From faults to earthquakes: scaling relations · From faults to earthquakes: brittle strain The geometric moment for faults is: Mf = U Af; where U is the mean geological displacement

From faults to earthquakes: scalingrelations

Displacement versus length of faults

What emerges from this data is a linear scaling

between displacement, U , and fault length, L:

U = ηL

Coseismic slip versus rupture length

What emerges from this data is that co-seismic

stress drop is constant over wide range of

earthquake sizes

Recall that the stress drop, ∆τ is given by:

∆τ ∝ γ∆U

r

Thus, the constancy of ∆τ implies linear

scaling between co-seismic slip, ∆U , and

rupture dimension, r:

∆U = ηr

From faults to earthquakes: brittlestrain

The geometric moment for faults is:

Mf = UAf ,

where U is the mean geological displacement

on a fault of area Af .

Similarly, the geometric moment for

earthquakes is:

Me = ∆UAe,

where ∆U is the mean co-seismic slip on a

rupture of area Ae. Thus, Me is the seismic

moment divided by the shear modulus.

Brittle strain can be expressed in two ways

[Kostrov, 1974]]:

From faults:

εij =1

2V

∑

k

[Mf ij]k

From earthquakes:

εij =1

2V

∑

k

[Meij]k

To illustrate the logic behind those equations,

consider the simple case of a plate of brittle

thickness W ∗ and length and width l1, l2 being

extended in the x1 direction by a population of

parallel normal faults of dip ϕ.

The mean displacement of the right-hand face

is:

U =∑

k

Uk cos ϕL2

k sin ϕ

W ∗l2

which may be re-arrange to give:

U

l1= ε11 =

cos ϕ sin ϕ

V

∑

k

UkL2

k

Geodetic data may also be used to compute

brittle strain.

εgeologic

Advantages:

long temporal sampling (Ka, Ma)

Disadvantages:

only fault that are exposed at the surface

cannot discriminate seismic from aseismic

εgeodetic

Advantages:

counts all contributing sources, buried or not

Disadvantages:

Short temporal window

εseismic

Advantages:

Spatial resolution better than that of the

geologic

Disadvantages:

Very short temporal window

Owing to their contrasting perspective, it is

useful to compare:

εgeologic versus εseismic

εgeodetic versus εseismic

εgeologic versus εgeodetic

Steven Ward has done exactly this for the

United States:

What does it mean?

For southern and northern California,

Mgeodetic/Mgeologic ≈ 1.2.

For California: Mseismic/Mgeologic ≈ 0.9.

For California:

Mseismic/Mgeodetic ≈ 0.86 − 0.73.

Recommended 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.

Paleoseismology

What can we learn from precariousrocks?

In the 1989 oblique-slip Loma Prieta

earthquake in California, there were

numerous reports of massive objects (e.g.,

cars) that were thrown into the air,

indicating that at least locally the ground

accelerations exceeded 1 g.

Precarious boulders may be used to place

limits on the amount of shaking in the past.

Field tests and modeling of the forces required

to topple the boulders indicate that

accelerations of greater than 0.2 g would

knock over the more precarious boulders,

whereas accelerations of 0.3-0.4 g would be

required for the ”semi-precarious” boulders.

Such numbers provide very useful

paleoseismological limits on the magnitude

of past shaking.

What can we learn from coral heads?

Trench

Fissure filling

Recurrence intervals

Stratigraphic and structural relationships

Offset of a channel

Coke can