Electronic Supplement to
Evidence for Truncated Exponential Probability Distribution of Earthquake Slip

by Kiran K. S. Thingbaijam and P. Martin Mai

This electronic supplement contains tables, figures, and references describing the rupture models and statistical analysis related to the probability distribution of earthquake slip.


Figures

Figure S1. The probability distribution of slip in terms of complementary cumulative distribution functions (CCDFs) based on the Source Inversion Validation (SIV) project (Mai et al., 2016; see also Data and Resources). The true rupture model is shown by red squares, whereas five proposed solutions obtained by different participants are shown by blue symbols. The black curves depict the fits of truncated exponential distribution to the true models. The results from two benchmark exercises are depicted here: (a) inv1 and (b) inv2a. The selection of these models is according to the approach outlined in the main article.

Figure S2. QQ plots for slip distributions of the SIV project, for which probability distributions are shown in Figure S1. The top row depicts the plots for benchmark exercise inv1, and the bottom row shows those for inv2a. The ordinate gives the quantiles of slip from the rupture models submitted by the different participants (numbered in Fig. S1). The abscissa gives the quantiles of slip from the true rupture model. The linear trend (red line) is obtained from the first and third quartiles of each distribution. Data points matching the linear trend suggest that, within these quantiles, both datasets can be described by the same statistical distribution (Wilk and Gnanadesikan, 1968). We observe that linearity in the QQ graphs is more pronounced for slip distributions with excellent-to-good similarity in their spatial heterogeneity structure (Razafindrakoto et al., 2015).

Figure S3. (Left column) Slip distributions resulting from inversions with variable smoothing strength (increasing from top to bottom). These slip-inversion results are obtained for the 1995 Kobe earthquake (Sekiguchi et al., 2000; their fig. 8). (Right column) CCDFs for the slip distributions are shown to the left. The data here are best described by a lognormal distribution (depicted by red lines on the CCDF plots; the truncated exponential fit is given by the dashed black lines) and are consistently observed even at increased smoothing, especially for the optimal level chosen by the authors (indicated by the red box on the left column). With increasing smoothing, the truncation of the upper tail becomes more prominent and leads to a truncated lognormal distribution.

Figure S4. Multidimensional scaling analysis (Razafindrakoto et al., 2015) for slip distributions derived with different degrees of smoothing: (a) for slip models by Sekiguchi et al. (2000), shown in Figure S3 and (b) for slip models developed in this study (Fig. 2 of the main article). This analysis shows that these models share common spatial features. The open red circle encloses models with excellent similarity. Notice how cases of extreme roughness or smoothness tend to be detached from the group.

Figure S5. Spatial distribution of static stress change on the rupture plane calculated for the slip distributions obtained with increasing levels of smoothing (from top to bottom; slip models are shown in Fig. S3). The on-fault stress change is computed following Ripperger and Mai (2004). Notice the decrease of maximum stress drop with increasing smoothing, as well as the diminishing spatial stress heterogeneity.

Figure S6. Same as Figure S5 but for progressively smoothed slip distributions shown in Figure 2 of the main article. Note the similarity in spatial characteristics of the stress changes (with increasing smoothing) to the stress-change function in Figure S5.

Figure S7. Summary of the variation of stress parameters with respect to the spatial smoothing of the slip distribution (Figs. S5 and S6). Here, stress drop corresponds to the positive stress change. (a) Maximum stress drop decreases with increasing spatial smoothing. (b) A similar trend can be observed for the average stress change (considering both negative and positive stress change) and average stress drop. We note that moderate smoothing produces average stress-drop values that are consistent with previous calculations (e.g., Fletcher and McGarr, 2006).

Figure S8. Scaling of total rupture area A(u ≥ 0) (in square kilometers) with seismic moment M0 (in newton-meters). Solid lines in red show the linear fits without any constraints on the slope parameter, and the confidence bounds are given by the thin dashed lines. Thicker dashed lines show the fits with self-similarity constraint of slope equal to 2/3. Notice that the data of reverse-faulting subduction zone earthquakes strongly support self-similar moment-area scaling. On the other hand, we find that the scaling relationships for the strike-slip and normal dip-slip events deviate from self-similar behavior. The intercept α, standard error σ, coefficient of determination R2, and correlation coefficient r are annotated on each plot. For the reverse dip-slip events, these are calculated for a slope of 2/3, whereas those for strike-slip and normal dip-slip events, the slope parameter is equal to 0.577 and 0.566, respectively.

