1. Coulomb Stress Changes and the Triggering of Earthquakes
King et al., 1994

2. Outline Mohr Circle Coulomb Failure Criterion
Procedures to Calculate Coulomb Stress Change
Results

3. Mohr Circle s1 s3 a b s3
Mohr Circle is a graphical representation of calculating shear and normal tractions on any plane in terms of the principal stress components.

4. Coordinate Transformation
To write matrices or vectors in a rotated coordinate frame, we need a coordinate transformation
Or more mathematically, we can write a vector in two coordinate frames and calculate the transformation matrix.

5. Coordinate Transformation of the Stress Tensor

6. Normal and Shear Tractionb
Normal and Shear Traction
Considering
= 90 –b, or 2a=180-2b
We get the eq (2) and (3) of the paper.

7. Coulomb Failure Criterionsb tb
δsf/ δb=0 => b=1/2. arctan(-m`) 2b
Tan(f)=m

8. Coulomb Stress on a Randomly Oriented Fault Plane
Given we can calculate shear and normal traction on any plane by transforming this vector with transformation matrix as before so s`=A.s.AT

9. Coulomb Stress Change Calculation procedure
Impose the initial displacement field and calculate deformation
Calculate stress tensor at all points of interest.
Rotate them to the pre determined “fault” orientations.
4. Calculate shear and normal tractions
5. Calculate coulomb stress change to see whether it would increase or decrease seismicity.

10. Adding Regional Stress and Optimally Oriented Faults
Assumption: There are many faults with various orientations on every part of the region. So faults exist at every point with angle b from the principal stress direction of the orientation of tan(2.b)=-m.
The orientation of the “fault” is determined from the summation of earthquake induced stress and the regional stress.
After determining the optimally oriented faults, the Coulomb stress changes are calculated using the stress changes due to earthquake only.

12. The significance of Regional Stress

13. 2003 Big Bear EQ
King et al., 1994

14. Results from Homestead Valley & Joshua Tree Earthquakes

15. Depth Profile and Fault Length Extent

16. Landers and pre-Landers Loading
The Coulomb failure stress change due to 5 largest earthquakes around landers, might have advanced the occurance of the Landers earthquake by couple of centuries.

18. Effects of Landers on SAF and SJF

19. Conclusions
Coulomb failure criterion estimate the aftershocks of Homestead Valley, Big Bear and Landers earthquakes.
0.5 bar of Coulomb stress change is sufficient to trigger earthquakes.
The forces on the adjacent fault will be higher than the ones estimated from Coulomb stress change calculations due to later inelastic stress relaxation in the lower crust
Regions of enhanced Coulomb stress are candidates for future events.

20. Problems with King et al., 1994 Coulomb Stress Triggering Study
Does not address the time dependence of the triggering.
(Tidal forces cannot trigger similar number of earthquakes).
Omori’s Law can not be obtained.
Optimally oriented faults might be a big assumption.
The background seismicity is ignored.
More robust statistical tests should have been done to understand the level of confidence.

21. Addressing optimally oriented faults Issue
Do aftershocks actually occur in the planes that are estimated by Coulomb stress changes?
This issue is addressed by Hardebeck et al., 1998.
They do a statistical test, to check if the orientations of aftershocks are consistent with the ones estimated from Coulomb stress changes.
Method: take events with aftershocks that have well determined source mechanisms (Landers 1992 and Northridge 1989).
Calculate a synthetic random mechanism catalog.
Compare whether the real catalog has a statistically more closer to “optimally oriented plane”.
This article assumes 0 tectonic stress.

22. Result of the test
For Landers, there is a statistically significant difference between the random mechanisms and the real mechanism orientations.
For Northridge the null hypothesis cannot be rejected.

23. Landers
Aftershocks with at least one plane consistent with static stress change triggering direction
Both planes inconsistent with static change triggering direction

24. Northridge

25. For events very close to the fault (<5km) or far from fault (>75 km), there is no statistical improvement of optimal orientations compared to random catalog.
Aftershocks are more consistent with regional stress field. Depending on how much of the regional stress field is released, the aftershock orientations become more or less consistent with maximum Coulomb stress change direction.
I think both of the issues are addressed in the King et al, 1994 paper.

26. Static vs. Dynamic Triggering (Kilb et al., 2002)

27. Static vs. Dynamic Triggering
Near the fault, dynamic stresses are much higher than the static stress changes.
Kilb et al., 2002, studies dynamic CFS change to understand whether dynamic stress change can explain earthquake triggering better.
Kilb et al., 2002

29. Removing Seismicity Rate Change

30. Seismicity Rate Change and peak of DCFS(t)>4 MPa
Seismicity Rate Change and DCFS>0.1MPa contours