Abstract
 Top of page
 Abstract
 1. Introduction
 2. Constant Q Analysis
 3. Spectral Ratio Analysis
 4. Discussion
 5. Conclusions
 Appendix A:: Coda Waves and Length of Time Window
 Acknowledgments
 References
 Supporting Information
[1] We reexamine the scaling of stress drop and apparent stress, rigidity times the ratio between seismically radiated energy to seismic moment, with earthquake size for a set of microearthquakes recorded in a deep borehole in Long Valley, California. In the first set of calculations, we assume a constant Q and solve for the corner frequency and seismic moment. In the second set of calculations, we model the spectral ratio of nearby events to determine the same quantities. We find that the spectral ratio technique, which can account for path and site effects or nonconstant Q, yields higher stress drops, particularly for the smaller events in the data set. The measurements determined from spectral ratios indicate no departure from constant stress drop scaling down to the smallest events in our data set (M_{w} 0.8). Our results indicate that propagation effects can contaminate measurements of source parameters even in the relatively clean recording environment of a deep borehole, just as they do at the Earth's surface. The scaling of source properties of microearthquakes made from deep borehole recordings may need to be reevaluated.
1. Introduction
 Top of page
 Abstract
 1. Introduction
 2. Constant Q Analysis
 3. Spectral Ratio Analysis
 4. Discussion
 5. Conclusions
 Appendix A:: Coda Waves and Length of Time Window
 Acknowledgments
 References
 Supporting Information
[2] For earthquakes above M_{w} 3, it has long been known that stress drop does not vary systematically with earthquake size. This results in wellknown scaling relationships of characteristic length and time with seismic moment [e.g., Kanamori and Anderson, 1975; Hanks, 1977]. Several studies have suggested that the scaling of small earthquakes is different from that of larger events. Archuleta et al. [1982] and Archuleta [1986] analyzed surface and shallow borehole records of earthquakes in the Mammoth Lakes, California and found that the stress drop decreases with decreasing moment for events smaller than M 3. Using deep borehole recordings at 2.5km depth in the Cajon Pass, California, Abercrombie [1995] concluded that there is no such breakdown in constant stress drop scaling. From the same analysis, however, she also concluded that the apparent stress (rigidity times the ratio between seismically radiated energy to seismic moment) decreases with decreasing seismic moment. More recently, Prejean and Ellsworth [2001] used data from a 2kmdeep borehole in Long Valley caldera, California, to determine the stress drop and apparent stress of earthquakes from M_{w} 0.5 to 5.0 and reached similar conclusions.
[3] There are many estimates of apparent stress and the energy/moment ratio across a wide range of earthquake size. Although these estimates seem to have a common upper limit defined by the average shear stress level [McGarr, 1999], there are often size dependencies within individual data sets [Gibowicz et al., 1991; Kanamori et al., 1993; Abercrombie, 1995; Mayeda and Walter, 1996; Jost et al., 1998; Prejean and Ellsworth, 2001] such that changes in scaling within each study are more rapid than the scaling changes across studies. Recording bandwidth limitations can severely affect the estimate of seismic energy [Boore, 1986; Di Bona and Rovelli, 1988; Singh and Ordaz, 1994; Hough, 1996], and Ide and Beroza [2001] have shown that most of the observed size dependence can be attributed to artifacts arising from bandwidth limitations. The artifacts they cite are (1) substantial missing contributions from waves with frequencies well above the corner frequency [e.g., Gibowicz et al., 1991; Jost et al., 1998] and (2) event selection with a constant upper cutoff on the corner frequency, which will bias event selection to lower stress drop events for smaller earthquakes in the sample [e.g., Abercrombie, 1995; Mayeda and Walter, 1996]. Even when these sources of bias are accounted for, however, there remains a size dependency that cannot be explained by either of these mechanisms in the results of Abercrombie [1995] and Prejean and Ellsworth [2001].
[4] Both of these studies modeled crustal attenuation with a frequency independent Q operator. They also did not account for the possibility of frequencydependent path or site effects in the data, both of which are thought to be reasonable assumptions given the extremely broad band nature of borehole recordings and the clean, pulselike nature of seismograms recorded in borehole environments. The validity of this assumption is to some extent untested, however, even in deep boreholes where seismic noise and path effects are clearly greatly reduced as compared to surface recordings [Abercrombie, 1998].
