, it would be necessary to consider at least a 2 D diffusion process and multiple sources Amplitude Duration Statistics [17] Amplitude duration statistics is a unique feature of tremor activity. Watanabe et al. [2007] first noted that the amplitude duration distribution of tremor resembles an exponential function. They calculated the RMS ground displacement with a fixed time window from tremor records that were band pass filtered between 2 and 10 Hz, and measured the duration for which the RMS amplitude exceeded a certain threshold. The logarithm of the duration is nearly proportional to the threshold level, which means that the duration is approximated by an exponential function of the threshold level. In other words, the cumulative distribution function (CDF) of the tremor envelope amplitude is exponential. Here, the CDF of variable X is defined as follows: PX> ð xþ ¼ Z 1 x p X ðx 0 Þdx 0 ; where p X (x ) is the probability density function of the random variable X. [18] Ide [2008] showed that the CDF of the moment rate in the BSE model is almost exponential, but its mathematical meaning remained unclear. In fact, the radius parameter r has a near normal distribution N(0, s 2 /2a) if we consider a long period relative to the characteristic time a 1. In this case, r 2 has an approximately c 2 distribution with one degree of freedom. The CDF of a random variable x obeying the c 2 distribution p ffiffi is expressed using a complementary error function, erfc( x ), which is close to (but not equal to) an exponential distribution for large x. Because the moment rate of the original BSE is proportional to r 2, its CDF resembles an exponential function. [19] However, it should be noted that Watanabe et al. [2007] measured RMS displacement in a narrow frequency range, which is almost proportional to the absolute velocity and the square root of the seismic energy rate. In the original BSE model, with constant slip, the seismic energy rate is also proportional to r 2, and its square root is proportional to r, reflecting the assumption of a constant slip rate. If we instead assume that the slip rate is proportional to r, the seismic moment rate, seismic energy rate, and envelope amplitude are proportional to r 3, r 4, and r 2, respectively. This outcome violates an observed feature of slow earthquakes; i.e., the proportionality between energy rate and moment rate [Ide et al., 2008]. The energy rate is proportional to the moment rate powered by 3/4, which is difficult to distinguish from the proportional relation in narrow size range. Indeed, the observed relation of Ide et al. [2008] can be explained reasonably well even with the new assumption. It should also be noted that the above relation between energy rate and moment rate hold only for small events for which the assumption of 1 D Brownian walk is valid. [20] The CDF of the absolute velocity of each tremor activity shown in Figure 3 is very close to the CDF of a c 2 ð3þ 5of13

6 Figure 5. Examples of numerically simulated moment rate functions with two assumptions. (a) Constant v slip. The moment rate is proportional to the square of the random parameter r. Moment rate functions and absolute velocity functions (square root of the seismic energy rate) are shown for characteristic times a 1 = 200 s and 2000 s. (b) With v slip proportional to r. As for Figure 5a but for the case in which the moment rate is proportional to the cube of r. (c and d) Fourier spectrum amplitude of moment acceleration, calculated for four time windows of 3600 s and averaged for each 0.05 decade of frequency, when v slip is constant (Figure 5c) and when v slip is proportional to r (Figure 5d). Amplitude is normalized at 10 Hz. distribution (Figure 6). The simulation results reveal that the new assumption, in which the moment rate is proportional to the cube of the random parameter r, performs better in reproducing the observed CDFs. All of the observed tremor records include background noise, which enhances the concave shape in the observed CDF. Although a convex shape is apparent in the simulated CDF with the new assumption and a 1 = 2000 s, this trend is weakened if we consider the effects of noise. However, even when considering the effect of noise, the convex shapes in the model with the old assumption are difficult to change. Therefore, the new model is considered an appropriate, and only the new assumption is used in the following analysis Tremor Spectra [21] The solution of equation (1) is represented using a Wiener process as rt ðþ¼ p ffiffiffiffiffiffi 2 W e 2t e t : ð4þ Using (4) with an initial condition, r(0) = 0, the autocorrelation function of r 2, which is proportional to the 6of13

