Comparison of direct and coda wave stress drop measurements for the Wells, Nevada, earthquake sequence

Authors

  • Rachel E. Abercrombie

    Corresponding author
    1. Department of Earth and Environment, Boston University, Boston, Massachusetts, USA
    • Corresponding author: Dr. Rachel E. Abercrombie, Department of Earth and Environment, Boston University, 685 Commonwealth A, Boston, MA 02215, USA. (rea@bu.edu)

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Abstract

[1] I calculate the corner frequencies and stress drops of the seven largest earthquakes (M4–6) in the Wells, Nevada, sequence (2008) using both coda and direct waves. I use spectral ratio, empirical Green's function (EGF), and methods to investigate whether differences and uncertainties in the analyses could affect the calculated source parameters. I find that the source spectral ratios from the coda and direct S waves are similar for the same pairs of earthquakes but that the source spectra and corresponding source parameters depend systematically on whether the earthquake of interest is the larger or smaller within the spectral ratio. Mayeda and Malagnini (2009a) calculated coda spectral ratios between the M6 main shock and six large aftershocks (M > 4), and performed a combined inversion to calculate source parameters. They found that the main shock had a higher stress drop than the large aftershocks. I model the identical spectral ratios individually in the manner that I use for direct waves. I find that the choice of source model and fitting procedure produces significant random scatter but no systematic biases. I calculate direct wave ratios for the same earthquake pairs and find higher variability but no systematic difference in the results. Lastly, I use M2.8–3.2 aftershocks as EGFs for the large aftershocks. These spectral ratios yield significantly higher corner frequencies and stress drops for the large aftershocks than when the same earthquakes are the denominator in ratios with the main shock. To improve the quality of EGF analyses, I propose strict objective criteria for acceptance of a spectral ratio fit: (1) observable source pulse, (2) the amplitude ratio of the low- to high-frequency limits of the fit is at least 5, and (3) the corner frequency only varies by 50% when the variance is within 5% of the minimum.

1 Introduction

[2] Stress drop scaling has been a controversial issue since first proposed by Aki [1967]. Are earthquakes self-similar with the stress-related growth of small earthquakes the same as large earthquakes from the rupture initiation? Alternatively, is there some breakdown or divergence from constant-stress drop scaling over some magnitude range? Assumptions based on scaling are fundamental to our understanding of earthquake rupture physics and also to seismic hazard and seismic monitoring. Unfortunately, in the decades since Brune [1970] and Madariaga [1976] proposed their simple circular source models for calculating source dimension and stress drop from frequency amplitude spectra, it has become clear that resolving the source process from attenuation and other propagation effects in real data is far from straightforward. Many studies have been published finding inconsistent results; for example, see Abercrombie and Rice [2005], Mayeda et al. [2005], and Kwiatek et al. [2011] for reviews. Measurement uncertainties essentially increase with frequency as propagation effects become stronger and less well known. Seismic moment is relatively well resolved as it is measured at relatively low frequencies. Stress drop depends on the cube of the corner frequency, and so it is less certain both because it is a higher frequency measurement and because of the effects of cubing the uncertainties. Seismic energy does not suffer from this problem, but it is an even higher frequency measurement and so is at least as problematic.

[3] The fundamental problem in resolving earthquake source scaling is that it requires comparing low- and high-frequency measurements, and differences could result from the source, attenuation, and site effects [e.g., Abercrombie and Leary, 1993; Ide et al., 2003], or data quality (e.g., bandwidth, Ide and Beroza [2001]; Viegas et al. [2010]). In addition, in any one study, large earthquakes radiating maximum energy nearest to the low-frequency bandwidth limit are compared to small ones radiating maximum energy near to the high-frequency bandwidth limit, making systematic fitting biases possible. Attempting to resolve the controversy is not helped by the fact that different studies use different approaches to calculate the spectra, extract and model the source, and then calculate the source parameters from the measurements. In this study, I focus on comparison of direct and coda wave measurements, on the source model used for fitting, and on whether there is any difference between measurements when an earthquake is the larger or smaller one in a spectral ratio.

[4] Two principal methods are used to extract source information from seismograms. One is to model the propagation effects or to perform a combined inversion for both source and propagation effects, for example, Prejean and Ellsworth [2001], Ide et al. [2003], Edwards et al. [2008], and Oth et al. [2011]. The former approach was followed by Abercrombie [1995], but in that instance the attenuation and site effects were minimized by the deep location of the borehole seismometer.

[5] A popular alternative is the spectral ratio or empirical Green's function (EGF) method in which two collocated earthquakes of different magnitudes are assumed to experience identical propagation and site effects and so can be used to cancel out these effects empirically. This method varies from a simple frequency spectral ratio [e.g., Izutani and Kanamori, 2001] to calculating source time functions [e.g., Mori and Frankel, 1990; Mori et al., 2003], inversions using clustered events [e.g., Ide et al., 2003], and stacking of many events with varying spacing [e.g., Shearer et al., 2006; Allmann and Shearer, 2007].

