Quantifying and characterizing crustal deformation • The geometric moment • Brittle strain • The usefulness of the scaling laws

Quantifying and characterizing crustal deformation: the geometric moment The geometric moment for faults is:

M f



U A f

.

where U is the mean geologic displacement over a fault whose area is A f .

 Similarly, the geometric moment for earthquakes is:

M e

 

U A e

.

where  U is the mean geologic displacement over a fault whose area is A e . Thus, the geometry moment is simply the seismic 

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

f

 1 2

V



k M f k

.

Seismic brittle strain:  

e

 1 2

V



k M e k

.



Quantifying and characterizing crustal deformation: brittle strain To illustrate the logic behind these equations, consider the simple l case of a plate of brittle thickness W* and length and width l 1 2 , respectively, that is being extended in the x population of parallel normal faults of dip  .

1 direction by a and The mean displacement of the right hand face is: ˆ  

U k

cos 

L k

2

Wl

2

k

sin  , which may be rearrange to give: 

l

1   11  cos  sin 

V



U k L k

2 .

k



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

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.

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:  geologic  seismic  geodetic  seismic  geologic  geodetic 

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

Quantifying and characterizing crustal deformation: brittle strain For Southern and Northern California: 

geo

det

ic



geo

log

ic



1.2 .

For California:  

seismic



geo

log

ic



seismic



geo

det

ic



0.9



0.86



0.73 .

What are the implications of these results?



 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:

U

 

L

.

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:

N

(

L

) 

aL



C

,  where a and C are fitting coefficients.

figure from Scholz San Andreas subfaults

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:

M f



U A f

, and since:

U

 

L

, 

M f

 

L

3

.

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   for the entire population.

Quantifying and characterizing crustal deformation: fault scaling relations Furthermore, recall that the geologic brittle strain is: 

f

 1 2

V



k M k f

 1 2

V

 k  L 3 k .

Using:

N

(

L

) 

aL



C

,  

f

 

2

V



a

L

3-C

dL

 

a

2

V

4

L

4 

C



C

.



 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: log

N

( 

M

) 

a



bM

.

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: 

U

 

r

.



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.