Measurements of spectral similarity for microearthquakes in western Nagano, Japan

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1 JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 111,, doi: /2005jb003834, 2006 Measurements of spectral similarity for microearthquakes in western Nagano, Japan Anupama Venkataraman, 1,2 Gregory C. Beroza, 1 Satoshi Ide, 3 Kazutoshi Imanishi, 4 Hisao Ito, 5 and Yoshihisa Iio 6 Received 12 May 2005; revised 24 October 2005; accepted 29 November 2005; published 4 March [1] We use P wave spectral ratios to estimate the seismic energy of 23 microearthquakes (M w ) recorded on a dense network of surface and borehole stations in western Nagano, Japan. These events were recorded at 45 surface stations and 2 borehole stations. The data set is unique in that most events have high signal-to-noise ratio at several surface stations and at the two borehole stations. The redundancy provided by the large number of stations recording these events allows us to assess the uncertainty of our results. We find that the seismic energy to moment (E R /M 0 ) ratios from P waves for the events in our data set are reliable, that they vary between and , and that smaller events tend to have smaller values of E R /M 0 because they are relatively deficient in radiated energy above the corner frequency. We observe two spectral effects: a variation in corner frequency, which accounts for most of the variation in E R /M 0, and steeper spectral falloff rates, which contribute to the small values of E R /M 0 for some of the smaller events. Our estimates of moment and corner frequency suggest that M 0 / f c (3+e) (e > 0), and this implies that either the static stress drop or rupture velocity or both change with earthquake size. Citation: Venkataraman, A., G. C. Beroza, S. Ide, K. Imanishi, H. Ito, and Y. Iio (2006), Measurements of spectral similarity for microearthquakes in western Nagano, Japan, J. Geophys. Res., 111,, doi: /2005jb Introduction [2] During an earthquake, slowly accumulated strain energy is suddenly converted into other forms, including: energy radiated as seismic waves, energy expended in enabling the fracture process and energy dissipated as frictional heat in the fault zone. Generation of seismic waves requires that the total energy exceed the energy required for fracture; however, there are considerable uncertainties in measurements of fracture energy and its dependence on the amount of slip. If fracture energy is independent of earthquake size, the partitioning of energy in large and small earthquakes ought to be very different. If fracture energy varies systematically with the amount of slip, then the energy balance during faulting may scale with earthquake size. A third possibility is that the fracture energy might be small compared to the radiated energy 1 Department of Geophysics, Stanford University, Stanford, California, USA. 2 Now at ExxonMobil Upstream Research Company, Houston, Texas, USA. 3 Department of Earth and Planetary Science, University of Tokyo, Tokyo, Japan. 4 Geological Survey of Japan, Tsukuba, Japan. 5 Japan Agency for Marine-Earth Science and Technology, Yokohama, Japan. 6 Research Center for Earthquake Prediction, Disaster Prevention Research Institute, Kyoto University, Kyoto, Japan. Copyright 2006 by the American Geophysical Union /06/2005JB003834$09.00 for earthquakes of all sizes, such that the effect of fracture energy on seismic waves would be difficult to observe. Accurate estimates of the radiated seismic energy, and how it might vary with earthquake size, is clearly critical to progress in understanding the energy balance during earthquake faulting [Kanamori and Heaton, 2000; Venkataraman and Kanamori, 2004; Kanamori and Rivera, 2004]. [3] Measuring radiated energy accurately requires high signal-to-noise ratio (SNR) data over a wide range of frequencies [Singh and Ordaz, 1994]. Obtaining the necessary bandwidth is particularly difficult for small earthquakes due both to the small amplitudes of seismic waves, and to the rapid attenuation of the high-frequency waves needed to analyze small events. Time domain methods [e.g., Kanamori et al., 1993; Mori et al., 2003], and frequency domain methods [e.g., Singh and Ordaz, 1994] have been used to estimate energy for smaller events at regional distances. Abercrombie [1995] demonstrated the tremendous potential of deep borehole data for determining source parameters and scaling properties of very small microearthquakes. With few exceptions [Ide et al., 2004], source studies using borehole data are based on data from a single downhole instrument, leading often to interesting results, but with uncertainties that are large and difficult to quantify. Uncertainties in measurements of radiated energy, especially for small earthquakes, has led investigators to interpret these results as indicative of constant versus nonconstant scaling of radiated energy with size [Ide and Beroza, 2001; Kanamori and Heaton, 2000; Kanamori and Rivera, 2004]. Measurements using wide-bandwidth, multiple-sta- 1of10

