This electronic supplement presents the explicit formulation of several parameters used in the FMNEAR inversion: maximum frequency, upper bound of the band-pass filtering, length of the time window used in the inversion, width of the triangular individual time windows used to discretize the local source time functions, minimization of the seismic moment, depth exploration, and computation of the index of confidence. It also contains complementary figures detailing some aspects of the FMNEAR method, such as flow charts (Figs. S1 and S2), computation of the distance between focal mechanisms (Fig. S3), waveform fit for six earthquakes among the test cases considered in the paper (Figs. S4–S9), the results of sensitivity tests discussed in the paper (Figs. S10 and S12), and the waveform fit for an inversion of the Chi-Chi, Taiwan, earthquake in which individual components were not discarded on the basis of waveform fit (Fig. S11). Finally, Table S1 describes the crustal velocity models used with the test cases.
Maximum frequency (fmax, the upper bound for the bandpass filtering) is basically fixed to 3 fmin (the lower bound for the bandpass filtering), but we apply some additional conditions:
If Mwi > 6: fmax cannot be higher than 0.1 + (6 − Mwi)/300.
If Mwi in the range 4–6: fmax cannot be higher than 0.2 + (5 − Mwi)/10.
If Mwi < 4: fmax cannot be higher that 1.5 + (2 − Mwi)/1.67.
Then we apply a correction as a function of hypocentral distance:
fmax = fmax − (dist/3500), where dist is in km.
Finally, we impose some minimum value for fmax, which is 0.04 Hz for very large earthquakes (Mwi > 8.5) and 0.07 Hz otherwise.
First we estimate a minimum duration (durmin) as a function of the propagation time of the S wave (tcalS) and the rupture length (L) estimated from Wells and Coppersmith (1994):
durmin = tcalS + L/1.5 (in seconds),
in which 1.5 represents a slow rupture velocity in km/s.
Then, we use different empirical formulas to obtain the final length of the time window (Mwi is the initial magnitude):
If Mwi < 4, final length = durmin + (dhypo/10) + 1.3/fmin + 4 (in seconds).
If Mwi > 4, final length = 1.3 durmin + (dhypo/8) + 0.35/ fmin + 5 (in seconds),
in which dhypo is hypocentral distance. The values of the coefficients in these formulas were progressively adjusted during the analysis of tens of earthquakes spanning a wide range of magnitudes.
This half width is 10 (Mwi − 6.5)/3 (in seconds), in which Mwi is the initial magnitude.
The minimization function, which is included in the total cost function of the simulated annealing algorithm (in addition to the root mean square [rms] misfit function), is the following:
0.01 EXP((M0/M0ref) − 1),
in which M0ref is the reference seismic moment obtained from the initial magnitude Mwi, M0 the seismic moment of the current solution explored, and EXP the exponential function.
Primary seven depths tested (in addition to initial depth) are shown below:
If initial depth < 20 km: 2, 5, 10, 20, 30, 50, 70 km (wide range of exploration since routine automatic locations often end up with a shallow depth, sometimes fixed)
If initial depth in the range 20–45 km: 12, 20, 28, 36, 44, 52, 60 km
If initial depth in the range 45–75 km: 30, 40, 50, 60, 70, 80, 90 km
If initial depth in the range 75–150 km: 60, 80, 100, 120, 140, 160, 180 km
If initial depth in the range > 150 km: 110, 150, 190, 230, 270, 310, 350 km
If the initial depth gave a better result, it is kept.
For the refined seach, four additional depths are tested, with a step of 1 km if depth < 15 km, a step of 2 km if depth is between 15 and 30 km, and a step of 10 or 20 km for deeper earthquakes.
We have 54 focal mechanism (FM) solutions that have been tested.
Let us consider the best solution, associated with the lowest rms misfit value (hereafter called "bestrms").
Then, let us consider a secondary solution among the 53 other ones, with its rms value (hereafter called “rms”).