Figure S9. Scaling of slip area A(u > ui) (in square kilometers) with seismic moment M0 (in newton-meters) for reverse dip-slip (subduction) events. To compute slip area, slip ui is centered at 12 uniformly spaced bins equal to fraction of maximum slip (k, such that ui = kumax). The analysis uses 41 events (with 76 slip models). For earthquakes with several rupture models, we use the ensemble average of the logarithm-transformed data. The dashed and solid lines indicate regressions with and without fixed slope (equal to 2/3, as given by self-similar scaling). For the former, the intercept α and standard error σ are given on each plot. Also, listed are the coefficient of determination R2 and correlation coefficient r. Notice that the data of reverse-faulting subduction zone earthquakes strongly support self-similar moment-area scaling.

Figure S10. Same as Figure S9 but for reverse dip-slip events in continental regions. The analysis uses 14 events (with a total of 27 slip models). The fixed slope is equal to 2/3, as given by self-similar scaling.

Figure S11. Same as Figure S9 but for strike-slip events. The regression analysis uses 34 events (with a total of 73 slip models). The fixed slope is equal to 0.577, which we estimated for the relationship between seismic moment and total rupture area (Fig. S8).

Figure S12. Same as Figure S9 but for normal-faulting dip-slip events. The regression analysis uses nine events (with a total of 13 slip models). The fixed slope is equal to 0.566, which we estimated for the relationship between seismic moment and total rupture area (Fig. S8).

Figure S13. Here, we provide QQ plots for slip distributions for selected rupture models to visualize how the tested statistical distributions compare with the data. Each model is indicated with the event/model tag used in the SRCMOD database (see Data and Resources). In each panel, the ordinate shows quantiles for the empirical slip values, whereas the abscissa displays the corresponding values for the assumed statistical distributions: truncated exponential (TEX), exponential (EXP), Weibull (WBL), gamma (GAM), and lognormal (LGN). In each panel, points located on the 45° reference line indicate a perfect fit of the corresponding statistical distribution.


Tables

Table S1. List of earthquakes and rupture models used in the present study.

Table S2. The fits of truncated exponential distribution using least-squares technique to the empirical probability distribution of slip derived for each rupture model.

Table S3. The fits of exponential distribution using least-squares technique to the empirical probability distribution of slip derived for each rupture model.

Table S4. The fits of Weibull distribution using least-squares technique to the empirical probability distribution of slip derived for each rupture model.

Table S5. The fits of gamma distribution using least-squares technique to the empirical probability distribution of slip derived for each rupture model.

Table S6. The fits of lognormal distribution using least-squares technique to the empirical probability distribution of slip derived for each rupture model.


Data and Resources

The rupture models used in this study were extracted from the SRCMOD database (http://equake-rc.info/srcmod, last accessed March 2016). The Source Inversion Validation (SIV) project is accessible at http://equake-rc.info/siv (last accessed March 2016).


References

Fletcher, J. B., and A. McGarr (2006). Distribution of stress drop, stiffness and fracture energy over earthquake rupture zones, J. Geophys. Res. 111, no. B03312, doi: 10.1029/2004JB003396.

Mai, P. M., D. Schorlemmer, M. Page, J.-P. Ampuero, K. Asano, M. Causse, S. Custodio, W. Fan, G. Festa, M. Galis, et al. (2016). The earthquake-source inversion validation (SIV) project, Seismol. Res. Lett. 87, no. 3, doi: 10.1785/0220150231.

Razafindrakoto, H. N., P. M. Mai, M. G. Genton, L. Zhang, and K. K. S. Thingbaijam (2015). Quantifying variability in earthquake rupture models using multidimensional scaling: Application to the 2011 Tohoku earthquake, Geophys. J. Int. 202, 17–40.

Ripperger, J., and P. M. Mai (2004). Fast computation of static stress changes on 2D faults from final slip distributions, Geophys. Res. Lett. 31, L18610, doi: 10.1029/2004GL020594.

Sekiguchi, H., K. Irikura, and T. Iwata (2000). Fault geometry at the rupture termination of the 1995 Hyogo-ken Nanbu earthquake, Bull. Seismol. Soc. Am. 90, 117–133.

Wilk, M. B., and R. Gnanadesikan (1968). Probability plotting methods for the analysis for the analysis of data, Biometrika 55, 1–17.

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