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Keywords:

  • apparent stress;
  • corner-frequency;
  • scaling

Abstract

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Coda Amplitude Ratios
  5. 3. Discussion and Conclusion
  6. Acknowledgments
  7. References
  8. Supporting Information

[1] Apparent stress and corner frequencies are measured for the Chi-Chi, Taiwan sequence beginning with the Mw 7.6 mainshock on 20 September 1999. Using the coda source ratio method, we obtained stable source ratio estimates using broadband stations on Taiwan. We find the following: (1) For the mainshock and 3 of the larger aftershocks, apparent stress is clustered around 0.8 MPa (±0.1 MPa); (2) Events below ∼Mw 5.5 exhibit lower apparent stress with larger scatter, ranging between ∼0.08 and 0.8 MPa and are spatially variable; 3) The Brune [1970a, 1970b] omega-square source model fits the spectral shape for events 4.7 < Mw < 7.6, however a step-wise break in self-similarity exists at ∼Mw 5.5. We hypothesize that larger events are subject to the regional state-of-stress, whereas smaller aftershocks are sensitive to the local state-of-stress from stress-field redistribution following the mainshock and/or fault zone lubrication that affects only larger events.

1. Introduction

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Coda Amplitude Ratios
  5. 3. Discussion and Conclusion
  6. Acknowledgments
  7. References
  8. Supporting Information

[2] For many years and particularly over the past decade, earthquake source scaling studies have been undertaken in a variety of tectonic settings and in some cases, on the same data sets using different methodologies. More often than not, there is usually no clear consensus on scaling behavior and hence the underlying dynamics of faulting for that particular region. In addition to the academic pursuit of earthquake physics, there are the societal applications of differentiating explosions from natural earthquakes, predicting strong ground motion, and hazard mitigation. Though seismic network quality and station coverage has improved, and significant advancements have been gained in fault rupture models and high-resolution 3-D structure, the accurate prediction of strong ground motion requires a priori knowledge of the regional source scaling [e.g., Bodin et al., 2001; Akinci et al., 2001]. A recent study by Malagnini et al. [2008] demonstrates the necessity of inclusion of non-self similar scaling in order to accurately predict strong ground motion for the Colfiorito earthquake on September 26, 1997 (Mw 6) in the central Apennines, Italy.

[3] Nonetheless, remaining questions still exist such as: What is the current state-of-the-art?; Are systematic variations region-dependent?; Is the observed scatter real and, if so, what are the causes?; Can spatial variations in apparent stress be reliably mapped?; Are slow earthquakes, super-shear earthquakes, and typical sub-shear tectonic earthquakes three different phenomena or part of a continuous spectrum of earthquake rupture behavior?

[4] Aside from long-period regional and teleseismic waveform modeling estimates for Mw, broadband studies that extrapolate back to the source have always been hampered by inadequate knowledge of scale-dependent path and site effects. Radiated energy and corner frequency estimation over a broad range of event sizes requires significant frequency-dependent corrections and assumptions, resulting in large error that makes interpretation of the results highly questionable. There are however a number of local and regional methods that circumvent the problem of path and site corrections, namely the empirical Green's function deconvolution method [e.g., Hough, 1997] and amplitude ratio techniques [e.g., Izutani, 2005]. These approaches have gained popularity because adjacent or co-located events recorded at common stations have shared path and site effects, which therefore cancel. In this paper we use a variant of the direct wave amplitude ratio, the coda ratio methodology of Mayeda et al. [2007] which has been shown to be roughly 3 times more stable than direct wave ratios and can be used with event pairs that are separated by several tens of kilometers, with little ill effects on amplitude ratio scatter. More recently, the method has been extended to datasets in Italy such as the San Giuliano di Puglia sequence in October–November 2002 [Malagnini and Mayeda, 2008] and Colfiorito sequence of 1997 and 1998 [Malagnini et al., 2008].