For that particular secondary solution, we compute an individual confidence index (Ci):
Ci = 150 [rms6 / (bestrms6.2 (1 + dist)4)]
Where dist is the distance between the best solution and the secondary solution, as defined in Figure S3 below. The value of 150 and the exponents were empirically adjusted after analyzing tens of inversions for different earthquakes over a wide range of magnitudes.
We see that Ci will be small if rms is close to bestrms and dist is large. The exponent 6.2 for bestrms is to give slightly more weight to the contribution of bestrms by itself; when bestrms < 1, which is generally the case, it will enhance the confidence in case of results displaying small values of bestrms.
In order to take into account the number of components used in the inversion (a larger number of component meaning a higher confidence), we modify Ci:
Ci = Ci [ncomp0.2 + 0.2]
The final confidence index, CI, is the minimum value among all the 53 individual Ci values. As a matter of fact, it is the most penalizing value, corresponding to the lowest Ci value, which has to be taken into account. In other word, CI = min (Ci).
Remark 1: In order to stabilize individual Ci toward large values, we apply an arc-tangent to it.
Remark 2: We do not allow individual Ci values to be larger than 100.
Remark 3: Because CI varies between 0 (lowest confidence) and 100 (highest confidence), we call it a confidence index in %.
Figure S1. Flow chart of the FMNEAR method. After a complete run of the two steps of the inversion (coarse and fine search on strike dip rake), a confidence index is computed (CI1). If CI1 is lower than 70%, the two steps are redone, keeping only those channels (components) for which the rms misfit value is smaller than 0.65. A new confidence index (CI2) is then computed, and the results of this last run are kept only if CI2 > CI1.
Figure S2. Flow chart detailing step 2 of the FMNEAR method. The left and right columns, corresponding to the exploration around the best dip-slip and best strike-slip solutions, respectively, are independent. In each step, the rake angle is inverted jointly with the rupture timing and local source time functions of each point source included in the source model with a simulated annealing algorithm.
Figure S3. Illustration of the distance between focal mechanisms (FM). The distance is defined as the sum of the differences, in the absolute sense, of the theoretical amplitude of the P wave (ampFM) between the two focal mechanisms, computed over m points sampling the focal sphere. In practice, m = 324. The distance varies between 0 (identical FM) to 1 (opposite FM). The relation giving ampFM in the figure is derived from the radiation pattern of the far-field P wave in Aki and Richards (1980). ABS is the absolute value.
Figure S4. Waveform modeling for the Tottori earthquake. The horizontal axes indicate time in seconds, and the vertical axes show displacement amplitude in centimeters. Observed data are shown as continuous lines, whereas computed values are shown as dashed lines. Components where the observed and computed signals are flat correspond to components that were discarded by the inversion.
Figure S5. Waveform modeling for the Chengkung earthquake. The descriptions are as for Figure S4.
Figure S6. Waveform modeling for the Saintes earthquake. The descriptions are as for Figure S4.
Figure S7. Waveform modeling for the Tarapaca earthquake. The descriptions are as for Figure S4.
Figure S8. Waveform modeling for the Miyagi-Oki earthquake. The descriptions are as for for Figure S4.
Figure S9. Waveform modeling for the Tokashi-Oki earthquake. The descriptions are as for Figure S4.
Figure S10. Sensitivity tests in the case of the Chi-Chi earthquake. Inversions with two different subsets of 6 stations (#1 and #2), with the original epicenter or the epicenter displaced 10 km northward.
Figure S11. Waveform fit for the Chi-Chi earthquake for an FMNEAR inversion in which the elimination of components as a function of waveform fit is disabled. Components that were discarded by the standard FMNEAR inversion (see Fig. 5 of the main paper) are indicated by a rectangular frame in the plots distance plots on the left side of each panel. The vertical component of C35 (marked by an star) was eliminated during data processing (no band-pass filtering found).
Figure S12. Sensitivity tests in the case of the Tohoku-Oki earthquake: (a) inversion with the JMA epicenter; (b) inversion with initial moment magnitude 8.3; (c) inversion with initial moment magnitude 8.5; and (d) inversion with initial moment magnitude 8.7.
Table S1. Velocity models used in the FMNEAR inversions for the different test cases.
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