2 Figure 1. Location map. Inset map shows the location of the western Nagano region in central Japan. The location and focal mechanism of the largest earthquake in each cluster are shown. Triangles indicate station locations. Solid triangles with labels represent stations that were used to compute energy. Though there were a total of 47 stations, not all of them could be used as explained in the text. The red star shows location of the 1984 JMA magnitude 6.8 earthquake. tion data are required both to obtain better constraints on seismic energy values and to assess the reliability of these values. More recent work by Yamada et al. [2005], Oye et al. [2005], and Mayeda et al. [2005] have started to address some of these measurement issues in using different data sets. [4] In this paper, we study microearthquakes using high-density and high SNR data recorded in the western Nagano region of central Japan. We use an Empirical Greens function (egf) spectral ratio technique that empirically accounts for propagation effects from closely spaced events. Although the size range of the earthquakes in this study is relatively small, we find a systematic decrease in E R /M 0 with decreasing seismic moment. The decrease in this ratio is manifest in the smaller than expected corner frequencies and/or the more rapid decay of amplitudes above the corner frequency for smaller earthquakes as compared to larger earthquakes. That is, we observe a change in the spectral decay characteristics of earthquakes as a function of size. 2. Data [5] The data set, obtained from National Research Institute for Earth Science and Disaster Prevention and Geological Survey of Japan, consists of records of microseismicity from 45 surface stations, one 800 m deep borehole station (OT0a), and one 145 m deep borehole station (OT01) all located within a km 2 area (Figure 1). Data from a third borehole (OT02) could not be used because of instrument malfunctions. The instruments are three-component velocity sensors with a natural frequency of 2 Hz and sampling rates of 10 khz [Iio et al., 1999]. The array is in a mountainous region that witnessed a steam eruption of the Ontake volcano in 1979, followed by a damaging shallow Japanese Meteorological Agency (JMA) magnitude 6.8 2of10

3 Figure 2. (a) Velocity waveforms of the largest event ( , M w 2.7) and a smaller event ( , M w 1.4) of cluster 5 recorded on 800 m deep borehole station (OT0a). Observe that data on all three components have the same polarity and similar amplitude ratios. (b) Signal (top curve) and noise (bottom curve) spectra at the deep borehole station (OT0a), the shallow borehole station (OT01), and a surface station (OT24) for event (c) Same as Figure 2b at three stations for event , M w 0.9 in cluster 5. We observe high SNR (>2) up to frequencies of Hz for borehole stations and up to Hz for surface stations. earthquake in 1984 and continued shallow seismicity [Horiuchi et al., 1992; Rydelek et al., 2002]. The borehole stations have high SNR up to frequencies of Hz. More unusual are the surface stations that have high SNR up to frequencies of Hz (Figure 2). The combination of a dense monitoring network and high SNR surface and borehole recordings provides a unique data set for the study of microearthquake source processes. Since the deployment of most of the surface and borehole stations was completed in 1999 and availability of the borehole data is important in this study, we only used data recorded between April and October 1999 (heavy snow necessitated the deactivation of most stations in winter [Iio et al., 1999]; subsequently, the borehole station OT0a was damaged in May 2000). The high data quality allowed a time domain analysis of the slow initial phase of some of these events [Iio et al., 1999]. [6] The events were relocated by S. Ito et al. (Complex fault patterns in western Nagano, Japan revealed by the double-difference method, manuscript in preparation, 