[5] Over the past decade a number of studies have either suggested that earthquake scaling is constant with high scatter [e.g., Ide and Beroza, 2001; Prieto et al., 2006] or that scaling increases as a function of magnitude [e.g., Kanamori et al., 1993; Mayeda and Walter, 1996; Izutani, 2005]. However, due to large errors and regional variations, it has been difficult to definitively tell which of these two ideas is correct. Knowledge of scaling, whether self-similar or not, is fundamental to earthquake rupture simulations, seismic hazard prediction, and seismic discrimination, especially if there exists region-dependent variations in apparent stress.

2. Coda Amplitude Ratios

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Coda Amplitude Ratios
  5. 3. Discussion and Conclusion
  6. Acknowledgments
  7. References
  8. Supporting Information

[6] For this study we selected 18 events ranging between Mw 4.7 and 7.6 recorded by 9 stations of the Broadband Array in Taiwan (BATS) (Figure 1). We applied the coda source methodology outlined by Mayeda et al. [2003] to compute stable Mw's and validated these results against independent estimates from the Harvard CMT catalog for the largest events and selected Mw's for the moderate-sized events from regional waveform modeling listed by Kao and Angelier [2001]. The coda ratio methodology is outlined by Mayeda et al. [2007] so we only give a brief processing description here. First, narrowband time-domain envelopes ranging between 0.03 and 8.0-Hz were made using the two horizontal components and log-averaged for additional stability and smoothed. Coda synthetic envelopes were then fit to the data for each station so that relative amplitudes could be measured using an L-1 fitting routine for each narrowband envelope [see Mayeda et al., 2003], then ratios were formed for all possible event pairs by subtracting the log10 amplitudes for each station that recorded the event pair. The end of the coda window was based upon signal-to-noise ratio as well goodness of fit between the synthetic and the observed envelope. For each frequency band we also required that we had a minimum of 3 stations recording the event pairs where the maximum event spacing was limited to ∼60 km. We note that the vast majority of the events were within 30 km distance and we normally had a minimum of 6 stations recording the event pairs. We visually checked each frequency-dependent envelope and those that were prematurely short due to aftershocks in the coda or had dropouts were rejected. In total, this left us with 18 suitable ratios that had a minimum of roughly 6 stations to average over.

image

Figure 1. Map of Chi-Chi aftershocks that were used in this study. Symbol size is proportional to moment magnitude, Mw, and shading is proportional to the log10[Apparent stress (Pa)].

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[7] Assuming a simple single corner frequency source model [Aki, 1967; Brune, 1970a, 1970b], the ratio of the moment-rate functions for two events (1 and 2) is given by,

  • equation image

where M0 is the seismic moment and ωc is the angular corner frequency (2πfc) and p is the high frequency decay rate. At the low frequency limit the source ratio shown in equation (1) is proportional to the ratio of the seismic moments image whereas at the high frequency limit, equation (1) is asymptotic to imageequation image under self-similarity. If we follow the usual Brune [1970a, 1970b] omega-square model and set p = 2, the exponent of the high-frequency ratio becomes 1/3. However, it has been proposed by Kanamori and Rivera [2004] that the scaling between moment and corner frequency could take on the form,

  • equation image

where ɛ represents the deviation from self-similarity and is usually thought to be a small positive number. For example, Walter et al. [2006] and Mayeda et al. [2005] found ɛ to be close to 0.5 for the Hector Mine mainshock and its aftershocks using independent spectral methods. Mayeda et al. [2007] used the source spectrum portion of the Magnitude Distance Amplitude Correction (MDAC) methodology of Walter and Taylor [2001], which allows for the variation of the corner frequency that does not have to be self-similar. For example,