[5] If colocated events are available, it is possible to cancel path and site effects by taking the spectral amplitude ratio between the spectra of the two events [e.g., Berckhemer, 1962; Bakun and Bufe, 1975; Mueller, 1985]. Hough [1997] proved the effectiveness of this approach and Hough et al. [1999] called it the multiple empirical Green function (MEGF) method. They used it to analyze the attenuation structure and source properties of small earthquakes (−0.4 < M < 1.3) at the Coso Geothermal area, California.
[6] In this study, we reexamine the data of Prejean and Ellsworth [2001] and additional data in the same area. We compare results from constant Q analysis and MEGF analysis to test whether the assumption of constant Q and negligible path and site effects for borehole recordings might affect estimates of source properties.
2. Constant Q Analysis
 Top of page
 Abstract
 1. Introduction
 2. Constant Q Analysis
 3. Spectral Ratio Analysis
 4. Discussion
 5. Conclusions
 Appendix A:: Coda Waves and Length of Time Window
 Acknowledgments
 References
 Supporting Information
[7] Figure 1 and Table 1 show the locations of the seismometer and the events analyzed by Prejean and Ellsworth [2001]. The event locations are relocated using the double difference earthquake location algorithm of Waldhauser and Ellsworth [2000]. The seismometer is located at 2054 m depth in the Long Valley Exploratory Well. At this level, the well deviates from vertical by only two degrees. The sensor is a threecomponent geophone with a pendulum frequency of 10 Hz and damping constant of 0.7. Events are recorded at three different sampling rates: 1000 (event 1–31), 250 (32–35), and 10,000 (36–41) sps. The data at frequencies higher than about 200 Hz show an unusual attenuation of the vertical component, so we limit our analysis to frequencies below 180 Hz.
Table 1. Earthquake AnalyzedID  Time^{a}  Latitude, °N  Longitude, °W  Depth, km  M_{w}  Group 


01  972541854  37.6480  118.9177  4.34  1.1  – 
02  972542323  37.5237  118.8088  5.33  1.5  – 
03  972561934  37.6630  118.8384  6.80  1.6  C1 
04  972582242  37.6639  118.8753  5.63  1.8  – 
05  972590251  37.6595  118.9259  5.57  1.3  – 
06  972590912  37.6591  118.9267  5.56  1.3  – 
07  972600247  37.6597  118.8415  5.79  1.6  – 
08  972602200  37.4888  118.8747  4.10  1.3  – 
09  972631047  37.5030  118.8695  5.44  1.7  – 
10  972640331  37.6582  118.9075  3.55  0.7  – 
11  972640949  37.6628  118.8476  5.49  1.7  – 
12  972641614  37.6470  118.8465  6.35  2.5  C1 
13  972641620  37.6565  118.8417  5.60  1.1  C1 
14  972641659  37.5745  118.8628  5.15  1.1  – 
15  972641856  37.6525  118.8390  6.59  2.2  C1 
16  972641910  37.6511  118.8391  6.46  1.3  – 
17  972641915  37.6502  118.8552  5.25  1.0  – 
18  972642050  37.6531  118.8387  6.40  2.7  C1 
19  972642102  37.6542  118.8404  5.95  1.7  C1 
20  972642107  37.6540  118.8401  6.00  2.0  C1 
21  972642115  37.6525  118.8401  5.94  1.3  C1 
22  972642301  37.6337  118.8701  8.47  0.7  – 
23  972642340  37.6544  118.8380  6.85  1.4  C1 
24  972650213  37.6498  118.8394  6.20  1.7  C1 
25  972660130  37.6134  118.9052  6.02  1.8  – 
26  972660256  37.4845  118.8432  7.98  1.3  – 
27  972660326  37.4860  118.8440  7.18  1.4  – 
28  972660631  37.6561  118.8757  3.37  0.7  – 
29  972660734  37.6629  118.8763  3.22  0.6  – 
30  972670150  37.6243  118.8520  4.77  1.0  – 
31  972670911  37.6320  118.9597  9.25  1.2  – 
32  973081302  37.6522  118.8558  4.73  2.9  – 
33  973111517  37.6280  118.8885  3.08  3.0  – 
34  973570219  37.6440  118.9446  7.40  3.5  – 
35  980381938  37.6415  118.9290  8.61  3.5  – 
36  981440310  37.6280  118.8553  7.93  2.4  – 
37  981590355  37.5893  118.7975  6.66  3.1  – 
38  981600524  37.5887  118.7955  6.75  5.0^{b}  – 
39  981600829  37.5848  118.7887  7.67  2.5  – 
40  981600845  37.5825  118.7813  7.02  3.1  – 
41  981601330  37.5862  118.8002  5.85  3.2  – 
A1  923130825  37.6467  118.8505  5.61  2.3  C2 
A2  923130842  37.6528  118.8458  4.25  0.7  C2 
A3  923130902  37.6487  118.8480  5.35  1.2  C2 
A4  923131038  37.6538  118.8512  3.74  1.0  C2 
A5  923131735  37.6492  118.8500  5.47  1.4  C2 
[8] In this paper, we also analyzed a small cluster of five earthquakes that occurred in 1992 (Table 1 and Figure 1). These events were recorded by a threecomponent set of very broad band Wilcoxon piezeoelectric accelerometers (0.05–200 Hz) in the same borehole. The depth of the instrument is almost the same (2046 m) as in the case of the 1997 data. The sampling rate is 500 Hz for these data.