7 Figure 6. Examples of CDFs calculated for simulated and observed tremor sequences. The horizontal axis in each panel is normalized by the maximum amplitude (absolute velocity). The thin dotted curve in each panel shows the CDF of the c 2 distribution. (top) Simulated moment rate function for different assumptions and a 1 values. Square and cube indicate different assumptions whereby the moment rates are proportional to the square and cube of the random parameter r, respectively. (middle) CDF values for Nankai tremor. (bottom) CDF values for Cascadia tremor. envelope amplitude of the new BSE model, is calculated as (see Appendix A) Er 2 ðþr t 2 ðt þ Þ ¼ 4 1 þ 2e 2j j =4 2 : ð5þ The Fourier transform of this function gives the power spectrum density (psd): psd r 2 ð! Þ ¼ 4 2 ð! Þþ8= 4 2 þ! 2 =4 2 : ð6þ The delta function in the first term corresponds to the direct component, and is negligible if the mean of the signal is subtracted before calculating the spectrum. The second term shows that the spectrum has a flat level and high frequency decay of f 2, divided by a corner frequency f c = a/p. The form of this analytic expression, without the first term, is shown in Figure 7, confirming with the results of numerically simulated psd of r 2, which is shown in the upper left panel of Figures 5a and 5b. Since the low frequency level is given as s 4 /2a 3, two examples with a 1 of 200 and 2000 s in Figure 7 are separated by three orders of magnitudes in the low frequency. [22] In the original BSE model proposed by Ide [2008], both the seismic moment rate and seismic energy rate are proportional to r 2. Therefore, the moment rate spectra are proportional to equation (6). It is unfortunate that the psd of r 3 is not expressed analytically, and that it is impossible to derive an analytic expression of the moment rate function 7of13

8 [23] To estimate psd for an actual tremor envelope, a multitaper spectrum analysis subroutine is employed [Thomson, 1982; Prieto et al., 2009]. Figures 8a and 8b show examples of estimated spectra. The spectra of the tremor envelope in Figure 3 are generally similar to the analytic form shown in Figure 7, being flat at the low frequency limit and decreasing rapidly above the corner frequency. The slope is close to 2, but there is slight deviation at high frequencies due to background noise. Figure 8c shows a psd of the noise velocity envelope calculated in a similar way to that in which the tremor envelope was calculated. If the noise were Gaussian white noise, we would expect a flat amplitude for the entire frequency range, but this noise spectrum has a slope close to 0.7. This is an example of typical noise psd; generally, the noise slope is between 0.5 and 1. The deviation in tremor psd at high frequencies is due to the inclusion of this noise. Therefore, to model the tremor envelope, it is necessary to take into account the contribution of background noise. Figure 7. Simulated tremor spectra (gray lines) and theoretical curves (solid lines) with characteristic times a 1 = 200 and 2000 s. The location of the corner frequency (a/p) is shown by arrows. for the new BSE model. Nevertheless, it is expected that the moment rate spectrum has a similar corner frequency to that of the original model and a flat level because the longterm average value of r is constant. In the new model, the value proportional to r 2 is the absolute velocity, or envelope, function. 4. Estimation of the Time Constant From the Source Spectrum [24] Here, a spectrum inversion method is developed to quantify the tremor time function based on the BSE model, assuming the analytic form in equation (6). The spectrum is characterized by a time constant a 1 and a low frequency constant level A S = s 4 /2a 3. As shown in section 3, tremor signal is contaminated by non Gaussian background noise; this also needs to be modeled by the inversion method. The noise is parameterized by amplitude A N at a reference frequency f ref and with a power law exponent g. Thus, the problem is to determine four unknown parameters (a, A S, A N, and g), as shown in Figure 9. [25] The observed psd is re sampled at equal intervals along the log frequency axis, with 20 data points in each order. The logarithm of the resampled psd at N logfrequency points, y i,(i =1,, N), constitutes the data for Figure 8. Power spectrum density (psd) values of the envelope examples shown in Figure 3. (a) and (b) Tremor recorded at stations KIS and B001, respectively, and (c) noise. The raw spectrum (thin gray curve) is smoothed at equal intervals along a logarithmic scale (bold black curve). Pairs of dashed curves show the 95% confidence interval for estimated psd [Prieto et al., 2009]. Thick gray line shows the slope of power 2or of13