2003), and we used the relocated catalog to choose the largest events (M w > 2.5) in the period range of interest. The double-difference algorithm used for the relocations results in uncertainties of <10 m in the relative location of the events. We searched the relocated catalog for events within 0.5 km of each of these large events in the magnitude range of interest. We found 11 clusters of which 6 clusters are located within 20 km of most of the stations (Figure 1) and have events with waveforms similar enough to be used in our analysis. [7] For each event, we picked the P and S wave arrivals and visually inspected the waveforms to reject stations with low SNR. The north and east components of the velocity data are rotated to the radial and transverse directions and windowed 3of10

4 to include 0.08 s before and 0.4 s after the P wave arrival times. The S wave data are rendered noisy by the P wave coda, and because of the low SNR (<2 in most cases) we could not use the S wave data to obtain reliable energy estimates. Matsuzawa et al. [2003] used P and S wave polarities and amplitudes to determine the focal mechanisms of some of these earthquakes. In addition, we determined the seismic moments and focal mechanisms of the largest events and some of the smaller events in each cluster using the amplitudes and polarities of P and S waves. For events common to both studies, the values of scalar seismic moment computed by Matsuzawa et al. [2003] agree with our estimates to within less than 10%. Determining the seismic moment of the large event accurately is important, since it directly affects the energy to moment ratio (E R /M 0 ). We rejected events for which the seismic moment of the largest event could not be determined well (usually as a result of very few stations or poor data quality). Since the focal mechanisms of several small earthquakes could not be reliably determined, the waveforms at each station for all the events in the cluster are compared to ensure similarity (Figure 3). The data are tapered using a Hanning taper of width 5% and then Fourier transformed. The radiated energy is estimated from the root-mean-square spectrum of all three components for P waves (shown on Figure 2). [8] Although there are 47 stations in the array, we could not use all the stations (see Figure 1) for all the events for the following reasons: [9] 1. Many events did not have data at several stations. Lack of data at stations common to the large and small events restricted the number of useful stations. [10] 2. Data quality at the station was poor, i.e., longperiod noise or clipped records. We did not use stations if the SNR for the large or small events were less than 5 at frequencies less than 100 Hz (since we would not be able to resolve the corner frequency of the smaller event well). [11] 3. Waveforms were not similar for all events in the cluster at some stations, probably due to differences in mechanisms between the events or very localized propagation effects. For these stations, we only use events that have similar waveforms. [12] Despite these stringent criteria, the sizable event and station distribution allows us to conduct a robust analysis of the remaining data. In most cases we have multiple stations recording each event. Cluster 10 is the best cluster in that it has the largest number of similar waveforms in our data set with 17 stations, 3 events, and 43 usable event pairs. Cluster 5 is also a good cluster with 9 stations, 8 events, and 39 usable event pairs. 3. Energy Estimation Using Spectral Ratios [13] Seismic energy is calculated by integrating squared velocity spectra after correction for source excitation and propagation effects. Methods based on the idea of empirical Green s function (egf) deconvolution [Hartzell, 1978], or spectral division, assume that a smaller event located close to a large event can be used to correct for propagation effects and thus recover the source spectrum and radiated energy of the larger event [Venkataraman et al., 2002]. A modification of the method used by Hough [1997] was used by Izutani and Kanamori [2001], Ide et al. [2003], and Ide et al. [2004] to Figure 3. All the events in cluster 5 that have similar displacement waveforms at station OT0a. Since the focal mechanisms of several small earthquakes could not be reliably determined, the waveforms at each station are compared to ensure similarity. calculate energy of both the small and large events. These methods require earthquakes with similar