[9] We first rotate the three component velocity seismograms to P (radial), SH (transverse), and SV wave directions, minimizing SH and SV wave energies before the S arrival. Then, using P and S wave windows of fixed lengths, we calculate the Fourier spectral amplitude for each wave. The window lengths are 0.4 s and 0.6 s for P and S waves, starting from P and S arrivals, respectively. The length of the window can be a possible source of uncertainty because the length determines how much of the coda waves are included in the energy estimation. However, as we show in Appendix A, the effect of coda waves is not significant in this analysis and various lengths of time window give almost identical results for this data set.
[10] After correcting instrumental response, we resample each spectral amplitude at equal intervals in log frequency at Δ log f = 0.05 and take a moving window average of length Δ log f = 0.3. By averaging we are able to estimate the standard deviation of the spectrum. The S wave amplitude spectrum is calculated as the vector summation of SH and SV spectra. Noise spectra are calculated using the same scheme on the same length of presignal record. We adjust the standard deviation based on the signaltonoise ratio. However, for data taken at 10,000 sps sampling rate the presignal length is too short to characterize the noise, and we made no adjustment for these data.
[11] In this study we used a spectral inversion method to estimate source parameters and a constant attenuation parameter. Our approach is similar to the method of Masuda and Suzuki [1982] and Anderson and Humphrey [1991]. We assume a simple omega square spectrum [Aki, 1967; Brune, 1970]. Following Boatwright [1978], we approximate the velocity amplitude spectrum as
where R^{c}, v^{c}, f_{c}^{c}, t^{c}, and Q^{c} are radiation pattern, wave velocity, corner frequency, travel time, and attenuation coefficients of wave type c (superscript), which may be either a P or S wave, and ρ, r, and M_{o} are density, hypocentral distance, and seismic moment, respectively. In this paper, we used v^{P} = 5.8 km/s, v^{S} = 3.3 km/s, ρ = 2700 kg/m^{3}. The hypocentral distances and travel times are measured from the SP time and these velocity values. For the radiation pattern, we used average values [Aki and Richards, 1980].
[12] Taking the logarithm of this equation,
The equations for the sampled frequencies comprise a linear inverse problem for M_{o} and 1/Q^{c} when f_{c}^{c} is fixed. We solve this problem for each f_{c}^{c}, and find f_{c}^{c} by a grid search that minimizes the residual
where σ_{i} is the standard deviation for each data point computed when we resampled the spectrum.
[13] Figure 2 shows examples of observed spectra and fitted omega square and constant Q model, together with the residual from equation (4). For some events, the minimum appeared at the edge of search area (Figure 2c). In these cases, we did not take that minimum and instead took the second minimum as the best estimate. Because of bandwidth limitations, some events have small residuals over a wide range of frequencies, suggesting that a reliable estimate of the corner frequency is difficult to define for these spectra (Figure 2c). Although the result is stable even for large events (Figure 2d), for the largest event (EV38) we could not obtain a reasonable solution, because the main frequency band of this event is far lower than the natural frequency of seismometer (10 Hz). Therefore we did not analyze this event further. To estimate the range of possible corner frequencies, we chose the upper and lower limits at which the residual increases to four times of data variance,
where N is the number of data points. This range corresponds approximately to twice the standard deviation of the corner frequency. The possible ranges for seismic moment and Q are estimated in the same manner. The spectrum of 40 events appear to be well explained by an omega square curve with a constant Q.