[6] In addition to these studies and many others that use direct P and S waves, a growing number of studies have used coda waves to extract earthquake source information. Although coda waves have experienced much more attenuation, they have been shown to provide stable, spatially averaged results [e.g., Mayeda et al., 2003; Mayeda et al., 2007], which is important given that the number of stations available in source studies of smaller earthquakes is often very limited.

[7] There has been relatively little work on quantifying the uncertainties in source parameter results. Most studies simply measure the inter-station variation but take no account of modeling assumptions, fitting constraints, limited-frequency bandwidth, etc. Others simply assume a factor of 10. Sonley and Abercrombie [2006] and Edwards et al. [2008] both demonstrated that there are significant trade-offs between attenuation and source effects when attempting to resolve both from recorded spectra. When using an EGF approach, the main question is how well the propagation effects cancel out. Mori and Frankel [1990] and Viegas et al. [2010] showed that the pulse width increases (and corner frequency decreases) by 30% or more as spacing between earthquakes exceeds 400 m. Kane et al. [2011] found that uncertainties from using different EGFs (~30% in stress drop) were larger than between stations for closely located stations. Prieto et al. [2006] investigated using multiple EGFs to increase the frequency bandwidth. For coda waves, Mayeda et al. [2007] found that the results were significantly less affected by location and focal mechanism differences between the pairs of earthquakes used to calculate spectral ratios. Prieto et al. [2007] investigated the uncertainties from simply calculating the amplitude spectrum.

[8] The EGF approach has been used to calculate source parameters of both the collocated earthquakes in a pair [e.g., Mayeda and Malagnini, 2009a, 2009b] and of groups of closely located events [e.g., Shearer et al., 2006; Allmann and Shearer, 2007]. Hough and Dreger [1995] and Viegas et al. [2010] compared the uncertainties in source parameters for the small and large earthquakes in pairs and found that those of the smaller earthquake are significantly less well resolved. This probably is partially because of limited signal at the higher frequencies needed to resolve the smaller source, but it also makes sense when the basis for the EGF assumption is considered. For earthquakes with collocated centroids (or hypocenters), each point of the small earthquake source is within one fault length of the large earthquake, but each point on the large earthquake source is not within one fault length of the small earthquake. Thus, it is possible that the source parameters of the smaller earthquake cannot be resolved well from spectral ratios.

[9] There are many different ways of calculating stress drop (Δσ), making it hard to compare studies directly. Almost all spectral source studies calculate a corner frequency, and so by using this measured parameter I can calculate a consistent stress drop for comparison. Stress drops calculated using the Brune [1970], Sato and Hirasawa [1973], and Madariaga [1976] models are based on the product of the seismic moment (M0) and the cube of the corner frequency (fc), and the Eshelby [1957] solution for a circular crack. They use different constants, but it is simple to account for the differences. Apparent stress, which is proportional to the ratio of energy and seismic moment, is calculated in two different ways: by integrating the velocity-squared source spectrum to estimate energy [e.g., Abercrombie, 1995] or by simply extrapolating from the corner frequency measurement using the relationships between moment, corner frequency, and energy for a model spectrum [e.g., Mayeda and Malagnini, 2009a, 2009b]. As long as the corner frequency values are calculated (and they usually are to apply model corrections for limited bandwidth [e.g., Ide and Beroza, 2001]) then it is possible to use them to obtain a corresponding stress drop to compare directly with the other studies. In this paper, I recalculate all mentioned stress drops (unless otherwise stated) using published corner frequency values and the Madariaga [1976] model, assuming Eshelby [1957]:

display math(1)

where k is 0.21 for S waves, and β is the S wave velocity.

[10] In this study, I calculate source parameters for aftershocks of the Wells, Nevada, earthquake using direct waves and compare them to published results using coda waves [Mayeda and Malagnini, 2009a]. I begin by refitting the coda wave spectral ratios using the same fitting approaches I use for direct waves to investigate fitting consistencies. I then calculate direct wave parameters using the same earthquake pairs as coda waves (main shock divided by each of six large aftershocks), and lastly I calculate direct wave parameters using smaller EGFs for the six large aftershocks. To quantify uncertainties, I develop further the approach of Viegas et al. [2010] to select only spectral ratios that meet clear objective criteria and then use a grid search around the preferred fit. I find that the source spectral ratios from the coda and direct waves are similar for the same pairs of earthquakes and that differences in spectral modeling approaches produce results within calculated uncertainties. I also find that the source spectra and corresponding source parameters depend systematically on whether the earthquake of interest is the larger or smaller within the spectral ratio.