9 Figure 9. Example of data and model parameters used for the inversion method. Data (black line) are explained by summation (thick gray line) of the noise and signal components (dashed lines). the inversion. Assuming that each data point has a Gaussian error, the four parameters are determined by minimizing a least squares residual: Res ¼ XN i¼1 "!# 2 y i log A S 2 þ ðf i Þ 2 þ A f 2 i N : ð7þ f ref This nonlinear least squares problem is linearized and solved iteratively using the Levenberg Marquardt algorithm. Because this algorithm involves a risk of falling into a local minimum, depending on the initial values of model parameters, the inversion is iterated starting from 10 random sets of initial values to find the best estimated values with the minimum Res among the results of all trials. [26] The four parameters (a, A S, A N, and g) can be determined for any data set, but a and A S are poorly constrained in the absence of tremor activity. The reliability of the result can be assessed using the covariance matrix of parameters, C m. Assuming that the 95% confidence interval of psd [Prieto et al., 2009] (Figure 8) corresponds to ±2 standard deviations of the data, s d, C m is estimated by mapping s d for the final iteration of the Levenberg Marquardt algorithm [e.g., Menke, 1989]. If the standard deviation of a calculated from the diagonal component of C m exceeds the parameter value a itself, the problem is considered to be unsolved. [27] This method is applied to the seven examples of tremor envelope shown in Figure 3. To assess the possible temporal variations in the parameters during tremor activity, a moving window of 16,384 s (about 4.55 h) is employed from the beginning to the end of each time sequence. For each time window, four parameters are estimated. Figure 10 summarizes the results of inversion, showing the temporal change in estimated a 1 and examples of spectrum fit. For most of the tremor periods, the time constant a 1 was successfully estimated. In some cases, a 1 is long, close to the minimum frequency of the data, and the low frequency flat level is almost missing. In such cases, the standard deviation is large, but it is confirmed that the constant has a large value based on the spectrum shape. Hence, it is possible to discuss the difference between stations as well as temporal change at a single station. [28] Time constants for Nankai tremor are generally shorter than those for Cascadia tremor, as expected from previous visual inspections of waveforms (Figures 1 and 2). The shortest value of about 100 s is obtained at the beginning of the record at KIS. a 1 increases over a period of 24 h, but remains below 600 s except for a short period with a very large value, which may represent unreliable data. At other stations in Nankai (NAA, OKW, and TSA), the values of a 1 are initially about 300 s and increase up to 2000 s. A similar increase in the time constant is commonly found at other stations located in the Nankai subduction zone. In other words, tremor in Nankai tends to start as successive impulsive LFEs with small a 1, becoming continuous over time. In contrast, tremor in Cascadia always show large values of a 1, between 1000 and 3000 s and the pattern of temporal change is unclear, although some change is apparent at much longer durations. The relation between the time constant and tremor amplitude remains unclear. [29] The power exponent of noise g is estimated to be between 0.5 and 1, similar to the values when background noise alone is analyzed. The difference between stations is not significant. The estimated amplitude of the noise component is much larger than the background level, suggesting that the noise includes not only mechanical and environmental components, but also coda waves generated along the propagation path and around the seismometer by incoming tremor waves. 