[14] Table 2 summarizes the estimated parameters. In Table 2, we also show radiated energy estimates for each spectrum. It is calculated using [Boatwright and Fletcher, 1984]
f_{0} and f_{1} are the lower and upper limits of integration. The upper limits are shown in Table 2.
Table 2. Source Parameter Determined Assuming Constant QID  M_{o}^{P}, N m  M_{o}^{S}, N m  f_{c}^{P}, Hz  f_{c}^{S}, Hz  Q^{P}  Q^{S}  E^{P} [f_{1}], J [Hz]  E^{S} [f_{1}], J [Hz]  Δσ_{B}, MPa  σ_{a}, MPa 

01  4.41e + 10  7.26e + 10  158  25  99  242  7.11e + 05 [281]  1.31e + 05 [281]  0.22  0.42 
02  2.26e + 11  2.44e + 11  15  12  1096  582  2.21e + 04 [177]  2.93e + 05 [177]  0.096  0.039 
03  1.41e + 11  5.00e + 11  15  7.9  447  389  3.80e + 04 [141]  1.49e + 05 [177]  0.037  0.017 
04  2.56e + 11  9.15e + 11  223  31  101  203  6.53e + 07 [223]  2.98e + 07 [281]  4.1  4.8 
05  1.31e + 11  9.01e + 10  28  25  276  293  2.61e + 04 [281]  4.16e + 05 [281]  0.41  0.12 
06  1.49e + 11  7.41e + 10  19  22  302  319  8.73e + 03 [281]  3.36e + 05 [281]  0.28  0.091 
07  2.56e + 11  3.00e + 11  28  22  166  195  1.66e + 05 [199]  2.26e + 06 [177]  0.70  0.26 
08  8.90e + 10  1.63e + 11  44  17  355  581  1.11e + 05 [158]  2.72e + 05 [199]  0.15  0.089 
09  8.42e + 10  7.05e + 11  63  15  338  586  4.56e + 06 [199]  1.98e + 06 [177]  0.31  0.49 
10  2.20e + 10  8.86e + 09  35  50  1311  407  1.04e + 03 [281]  8.98e + 04 [281]  0.46  0.17 
11  3.33e + 11  5.90e + 11  35  15  122  184  8.73e + 05 [199]  2.25e + 06 [177]  0.37  0.20 
12  9.11e + 12  4.25e + 12  10  12  169  230  4.21e + 06 [199]  2.35e + 08 [177]  2.7  1.1 
13  4.76e + 10  5.33e + 10  25  15  339  477  4.04e + 03 [199]  2.46e + 04 [199]  0.040  0.017 
14  5.14e + 10  6.27e + 10  35  17  526  677  1.35e + 04 [199]  4.49e + 04 [223]  0.066  0.030 
15  1.16e + 12  4.01e + 12  15  7.9  244  232  3.38e + 06 [199]  9.44e + 06 [158]  0.30  0.15 
16  5.59e + 10  1.47e + 11  39  15  260  332  6.16e + 04 [199]  1.15e + 05 [223]  0.081  0.051 
17  2.97e + 10  5.35e + 10  39  35  791  263  1.08e + 04 [281]  1.97e + 05 [281]  0.42  0.15 
18  6.69e + 12  2.60e + 13  12  7.1  147  158  6.46e + 07 [177]  2.50e + 08 [125]  1.4  0.57 
19  4.15e + 11  6.43e + 11  28  22  178  178  6.92e + 05 [199]  8.84e + 06 [177]  1.3  0.53 
20  1.00e + 12  1.83e + 12  19  12  179  210  1.81e + 06 [141]  1.15e + 07 [158]  0.58  0.28 
21  6.63e + 10  1.50e + 11  39  17  250  296  7.47e + 04 [199]  1.78e + 05 [199]  0.13  0.069 
22  4.76e + 09  2.51e + 10  50  25  Inf  895  2.60e + 03 [251]  1.02e + 04 [199]  0.055  0.025 
23  1.61e + 11  1.57e + 11  31  25  194  255  7.89e + 04 [199]  1.11e + 06 [223]  0.59  0.22 
24  2.82e + 11  4.95e + 11  35  17  212  257  7.63e + 05 [223]  2.94e + 06 [199]  0.45  0.28 
