The two most commonly used techniques for inverting earthquake focal mechanisms for stress orientation, FMSI [Gephart and Forsyth, 1984; Gephart, 1990a] and LSIB [Michael, 1984, 1987b], were tested on noisy synthetic data sets. Both techniques determine stress orientation accurately. FMSI is generally more accurate for high- quality data, while LSIB is more accurate for very noisy data. The confidence regions produced by LSIB are usually approximately the right size, while those of FMSI are usually too large.
The results for the synthetic data sets indicate that the confidence regions pro- duced by LSIB should be appropriate for real data sets as well, as long as the model assumptions of homogeneous stress and slip in the direction of resolved shear stress generally hold. The numerous stress field variations which have previously been ob- served in southern California [Michael, 1987b; Jones, 1988; Hauksson, 1990, 1994;
Kerkela and Stock, 1996], and the stress field variations reported in Chapters 1, 2, and 3, which are significant with respect to LSIB's uncertainty estimates, are therefore
larger than the true inversion uncertainty and probably represent real signals.
50 40
c
~ 30<D
a.
20
10
0 5 15 25 35 45 55 65 75 85 mechanism error (degrees)
Figure A.1: Histogram of the errors in first-motion focal mechanisms resulting from polarity and takeoff angle errors. Forty-one diverse focal mechanisms were chosen, and first-motion polarities for each were assigned to stations using the station distri- bution of Southern California Seismic Network (SCSN) stations for 41 actual southern California events. Random errors in polarity and in takeoff angle (i.e., in event loca- tion or velocity model) were added. Each first-motion observation has a 20% chance of being reversed. The error in azimuth to each station is normally distributed with a standard deviation of 2°, and the error in takeoff angle is normally distributed with a standard deviation of 10°. New focal mechanisms were determined using the FP- FIT software package [Reasenberg and Oppenheimer, 1985], and the mechanism error computed by determining the minimum rotation angle between the computed and correct mechanisms. This was repeated 100 times for the 41-event data set, and a histogram of mechanism error compiled. Note that the mechanism errors resemble an exponential distribution, not a normal distribution.
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confidence region (percent)
Figure A.2: An illustrative plot of the number of times the correct stress state falls within a given confidence region versus the percent confidence level. If the uncertainty estimates are correct, the plot should approximate a straight line. If the plot falls above this line, this means that the correct answer falls within the X% confidence region » X% of the time and the confidence regions are too big. Similarly, if the plot falls below the line, the correct answer falls within the X% confidence region «X%
of the time and the confidence regions are too small.
LSI B. A=0.5, error=5,1 0,15,20
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data error (degrees) data error (degrees)
Figure A.3: Accuracy of LSIB, for synthetic data sets with the given values of N (data set size), R, and mechanism error. There are 50 data sets for each combination of parameter values listed. The circles connected by the solid line represent the mean error for the suite of data sets, the crosses connected by the dashed lines encompass the middle 80%.
FMSI, A=0.5, error=5,1 0,15,20
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data error (degrees)
Figure A.4: Accuracy of FMSI, symbols as in Figure A.3.
c R=0.5, err=5,10,15,20, N=20 c R=0.5, err=5, 1 o, 15,20, N=50
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confidence region (percent) confidence region (percent)
Figure A.5: Confidence region appropriateness for both inversion methods, for vary- ing N (data set size.) The number of times the correct stress state falls within the X% confidence region is plotted as a function of X . If the confidence regions are appropriate, the function should fall within the shaded area. Results from an exper- iment with perfect confidence regions would have a 95% probability of falling within the shaded zone, which is the 95% confidence region of the binomial probability dis- tribution for 200 trials with an X% probability of success each trial. If the confidence regions are too big or too small, the function will fall above or below the shaded area, respectively (see Figure A.2). There are 50 data sets for each combination of parameter values listed. The label 4D indicates that all four model parameters were considered; 3D, only principal axis orientations.
R=0.5, err=S, N=20,50,1 00,300
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Figure A.6: Confidence region appropriateness, for varying focal mechanism error.
Symbols as in Figure A.5.
g 100 R=O. err=5, 10, 15,20, N=50
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confidence region (percent) confidence region (percent)
Figure A.7: Confidence region appropriateness, for varying R. Symbols as in Figure A.5.
c: LSIB, A=0.5, err=5, 10, 15,20, N=300
0
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Figure A.8: Confidence region appropriateness for LSIB for N = 300 when a modified resampling method is used. The focal mechanisms are sorted into 10°-wide bins based on strike, dip and rake. Only one mechanism from each non-empty bin is used in the base data set which is resampled during bootstrap error estimation. Although the base data set contains <300 events, each resampled data set still contains 300 events.
Symbols as in Figure A.5.
FMSI, N=50, R=O FMSI, N=50, R=1
0.4 0.4
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mechanism error (degrees) mechanism error (degrees)
Figure A.9: Average accuracy of R determined by FMSI, for axisymmetric stress tensors (R =0 and R =1) when the fault plane is ambiguous (circles and solid line) and known (crosses and dashed line). There are 50 data sets for each level of error.
§ N=O, err=5,1 0,15,20, R=O, fault planes known -~100r---~----~----~----~----.
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0 0
~ 0 0 20 40 60 80 100 confidence region (percent)
§ N=O, err=5,1 0,15,20, R=1, fault planes known -~100.---~----~----~----~----.
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~ 0 0 20 40 60 80 100 confidence region (percent)
Figure A.lO: Confidence region appropriateness for FMSI for the axisymmetric case when the fault plane in known. Solid line, 3D; dashed line, 4D. All other symbols as in Figure A.5. There are 50 data sets for each level of error.