1. Kinematics VI: Quantifying and characterizing crustal deformation
The geometric moment
Brittle strain
The usefulness of the scaling laws

2. Quantifying and characterizing crustal deformation: 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 co-seismic displacement over an earthquake rupture whose area is Ae. (The geometry moment is in fact the seismic moment divided by the shear modulus.)

3. Quantifying and characterizing crustal deformation: brittle strain
Brittle strains are a function of the geometric moment as follows [Kostrov, 1974]:
Geologic brittle strain:
Seismic brittle strain:

4. Quantifying and characterizing crustal deformation: brittle strain
To illustrate the logic behind these equations, 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:

5. Quantifying and characterizing crustal deformation: brittle strain
Geodetic data may also be used to compute brittle strain:

6. Quantifying and characterizing crustal deformation: brittle strain
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.

7. Quantifying and characterizing crustal deformation: brittle strain
Seismic brittle strain:
Advantages:
Spatial resolution is better than that of the geologic brittle strain.
Disadvantages:
Short temporal window.
Owing to their contrasting perspective, it is interesting to compare:

8. Quantifying and characterizing crustal deformation: brittle strain
Ward (1997) has done exactly this for the United States:

9. Quantifying and characterizing crustal deformation: brittle strain
For Southern and Northern California:
What are the implications of these results?
“The near unit ratio points to the completeness of the region’s fault data and to the reliability of the geodetic measurements there.” (Ward, 1998)

10. Quantifying and characterizing crustal deformation: brittle strain
In the Basin and Range, northwest and central USA:
“Of possible causes, high incidences of understated and unrecognized faults…” (Ward, 1998)

11. Quantifying and characterizing crustal deformation: brittle strain
Everywhere:
The ratio runs systematically from 70-80% in the fastest straining regions, to 2% in the slowest.
“Although aseismic deformation may contribute to this shortfall, I (Steven Ward) argue that existing seismic catalogs fail to reflect the long-term situation.”
“Slowly straining regions require a proportionally longer period of observations.” (Ward, 1998)

12. Quantifying and characterizing crustal deformation: fault scaling relations
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:

13. Cumulative length distribution of faults:
Quantifying and characterizing crustal deformation: fault scaling relations
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

14. Quantifying and characterizing crustal deformation: fault scaling relations
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.

15. Quantifying and characterizing crustal deformation: fault scaling relations
Furthermore, recall that the geologic brittle strain is:
Using:
one can write:

16. Quantifying and characterizing crustal deformation: earthquake scaling relations
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:

17. Quantifying and characterizing crustal deformation: earthquake scaling relations
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:

18. Quantifying and characterizing crustal deformation: brittle strain
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.