25  5.32e + 11  8.13e + 11  39  25  168  233  2.68e + 06 [251]  2.28e + 07 [199]  2.5  1.1 
26  1.43e + 11  7.61e + 10  15  35  867  488  4.15e + 03 [177]  2.86e + 06 [177]  1.1  0.77 
27  2.55e + 11  6.60e + 10  281  35  209  530  1.34e + 07 [281]  5.00e + 06 [177]  1.6  3.4 
28  1.04e + 10  1.83e + 10  89  56  211  249  1.03e + 04 [281]  1.10e + 05 [251]  0.60  0.25 
29  1.07e + 10  6.96e + 09  35  22  200  214  3.14e + 02 [199]  2.57e + 03 [199]  0.022  0.010 
30  4.54e + 10  2.40e + 10  39  44  2212  347  6.93e + 03 [177]  2.53e + 05 [251]  0.70  0.22 
31  9.05e + 10  5.81e + 10  12  10  497  1178  1.07e + 03 [158]  1.34e + 04 [199]  0.018  0.006 
32  2.29e + 13  4.23e + 13  7.9  5.6  928  523  5.15e + 07 [70]  4.54e + 08 [70]  1.4  0.46 
33  6.63e + 13  9.43e + 12  7.1  12  133  161  4.76e + 07 [70]  7.13e + 09 [70]  15.5  5.6 
34  3.02e + 14  1.39e + 14  2.5  2.5  158  382  8.80e + 07 [70]  2.03e + 09 [70]  0.81  0.28 
35  3.07e + 14  9.92e + 13  2.2  2.8  Inf  2941  4.85e + 07 [70]  2.43e + 09 [70]  1.1  0.36 
36  4.13e + 12  6.70e + 12  15  14  275  356  1.14e + 07 [223]  2.63e + 08 [223]  3.5  1.5 
37  8.27e + 13  2.98e + 13  8.9  8.9  219  292  2.25e + 08 [70]  5.56e + 09 [70]  9.4  3.0 
38  8.03e + 13  3.31e + 14  12  4.5  322  413  8.97e + 09 [199]  1.12e + 10 [141]  4.4  2.9 
39  9.35e + 12  5.49e + 12  4.0  6.3  660  467  3.17e + 05 [177]  3.12e + 07 [177]  0.44  0.12 
40  7.02e + 13  5.04e + 13  7.1  6.3  343  355  2.01e + 08 [177]  3.14e + 09 [125]  3.6  1.6 
41  1.27e + 14  1.05e + 13  6.3  14  297  316  1.10e + 08 [199]  5.04e + 10 [125]  44.6  21.6 
A1  3.35e + 12  3.00e + 12  17  12  119  189  6.51e + 06 [112]  4.99e + 07 [89]  1.3  0.52 
A2  1.29e + 10  1.55e + 10  50  70  240  161  2.01e + 03 [141]  1.22e + 05 [141]  1.2  0.26 
A3  7.88e + 10  8.81e + 10  50  14  138  4804  7.22e + 04 [141]  4.61e + 04 [100]  0.054  0.042 
A4  5.81e + 10  2.35e + 10  31  89  360  170  4.83e + 03 [141]  8.98e + 05 [100]  6.8  0.65 
A5  1.39e + 11  1.30e + 11  19  14  224  310  1.35e + 04 [112]  1.24e + 05 [141]  0.087  0.030 
[15] Usually the estimates of seismic moment from P and S, M_{o}^{P} and M_{o}^{S} respectively, are found to be slightly different. We calculate the average seismic moment, _{o}, as
Energy values are calculated using amplitude spectra whose lower limit is adjusted to this average seismic moment value and can be written
[16] Brune stress drop [Brune, 1970] Δσ_{B} and apparent stress [Wyss and Brune, 1968] σ_{a} are calculated using
where μ is the rigidity. In this expression, we include correction of seismic moment to account for radiation pattern difference. These values are also shown in Table 2.