focal mechanisms located close to each other, with high SNR over frequencies at least three times the corner frequency of the smaller event. [14] To compute radiated energy, we use a method similar to the spectral ratio method described by Ide et al. [2003] and Ide et al. [2004]. The spectral ratio, ^R (i,j) (w) of two events located close to each other (assuming that the focal mechanisms and path effects are similar for both events) can be given as ^R ði;jþ ðwþ ¼ ^O i ^O j ðþ w ð Þ ^_M ð Þ ð Þ ¼ ^_M ðþ j ðwþ ðþ w ðþ i w where ^O(w) and ^_M(w) are the observed spectrum and moment rate spectrum (source spectrum); the hat is used to denote a quantity in the frequency domain and the superscripts are the event indices. We take the ratio of the larger event and egf event for all stations (Figure 4a) for all event pairs for which the magnitude difference between the large and egf events is at least 1 unit. For P waves the highest frequency used for each station is determined by the frequency up to which the amplitude of the signal spectrum is more than five times that of the noise spectrum for both events. This criterion imposes a strict high-frequency cutoff for the data used at each station. For example, though a larger events could have SNR greater than 5 for frequencies up to 250 Hz at a particular station (say OT0a), if its egf pair has SNR less than 5 at frequencies above 150 Hz, the high-frequency cutoff for this pair at that particular station would be 150 Hz. The ratio spectra are resampled at logarithmically uniform spacing and smoothed using a moving window of width 5% of the total frequency range (Figure 4b). [15] A general form of the source spectrum can be expressed as ^_M(w) = M 0 /(1 + (w/w c ) gn ) 1/g, where w c = 2pf c, f c is the corner frequency, gn and is the high-frequency falloff rate. The classical Brune spectrum [Brune, 1970], has ð1þ 4of10

5 Figure 4. (a) Spectra of a large event (solid) and egf event (dashed) for three events in cluster 5. Steeper spectral falloff of the smaller event at high frequencies is outlined by the dotted ellipses. (b) Ratio spectra obtained by dividing the large event spectra by the egf event spectra (gray dashed), the smoothed spectra (solid black), and the spectral fit (bold dashes). The ratio spectra are not flat above the egf corner frequency but turn upward, indicating a steeper spectral falloff for the egf event. frequencies w (i) c and w (j) c of the two events at each station (Figure 4b). This estimation is performed using all stations and all event pair combinations for each cluster. The set of equations is solved as a linearized least squares problem. The high-frequency falloff rate (2n) was allowed to vary between 3.1 and 8. Corner frequencies are allowed to vary between zero and 10 times the corner frequency predicted by the constant stress drop scaling; the corner frequency is also allowed to vary for each station. [17] The difference between the observed and the calculated spectral ratios (Figures 4b and 5b) is assumed to represent the path and site effects. The observed spectra for each event at a station are corrected to determine the moment rate spectrum for each event by dividing them by the average of the path and site effects over all events of the cluster at that station. We determined the seismic moments of the larger events independently using the amplitudes and polarities of the P and S waves (details in section 2). For the smaller events seismic moments are determined from the ratio of the moment spectra at low frequency. [18] Radiated energy can be calculated from the moment rate spectra using (modified from Vassiliou and Kanamori [1982]) Z 1 E R ¼ 15p 2 ra 5 þ p 2 rb 5 w 2 ^_M ðwþ 2 dw ð3þ where r is the density, a and b are the P and S wave velocities of the medium. We set r = 2.7 kg/m 3, a = 0 n =2,g = 1, and the spectrum used by Boatwright [1978] has n = 2, g = 2. For an omega-squared moment rate spectrum [Boatwright, 1978], ^_M(w) =M 0 /(1 + (w/w c ) 4 ) 1/2, and frequencies up to 3 times the corner frequency carry about 70% of the energy. Since the highest frequencies recorded with good signal-to-noise ratios are 250 Hz, the highest corner frequency that can be resolved is f C 83 Hz. Assuming a stress drop of 3 MPa (typical crustal values), this corresponds to M w 1.0. [16] In this data set, we observe that the velocity spectra of events belonging to the same cluster have different highfrequency falloff rates (Figures 4a and 5a) and that the ratio of the velocity