Electronic Supplement to
Resolution of non-double-couple-mechanisms: simulation of hypocenter mislocation and velocity
structure mismodeling
by Jan Šílený
1.
Summary of the values of the Normalized Root Mean Square (NRMS) in the data inversion for the complete moment tensor (MT)
and pure double couple (DC), and of the values of the condition number in the SVD linear inversion for the MT.
Capital letters A, B, C, D, BC and BD indicate types of inaccuracies in the inversion problem. In each of the types, NRMS corresponding to the MT model is
in the left-hand column, NRMS for the DC model in right-hand column. Shear-tensile sources with slip vector off the fault plane by angle =50, 100, 200, 300,
450 and 900 are considered.
NRMS is L2 norm of the differences of the data and the synthetics normalized by the L2 norm of the data. The variance reduction can be calculated as 100*(1NRMS).
Shear-tensile source:  = 5o
mislocation
NRMS
MT vs. DC
A
B
23: P+S
0.0097 0.017 0.0107
23: P
0.0119 0.104
0.012
10a: P+S
0.017 0.0288 0.0175
10a: P
0.0132 0.151
0.009
10b: P+S
0.00925 0.0209 0.00897
10b: P
0.0049 0.0278 0.00558
5a: P+S
0.00958 0.0206 0.0107
5b: P+S
0.0108 0.026 0.00734
mismodeling
C
0.0189
0.106
0.03
0.017
0.017
0.0262
0.0186
0.0237
mislocation mismodeling
both
A
B
C
D
BC BD
23: P+S
7.8
7.8
7.0
7.0
6.9 6.9
23: P
4.7
4.9
6.7
6.7
6.0 6.0
10a: P+S 7.7
7.8
6.8
6.8
6.7 6.7
10a: P
3.7
9.4
4.7
4.7
4.3 4.3
10b: P+S 9.5
9.4
8.4
8.4
8.2 8.2
10b: P
21.6 20.1 30.6 30.6 25.6 25.6
5a: P+S
9.5
9.3
9.0
9.0
8.6 8.6
5b: P+S 12.6 12.4 13.2 13.2 12.4 12.5
cond.no.
0.0241
0.0431
0.0408
0.0652
0.0262
0.0492
0.031
0.0276
both
D
BC
BD
0.0298 0.0548 0.0576 0.0231 0.0292 0.0572 0.06
0.114 0.127 0.159 0.0439 0.115
0.13
0.161
0.0454 0.0824 0.0839 0.0391 0.0441 0.0857 0.0872
0.18
0.214 0.261 0.0713 0.182 0.223 0.267
0.042 0.0833 0.0911 0.0233 0.0371 0.083 0.0895
0.0554 0.177 0.180 0.0486 0.0536 0.177 0.179
0.052 0.106 0.114 0.0279 0.0453 0.106 0.113
0.0345 0.087 0.0886 0.0244 0.0301 0.0828 0.0838
Shear-tensile source:  = 10o
NRMS
MT vs. DC
23: P+S
23: P
10a: P+S
10a: P
10b: P+S
10b: P
5a: P+S
5b: P+S
mislocation
A
0.00937
0.00945
0.0161
0.00971
0.00907
0.00453
0.00963
0.0104
mismodeling
B
0.0282
0.159
0.0438
0.225
0.035
0.0381
0.0404
0.0369
0.0104
0.0101
0.0167
0.00892
0.00868
0.005
0.0105
0.00765
both
BC
BD
0.023 0.0377 0.0529 0.06 0.0222 0.0372 0.0552 0.0621
0.0376 0.152 0.117 0.168 0.0387 0.150
0.12
0.168
0.038 0.0534 0.0804 0.0861 0.037 0.0528 0.084 0.0894
0.0554 0.225 0.189 0.254 0.0604 0.232 0.198 0.261
0.0251 0.0537 0.0796 0.0927 0.0224 0.0488 0.0788 0.0913
0.0463 0.057 0.167
0.17 0.0446 0.0551 0.163 0.166
0.0303 0.0677 0.101 0.116 0.0271 0.0616 0.101 0.114
0.0278 0.0371 0.0936 0.0955 0.0246 0.0376 0.089 0.911
C
0.0291
0.155
0.0441
0.227
0.0327
0.0346
0.0378
0.0364
D
mislocation mismodeling
both
A
B
C
D
BC BD
23: P+S
7.8
7.8
7.0
7.0
6.9 6.9
23: P
4.7
4.9
6.7
6.7
6.0 6.0
10a: P+S 7.7
7.8
6.8
6.8
6.7 6.7
10a: P
3.7
3.7
4.7
4.7
4.3 4.3
10b: P+S 9.5
9.4
8.4
8.4
8.2 8.2
10b: P
21.6 20.1 30.6 30.6 25.6 25.6
5a: P+S
9.5
9.3
9.0
9.0
8.6 8.6
5b: P+S 12.6 12.4 13.2 13.2 12.4 12.5
cond.no.