[17] Figure 3 shows the relationship between seismic moment and corner frequency f_{c}, Brune stress drop Δσ_{B}, and apparent stress σ_{a}, respectively, together with the possible range of their values. The uncertainties of corner frequency and seismic moment are reflected in the large error bars. Considering the uncertainties, we can find some small events with stress drop and apparent stress that are quite small. For these events, corner frequencies are estimated to be smaller than expected from cube root scaling and the values for larger events. It is possible that some small events have large stress drop and apparent stress. However, based on these results, which are obtained with the assumption of constant Q, we would conclude that the lower limits of both the stress drop and apparent stress decrease as seismic moment decreases.
[18] This result is essentially the same as the previous result of Prejean and Ellsworth [2001]. The only significant difference is the values of Q; the values in this paper are systematically larger than those of Prejean and Ellsworth [2001] by a factor about 2. This arises from the difference of the shape of the assumed omega square model. We use a modified version [Boatwright, 1978] of the original omega square model [Brune, 1970], which was used by Prejean and Ellsworth [2001]. We carried out the same analysis for the original model, too. In fact, the residuals and all scaling relationships shown in Figure 3, do not differ significantly between the two models. The reason we used the modified version is that it tends not to create small residuals for a high corner frequency that sometimes results in ambiguity of corner frequency (Figure 2c). We would like to emphasize that the calculated value of Q depends on the model.
[19] Figure 4 shows that there is a strong relationship between stress drop and apparent stress. We find that the ratio of apparent stress to stress drop does not vary with seismic moment. For the omega square spectral model of Brune [1970], Singh and Ordaz [1994] showed that
If the observed spectra follow equation (1), then the corresponding relation is
We find values that are very close to this line. This result is not surprising because each spectrum is well represented by an omega square model and estimates for the apparent stress to Brune stress drop ratio far from this line would imply a different spectral shape. There are four outliers (events 1, 4, 9, 27). The estimates of P wave energy for these events are higher than those of S wave due to the high values of P wave corner frequency with large errors, and they are unlikely to be reliable. There are many events far from this line in the analysis of Prejean and Ellsworth [2001], but that appears to be an artifact of their treating Q estimates differently when calculating the corner frequency and the radiated seismic energy. Specifically, they used singleevent estimates of Q when estimating the stress drop and averaged values of Q when estimating apparent stress. In fact, our results indicate that stress drop scales with seismic moment in the same way that the apparent stress does. The same observation might also hold true for the results of Abercrombie [1995] when the missing events in her apparent stress estimates [Ide and Beroza, 2001] are included.
3. Spectral Ratio Analysis
 Top of page
 Abstract
 1. Introduction
 2. Constant Q Analysis
 3. Spectral Ratio Analysis
 4. Discussion
 5. Conclusions
 Appendix A:: Coda Waves and Length of Time Window
 Acknowledgments
 References
 Supporting Information
[20] Though it is frequently used, the validity of a constant Q model is not assured, especially at high frequencies where not much is known about attenuation. However, we can determine the source parameters without assuming path and site effects if we have a set of colocated events [e.g., Berckhemer, 1962; Bakun and Bufe, 1975; Mueller, 1985; Hough, 1997]. Using the multipleempirical Green's function (MEGF) method [Hough, 1997] with a slight modification about the assumption on source spectral shape and attenuation behavior, we determine the corner frequencies and relative seismic moments for a cluster of events from 1997 to 1998 located 7 km SE of the borehole and identified as “C1” in Table 1 and another set of events in 1992 (C2). The 10 events in C1 are 1 < M < 3 and the five events in C2 are 0.5 < M < 2.5. They show the basic characteristics of size dependencies of stress drop and apparent stress evident in Figure 3.
[21] The ratio between spectra of two events are expressed as
If the path is common to the two spectra, the right hand side of the above equation can be written as
and the unmodeled effects of attenuation are removed from the problem.
[22] When there are N points of frequency data for each spectra and M events, we have N * M(M − 1)/2 equations to determine M seismic moments and M corner frequencies. However, these equations do not supply information about the absolute values of seismic moments. We add another equation to equalize the logarithmic average of seismic moments to that determined by the previous constant Q analysis for events in the C1 cluster.