  • equation image

where σa is the apparent stress [Wyss, 1970], σa and M0 are the apparent stress and seismic moment of the reference event, ψ is a scaling parameter, k is a constant related to the P and S- wave velocities at the source (αS, βS), radiation pattern coefficient for P and S waves (RθϕP and RθϕS), and scale factor (ζ) which is the ratio of the wave velocities. For the case of constant apparent stress, ψ = 0 and ɛ = 0 in the previous equations, however, Mayeda and Walter [1996] found ψ = 0.25 for moderate-to-large earthquakes in the western United States and Mayeda et al. [2007] found ψ = 0.25 for the Hector Mine sequence using the coda ratio methodology. By using the corner frequency defined in equation (3) into equation (1), we can apply a grid search to find the parameters that best fit the individual spectral ratio data. We note however, the decay parameter p must be greater than 1.5 to keep the energy finite [e.g., Walter and Brune, 1993]. If the high-frequency decay value p were close to 1.5 it would be possible to nearly match the spectral ratios observed for Hector Mine, however, given that such shallow falloff is not consistent with most earthquake observations [e.g., Hough, 2001] and independent methods [e.g., Mayeda et al., 2005; Walter et al., 2006], the preferred interpretation of Mayeda et al. [2007] was that the apparent stresses were systematically lower for the aftershocks than the mainshock.

[8] Using as many as 9 stations, we formed the average spectral ratio for all Chi-Chi event pairs so long as the Mw difference was at least 1.0. Next, we grid-searched using equations (1) and (3) assuming that the reference moment corresponded to an Mw 5.0 event and the reference apparent stress was varied between 0.1 and 10 MPa. We note that our results are not dependent upon the choice of the reference moment. For every source pair, we obtain an estimate of the corner frequency for both events, then form averages and compute standard deviations. Next, we obtain estimates of the apparent stress from the spectral fits and the Brune stress drop from the corner frequency (see Table 1). As observed for other regions where this method has been applied, the coda spectral ratios for Chi-Chi are very stable, with average standard deviations around 0.1–0.2 for all frequencies. Though the Chi-Chi mainshock is complex with multiple sub-events [e.g., Kao and Chen, 2000; Chi et al., 2001], the broadband coda represents a time-domain convolution over the entire source duration. Due to crustal heterogeneity, a homogeneous scattered wavefield is quickly formed that is virtually free of source radiation and directivity effects. Figure 2 shows example spectral ratios between the mainshock and selected aftershocks. The coda-derived source ratios exhibit little scatter and thus source parameters, such as corner frequency, will be better constrained when we fit the observed data with theoretical source models, such as the commonly used omega-square model [Aki, 1967; Brune, 1970a, 1970b]. As found in the Hector Mine sequence [e.g., Mayeda et al., 2007], there is an increase in the corner-frequency versus moment relation (see Figure 3). However, unlike Hector Mine, the Chi-Chi sequence was very rich in large magnitude aftershocks, in the Mw ∼6+ range. From Figure 3 we see that the large events above ∼Mw 5.5, behave self-similarly whereas below this threshold, the event scatter is larger and the mean of the population decreases, representing a clear break in similarity. In an attempt to understand the observed variations, we also show in Figure 1 the apparent stress values for the aftershocks to see if there is any spatial pattern. Though there appears to be a hint that lower apparent stresses exist for events in the north, the region defined by high slip by finite-fault inversions by Chi et al. [2001], we cannot say anything definitively due to the complex deformation on Taiwan [Kao and Chen, 2000].

image

Figure 2. Example of coda-derived source ratios between the Mw 7.6 mainshock and selected aftershocks. In each plot, we show the low and high frequency asymptotes from equation (1) as solid black horizontal lines and dashed lines represent the case when p = 1.5. Red lines represent ±1σ about the average over a minimum of 6 stations and blue lines represent the best-fit theoretical ratio using equation (1).

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image

Figure 3. Seismic moment (Mo) versus corner frequency (fc) for the Chi-Chi events used in this study along with error estimates based on a bootstrap method. Lines of constant Brune stress drop are shown ranging between 0.1 and 100.0 MPa. We note that the mainshock and the three largest aftershocks appear to scale self-similarly, however a number of the aftershocks around Mw 5 are quite variable, as evidenced in the spectral ratios in Figure 2.