Shear-tensile source:  = 20o
NRMS
MT vs. DC
23: P+S
23: P
10a: P+S
10a: P
10b: P+S
10b: P
5a: P+S
5b: P+S
mislocation
A
0.00885
0.00574
0.0148
0.00642
0.00855
0.00349
0.00897
0.00998
0.0489
0.133
0.0704
0.233
0.0612
0.0495
0.0756
0.0524
B
0.00979
0.00671
0.0155
0.00644
0.00812
0.00346
0.00929
0.0083
mismodeling
both
BC
BD
0.0492 0.0213 0.0548 0.0509 0.0689 0.0209 0.0546 0.053 0.0702
0.129 0.0308 0.103 0.105
0.13 0.0323 0.107 0.109 0.134
0.0701 0.0346 0.0733 0.0804 0.0967 0.0346 0.0735 0.0843 0.0995
0.232 0.0486 0.181 0.173 0.218 0.0525 0.194 0.181 0.229
0.0587 0.0236 0.0728 0.0757 0.0983 0.0209 0.0698 0.0735 0.0965
0.0491 0.0465 0.0584 0.169 0.171 0.0444 0.0584 0.163 0.166
0.0712 0.028 0.0925 0.0927 0.121 0.0249 0.0876 0.0906 0.118
0.0522 0.0286 0.496 0.104 0.109 0.0254 0.0485 0.099 0.104
C
D
mislocation mismodeling
both
A
B
C
D
BC BD
23: P+S
7.8
7.8
7.0
7.0
6.9 6.9
23: P
4.7
4.9
6.7
6.7
6.0 6.0
10a: P+S 7.7
7.8
6.8
6.8
6.7 6.7
10a: P
3.7
3.7
4.7
4.7
4.3 4.3
10b: P+S 9.5
9.4
8.5
8.4
8.2 8.2
10b: P
21.6 20.1 30.6 30.6 25.6 25.6
5a: P+S
9.5
9.3
9.0
9.0
8.6 8.6
5b: P+S 12.6 12.4 13.2 13.2 12.4 12.5
cond.no.
Shear-tensile source:  = 30o
NRMS
MT vs. DC
23: P+S
23: P
10a: P+S
10a: P
10b: P+S
10b: P
5a: P+S
5b: P+S
mislocation
A
0.0086
0.00392
0.0143
0.0155
0.00797
0.00251
0.00768
0.01001
0.0646
0.121
0.0937
0.216
0.0772
0.0488
0.0919
0.0656
B
0.00927
0.00485
0.0149
0.00536
0.00765
0.00235
0.00765
0.00893
mismodeling
C
0.0645
0.118
0.093
0.214
0.0753
0.0485
0.082
0.0662
mislocation mismodeling
both
A
B
C
D
BC BD
23: P+S
7.8
7.8 7.0
7.0
6.7 6.9
23: P
4.7
4.9 6.7
6.7
6.9 6.0
10a: P+S 7.7
7.8 6.8
6.8
6.7 6.7
10a: P
3.7
3.7 4.7
4.7
4.3 4.3
10b: P+S 9.5
9.4 8.5
8.4
8.2 8.2
10b: P
21.6 20.1 30.6 30.6 25.6 25.6
5a: P+S
9.5
9.3 9.0
9.0
8.6 8.6
5b: P+S 12.6 12.4 13.2 13.2 12.5 12.5
cond.no.