5. Discussion [30] A difference in the time constant is found for tremor observed at seven stations. This represents a small number of examples; further long term systematic investigations of tremor activity are required at many stations in order to arrive at a general conclusion, which would be discussed in a separate report. At this stage, it is assumed that this difference reflects the characteristics of each station. If this is the case, what can be said about the source of this difference? At Nankai, station KIS has an especially small value of a 1. This value appears to be robust because it is based on 13 years of records. Of note, SSEs have not been detected by the Hi net tiltmeter network [Obara, 2009]. Strainmeters with greater sensitivity than those of the Hi net network have detected signals related to smaller SSEs [Fukuda and Sagiya, 2007]. This means that SSEs in this area are smaller than the detection threshold of M w In contrast, many SSEs have been reported, with the largest located near station TSA [Hirose and Obara, 2005], where long term SSEs occur without visible tremor. This finding may reflect the fact that the average time constant of TSA is the largest among the stations in Nankai. Thus, the time constants appear to correlate with the size of SSEs. [31] This potential correlation is supported by the large values of a 1 observed at stations above the Cascadia subduction zone, where the magnitudes of SSEs are about M w [Dragert et al., 2004; Szeliga et al., 2008]. One of the major differences between the Nankai and Cascadia 9of13

10 Figure 10. Result of spectrum inversion to measure the time constant in the BSE model. For each station, (top) the tremor envelope, which is the same as in Figure 3, (middle) the temporal change in a 1 (black) with the standard deviation shown by gray lines, and (bottom) examples of a comparison between observed (black) and calculated (gray) spectra. The time of each spectrum comparison is shown by the star in Figure 10 (middle). subduction zones is the width of the tremor zone, as measured in the direction of plate motion: the Cascadia tremor zone is twice as wide as the Nankai LFE zone (Figures 1 and 2) The widest zone at Nankai is found near station TSA, where large SSE and a 1 are observed. Although it is difficult to identify the origin of this difference, the consistent variation among a 1, the magnitude of SSEs, and size of the tremor zone would indicate the existence of a common controlling mechanism, which may be related to environmental conditions such as relative plate velocity, amplitude and temperature at tremor depth, composition of the surrounding crust and mantle, and the amount of fluid present. [32] The large values of a 1 at the Cascadia subduction zone may explain the lack of reports of LFEs and VLFs for this region. Brown et al. [2009] detected and determined LFEs using a network autocorrelation method; however, these LFEs are not isolated and are difficult to detect using ordinary methods. In fact, a long lasting tremor simulated by the BSE model with large a 1 is difficult to separate into distinct LFEs. The BSE model also predicts that the typical 10 of 13

11 Figure 11. Comparison between (top) the history of the time constant a 1 estimated for the vertical seismogram at station TSA and (bottom) tremor source location determined by Ide [2010], for a tremor activity from 15 to 21 April 2006 in western Shikoku. Map view and space time plot are shown. At least three periods of increasing a 1, shown by arrows, correspond to the expansion of the source area. event duration recognized as a VLF is determined by a 1 [Ide, 2008]. The amplitude of VLF waves is greatest for an event with duration a 1 and rapidly decreases with decreasing event duration. Therefore, for a 1 above 1000 s, it is possible that the amplitude of VLF waves in the seismologically observable frequency range is too small to be detected. Thus, VLF may be difficult to find in the condition with very large a 1, although the detectability may be also dependent on the spatiotemporal difference in a 1 as we will see below. [33] The increase in the time constant during tremor activity, or the existence of a small initial time constant, is another interesting finding of this study. Because the governing equation of the BSE model is a diffusion equation, something, for example shear stress or fluid, is expected to be concentrated at the beginning of tremor activity. The increase in a 1 corresponds to a weakening of the damping term in equation (1), possibly reflecting a different type of mechanism that is not included in our 1 D Brownian model (e.g., migration of the source area). There is some evidence suggesting that the increase reflects the spatial migration of the source area. Figure 11 shows an example, comparison between the history of the time constant a 1 and the