2 Wells, Nevada, Earthquake

[11] A M6 earthquake occurred on 21 February 2008, NE of the town of Wells, Nevada. The normal faulting earthquake and its aftershock sequence were located within the Earthscope Transportable Array (TA) of seismometers and so were recorded with exceptional azimuthal coverage at regional distances (Figure 1). Mendoza and Hartzell [2009] used TA recordings of eight M ~ 4 aftershocks as empirical Green's functions to invert for the slip distribution of the main shock. They found it to be a relatively compact earthquake with a high stress drop compared to previous estimates for earthquakes in the region (40–45 MPa, again I note that I recalculate all stress drops using equation ((1)) to ease comparison). Mayeda and Malagnini [2009a] calculated coda wave spectral ratios of the main shock to the largest aftershocks (mostly normal faulting but two have strike-slip mechanisms; Herrmann et al. [2011]) and modeled them to obtain a very similar stress drop for the main shock. They also calculated the stress drops for the large aftershocks, which they used as EGFs, and obtained systematically smaller stress drops of 5–10 MPa. Baltay et al. [2010] used an alternative approach to calculating source spectra from coda waves and obtained similar stress drops of around 100–200 MPa for both the main shock and aftershocks.

Figure 1.

Map of Wells, Nevada, showing the location of the Wells main shock (star) and global centroid moment tensor (CMT) mechanism, the six large aftershocks (pale gray circles) and the small aftershocks (dark gray circles) used in the direct wave analysis, The solid triangles are the nearby TA stations used in the direct wave analysis, and the open triangles are the TA stations used in the coda analysis.

3 Coda Wave Analysis

[12] Mayeda and Malagnini [2009a, herein after MM09] modeled coda spectral ratios of the Wells main shock and aftershocks to estimate corner frequency and stress drop. They calculated mean spectral ratios of the main shock with six large (M > 4) aftershocks (Figure 1 and Table 1), following the procedure developed by Mayeda et al. [2007]. They used TA horizontal component recordings (40 samp/s) at up to 11 stations within 200 km (Figure 1) and both coda and Lg waves. The coda wave time windows were 150–300 s, and the minimum signal-to-noise ratio is 3 [Mayeda, personal communication, 2012]. The coda wave ratios exhibited much less azimuthal variation than the direct Lg waves, because they average out the radiation pattern and any directivity in the source. The mean ratios, however, are very similar, leading MM09 to conclude that they are able to correctly retrieve the source spectra from the coda waves. For direct waves, the EGF assumptions of colocation and similar mechanism are much more stringent than for coda waves, as noted by Mayeda et al. [2007], and the variability of the direct wave spectral ratios may result partly because they are not so well met for the Wells main shock and large aftershocks.

Table 1. Hypocentral Parameters of the Wells Earthquakes Used in This Studya
IDDate (in 2008)Hour:MinuteLatitude (°)Longitude (°)Depth (km)Magnitude
  1. aLocations and ML from UNR Catalogue, Mw from Saint Louis University earthquake Center moment tensor catalog (SLU), R. Herrmann.
MS21 Feb14:1641.16−114.888Mw 5.88
A121 Feb14:3441.00−114.795Mw 4.48
A222 Feb01:5041.14−114.929Mw 4.05
A322 Feb23:2741.10−114.9216Mw 4.41
A427 Feb07:5941.18−114.839Mw 4.28
A51 Apr13:1641.23−114.8411Mw 4.21
A622 Apr20:4041.22−114.818Mw 3.97
EGF129 Apr12:0741.11−114.899ML 2.9
EGF222 Feb03:5641.10−114.9111ML 2.8
EGF329 Feb13:0941.19−114.904ML 3.0
EGF425 Feb02:2241.17−114.828ML 2.9
EGF529 Feb09:3441.10−114.9010ML 3.2
EGF626 Feb20:4741.12−114.907ML 3.0

3.1 Joint Inversion of Coda Ratios—MM09

[13] MM09 modeled the spectral ratios of the main shock with each of the six large aftershocks, using the spectral source model [Brune 1970, Boatwright, 1980]:

display math(2)

where f is frequency, fc1 and fc2 are the corner frequencies of the large and small earthquakes, respectively, M01 and M02 are the seismic moments of the large and small earthquakes, respectively, n is the high-frequency falloff (they and I assume n = 2), and γ is a constant controlling the shape of the corner. MM09 used the original Brune [1970] model with a gradual corner in which γ = 1. MM09 also constrained fc1 to be the same in all six ratios, and fixed the moment ratio (M01/M02) using the seismic moments determined independently by Herrmann et al. [2011] using regional surface waves. They did this by adding the moment ratio as an extra data point in the spectral ratios at 0.01 Hz. The spectral ratios and fits are shown in Figure 2.

Figure 2.