Shear-tensile source:  = 45o
0.0205
0.0272
0.0335
0.0444
0.0226
0.0419
0.0254
0.03
D
0.069 0.0511 0.0783
0.0844 0.0957 0.113
0.095 0.084 0.112
0.161
0.16
0.198
0.0832 0.075 0.102
0.0513 0.153 0.154
0.0923 0.0872 0.121
0.0578 0.113 0.120
both
BC
0.0201 0.0687
0.0286 0.0917
0.034 0.0953
0.0476 0.176
0.0942 0.0822
0.0404 0.0522
0.0227 0.097
0.0267 0.0586
BD
0.0526
0.0994
0.088
0.168
0.0718
0.148
0.0834
0.107
0.079
0.119
0.115
0.21
0.101
0.15
0.117
0.115
mislocation
NRMS
MT vs. DC
A
B
23: P+S
0.00872 0.0821 0.00889
23: P
0.0031 0.119 0.00387
10a: P+S
0.0147 0.124 0.0151
10a: P
0.00429 0.208 0.00492
10b: P+S
0.0075 0.0912 0.00741
10b: P
0.00206 0.0437 0.00191
5a: P+S
0.0061 0.087 0.00591
5b: P+S
0.0104 0.0814 0.0098
mismodeling
C
0.0814
0.115
0.124
0.206
0.09
0.0434
0.0897
0.0814
0.0208
0.0259
0.0352
0.0426
0.0226
0.0406
0.0236
0.0321
0.0853
0.0798
0.125
0.153
0.0894
0.0468
0.0783
0.0695
both
D
BC
BD
0.0541 0.0912 0.0201 0.0848 0.0549 0.0911
0.0925 0.108 0.0273 0.0882 0.0958 0.114
0.0939 0.138 0.0359 0.125 0.0979
0.14
0.154 0.188 0.0453 0.168 0.161
0.2
0.0782 0.107 0.0201 0.0887 0.0742 0.105
0.148 0.149 0.0395 0.048 0.145
0.146
0.0878 0.111 0.0213 0.0811 0.0826 0.109
0.124 0.134 0.0287 0.708 0.118
0.129
mislocation mismodeling
both
A
B
C
D
BC BD
23: P+S
7.8
7.8
7.0
7.0
6.7 6.9
23: P
4.7
4.9
6.7
6.7
6.0 6.0
10a: P+S 7.7
7.8
6.8
6.8
6.7 6.7
10a: P
3.7
3.7
4.7
4.7
4.3 4.3
10b: P+S 9.5
9.4
8.5
8.4
8.2 8.2
10b: P
21.6 20.1 30.6 30.6 25.5 25.6
5a: P+S
9.5
9.3
9.0
9.0
8.6 8.6
5b: P+S 12.6 12.4 13.2 13.2 12.4 12.5
cond.no.
Tensile crack:  = 90o
NRMS
MT vs. DC
23: P+S
23: P
10a: P+S
10a: P
10b: P+S
10b: P
5a: P+S
5b: P+S
cond.no.
