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

The seismic cycle - the simplest view

The seismic cycle - the third dimension

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

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

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.

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.

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

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:

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:

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

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.

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

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:

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:

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

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.

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:

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

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

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.