1. Second degree moments – a tool for the fault plane detection?
Petra Adamová
Jan Šílený
Geophysical Institute, Academy of Sciences, Prague, Czech Republic

2. Introduction to second degree moments
Standard moment tensor
Point source: couple of planes

3. Introduction to second degree moments
“Finite source” parameters from point source approximation
traditional modeling of slip on fault plane is costly and data are not available (no stations near the fault)
2nd degree moments are advantageous alternative
geometry of the source
duration of the source process
spatial and temporal centroid
rupture velocity vector

4. Second degree moments, Doornbos (1982)
Theory
Zero degree moment tensor (standard MT)
Second degree moments, Doornbos (1982)
Temporal centroid
Spatial centroid
Rupture propagation
Source process duration
Source ellipsoid

5. Inverse scheme: full waveform inversion
Standard MT
Estimation of first and second degree moments
exclusion of the non-physical solutions  inversion is faster

6. Interpretation of second degree moments
Centroid
Rupture propagation vector
Source ellipsoid

7. Detection of the fault plane
Moment tensor (DC part)  couple of planes (fault and auxiliary)  ambiguity
2nd degree moments  source ellipsoid (volume of the focus)
Removing of the ambiguity:
Plane containing the source ellipsoid (large axis)  fault plane
Convenient geometry
Inconvenient geometry

8. Ridgeway Mine, Australia
Seismicity in December (ISS International)
Malovichko, D. [2010] Ridgeway. Routine estimation of the source mechanisms: December 2009.
ISSI Document Number: ISSREP_RWM_MECHANISMS

9. Location of selected events2 5 3 4 2 1 5 3 4 1
Location of 5 events from Ridgeway mine from December 2009

10. Moment tensor solution 2009 Dec 04 07:35:06, M = 2.1
IMS solution (spectral amplitudes)
P and S wave amplitude inversion
Full waveform inversion
Frequency range Hz

11. Second degree moments (1) 2009 Dec 04 07:35:06, M = 2.1
Source ellipsoid
3-D view

12. Second degree moments (2) 2009 Dec 04 07:35:06, M = 2.1
Temporal centroid = 0.01 sec
Duration of the source process = 0.1 sec
Average rupture velocity 2.0 km/s
H – hypocenter
C – spatial centroid position

13. Moment tensor solution 2009 Dec 04 03:19:07 M = 1.3
DC = 32 %
CLVD = 23 %
ISO = 46 %
IMS solution (spectral amplitudes)
P wave amplitude inversion
DC = 38 %
CLVD = 25 %
ISO = 37 %
Full waveform inversion
Frequency range Hz
(Corner frequency ~ 35 Hz)

14. Second degree moments 2009 Dec 04 03:19:07 M = 1.3
Temporal centroid = sec
Duration of the source process = 0.15 sec
Average rupture velocity 1.8 km/s
Inconvenient geometry for estimation of the remaining parameters: source ellipsoid and rupture velocity vector lie on the intersection of 2 nodal planes
We estimated second degree moments successfully but we cannot recognize which plane is the fault plane

15. Jack-knife test, event 3 Removing one or 2 stations from the data set
All jack-knife trials
All jack-knife trials except removal of stations 25 and 39

16. Jack-knife test, second degree moments
Temporal centroid sec For all jack-knife trials Source duration 0.1 sec nearly the same values
Rupture velocity 2.1 km/sec
Spatial centroid 1 m NS, 1 m EW, -10 m Z from the hypocenter
Source ellipsoid
Fault orientation
Gray ellipsoid: inversion from all stations
Dashed ellipsoids: jack-knife trials with extreme deviations

17. Standard MT for selected events
Fault plane

18. Source ellipsoids Removal of ambiguity of the two nodal planes
Correlation with geological faults

19. Conclusions a chance to remove ambiguity of the two nodal planes
standard moment tensor: dot
second degree moments: beyond dot:
estimate of source shape, rupture propagation vector, …
second degree moments bring additional information to standard MT: source ellipsoid, duration of the source process, average rupture velocity
a chance to remove ambiguity of the two nodal planes