Coda spectral ratios of the main shock with the six large aftershocks, from MM09. Each panel shows a separate pair. The data (solid lines and black circles) are shown twice, offset by a factor of 10 to compare fits using the Brune (equation ((2)), γ = 1) and sharper-cornered Boatwright (γ = 2) source models. Fit 1 includes the independent moment ratio (i), fit 2 constrains the moment ratio (ii), and fit 3 omits any moment constraint (iii). Fit 4 are the fits from MM09, both using γ = 1, and constraining all ratios to have the same fc for the main shock. Large and small corner frequencies are indicated by the diamonds (individual fits) and stars (MM09 fits).

[14] I use the simple but dynamic model of Madariaga [1976] and Eshelby [1957] to calculate stress drop (Δσ) from the moment and corner frequency measurements (equation ((1))). The stress drops are not the same as those published by MM09, but by using the same method to calculate all stress drops in this paper, I can compare the different methods directly. Converting to other models simply involves multiplication by a scalar [e.g., Abercrombie and Rice, 2005].

3.2 Individual Fitting of Coda Ratios

[15] To investigate how much variation there is in modeling the same data, I refit the coda spectral ratios calculated by MM09 following the methods used for the subsequent direct wave analyses. I use both the original Brune [1970] version of equation ((2)) (γ = 1) and the sharper-cornered version (γ = 2) proposed by Boatwright [1980]. I fit the six ratios independently and also investigate the effect of the long-period moment constraint. I fit the spectral ratios using the Nelder-Meade inversion in MATLABTM [Abercrombie and Rice, 2005]. I (i) include the moment-ratio data point but treat it as any data point, (ii) include the data point and constrain the inversion to have that amplitude ratio, and (iii) omit the moment-ratio data point and fit the coda spectra with no constraint on moment (Figure 2). The results of (i) and (ii) are essentially identical, and so only the results of (ii) and (iii) are included in Table 2. Following Viegas et al. [2010], I perform a grid search around the preferred corner frequency of the larger event (fc1) to determine the range of fc1 and fc2 in which the variance of the fit is within 5% of the minimum value.

Table 2. S Wave Corner Frequency Results From Fitting the Direct Wave and Coda Wave Ratiosa
 A/EGFMS/AMS/AMS/AMS/A
IDDirect WaveDirect WaveDirect Wave M0 fixCoda Wave (no M0 point, (iii))Coda Wave M0 fix (ii)c
  1. aThe main shock values Are the mean and 1 standard deviation over all EGF pairs. The fitting uncertainties Are smaller than the inter-EGF variation.
Brune spectral model (γ = 1)
MS 0.53 ± 0.080.45 ± 0.140.30 ± 0.070.33 ± 0.04
A12.5−0.4/+0.51.5−0.2/+0.31.0−0.2/+0.31.0−0.1/+0.041.1−0.1/+0.1
A24.4−0.5/+0.62.4−0.3/+0.31.4−0.2/+0.31.6−0.03/+0.031.6−0.03/+0.03
A33.6−0.4/+0.52.3−0.4/+0.50.47−0.08/+0.081.0−0.07/+0.011.0−0.04/+0.05
A43.8−0.6/+0.72.3−0.3/+0.41.5−0.2/+0.21.8−0.1/+0.11.9−0.1/+0.1
A52.8−0.4/+0.61.4−0.1/+0.11.0−0.2/+0.21.3−0.1/+0.11.4−0.1/+0.1
A63.9−0.7/+0.93.0−0.4/+0.52.0−0.3/+0.32.3−0.1/+0.22.1−0.1/+0.1
Boatwright Spectral Model (γ = 2)
MS -0.34 ± 0.100.41 ± 0.090.32 ± 0.04
A12.4−0.3/+0.3-0.91−0.1/+0.21.2−0.1/+0.11.1−0.1/+0.1
A24.1−0.4/+0.4-1.2−0.2/+0.21.7−0.1/+0.11.5−0.1/+0.1
A33.4−0.3/+0.3-1.4−0.3/+0.40.97−0.1/+0.20.94−0.1/+0.1
A45.0−0.5/+0.6-1.2−0.2/+0.21.3−0.1/+0.11.7−0.1/+0.1
A52.8−0.3/+0.3-0.87−0.1/+0.11.2−0.1/+0.11.2−0.1/+0.1
A63.4−0.3/+0.3-1.6−0.2/+0.22.2−0.1/+0.11.9−0.1/+0.2

[16] Figure 2 shows that there is very little variation between the different fitting approaches, which is encouraging. The uncertainties in corner frequency using the different methods all overlap with one another and those of MM09 (Figure 2 and Table 2). The largest differences in fc1 are for the individual fits with no moment-ratio constraint, which is reasonable given how close the corner frequency is to the lower limit of the bandwidth. With no moment-ratio information, fc1 is systematically higher when using the sharper-cornered source model, as is expected given the limited-frequency bandwidth. For these spectral ratios, the variances are slightly smaller for the Brune [1970] spectral model than for the sharper-cornered fit. I conclude that the fitting method is not a significant source of systematic uncertainty, unless either corner frequency is very close to the frequency band limits.

