02/11/11 South Carolina Event

Arrival Times (from USGS)

Felt Map

Focal Mechanism

The focal mechanism was determined using broadband seismic waveforms. The location of the event and the station distribution are given in Figure 1.

Figure 1. Location of broadband stations used to obtain focal mechanism

Focal Mechanism

Strike = 350	Strike = 170
Dip = 40	Dip = 50
Rake = 90	Rake = 90
Mw = 4.00	Depth = 9 km

Surface-wave analysis

Surface wave analysis was performed using codes from Computer Programs in Seismology, specifically the multiple filter analysis program do_mft and the surface-wave radiation pattern search program srfgrd96.

Data preparation

Digital data were collected, instrument response removed and traces converted to Z, R an T components. Multiple filter analysis was applied to the Z and T traces to obtain the Rayleigh- and Love-wave spectral amplitudes, respectively. These were input to the search program which examined all depths between 1 and 25 km and all possible mechanisms. The figure

Best mechanism fit as a function of depth

          Depth  Stk   Dip  Rake  RR     RL     MWR    MWL  Best
SRFGRD96    1.0  120.   10.   30. 0.816 0.944   4.31   4.31 0.7685
SRFGRD96    2.0  170.   70.  -80. 0.802 0.944   4.12   4.11 0.7489
SRFGRD96    3.0  160.   70.  -70. 0.828 0.917   4.09   4.09 0.7429
SRFGRD96    4.0   60.   60.   80. 0.845 0.942   4.05   4.05 0.7911
SRFGRD96    5.0  130.   30.   60. 0.882 0.936   4.07   4.06 0.8192
SRFGRD96    6.0  150.   30.   80. 0.903 0.944   4.07   4.07 0.8484
SRFGRD96    7.0  340.   60.  -70. 0.906 0.926   4.06   4.05 0.8378
SRFGRD96    8.0  260.   40.  -70. 0.905 0.941   4.04   4.04 0.8486
SRFGRD96    9.0   60.   50.   70. 0.903 0.933   4.04   4.04 0.8414
SRFGRD96   10.0  240.   50.   80. 0.895 0.938   4.06   4.06 0.8389
SRFGRD96   11.0  230.   50.   70. 0.883 0.936   4.05   4.05 0.8264
SRFGRD96   12.0  260.   50.  -60. 0.863 0.930   4.05   4.04 0.7932
SRFGRD96   13.0  320.   50.   60. 0.833 0.931   4.05   4.05 0.7738
SRFGRD96   14.0  100.   50.  -50. 0.826 0.928   4.04   4.04 0.7584
SRFGRD96   15.0   90.   50.  -50. 0.807 0.929   4.04   4.04 0.7447
SRFGRD96   16.0   40.   60.   50. 0.785 0.927   4.04   4.03 0.7233
SRFGRD96   17.0  210.   60.   50. 0.765 0.919   4.04   4.04 0.6906
SRFGRD96   18.0   90.   50.  -40. 0.744 0.916   4.04   4.05 0.6812
SRFGRD96   19.0  290.   50.  -30. 0.717 0.912   4.04   4.04 0.6461
SRFGRD96   20.0  100.   50.  -40. 0.701 0.913   4.07   4.06 0.6311
SRFGRD96   21.0  310.   50.   40. 0.671 0.905   4.07   4.08 0.6059
SRFGRD96   22.0  110.   50.  -30. 0.658 0.899   4.07   4.07 0.5892
SRFGRD96   23.0  200.   60.   40. 0.640 0.889   4.08   4.08 0.5668
SRFGRD96   24.0  100.   50.  -30. 0.619 0.894   4.08   4.08 0.5511
SRFGRD96   25.0   40.   70.   40. 0.602 0.883   4.09   4.09 0.5243

The following plots present the displacement spectrum radiation patterns. The solid curves are the predicted patterns for the mechanism, depth and seismic moment given above. Each figure is fo a given period (T= ) and the scale indicates the displacement spectrum in units of cm-sec. The colored dots are the observations, with red indicating that the observed and predicted values are within a factor of 2, green that they are within the factor of 2 - 3 and blue that they differ by a factor greater than 3.

Love-wave radiation patterns

Rayleigh-wave radiation patterns

Waveform grid search

Since the analysis of the surface-wave radiation patterns uses only spectral amplitudes and because the surface-wave radiation patterns have a 180 degree symmetry, each surface-wave solution consists of four possible focal mechanisms corresponding to the interchange of the P- and T-axes and a rotation of the mechanism by 180 degrees. To select one mechanism, P-wave first motion can be used. Only the station NHSC at a distance of 83 km and an azimuth of 353 degrees had a good sharp arrival - a compression. To assist in selecting the mechanism the waveforms at NHSC were inverted using a locally designed model.