[23] These N * M(M − 1)/2 + 1 equations are solved by nonlinear inversion using the LevenbergMarquart method. In such a nonlinear inversion it can be difficult to find a global minimum solution; however, as shown in the constant Q analysis, the residual changes gradually with corner frequency (Figure 2). This means that the nonlinearity is weak and that our linearized method should readily converge to the global minimum. We solved the system from an extreme initial condition in which all seismic moments are the average moment value and all corner frequencies are 10 Hz.
[24] In our analysis we have 10 events and 45 spectral ratio curves for C1 and five events and 10 curves for C2, for both P and S waves. Some of these curves are shown in Figure 5 together with the best fitting model. Table 3 summarizes the estimated parameters and the values of stress drop calculated from these values using equations (9). Figure 5 suggests that the corner frequencies for EV13, EV21, and EVA2 may not be well resolved since the upper limit of analysis range is close and there is little flat part of spectral ratio. That the corner frequencies of these events are near the upper limit of our observation band means that we can only place a lower bound on the corner frequencies. Thus, if anything, the stress drops of these very small events are higher than we estimate.
Table 3. Source Parameter by Spectral Ratio AnalysisID  M_{o}^{P}, N m  M_{o}^{S}, N m  f_{c}^{P}, Hz  f_{c}^{S}, Hz  Δσ_{B}, MPa 

03  1.06e + 11  3.30e + 11  38  25  0.81 
12  4.34e + 12  3.93e + 12  16  22  10.4 
13  5.34e + 10  5.71e + 10  44  58  2.5 
15  1.12e + 12  3.07e + 12  22  13  1.1 
18  6.69e + 12  1.85e + 13  12  8.8  2.0 
19  4.79e + 11  8.09e + 11  31  24  2.1 
20  1.07e + 12  1.84e + 12  21  18  2.0 
21  9.37e + 10  1.92e + 11  56  37  1.7 
23  1.83e + 11  2.10e + 11  36  41  3.2 
24  3.69e + 11  6.43e + 11  41  28  2.6 
A1  2.24e + 12  2.18e + 12  21  17  2.6 
A2  2.08e + 10  2.14e + 10  95  55  0.83 
A3  8.75e + 10  8.57e + 10  55  48  2.3 
A4  6.84e + 10  3.47e + 10  87  67  3.7 
A5  9.81e + 10  8.95e + 10  40  34  0.87 
[25] Figure 6 compares the estimated corner frequencies and stress drops to the seismic moments. The corner frequencies follow cube root scaling with seismic moment and a 1 MPa stress drop (Figure 6b), while those of constant Q analysis tend to be small for small events (Figure 6a). The stress drops show almost no size dependence and are generally higher than 1 MPa (Figure 6d), which is substantially higher than the stress drops found from the constant Q analysis of the same events (Figure 6c). This difference arises from the difference in estimates of the corner frequencies (Figure 7). Estimated seismic moments from the two methods are consistent. For some events, the difference in corner frequency between the two analyses is more than a factor of two, which results in a difference of more than an order of magnitude in the stress drop.
4. Discussion
 Top of page
 Abstract
 1. Introduction
 2. Constant Q Analysis
 3. Spectral Ratio Analysis
 4. Discussion
 5. Conclusions
 Appendix A:: Coda Waves and Length of Time Window
 Acknowledgments
 References
 Supporting Information
[26] We have shown how two different methods for recovering source properties can give very different results. The cause of the difference may be a propagation effect that can affect source estimates even in the relatively clean recording environment of a deep borehole. Using the set of estimated parameters, we can investigate what attenuation behavior is required to explain the differing results. By taking the ratio between observed and calculated spectra, we can derive attenuation curves for P and S waves for two groups of events (Figure 8). Except for low frequencies where the noise is large, the curves all have similar forms in every figure. At frequencies between 30 and 100 Hz, these curves are well explained by a constant Q model of Q = 150 for all cases. However, we can see departure from these curves at higher and lower frequencies. The attenuation is not as high as predicted by Q = 150 over 100 Hz for the C1 group. The attenuation is also small around 8–10 Hz, so amplification is required to explain the bump. On the other hand, stronger attenuation exists at around 20 Hz, to particularly for the S waves.