spectra is not constant at high frequencies (Figures 4b and 5b), indicating that the high-frequency spectral decay of small and large events differs. To explore this possibility, we use a model spectra of the form ^_M(w) = M 0 /(1 + (w/w c ) 2n ) 1/2 and fit the log of the ratio spectra using log ^R ði;jþ ðwþ ¼ log M ðþ i 0 log M j ðþ 0 þ 1 1 þ w=w j 2 log 1 þ w=w i ðþ c 2n ðþ j ðþ 2n ðþ i c to determine the ratio of the seismic moments M 0 (i) /M 0 (j), the high-frequency falloff rates n (i) and n (j) and the corner ð2þ Figure 5. (a) Spectra of a large event (solid) and egf event (dashed) for 3 events in cluster 10. The high-frequency spectral falloff rates are about the same for the large and egf events; (b) The ratio spectra obtained by dividing the large event spectra by the egf event spectra (gray dashed), the smoothed spectra (solid black) and the spectral fit (bold dashes). 5of10

6 differences in path and/or directivity effects, but these effects are difficult to quantify for this data set. Figure 6. (a) Mean values of E R /M 0 derived from the bootstrap distribution of all events from P wave data. Also plotted are the 95% confidence intervals. Events belonging to set 1 are represented by the open diamonds, and events in set 2 are shown using the solid diamonds. (b) Bootstrap values of energy to moment ratios (E R /M 0 ) of events in cluster 5 (solid diamonds) and cluster 1 (open diamonds). (c) Variation in high-frequency falloff rate (2n) with earthquake size. 6.4 km/s, and b = 3.7 km/s. P wave data are used to determine the moment rate spectra, but P waves carry less than 5% of the total radiated energy. Thus we use the second term of the above equation to calculate radiated energy; this assumes that the ratio of S wave energy to P wave energy is 1.5(b/a) 5. The radiated energy of each event is calculated by integrating the moment rate spectrum up to the highest frequency used for that station. We use the model spectra to determine the percentage of radiated energy in the frequency range of analysis. The missing fraction of radiated energy is added to the energy calculated from the moment rate spectrum. [19] The data set used in this study is unique because of the dense network and high SNR at surface stations. However, useful data were limited because of the small number of colocated events with similar focal mechanisms and the small range in event size. On the basis of this study and earlier work on using egf methods [e.g., Mori and Frankel, 1990; Miyake et al., 2001], we conclude that the method works best when the magnitude difference between large and egf event is between 1.0 and 3 units, the distance between the main and egf events is less than the length of the main shock rupture and the focal mechanisms are similar. Despite careful data selection, we observe a difference in the energy estimates between different stations for the same event, and these variations could be due to slight 4. Results [20] We determined radiated energy estimates for 23 events using the method outlined above and have plotted E R /M 0 versus M w in Figure 6. Since we recover at least 60% of the radiated energy directly from the spectral ratio measurements at all stations and for all events, our results should not have a systematic bias in event selection that could potentially affect some previous studies as pointed out by Ide and Beroza [2001]. We could not use spectral ratios to estimate the energy of the largest event in the data set (M w = 3.8), because the S-P time is too short to recover the P wave corner frequency of the event reliably. Also, we could not use the S waves for this event as they are clipped at most stations. This event was recorded at broadband (JMA) stations; however, data for the smaller events are not available at these stations, so the spectral ratio method cannot be used. [21] A bootstrap procedure is used to estimate the radiated energy and its uncertainty. For each event, the estimate of radiated energy based on the mean of the bootstrap-derived distribution and the 95% confidence intervals calculated from P waves is plotted in Figure 6a. These values are very close to the mean values and are listed in Table 1. For the earthquakes analyzed in this study, the mean E R /M 0 calculated from P waves varies between and (Figure 6). [22] Table 1 lists the moments, average corner frequencies and the high-frequency falloff rates (2n) for the 23 events that satisfied all our selection criteria. The events are grouped by cluster. We