Since the station NHSC is the closest station and since this station has good Rayleigh wave signals, the waveform grid search program wvfgrd96 was used, initially with with the CUS earth model and later with a path specific model to the station. To derive the model for this path, the multiple filter analysis program do_mft was used to obtain the Rayleigh wave dispersion; the Love wave dispersion obtained was not realistic. The CUS model was modified to make a model with a 30 km crust and a 1 km thick low velocity layer at the surface. This model was used as a starting model in the program surf96 to use the Rayleigh-wave dispersion data to improve the model. The The model used for the waveforms is

MODEL.01
Preliminary model for NHSC
ISOTROPIC
KGS
FLAT EARTH
1-D
CONSTANT VELOCITY
LINE08
LINE09
LINE10
LINE11
      H(KM)   VP(KM/S)   VS(KM/S) RHO(GM/CC)    QP-INV     QS-INV   ETAP   ETAS  FREFP  FREFS
     1.0000     2.3695     1.0544     2.3184  0.172E-02  0.387E-02  0.00   0.00   1.00   1.00    
     9.0000     5.9401     3.4277     2.7020  0.160E-02  0.363E-02  0.00   0.00   1.00   1.00    
    20.0000     6.6868     3.8623     2.8986   0.00       0.00      0.00   0.00   1.00   1.00    
     0.0000     8.1461     4.6977     3.3626  0.194E-02  0.431E-02  0.00   0.00   1.00   1.00

The waveform preparation for use with wvfgrd96 was to bandpass filter the observed and theoretical ground velocities in m/sec by a 3-pole Butterworth bandpass filter with corners at 0.02 and 0.10 Hz. The following figure and table summarizes the grid search results.

Best mechanism fit as a function of depth

complete wavenumber integration synthetic Green's functions ( P, S and surface waves are all included) to find the mechanism and depth that best fit the NHSC waveforms. For compatibility with the previously displayed traces, the Green's functions and observed ground velocities were filtered as follows: 3-pole Butterworth highpass at 0.01 Hz and a 3-pole Butterworth lowpass at 0.06 Hz. The grid search looked at all possible focal mechanisms for depths between 0.5 and 29 km. The wvfgrd96 results for the best fit at each depth are

         Depth    Stk   Dip  Rake    Mw    Fit
WVFGRD96    0.5    40    80   175   3.07 0.0704
WVFGRD96    1.0   305    80   -10   3.48 0.2337
WVFGRD96    2.0   305    80    -5   3.63 0.2376
WVFGRD96    3.0    35    75   170   3.66 0.1422
WVFGRD96    4.0   170    50    90   3.85 0.2309
WVFGRD96    5.0   170    45    90   3.90 0.3732
WVFGRD96    6.0   165    45    85   3.91 0.4710
WVFGRD96    7.0   165    45    85   3.92 0.5343
WVFGRD96    8.0   170    45    90   3.92 0.5401
WVFGRD96    9.0   170    45    90   3.93 0.5584
WVFGRD96   10.0   170    45    90   4.02 0.5618
WVFGRD96   11.0   165    45    85   4.02 0.5619
WVFGRD96   12.0   165    45    85   4.02 0.5539
WVFGRD96   13.0   165    45    85   4.02 0.5385
WVFGRD96   14.0   275    45    95   3.85 0.5180
WVFGRD96   15.0   275    45    95   3.85 0.4949
WVFGRD96   16.0   275    45    95   3.85 0.4679
WVFGRD96   17.0   355    45   100   4.03 0.4396
WVFGRD96   18.0   355    45   100   4.03 0.4105
WVFGRD96   19.0   355    45   100   4.03 0.3795
WVFGRD96   20.0   355    45   100   4.03 0.3469
WVFGRD96   21.0   355    45   100   4.02 0.3134
WVFGRD96   22.0   360    45   105   4.02 0.2817
WVFGRD96   23.0   360    45   105   4.02 0.2509
WVFGRD96   24.0   155    45    70   4.01 0.2227
WVFGRD96   25.0   150    50    65   4.02 0.2157

Pareto - MOOP Analysis

Because of the small event size, a waveform fit is absolutely required to resolve the ambiguity of the surface-wave amplitude spectrum radiation pattern solutions. In addition the surface-wave solution favors a shallow depth and the waveform a deeper, with an overlap at a dep of about 10 km.

the output of srfgrd96 and wvfgrd96 programs were merged to form a new data set which look as like the following:



 Depth   Stk Dip Rake   FIT-SW  FIT-WF
5.000000 100  30   0  0.454300 0.996100
5.000000 100  30 100  0.494800 0.910000
5.000000 100  30 110  0.432100 0.921000
5.000000 100  30 120  0.390600 0.936200
5.000000 100  30 150  0.409900 0.987000
5.000000 100  30 160  0.444000 0.996900

The FIT is defined as '1-BEST' for the srfgrd96 output and '1-Reduction of Variance' for the wvfgrd96 output. The worst possible solution is one with a value of 1.0 and the best is one with a value of 0.0. The surface-wave data set was already seived by rejecting those solutions with a 'BEST < 0.5'.