4 Direct Wave Ratios

[17] I perform two separate analyses using the direct S waves recorded on the horizontal components of four of the nearest stations of the TA (40 samp/s), with a good azimuthal distribution (Figure 1). The nearest stations are typically preferred for direct wave analysis because these recordings have the highest signal amplitude, widest signal frequency bandwidth (especially important for small earthquakes), and also have experienced the least attenuation. I begin by calculating the spectral ratio of the main shock to the same six large aftershocks as before to form the same earthquake ratios used in the coda analysis. These large aftershocks are not optimal for direct wave analysis of the main shock because they only loosely meet the criteria for well-constrained EGF analysis. The aftershock waveforms are not highly correlated with those of the main shock; nor are they closely collocated. They are, however, sufficiently good to resolve directivity and a slip distribution for the main shock, and most are within about one fault length of the main shock [Mendoza and Hartzell, 2009] and so are of similar quality to those used in other studies.

[18] Although the aftershocks are within one fault length of the main shock and so are consistent with the EGF assumptions at frequencies and wavelengths corresponding to the main shock source, all the radiation from the main shock is not within one fault length of each aftershock. Hence, it is more doubtful whether the EGF assumptions hold at the frequencies and wavelengths corresponding to the size and duration of the large aftershocks and whether the spectral ratios of the main shock to the large aftershocks can be used to resolve reliable parameters for the large aftershocks. My preferred approach for direct wave EGF analysis of such earthquakes as the six large aftershocks is to find smaller EGF events for each one, which meet strong selection criteria implying that the EGF assumptions are reliable at the frequencies and wavelengths corresponding to the size and duration of the larger events. I then determine the source parameters of the large aftershocks from these ratios. I select M ~ 3 earthquakes to make EGFs for the six large aftershocks and use these to calculate spectral ratios.

4.1 Main Shock and Large Aftershocks

[19] I calculate the direct wave spectral ratios between the main shock and the six large aftershocks used in the coda analysis by MM09. I select 6 s time windows around the S wave arrival at TA stations M11, M13, and N12; I could not use the nearest station, M12, as it was clipped for the main shock S waves (Figure 1). I use both horizontal recordings at each station, and the windows start 5% before the picked S wave arrival. I calculate the complex spectra and the ratios between the main shock and the large aftershocks using the multitaper method developed by Prieto et al. [2009] and used by Viegas et al. [2010]. This approach produces the most reliable amplitude spectra [e.g., Prieto et al., 2009; Park et al., 1987], and the complex division enables me to extract the relative source time function of the main shock from each ratio. Although I do not make measurements on the source time function, deconvolution of a clear source pulse confirms the EGF assumptions, since the approximation is good enough to work in phase as well as amplitude. The cross-correlations of the main shock and aftershock seismograms are relatively low (between 0.13 and 0.39), and the relative source time functions are noisy (Figures 3 and 4). This suggests that these aftershocks are not ideal EGFs for the main shock when using direct waves and lack the low-frequency signal needed to resolve the large earthquake source time function.

Figure 3.

Direct wave analysis: Main shock with large aftershock A4, at N12, BHN: (a) Raw S waveforms (normalized) of the main shock (black) and A4 (gray); (b) spectra of S waves (solid) and of preceding noise window (dotted); (c) S wave spectral ratio with fits using the Brune model and moment ratio fixed, within variance 5% of the minimum (gray shades); (d) relative source time function, no filtering; (e) variance increase compared to minimum for varying large earthquake corner frequency; and (f) relative moment (diamonds) and small earthquake corner frequency (stars) for fits at varying large earthquake corner frequency. The waveform cross-correlation is only 0.3, and the source time function is not well resolved. The variance curve is a tight parabola, implying good constraints on the fits to the corner frequency of the large earthquake even though it is outside the signal bandwidth of the data, given the fixed moment ratio.

Figure 4.

Direct wave analysis: Main shock with large aftershock A5, at N12, BHN: See Figure 3 for details. Note that the moment ratio is fixed in the fitting and that the main shock corner frequency is out of the range of the data. The waveform cross-correlation is only 0.3, and the source time function is not well resolved, partly because of the lack of low-frequency signal, as in Figure 3.