observe that events in clusters 5 and 12 had best fitting solutions (i.e., minimum residuals) when the high-frequency falloff rates of the smaller events are about 2 units larger than those of the larger events. Thus, for these events the smaller events have a much higher highfrequency falloff rate as compared to the larger events. This effect is visible in the raw velocity spectra as well as the ratio spectra (Figure 4). In contrast, events in clusters 1, 4, 10, and 13 do not show such a large difference between the high-frequency falloff rates of the small and large events (the rates differ by less than 1). Because we use a linearization of a nonlinear problem, we performed our estimation for different initial values of the model parameters to find the global minimum. The use of additional parameters in a least squares solution would result in models that better fit the data. To determine if the addition of extra parameters for the falloff rates (n) is required, we performed an Akaike information criterion (AIC) test where for a least squares problem, the AIC criterion is given by AIC ¼ N logðresidual=nþþ2m where N is the number of independent data points, M is the number of parameters, and the residual equals the least squares misfit (L2 norm). All clusters except cluster 13 satisfy the AIC criterion. So, for cluster 13, we fix the falloff rate at n = 2 for all events. We also note that inverting for the falloff rates results in better fits (lower residuals) but does not affect the estimates of corner frequencies and 6of10

7 Table 1. Events Used in This Study: Results of Spectral Fit a Origin Time Latitude Longitude Depth, km Strike, deg Dip, deg Rake, Radiated deg M w Energy, J Moment Nm f c 2n Cluster NA NA NA E E E E E E NA NA NA E E Cluster E E E E E E Cluster E E E E E E Cluster E E E E Cluster E E E E E E E E E E E E NA NA NA E E NA NA NA E E Cluster E E E E E E a NA means not available. Read 3.59E+08 as Number of Stations radiated energy significantly (the values change by less than 15% in most cases). 5. Discussion [23] We group the clusters 1, 4, 10, and 13 as a set (set 1) and the clusters 5 and 12 as another set (set 2). We believe that the difference in the high-frequency spectral falloff rates for the data in set 2 is a source effect, rather than a path effect, since it is observed at all stations that have good signal-to-noise ratio up to 150 Hz and for more than one cluster. We do not think that nonlinear attenuation effects provide a reasonable explanation for our observations because these earthquakes are too small to produce such effects; moreover there is no systematic temporal change in the falloff rates. Iio [1992] used Q corrections to determine the source spectra for small events in the same region and observed that high-frequency falloff rates of these microearthquakes are larger than 2. [24] We observe that events in set 2 have falloff rates that increase significantly below M w 1.5 (Figure 6c). The difference in falloff rates between the large and small events in set 2 could be indicative of a fundamental difference in the source processes. These events may have slower particle/rupture velocities as compared to the larger events. The events in set 1, however, do not show a significant difference in falloff rates with size. We ignore differences in falloff rates that are less than 1.5 since small changes in falloff rates do not affect the least squares residuals (spectral fit) significantly. Gravity data [Shichi et al., 1992] and resistivity data [Kasaya et al., 2002] suggest that in the northeastern section of the area shown in Figure 1, the fault that ruptured the 1984 earthquake separates regions with different properties. Events in set 1 fall in the northeastern region and are to the north of the rupture zone, while events in set 2 are located to the south of the rupture zone. It s possible that the difference in falloff rates that we observe between the two sets is due to differences in properties (maybe fluid pressure) in the two regions Scaling of E R /M 0 [25] Our estimates of corner frequency are fairly robust since we have more than one station with similar energy estimates and corner frequencies. Moreover, high SNR data with sufficient bandwidth ensure that the corner frequencies of the small events are well resolved (Figure 2c). Accurate estimates of seismic moment are important in understanding the scaling of E R /M 0 with earthquake size. The western Nagano data set was also analyzed by Stork and Ito [2004]; however, their seismic energy estimates for two events common to this study are more than an order of magnitude smaller than our estimates. They fit the velocity spectrum of the 800 m borehole station (OT0a) to estimate the spectral level, Q and corner frequency and then determine the 7of10

