Quantifying and characterizing crustal deformation

1. 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 geologic displacement over a fault whose area is Ae. Thus, the geometry moment is simply 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: For California: What are the implications of these results?

10. 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:

11. 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

12. 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.

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

14. 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:

15. 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:

16. 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, 172-186, 1998.