[20] I model these spectral ratios using the same approach as for the coda ratios. First I log-sample the spectra in frequency so as to decrease the weighting towards the higher frequency part of the spectra. I use only the frequency range in which the signal is at least 3 times the pre-P-wave noise level. I use both the simple Brune [1970] and the sharper-cornered Boatwright [1980] source models (equation ((2))). I also fit with both the moment ratio of the two earthquakes free or fixed to the independent values used by MM09. Following Viegas et al. [2010], in each case I investigate how well resolved the models are by performing a grid search over the large earthquake corner frequency to find the range within which the variance does not increase more than 5% (Figures 3 and 4).

[21] To calculate the mean values per earthquake over all horizontal components at all stations, I use further selection criteria to use only the highest quality measurements. Following Viegas et al. [2010], I require the variance to have a parabola shape with a clear minimum (≤0.005) at the preferred corner frequency. I then impose two further limits. The first is to use only measurements where the large earthquake corner frequency minimum is relatively tight (variance increases to 5% within 50% of the corner frequency measurement, fc1_err ≤ 0.5). The second is to ensure that the fits are not affected by bumps and irregularity in the spectral ratios but that I really am observing the corner frequencies. I use only measurements where the difference in amplitude of the high and low-frequency levels in the fit is greater than 5 (fit_amp_ratio ≥ 5). In addition, I only include measurements if the small earthquake corner frequency (that of the large aftershock) is also well constrained (variance increases to 5% within 100% of the small earthquake corner frequency measurement, fc2_err ≤1). The constraint on fc1 is looser than on fc1 because the higher frequency fc is typically less well constrained [e.g., Viegas et al., 2010]. Although these quality criteria are objective, the choice of cutoff value is simply a balance between quantity and quality of results. I select the values to maximize the azimuthal distribution without including obviously poorly resolved or outlying readings; both included and less well-constrained fits are included in Figure 5.

Figure 5.

Comparison of coda and direct wave spectral ratios of the main shock with the six large aftershocks. Thicker lines are the stations and components that meet the selection criteria (fit_amp_ratio ≥ 5, fc1_err ≤ 0.5, fc2_err ≤ 1, variance ≤ 0.005), and thinner lines are those that do not. Gray shades represent different stations and components. Solid lines are data, and dashed lines are best fits assuming a moment constraint and a Brune [1970] source spectral shape. The thick black solid line is the MM09 fit to the coda wave ratios, and the black dotted line is the fit calculated using the same moment ratio and main shock corner frequency and the large aftershock corner frequency from the direct wave fits using small earthquake EGFs.

[22] Comparing these direct wave ratios with the coda spectral ratios confirms the results of MM09 that the direct wave ratios scatter about the coda ratios. The relatively poor performance of the large aftershocks as EGFs for the main shock, demonstrated by their low cross-correlations and poor source time functions, explains some of the scatter. Only the Brune [1970] source spectrum fits are shown in Figure 5, but the results for both corner shapes are included in Table 2. The corner frequency of the main shock is so close to the low-frequency bandwidth limit that the results with the moment ratio fixed are preferred. In fact, it was not possible to obtain well-constrained fits using the sharper-cornered spectral model without a moment constraint. With the moment ratio constrained, the results from the direct wave spectral ratios are within the uncertainties of the values obtained from the coda wave ratios. So, although the direct wave ratios are far from ideal, this comparison of coda to direct waves again supports the use of coda waves to obtain source information.

4.2 Large Aftershocks and Smaller EGF

[23] I search the earthquake catalog for earthquakes of 2.8 ≤ ML ≤ 3.3 within 3 km of each of the six large aftershocks and find 52 candidates. The best EGFs are about 1 order of magnitude smaller than the earthquake of interest. Larger and they cannot be considered a delta function, smaller and the signal is only above noise for too small a frequency range. I use a shorter time window of 3 s, with the S wave arrivals at the 5% point. I calculate spectral ratios and source time functions using all 52 EGFs for all six large aftershocks at the four stations (M11, M12, M13, and N12) and select one EGF for each large aftershock, having a clear source time function and a relatively high cross-correlation (Table 1 and Figures 6 and 7). Cross-correlation alone cannot be used to determine the best EGF because the highest values are for earthquakes of similar magnitude, whereas the EGF approach requires a difference in magnitude and frequency content in the two events. The spectral ratios for all the EGFs that produced source time functions were very similar. I log-sample the spectral ratios within the frequency range where the signal is at least three times the noise level and fit the ratios using both the original Brune and the sharper-cornered Boatwright source models. In all models, the moment ratio is a free parameter as independent estimates of the seismic moments of the small EGF earthquakes are not available (Figures 6 and 7).

Figure 6.

Direct wave analysis: Large aftershock A4 with EGF4, at N12, BHN. The waveforms have a maximum cross-correlation of 0.6, the source time function is clearly resolved, and the variance curve is a tight parabola, showing good constraint on the fits to the corner frequency of the large earthquake. The corner frequency of the small earthquake is much less well resolved and is not considered reliable.

