Waveform Inversion using wvfgrd96

The focal mechanism was determined using broadband seismic waveforms. The location of the event and the and stations used for the waveform inversion are shown in the next figure.
Location of broadband stations used for waveform inversion

The program wvfgrd96 was used with good traces observed at short distance to determine the focal mechanism, depth and seismic moment. This technique requires a high quality signal and well determined velocity model for the Green functions. To the extent that these are the quality data, this type of mechanism should be preferred over the radiation pattern technique which requires the separate step of defining the pressure and tension quadrants and the correct strike.

The observed and predicted traces are filtered using the following gsac commands:

cut o DIST/3.3 -20 o DIST/3.3 +50
rtr
taper w 0.1
hp c 0.03 n 3 
lp c 0.10 n 3 
The results of this grid search from 0.5 to 19 km depth are as follow:

           DEPTH  STK   DIP  RAKE   MW    FIT
WVFGRD96    1.0   240    85    90   4.10 0.3487
WVFGRD96    2.0    30     5    55   3.96 0.4096
WVFGRD96    3.0    40    10    65   3.90 0.4386
WVFGRD96    4.0    50    10    75   3.87 0.4468
WVFGRD96    5.0    50    10    75   3.85 0.4414
WVFGRD96    6.0    65    10    85   3.84 0.4292
WVFGRD96    7.0   240    15  -100   3.85 0.4207
WVFGRD96    8.0    50    60   -80   3.89 0.4185
WVFGRD96    9.0    45    55   -80   3.91 0.4152
WVFGRD96   10.0    50    60   -80   3.91 0.3944
WVFGRD96   11.0    40    55   -85   3.92 0.3844
WVFGRD96   12.0    40    55   -85   3.92 0.3702
WVFGRD96   13.0    40    55   -85   3.91 0.3537
WVFGRD96   14.0   215    35   -90   3.91 0.3361
WVFGRD96   15.0   215    35   -90   3.91 0.3181
WVFGRD96   16.0   220    40   -80   3.92 0.3013
WVFGRD96   17.0   225    45   -75   3.92 0.2852
WVFGRD96   18.0   225    45   -75   3.92 0.2719
WVFGRD96   19.0   225    45   -75   3.93 0.2588
WVFGRD96   20.0   225    50   -75   3.94 0.2412
WVFGRD96   21.0   225    50   -75   3.95 0.2312
WVFGRD96   22.0    35    65   -90   3.95 0.2230
WVFGRD96   23.0   215    30   -85   3.96 0.2168
WVFGRD96   24.0    35    60   -90   3.96 0.2101
WVFGRD96   25.0    35    60   -90   3.97 0.2029
WVFGRD96   26.0   215    30   -85   3.97 0.1957
WVFGRD96   27.0   215    30   -85   3.98 0.1879
WVFGRD96   28.0   295    20   -85   4.02 0.1879
WVFGRD96   29.0   300    20   -75   4.03 0.1885

The best solution is

WVFGRD96    4.0    50    10    75   3.87 0.4468

The mechanism correspond to the best fit is
Figure 1. Waveform inversion focal mechanism

The best fit as a function of depth is given in the following figure:

Figure 2. Depth sensitivity for waveform mechanism

The comparison of the observed and predicted waveforms is given in the next figure. The red traces are the observed and the blue are the predicted. Each observed-predicted component is plotted to the same scale and peak amplitudes are indicated by the numbers to the left of each trace. A pair of numbers is given in black at the right of each predicted traces. The upper number it the time shift required for maximum correlation between the observed and predicted traces. This time shift is required because the synthetics are not computed at exactly the same distance as the observed and because the velocity model used in the predictions may not be perfect. A positive time shift indicates that the prediction is too fast and should be delayed to match the observed trace (shift to the right in this figure). A negative value indicates that the prediction is too slow. The lower number gives the percentage of variance reduction to characterize the individual goodness of fit (100% indicates a perfect fit).

The bandpass filter used in the processing and for the display was

cut o DIST/3.3 -20 o DIST/3.3 +50
rtr
taper w 0.1
hp c 0.03 n 3 
lp c 0.10 n 3 
Figure 3. Waveform comparison for selected depth. Red: observed; Blue - predicted. The time shift with respect to the model prediction is indicated. The percent of fit is also indicated.
Focal mechanism sensitivity at the preferred depth. The red color indicates a very good fit to thewavefroms. Each solution is plotted as a vector at a given value of strike and dip with the angle of the vector representing the rake angle, measured, with respect to the upward vertical (N) in the figure.

A check on the assumed source location is possible by looking at the time shifts between the observed and predicted traces. The time shifts for waveform matching arise for several reasons:

Assuming only a mislocation, the time shifts are fit to a functional form:

 Time_shift = A + B cos Azimuth + C Sin Azimuth

The time shifts for this inversion lead to the next figure:

The derived shift in origin time and epicentral coordinates are given at the bottom of the figure.