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.  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 . The coda ratio methodology is outlined by Mayeda et al.  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.
 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,
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 whereas at the high frequency limit, equation (1) is asymptotic to 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  that the scaling between moment and corner frequency could take on the form,
where ɛ represents the deviation from self-similarity and is usually thought to be a small positive number. For example, Walter et al.  and Mayeda et al.  found ɛ to be close to 0.5 for the Hector Mine mainshock and its aftershocks using independent spectral methods. Mayeda et al.  used the source spectrum portion of the Magnitude Distance Amplitude Correction (MDAC) methodology of Walter and Taylor , which allows for the variation of the corner frequency that does not have to be self-similar. For example,
where σa is the apparent stress [Wyss, 1970], σ′a and M′0 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  found ψ = 0.25 for moderate-to-large earthquakes in the western United States and Mayeda et al.  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.  was that the apparent stresses were systematically lower for the aftershocks than the mainshock.
 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. , we cannot say anything definitively due to the complex deformation on Taiwan [Kao and Chen, 2000].