[27] The behavior between 5 and 100 Hz is common for both groups, which suggests that this is not an instrumental effect. This kind of frequency dependent amplification/attenuation is not usually expected for deep borehole observation. One possible explanation is that the sensor is only about 25 m below the contact between the precaldera basement rocks and the overlying tuff. We cannot separate reflection or backscattering from this boundary from the direct arrival, so this effect might create a complex path/site effect. The difference of the behavior above 100 Hz might be an instrumental effect or arise from the difference in path between the two clusters. Since the location of the C2 cluster is just 1 km closer to the borehole and 1 km shallower than cluster C1, the path of C2 is not much different from that of C1. However, the wavelength is about 30–50 m at a frequency of 100 Hz, so the different locations could have an important effect.
[28] By assuming that the path and site effects are given by the logarithmic average of these attenuation curves, A^{c}(f), we can calculate the radiated energy and apparent stress (Figure 9, Table 4) using
and equation (8). In this calculation we added missing energy above 180 Hz by extrapolation of the omega square model [Ide and Beroza, 2001]. Again, there is little size dependence in apparent stress. In fact, the estimates for EV13, EV21, and EVA1 are the lower limit of apparent stress since the upper limit is close to the corner frequency. Nevertheless, these values are much higher than the estimates of constant Q analysis. The smallest value is 0.3 MPa, which is more than 10 times larger than the estimate from the constant Q analysis.
Table 4. Energy and Apparent Stress by Spectral Ratio AnalysisID  _{o}, N m  ^{P} + ^{S}, J  σ_{a}, MPa 

03  2.18e + 11  2.43e + 06  0.33 
12  4.14e + 12  6.04e + 08  4.3 
13  5.52e + 10  1.46e + 06  0.78 
15  2.10e + 12  3.56e + 07  0.50 
18  1.26e + 13  4.71e + 08  1.1 
19  6.44e + 11  2.95e + 07  1.3 
20  1.46e + 12  4.68e + 07  0.94 
21  1.43e + 11  3.38e + 06  0.69 
23  1.96e + 11  9.04e + 06  1.4 
24  5.06e + 11  2.50e + 07  1.5 
A1  2.21e + 12  1.04e + 08  1.4 
A2  2.11e + 10  2.26e + 05  0.31 
A3  8.66e + 10  1.94e + 06  0.66 
A4  5.16e + 10  1.37e + 06  0.78 
A5  9.38e + 10  1.15e + 06  0.36 
[29] Figure 10 illustrates why such a large difference in apparent stress arises using EV13, the event with the largest difference. Though Q = 150 is a good average of overall attenuation as shown in Figure 8, small values of Q produce anonymously large amplitude for frequencies higher than 100 Hz. Hence we get a solution at Q = 480 for constant Q analysis. When we use the average attenuation curves obtained by spectral ratio analysis, the modeled spectrum has a higher corner frequency. Since the energy is given as an integral over frequency, the discrepancy at high frequency is more significant. The difference shown in Figure 10 corresponds to a factor of 50 in seismic energy. For most events of the clusters C1 and C2, the estimated values of Q are larger than 150, and this is the explanation for overall underestimation of energy by constant Q analysis.
[30] The values of apparent stress are comparable to previous estimates for larger earthquakes [e.g., Kanamori et al., 1993; Mayeda and Walter, 1996]. Though there are no estimates of energy in the MEGF study of small events (0 < M < 1.5) by Hough et al. [1999], most events have stress drops larger than 1 MPa, which are consistent with our estimates since there is a strong relationship, σ_{a} = 0.33Δσ_{B} (Figure 4).
[31] Many previous studies have reported a size dependence of stress drop even after correction for constant Q [e.g., Masuda and Suzuki, 1982; Fletcher and Boatwright, 1991]. Such size dependence is observed for surface observations and is usually explained by attenuation, especially sitecontrolled f_{max} [Hanks, 1982]. Our results indicate that frequency dependent amplification and attenuation should be considered even for borehole observations.
[32] Seismograms for two of the events in the C1 cluster (IDs 20 and 23) appear in Figure 4 (displacement) and Figure 5 (velocity) of Prejean and Ellsworth [2001]. The recordings display welldeveloped coda waves in the wake of the body wave arrivals, indicating that the homogeneous whole space assumption contained in (1) and used in the spectral method is not entirely justified. Indeed, it would be more appropriate to include in (1) either a frequency dependent “path” amplitude factor or frequency dependent Q, or both. While these factors cancel in the MEGF method, we ultimately want to be able to interpret borehole seismograms in the time domain, for which new approaches may be required.