1. Plate tectonics: Quantifying and characterizing crustal deformation
The seismic cycle
The seismic moment and magnitude
The geometric moment
Brittle strain
The usefulness of the scaling laws

2. The seismic cycle - the simplest view

3. The seismic cycle - the third dimension

4. The seismic cycle - evidences for aseismic slip
Relocated seismicity along the Calaveras fault in CA (Rubin, 2002):
creep?

5. The seismic cycle - evidences for aseismic slip
Relocated seismicity near the southern end of the Loma Prieta rupture (Schaff and Beroza, 2004):
creep
stick-slip

6. The seismic moment
The seismic moment is a physical quantity (as opposed to earthquake magnitude) that measures the strength of an earthquake. It is equal to:
where:
G is the shear modulus
A is the rupture area
U is the average co-seismic slip
face on rupture plane U
And by the way, the (moment) magnitude is related to the seismic moment, M0, as:
where M0 is in dyne-cm.

7. The geometric moment
The geometric moment for faults is:
where U is the mean geologic displacement over a fault whose area is Af.
Similarly, the geometric moment for earthquakes is:
where U is the mean geologic displacement over a fault whose area is Ae. Thus, the geometry moment is simply the seismic moment divided by the shear modulus.

8. The brittle strain
Brittle strains are a function of the geometric moment as follows [Kostrov, 1974]:
Geologic brittle strain:
Seismic brittle strain:

9. The brittle strain
To see the logic behind these equations, it is useful to consider the simple case of a plate of brittle thickness W* and length and width l1 and l2, respectively, that is being extended in the x1 direction by a population of parallel normal faults of dip .
The mean displacement of the right-hand face is:
which may be rearrange to give:

10. The use of fault scaling relations to calculate the geologic brittle strain
The use of scaling relations allows one to extrapolate beyond one’s limited observational range.
Displacement versus fault length
What emerges from this data is a linear scaling between average displacement, U, and fault length, L:

11. Cumulative length distribution of faults:
The use of fault scaling relations to calculate the geologic brittle strain
Cumulative length distribution of faults:
Normal faults on
Venus
Faults statistics obeys a power-law size distribution. In a given fault population, the number of faults with length greater than or equal to L is:
where a and C are fitting coefficients.
San Andreas
subfaults
figure from Scholz

12. The use of fault scaling relations to calculate the geologic brittle strain
These relations facilitate the calculation of brittle strain. Recall that the geometric seismic moment for faults is:
and since:
the geometric seismic moment may be written as:
This formula is advantageous since: 1. It is easier to determine L than U and A; and 2. Since one needs to measure U of only a few faults in order to determine  for the entire population.

13. The use of fault scaling relations to calculate the geologic brittle strain
Furthermore, recall that the geologic brittle strain is:
Using:
one can write:

14. The use of earthquake scaling relations to calculate the seismic brittle strain
Similarly, in order to calculate the brittle strain for earthquake, one may utilize the Gutenberg-Richter relations and the scaling of co-seismic slip with rupture length.
Gutenberg-Richter relations:

15. The use of earthquake scaling relations to calculate the seismic brittle strain
Seismic moment versus source radius
What emerges from this data is that co-seismic stress drop is constant over a wide range of earthquake sizes.
The constancy of the stress drop, , implies a linear scaling between co-seismic slip, U, and rupture dimensions, r:

16. The geodetic brittle strain
Geodetic data may also be used to compute brittle strain:

17. Comparing geodetic, geologic and geodetic brittle strains
Geologic brittle strain:
Advantages:
Long temporal sampling (Ka or Ma).
Disadvantages:
Only exposed faults are accounted for.
Cannot discriminate seismic from aseismic slip.
Geodetic brittle strain:
Advantages:
Accounts for all contributing sources, whether buried or exposed.
Disadvantages:
Short temporal window.

18. Comparing geodetic, geologic and geodetic brittle strains
Seismic brittle strain:
Advantages:
Spatial resolution is better than that of the geologic brittle strain.
Disadvantages:
Short temporal window.
Does not account for the aseismic strain.
Owing to their contrasting perspective, it is interesting to compare:

19. Comparing geodetic, geologic and geodetic brittle strains
Ward (1997) has done exactly this for the United States:

20. Comparing geodetic, geologic and geodetic brittle strains
For Southern and Northern California:
For California:
What are the implications of these results?

21. Further reading:
Scholz C. H., Earthquake and fault populations and the calculation of brittle strain, Geowissenshaften, 15, 1997.
Ward S. N., On the consistency of earthquake moment rates, geological fault data, and space geodetic strain: the United States,
Geophys. J. Int., 134, , 1998.