A program by C. J. Ammon ( http://www.essc.psu.edu/~ammon/) was used to search for the set of solutions with optimized both observables. Because of the nest fit plots shown above, only the optimal solution(s) in the depth range of 5 - 15 km were searched for. The results of the analysis are:

Reading data from allfits
Mxpts 12182
Searching for non-dominated solutions


 The Complete Pareto Set of Solutions:
   8.0  170.0   50.0   90.0  0.1536  0.4706  0.4246  0.5754
   8.0  320.0   40.0   60.0  0.1490  0.6808  1.0000  0.0000
   8.0  350.0   40.0   90.0  0.1536  0.4706  0.4246  0.5754
   9.0  160.0   50.0   90.0  0.1512  0.4806  0.4936  0.5064
   9.0  340.0   40.0   90.0  0.1512  0.4806  0.4936  0.5064
  10.0  170.0   50.0   90.0  0.1629  0.4490  0.0000  1.0000
  10.0  350.0   40.0   90.0  0.1629  0.4490  0.0000  1.0000




 Balance is 0.33 for 2 functions
 Balanced Solutions:
   8.0  170.0   50.0   90.0  0.1536  0.4706  0.4246  0.5754
   8.0  350.0   40.0   90.0  0.1536  0.4706  0.4246  0.5754
   9.0  160.0   50.0   90.0  0.1512  0.4806  0.4936  0.5064
   9.0  340.0   40.0   90.0  0.1512  0.4806  0.4936  0.5064


Nadir Vector: 0.162900 0.680800
Ideal Vector: 0.149000 0.449000
   8.0  320.0   40.0   60.0  0.1490  0.6808  1.0000  0.0000
  10.0  170.0   50.0   90.0  0.1629  0.4490  0.0000  1.0000

The balanced solutions are subset of the Pareto set that have more-or-less equal importance to each objective function or data type. Requiring the equal weights removes solutions with a depth of 10 km in favor of those with depths of 8 and 9 km.

The Nadir vector is vector contains the largest misfits of any solutions in the Pareto set. The ideal vector contains the best possible fits for each set of observations. Note that no solution has the best fits for both data sets - the data are to some degree inconsistent because of noise in the observations or under the assumed earth model.

The last two entries list the best fitting SW and the best fitting WF solutions, for reference.

On the basis of this study the solution at 9 km is preferred. As a final check, the goodness of fit output for the two programs is listed:


Program   Depth   STK   DIP  RAKE       RR   RL     MWR    MWF  BEST-SW

SRFGRD96    9.0   170    50    90     0.905 0.917   4.07   4.08 0.8146


Program   Depth   STK   DIP  RAKE    MW  BEST-WV  RZ      RR      RT

WVFGRD96    9.0   170    50    90   3.93 0.5468  0.9155  0.7298  0.1671

Since both solutions give the same moment, this solution will be accepted with an M_W = 4.00.

Comparison of observed and predicted waveforms at NHSC for a source depth of 10km, strike of 170, dip of 50 and rake of 90 degrees. The same amplitude scale is used for all traces. The amplitudes have units of m/sec and observed and predicted waveforms are 3 pole bandpass filtered between 0.02 and 0.10 hz.

Comparison of observed and predicted waveforms at NHSC for a source depth of 10km, strike of 170, dip of 50 and rake of 90 degrees. The same amplitude scale is used for all traces. The amplitudes have units of m/sec and observed and predicted waveforms are 3 pole bandpass filtered between 0.02 and 0.50 hz.

It is very satisfying that the peak filtered velocities of the solution and earth model predictions at the high frequencies agree very well with the observed values. In addition the waveform timing is reasonably correct. An exact match is not expected since the earth model is crude at this point.

The Future

Should the national backbone of the USGS Advanced National Seismic System (ANSS) be implemented with an interstation separation of 300 km, it is very likely that an earthquake such as this would have been recorded at distances on the order of 100-200 km. This means that the closest station would have information on source depth and mechanism that was lacking here.

Acknowledgements

Dr. Harley Benz, USGS, provided the USGS USNSN digital data. Dr. Chuck Ammon of Penn State provide the MOOP code. Felipe Leyton of SLU helped implement the local use of MOOP.

References

Ammon, C. J. (2002). Seismic source inversion as a multiple-objective optimazation problem, GJI, submitted.

Appendix

The figures below show the observed spectral amplitudes (units of cm-sec) at each station and the theoretical predictions as a function of period for the mechanism given above. The CUS earth model was used to define the Green's functions.

Last Changed 02/11/14


Comparison of observed and predicted waveforms at NHSC for a source depth of 10km, strike of 170, dip of 50 and rake of 90 degrees. The same amplitude scale is used for all traces. The amplitudes have units of m/sec and observed and predicted waveforms are 3 pole bandpass filtered between 0.02 and 0.10 hz.

Comparison of observed and predicted waveforms at NHSC for a source depth of 10km, strike of 170, dip of 50 and rake of 90 degrees. The same amplitude scale is used for all traces. The amplitudes have units of m/sec and observed and predicted waveforms are 3 pole bandpass filtered between 0.02 and 0.50 hz.