location of tremor sources near the station TSA [Ide, 2010]. At least three sequences of increasing a 1 correspond to the expansion of the source area. Although this is just one example and further detailed studies are required, this comparison suggests that a 1 is related to the spatial extent of tremor source. [34] A new assumption (v slip r) is included in the BSE model to explain the observed CDF of tremor amplitude. This also implies that the strain drop rate is almost constant, since the strain drop of a crack is roughly proportional to the ratio of slip to length. It is known that tremor is activated by small fluctuations in stress resulting from long period waves from distant earthquakes [Miyazawa and Mori, 2005; Rubinstein et al., 2007] or even by tides [Shelly et al., 2007b; Nakata et al., 2008; Rubinstein et al., 2008]. Therefore, the driving stress of tremor is small and probably has magnitudes within a small range. These two considerations lead to the idea that both strain rate and stress are nearconstant in the slow earthquake source, indicating a viscous flow like mechanism. This is a candidate in terms of providing a diffusion mechanism in future models. 6. Conclusion [35] Tremor and similar phenomena observed worldwide probably share a common mechanism, with slightly different control parameters depending on regional geophysical and tectonic conditions. Because tremor behavior appears random and is complex, it is not straightforward to extract quantitative measures from observed data. In this study, tremor is quantified based on a BSE model governed by a stochastic differential equation [Ide, 2008]. Although this is a simple kinematic model, it explains many of the features of observed tremor and slow earthquakes. One shortcoming of the original model was that its CDF is not exponential. This limitation is overcome by introducing a new assumption that the slip rate is proportional to event size. This assumption makes it difficult to express the moment rate spectra and energy rate spectra in simple analytic forms, but it yields a simple analytic form for the spectrum of the absolute velocity, or envelope. [36] The observed tremor psd resembles the prediction of the stochastic model, with a corner frequency and f 2 decay, except for non Gaussian background noise that is dominant at high frequencies. To measure the corner frequency, an inversion method was developed and applied to seven 1 day records of tremor (four in Nankai and three in Cascadia). The Nankai tremor has time constants of s, 11 of 13

12 whereas the Cascadia tremor has longer constants of s. At Nankai, a temporal increase in the characteristic time was observed over the course of 1 day of tremor activity. Comparison with tremor source location suggests that the increase reflects temporal change of the source area, which is not accommodated in the current 1 D model. [37] The time constant is related to other differences in tremor activity between the Nankai and Cascadia subduction zones, including duration, recurrence time of activity, and the size of the tremor area, as measured in the direction of plate motion. All these differences may be explained in terms of environmental conditions such as relative plate velocity, amplitude and temperature at tremor depth, composition of the surrounding crust and mantle, and the amount of fluid present. To identify the governing conditions in this regard requires comparative studies and knowledge of the time constant discussed in this paper. Appendix A: Derivation of the Power Spectrum Density of Brownian Slow Earthquake Model [38] We can confirm that the parameter r in equation (4) satisfies the stochastic differential equation (1) by substitution, as drðþ¼ t rffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffi dw e 2t d 2 dt e2t þ pffiffiffiffiffiffi W e 2t d 2 dt e t e t dt ¼ dw rdt: ða1þ From the definition, the pdf of the increment of a Wiener process from W(s) to W(s + t) has a normal distribution N(0, t). For a random variable x N(0, s 2 ), the expectations E(x 2 ) and E(x 4 ) are s 2 and 3s 4, respectively. Then, with an initial condition, r(0) = 0, the expectation of r 2 (t)r 2 (t + t), for t > 0, is given as Er 2 ðtþr 2 4 ðt þ Þ ¼ 4 2 e 4t 2t E 2 W 2 ðe 2t Þ W ðe 2t ÞþWðe 2ðtþÞ e 2t Þ 4 2 e 4t 2t EW 4 ðe 2t Þ h i þ E W 2 ðe 2t ÞW 2 ðe 2ðtþÞ e 2t Þ 3e 4t þ e 2t ðe 2ðtþÞ e 2t Þ ¼ 4 ¼ e 4t 2t ¼ þ 2e 2 : ða2þ The expectation for t < 0 is similarly calculated, and we obtain the equation (5). 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