8 Figure 7. Cumulative energy as a function of frequency for different moment rate spectra. The thick gray curve is for a moment rate spectrum with a spectral form given by Brune [1970]; the solid line is for a moment rate spectrum when the model corner frequency determined in this study and spectral falloff rate of 4 is used; the dashed line is for a moment rate spectrum when both corner frequency and the highfrequency falloff rate determined in this study are used. (a) Event in cluster 10 as a typical example of the small events in set 1. (b) Event in cluster 5 as a typical example of the small events in set 2. seismic moment from the spectral level. Their use of data at a single station to determine the spectral ratio, corner frequencies, and Q can result in uncertainties in their estimates, which would then affect the seismic moments significantly. Their seismic moment estimates are a factor of 2 3 smaller than our estimates, which by itself would lead to a factor of 4 9 difference in the seismic energy estimates. [26] Despite the limited magnitude range of the data, we observe a variation of E R /M 0 with size (Figures 6a and 6b). For both sets, we observe that E R /M 0 drops significantly for earthquakes below M w 1.0. We also note a variation in E R /M 0 within the clusters (Table 1 and Figure 6b). Much of this variation is attributable to changes in the corner frequency. For most earthquakes in set 1, the small E R /M 0 values of the smaller events can be explained by changes in the corner frequency (Figure 7a). For the smaller events in set 2, however, the corner frequency alone only explains a decrease of a factor of 1.5 in E R /M 0 ; the remainder of the difference, a factor of 2 ine R /M 0, requires changes in the spectral decay rate (Figure 7b). For both sets, the observed variation in E R /M 0 with size is mostly due to variation in corner frequency. Our estimates of E R /M 0 compare well with other estimates obtained for earthquakes in the same size range (Figure 8). Figure 8. E R /M 0 as a function of moment magnitude (M w ) obtained from different studies that used regional earthquake data. The methods used to estimate energy are different, and the tectonic environments are also different. However, there is a general decrease in E R /M 0 for smaller earthquakes. 8of10

9 Figure 9. Moment, M 0, as a function of P wave corner frequency, f c, for the events in this study. The 95% confidence intervals of the corner frequencies are shown. Events belonging to set 1 are represented by the open diamonds, and events in set 2 are shown using the solid diamonds. Dark lines follow constant static stress drop assuming M 0 / f 3 c. Gray lines represent M 0 / f (3+e) c for e =1. [27] For events in set 1, our results suggest that the highfrequency falloff rates do not change significantly with size. However, for events in set 2 the change in the highfrequency falloff rates with size suggests that the particle/ rupture velocity is smaller for the smaller earthquakes. For the same seismic moment, less energy would be radiated if we have longer rupture durations (lower corner frequencies in set 1) or smoother rupture (as implied by the steeper falloff rates in set 2). Before we interpret these results, we first consider the variation of moment with corner frequency for the events in our study Static Stress Drop and Rupture Velocity [28] To understand the variation in faulting processes with earthquake size, we require estimates of static stress drop or rupture velocity in combination with estimates of E R /M 0 [Kanamori and Heaton, 2000]. This information can 0 be used to calculate apparent radiation efficiency, h R = 2m(E R /M 0 )/Ds s, which for a simple slip weakening model is equal to the radiation efficiency, h R = E R /E R + E G, where E G is the fracture energy [Husseini, 1977; Venkataraman and Kanamori, 2004]. Thus radiation efficiency can be used to understand the partitioning of energy in earthquakes. To calculate apparent radiation efficiency, we require estimates of static stress drop. Static stress drop can be written as efficiency. However, the variation of seismic moment with corner frequency can nevertheless provide helpful