Figure 7.

Direct wave analysis: Large aftershock A5 with EGF 5, at N12, BHN. See Figure 3 for details. Maximum cross-correlation is 0.3, but the source time function is well resolved.

[24] I use the same criteria to select the higher-quality measurements as for the main shock/aftershock direct-wave ratios, namely, a clear minimum in variance for the corner frequency, an amplitude ratio of greater than 5 between high and low levels of the spectral ratio fit, and corner frequency only varying by 50% with variance within 5% of the minimum (Figure 8). I do not impose a constraint on the uncertainty in the small earthquake corner frequency because I do not use it. These small event corner frequencies are typically near or outside the frequency range and poorly resolved. The parameters of the large earthquake in the ratio can be well resolved, even if the small earthquake corner frequency is out of the frequency bandwidth [e.g., Viegas et al. 2010].

Figure 8.

Direct wave (a) spectral ratios and (b) source time functions of six the large aftershocks with smaller EGFs. Thicker lines are the stations and components that meet the selection criteria (fit_amp_ratio ≥ 5, fc1_err ≤ 0.5, variance ≤0.005), and thinner lines are those that do not. Gray shades represent different stations and components. Solid lines are data, and dashed lines are best fits assuming no moment constraints and a Brune [1970] source spectral shape. The thick black dotted line is the average best fit to the direct wave ratios of the large aftershocks with smaller EGFs. The thick solid black line is the fit calculated using the same moment ratio, the large aftershock corner frequency from MM09, and an average small earthquake corner frequency of 10 Hz. The poorer quality source time functions are offset for clarity.

[25] The spectral ratios and source time functions shown in Figure 8 reveal that the EGF assumption is working well for these earthquake pairs. The range of moment ratios is significantly smaller over the different stations and components (Figure 8b) than it is for the main shock to large aftershock ratios (Figure 5), demonstrating the difference in EGF quality. The spectral shapes are also very similar, suggesting that none of these large aftershocks exhibit strong directivity.

[26] Figure 8 also compares the results of MM09 to the spectral ratios of the large aftershocks to the smaller EGF. I use the corner frequencies determined for the large aftershocks by MM09 and combine them with a smaller earthquake corner frequency of 10 Hz (an average of the direct wave values) and the mean direct wave moment ratio to calculate mean spectral models. The corner frequencies of the large aftershocks, calculated when they are the smaller earthquake in the ratio, are systematically smaller than when they are the large earthquake in the ratios. Since the direct wave ratios of the main shock and large aftershocks are in good agreement with the coda wave results, this does not appear to be a difference between the coda and direct wave approaches. I think it most likely to result from the nature of the EGF assumptions as a function of frequency. For earthquakes with collocated centroids (or hypocenters), each point of the small earthquake source is within one fault length of the large earthquake, but each point on the large earthquake source is not within one fault length of the small earthquake. Therefore, although the smaller earthquake can be considered a reliable EGF at frequencies and wavelengths corresponding to the source size and duration of the large earthquake, the reverse is not true. Thus, it is possible that the source parameters of the smaller earthquake cannot be resolved well from spectral ratios, and the poorer resolution leads directly to an overestimate of the source size and duration.

5 Source Parameter Results and Discussion

[27] The resulting measurements of corner frequency, seismic moment, and stress drop for the Wells main shock and the six large aftershocks from all the different data and analyses are shown in Figure 9. I also add the results of Baltay et al. [2010] for the same earthquakes. They use an alternative approach to Mayeda et al. [2007] and MM09 to calculate the source spectra from coda waves. Essentially, they correct the spectra by assuming that the smallest earthquake in their cluster can be considered a delta function. Their approach is thus closest to my fitting of the ratio of large aftershocks to smaller EGFs. There is a lot of scatter, but two principal observations are clear.

  1. [28] The corner frequencies and stress drops for the large aftershocks from the different approaches do not lie within the calculated uncertainties of the other methods. The stress drops of the large aftershocks from the different approaches are not highly correlated implying that interevent differences of this magnitude are not resolvable.

  2. [29] The corner frequencies and stress drops from the two methods in which the large aftershocks are the large events in the ratios, are higher than the methods which ratio these aftershocks to the larger mainshock. This suggests that there may be a tendency when fitting a real spectral ratio either to overestimate the lower frequency corner, underestimate the higher frequency corner, or both.

Figure 9.