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Table 1. Source Parameter Information for Events Used in This Study
EventMwfcError fclog10Δσapp MPalog10Δσapp Error
9926317477.600.0379640.0058575.858170.182852
9926323124.850.7112070.1431675.543590.226294
9926418125.050.3848630.0445405.057240.127113
9926500136.300.1699090.0262885.860570.184166
9926500485.850.3020910.0509705.933130.198450
9926512175.050.7615610.1682195.928020.252971
9926612445.050.5984170.1213415.618040.230245
9926808425.150.5319520.0899535.620350.198095
9927003554.700.6902110.1177065.284300.200888
9927007204.950.5055620.1035105.247420.236861
9927105464.850.6939660.1232475.515270.207271
9927115374.850.4258850.0582484.886490.151852
9927412475.150.4679060.0941005.446770.237266
9929502175.900.2643080.0437205.834670.194255
9930308265.200.4532240.0729765.488060.188683
9932107345.150.4794380.0791935.485730.192368
0017121555.000.6559230.1132205.668640.192658
0024509244.950.4316700.0707155.049820.184263

3. Discussion and Conclusion

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Coda Amplitude Ratios
  5. 3. Discussion and Conclusion
  6. Acknowledgments
  7. References
  8. Supporting Information

[9] The averaging nature of coda waves has been shown to provide significantly lower amplitude variance than any traditional direct phase method. We have obtained stable source ratio estimates using broadband local and regional stations on Taiwan. We find the following: (1) For the Mw 7.6 mainshock and 3 of the larger aftershocks (>Mw 5.5), apparent stress is tightly clustered around 0.8 MPa (±0.1 MPa); (2) In contrast, events below moment magnitude ∼Mw 5.5 exhibit lower average apparent stress as well as larger scatter, ranging between ∼0.08 and 0.8 MPa and are spatially variable; (3) For this dataset, the Brune [1970a, 1970b] omega-square source model fits the spectral shape for events 4.7 < Mw < 7.6, however a clear, step-wise break in self-similarity exists at around ∼Mw 5.5; (4) ψ = 0 is appropriate only for the mainshock and the largest aftershocks; (5) Using only the smaller events, extrapolation to larger events would yield unrealistically high apparent stresses and corner frequencies for the largest events; (6) We hypothesize that the larger events are subject to the average state-of-stress over a broader region, whereas the smaller aftershocks are more sensitive to the local state-of-stress resulting from stress-field redistribution following the mainshock. Recently, dynamic phenomena such as fault zone lubrication [e.g., Brodsky and Kanamori, 2001] as well as pore fluid pressurization (L. Malagnini, personal communication, 2008) have been proposed that may help to explain the observed step-wise change, though much work is needed. We note that other sequences where this method has been applied also suggests a step-wise break, albeit not as obvious due to insufficient numbers of large magnitude aftershocks.

[10] In conclusion, the scaling results from this dataset are consistent with those from other sequences where this same method has been applied such as Hector Mine (CA), Wells (NV) (unpublished), San Giuliano (Italy), and Colfiorito (Italy). Rather than invoking a linear increase in Brune stress or apparent stress over the entire magnitude range, there appears to be a linear increase for events below ∼Mw 5.5 (non-self-similar), then constant above this magnitude (self-similar). However, that being said, there is strong evidence that the absolute levels of the stress are region dependent, as Hector Mine and Chi-Chi are roughly 5 and 10 times smaller, respectively, in Brune stress drop than the other 2 Italian sequences and Wells, NV. The compilation of more sequences is needed in conjunction with models that can account for the observed dynamic phenomena.

Acknowledgments

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Coda Amplitude Ratios
  5. 3. Discussion and Conclusion
  6. Acknowledgments
  7. References
  8. Supporting Information

[11] K. Mayeda was supported under Weston Geophysical subcontract GC19762NGD and AFRL contract FA8718-06-C-0024. Work by L. Malagnini was performed under the auspices of the Dipartimento della Protezione Civile, under contract S3 – INGV-DPC (2007–2009), project: “Valutazione rapida dei parametri e degli effetti dei forti terremoti in Italia e nel Mediterraneo”.

References

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Coda Amplitude Ratios
  5. 3. Discussion and Conclusion
  6. Acknowledgments
  7. References
  8. Supporting Information

Supporting Information

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Coda Amplitude Ratios
  5. 3. Discussion and Conclusion
  6. Acknowledgments
  7. References
  8. Supporting Information
FilenameFormatSizeDescription
grl25767-sup-0001-t01.txtplain text document1KTab-delimited Table 1.

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