constraints. [29] Figure 9 plots the values and the 95% confidence intervals of the corner frequencies as a function of seismic moment. Also plotted are lines of constant stress drop [Brune, 1970], where M 0 / f c 3. We observe that for events in both sets M 0 / f c 4. A similar result for the scaling of seismic moment with corner frequency was obtained by Iio [1986], for earthquakes in western Nagano. Kanamori and Rivera [2004] showed that if the small E R /M 0 ratios observed in some studies (Figure 8) are correct, then M 0 / f c (3+e) (e > 0), and this implies that either the static stress drop or rupture velocity or both change with earthquake size. For this data set, E R /M 0 decreases with size and M 0 / f c 4. [30] To interpret our observations fully, we require independent estimates of rupture length and rupture velocity (say from source inversion or study of stopping phases) so that we can determine apparent radiation efficiency. Imanishi et al. [2004] estimated static stress drops (0.1 to 2MPa) and rupture velocities (0.4 to 0.9 times shear wave velocities) of earthquakes in the western Nagano region using stopping phases, but none of the events are common to both studies. While Figure 10 illustrates that Ds S (V R /b) 3 is strongly variable in their study, it does not change systematically with earthquake size. Our study, in contrast, requires that Ds S (V R /b) 3 change systematically with earthquake size. 6. Conclusions [31] The extensive network that recorded the events we have studied in the western Nagano region allows us to use spectral ratios to determine robust energy estimates. Though the magnitude range of the earthquakes in this study is Ds S / M 0 L 3 ¼ M f 3 c 0 ð4þ Models used for static stress drop estimates [e.g., Brune, 1970; Madariaga, 1977, 1979] assume a fixed value for rupture velocity. From the formula it is clear that unless we have independent estimates of rupture length or rupture velocity and corner frequency, we cannot obtain estimates of static stress drop that can be used to understand radiation V R Figure 10. Ds S (V R /b) 3 as a function of moment magnitude (M w ) from estimates of static stress drops and rupture velocities determined by Imanishi et al. [2004] for earthquakes in the western Nagano region. It illustrates that Ds S (V R /b) 3 is strongly variable in their study and does not change systematically with earthquake size. Our study, in contrast, requires that Ds S (V R /b) 3 change systematically with earthquake size. 9of10

10 limited, we see a systematic decrease in E R /M 0 with decreasing earthquake size and M 0 / f 4 c. For the smallest earthquakes in this data set, the decrease can be attributed to anomalously low seismic radiation above the corner frequency. With better estimates of static stress drops and rupture velocities we can quantify how efficiently these earthquakes generate seismic waves. [32] Acknowledgments. We thank AIST, Japan, for facilitating the transfer of data. Funding for this work was provided by NSF grant EAR and the George Thompson Fellowship at Stanford University. We thank Jeff McGuire, Rachel Abercrombie, and Kevin Mayeda for their comments and suggestions which helped us clarify and improve the manuscript. References Abercrombie, R. E. (1995), Earthquake source scaling relationships from 1 to 5 ML using seismograms recorded at 2.5-km depth, J. Geophys. Res., 100, 24,015 24,036. Boatwright, J. (1978), Detailed spectral analysis of two small New York state earthquakes, Bull. 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Ide, H. Kawakata, Y. Iio, and H. Ogasawara (2005), Radiation efficiency and apparent stress of small earthquakes in a South African gold mine, J. Geophys. Res., 110, B01305, doi: / 2004JB G. C. Beroza, Department of Geophysics, Stanford University, 397 Panama Mall, Stanford, CA , USA. S. Ide, Department of Earth and Planetary Science, University of Tokyo, 7-3-1, Hongo, Bunkyo, Tokyo, , Japan. Y. Iio, Research Center for Earthquake Prediction, Disaster Prevention Research Institute, Kyoto University, Gokajyo, Uji, Kyoto , Japan. K. Imanishi, Geological Survey of Japan, AIST Tsukuba Central 7, Earth Science Information, 1-1 Higashi 1-Chrome, Tsukuba, Ibaraki, , Japan. H. Ito, Japan Agency for Marine-Earth Science and Technology, CDEX, Showa-machi , Kanazawa-ku, Yokohama, Kanagawa , Japan. A. Venkataraman, ExxonMobil Upstream Research Company, P.O. Box 2189, GW3-917A, Houston, TX 77063, USA. (anupama.venkataraman@ exxonmobil.com) 10 of 10