23: P+S
mislocation
A
0.0106
0.00308
0.0175
0.0044
0.00794
0.00182
0.00433
0.0117
0.0993
0.114
0.16
0.207
0.102
0.04
0.083
0.114
mismodeling
B
0.00972
0.00333
0.0171
0.00502
0.0085
0.00183
0.00524
0.0116
mislocation mismodeling
A
B
C
D
7.8
7.8
7.0
7.0
C
0.0986
0.11
0.16
0.204
0.121
0.0402
0.0823
0.114
both
BC BD
6.7 6.9
0.0264
0.0261
0.046
0.0421
0.0272
0.0409
0.027
0.037
both
D
BC
BD
0.109 0.0693 0.118 0.0245
0.106 0.0691 0.117
0.0754 0.0926 0.106 0.0274 0.0833 0.0959 0.112
0.172 0.117 0.190 0.00962 0.169
0.12
0.19
0.152 0.151 0.184 0.0445 0.167 0.158
0.196
0.0893 0.0989 0.124 0.0245 0.0889 0.0944 0.121
0.0444 0.149 0.150 0.0403 0.456 0.148
0.149
0.0698 0.113 0.124 0.0252 0.0712 0.108
0.12
0.095
0.145 0.162 0.0334 0.0975 0.138
0.158
23: P
4.7
10a: P+S 7.7
10a: P
3.7
10b: P+S 9.5
10b: P
21.6
5a: P+S
9.5
5b: P+S 12.6
2.
4.9
7.8
3.7
9.4
20.1
9.3
12.4
6.7
6.8
4.7
8.5
30.6
9.0
13.2
6.7
6.8
4.7
8.4
30.6
9.0
13.2
6.0 6.0
6.7 6.7
4.3 4.3
8.2 8.2
25.6 25.6
8.6 8.6
12.5 12.5
The moment tensor (MT) resolution in the synthetic experiments with a shear-tensile source.
Station configurations from Fig. 3 are considered: 23, 10a, 10b, 5a, 5b. In the configurations 23, 10a, 10b two inversion options are considered: (i) P+S indicates that the
inversion includes both P and S amplitude data, while (ii) P indicates that only P amplitudes are included in the inversion. In the configurations 5a and 5b the case (i) only is
considered, as the case (ii) represents an underdetermined inverse problem. For each configuration, in turn, the inversions were performed using GF constructed for
(A) the hypocenter mislocated in depth,
(B) the hypocenter mislocated both in the depth and the horizontal coordinates,
(C) a simplified velocity model – layered 1-D model replaced by a homogeneous halfspace,
(D) even more simplified velocity model – the 1-D velocity model replaced by a homogeneous space,
(BC) combination of (B) and (C),
(BD) combination of (B) and (D).
For quantitative details about the cases (A-D) see the last paragraph of the section Set-up of the experiment.
The four tables composing the Figure summarize the quantities characterizing the MT resolution:
(1) the percentage of non-DC components (top left), for the theoretical values see Tab. 1,
(2) the ratio of the ISO percentage and CLVD percentage ISO/CLVD (bottom left), the theoretical value being 5/4 for Lame’s constants =,
(3) the probability (in percents) that the MT model is significant, obtained in the F-test comparing the unconstrained MT and pure DC model, and
(4) the deviation (in degrees) of the orientation of the retrieved mechanism in the MT representation from the theoretical model used to synthesize the data, expressed as
the sum of deviations for all 3 principal axes (bottom right).
Grades of shade mark qualitatively the size of the deviation of the quantities (1-4) from their theoretical values: the bigger deviation, the darker shade.
Shear-tensile source: slip-vector 100 off fault plane.
Shear-tensile source: slip-vector 300 off fault plane.
3.
Fault-plane solutions corresponding to synthetic experiment with combined shear-tensile model.
MT – moment tensor model, DC – pure double-couple model. Station configurations are from Fig.3. A, B, C, D, BC and BD indicate types of inaccuracies in the inversion
problem. Fault-plane solutions displayed in the equal-area projection of the lower focal hemisphere, compression zone – green, principal axes – red.
Slip vector 200 off the fault plane.
Slip vector 900 off the fault plane (corresponding to a tensile-crack).