Stress drops and corner frequencies from (a) the Brune model and (b) the sharper-cornered Boatwright model for spectral ratios modeled in this work. The SLU (Saint Louis University earthquake Center moment tensor catalog) moments are used for all earthquakes. The solid symbols are the event means, and the open smaller symbols are the individual measurements. The error bars for the individual measurements are minimum and maximum values within 5% of the minimum variance. The error bars of the event means are the mean of the individual measurement maximum and minimum values. Also shown are the results of MM09 and Baltay et al. [2010] for comparison. The direct wave results for the main shock are offset in moment for plotting clarity. Note that the preferred direct wave corner frequencies are higher than those from the coda waves for the large aftershocks, but when the large aftershocks are the denominator in the ratios, the direct wave measurements are smaller. All stress drops are calculated using the Madariaga [1976] relationship between corner frequency and earthquake source radius. To convert to the Brune model, divide stress drop values by 5.5.

[30] The different approaches to fitting the coda spectral ratios do have overlapping uncertainties implying that the fitting approach does contribute to the uncertainties but is relatively minor and not systematic. The exception to this is when the data quality is poor and the corner frequency is close to the high- or low-frequency limit. On the whole, the original Brune [1970] source model fits the spectral ratios better than the sharper-cornered Boatwright, 1980 model. Perhaps the fact that the latter model was needed for the unusually unattenuated Cajon Pass borehole recordings [Abercrombie 1995] reflects how well the attenuation effects are removed.

[31] Comparing the coda wave and direct wave results from spectral ratios of the same pairs of earthquakes (main shock to large aftershocks) reveals no systematic differences. The corner frequencies and stress drops from the direct waves tend to be higher when the moment ratio is free and lower when it is fixed. The range in moment ratios in the spectral ratios (Figure 5) implies differences in radiation pattern between the event pairs, making it unclear whether the moment-ratio constraint is a good one. Unfortunately, many of the ratios do not have sufficient low-frequency signal to constrain the long-period level of the fit without a moment-ratio constraint. There is no clear dependence of any parameter on the epicentral difference between the event pairs.

[32] In this study, I have developed objective criteria for selecting spectral ratios for use in calculating source parameters. Any pair of earthquakes will produce a spectral ratio that can be modeled using equation ((2)), but whether the results are of any value depends on how well the EGF assumptions are really met. Following Viegas et al. [2010], I propose the calculation of a source time function as a test of the validity of the EGF assumption. In addition, I require that the ratio of the long-period level of the fit to the high-frequency level exceeds a limit (I use 5). This ensures that the modeling is not fitting random fluctuations in the spectra. I then perform a grid search around the preferred corner frequency of the larger event. I discard results for which there is not a well-constrained minimum. I require there to be a 5% increase in variance if the corner frequency is altered by more than 25% of the minimum value. These criteria are objective, but the actual values used are constrained by the balance between quality and quantity of the observations. Limiting the number of observations too severely leads to the possibility of bias by losing any azimuthal distribution.

[33] The results of this study imply that observations of scaling breakdown from comparing earthquakes that are the larger ones in spectral ratios with those that are the smaller may not be reliable. This clearly affects the results of Mayeda et al. [2007], Mayeda and Malagnini [2009a, 2009b] etc., but is also relevant to direct waves. Izutani and Kanamori [2001] and Izutani [2005] calculated source parameters for large and small earthquakes in direct wave spectral ratios and found a breakdown in scaling. This could also be the result of the same effect observed here. Viegas et al. [2010] found that the uncertainties in the higher frequency corner were a factor of ten higher than those for the lower frequency corner for the two earthquakes for which they had both larger and smaller collocated earthquakes. They preferred using only corner frequencies calculated for the larger earthquake in the ratio. This was also the corner frequency nearer the center of the frequency bandwidth.

[34] The question remains as to what differences in stress drop are resolvable if a consistent method is used and what quality of data is required. This is clearly an important question and requires method comparison over much larger numbers of earthquakes.

6 Conclusions

[35] This comparative analysis of different methods of obtaining source parameters demonstrates the following:

  1. [36] The systematic and random differences between different methods of obtaining source parameters are larger than the published uncertainties.

  2. [37] Corner frequencies determined for an earthquake that is the smaller in a spectral ratio are systematically lower (and typically less well constrained) than when the same earthquake is the larger in the ratio.

  3. [38] Improved quantification of uncertainties and comparison of methods is needed to improve our resolution of earthquake source parameters and scaling.

Acknowledgments

[39] I am grateful to Kevin Mayeda for providing the coda spectral ratios and answering many questions about his analysis. Reviews and comments by J. Mori, the Associate Editor, and two anonymous reviewers significantly improved the manuscript. Data from the TA network were made freely available as part of the EarthScope USArray facility supported by the National Science Foundation, Major Research Facility program under Cooperative Agreement EAR-0350030. This material is based on research sponsored by the Air Force Research Laboratory under agreement number FA8718-06-C-0024. The views and conclusions contained herein are those of the author and should not be interpreted as necessarily representing the official policies or endorsements, either expressed or implied, of the Air Force Research Laboratory or the U.S. Government. Maps were drawn using the Generic Mapping Tools [Wessel and Smith, 1998].