Source parameters II Stress drop determination Energy balance Seismic energy and seismic efficiency The heat flow paradox Apparent stress drop

Source parameters II

• Stress drop determination• Energy balance• Seismic energy and seismic efficiency• The heat flow paradox• Apparent stress drop

Source parameters II: use of empirical Green function for source-time function retrieval

• Recall that:

• A way to retrieve s is by deconvolving an empirical .

• This is done by use of a seismic recording from a small earthquake located near a larger event of interest.

• If the the source depth and focal mechanism of the two events are identical, the earth response to each station will be identical.

Note the directivity effect!

Source parameters II: stress drop

Seismic moment, stress drop and rupture dimensions are related.


How can the stress drop be estimated?

Rupture dimensions may sometimes be estimated from:

•the aftershock distribution

•geodetic observations.

•the corner frequency of the amplitude spectra.


Use of aftershock distribution for stress drop determination

Figures from Lay and Wallace


Even in cases where the spectra is of excellent quality, the precision of the corner frequency is modest.

Use of corner frequency for stress drop determination


The other problem is that the use of amplitude spectra for stress drop determination is extremely model-dependent.

Note that small differences in time function duration correspond to large differences in stress drop, even for assumed rupture velocity and fault geometry.

How uncertain is the stress drop?

The uncertainty is related to the uncertainties of each parameter in this expression:

A common approach for the standard deviation is to use the propagation of error relation:

It seems that the uncertainty of a stress drop is often a factor of 2-3.



Despite this uncertainty, the result that earthquake stress drops are typically 10-100 bars over a very wide range of seismic moments is convincing.


The near constancy of the stress drop implies that the ratio of average slip to fault length, i.e. the strain release is constant.

Taking a stress drop of 50 bars, shear modulus of 50x1011 dyn/cm2 yields:

Source parameters II: seismic energy

• The physical size of an earthquake is often described in terms of its seismic moment.

• An alternative measure of earthquake size is the energy release.

• To calculate the energy release, we consider the kinetic and potential energies of a material particle during the passage of the seismic waves.

• Consider a monochromatic source of seismic energy. In that case, the ground displacement at the station is given by:

where A is the amplitude of a wave whose period is T.


• The ground velocity is then:

• The kinetic energy is just:

• The average of this over one cycle gives the kinetic energy density:


• And since the kinetic and the potential energies are equal, we can write:

• To integrate over a spherical wavefront, the particle motion should be corrected for amplitude attenuation due to geometrical spreading:

• This can be recast as:

• Thus, if F is know, the seismic energy may be related to magnitude.


• Gutenberg-Richter obtained the following empirical relation between the seismic energy and the surface wave magnitude:

• This result highlights the tremendous range of earthquake size!

Source parameters II: a long-standing question

• Are faults weaker or stronger than the surrounding crust?

• Do earthquakes release most, or just a small fraction of the strain energy that is stored in the crust?

Source parameters II: Griffith criteria

The static frictionless case:

• UM is the mechanical energy.• US is the surface energy.

crack extends if:crack at equilibrium ifcrack heals if:

Source parameters II: dynamic shear crack

Dynamic shear crack:

Here, in addition to UM and US:• UK is the kinetic energy.• UF is the work done against friction.

During an earthquake, the partition of energy (after less before) is as follows:

where ES is the radiated seismic energy.

Source parameters II: dynamic shear crack

Question: what are the signs of UM, US and UF?

Let us now write expressions for UM , US and UF .

Source parameters II: elastic strain energy

The reduction in the elastic strain energy stored in the spring during a slip episode is just the area under the force versus slip curve.

For the spring-slider system, UE is equal to:

To get a physical sense of what UM is, it is useful to consider the spring-slider analog.

Source parameters II: elastic strain energy

where 1 and 2 are initial and final stresses, respectively, and the minus sign indicates a decrease in elastic strain energy.

Similarly, for a crack embedded within an elastic medium, UM is equal to:

Source parameters II: frictional dissipation and surface energy

where F is the friction, V is sliding speed, and t is time. Frictional work is converted mainly to heat.

The frictional dissipation:

The surface energy:

where is the energy per unit area required to break the atomic bonds, and A is the rapture dimensions. Experimental studies show that is very small, and thus its contribution to the energy budget may be neglected (but not everyone agrees with this argument).

Source parameters II: the simplest model

Consider the simplest model, in which the friction drops instantaneously from 1 to 2.

In such case: F=2, and we get:

Source parameters II: seismic efficiency

We define seismic efficiency, , as the ratio between the seismic energy and the negative of the elastic strain energy change, often referred to as the faulting energy.

which leads to:

with being the static stress drop. While the stress drop may be determined from seismic data, absolute stresses may not.


The static stress drop is equal to:

where G is the shear modulus, C is a geometrical constant, and the tilded L is the rupture characteristic length.

The characteristic rupture length scale is different for small and large earthquakes.

For small earthquakes, and . Combining this with the expression for seismic moment we get:

Both M and r may be inferred from seismic data.


Figure from: Hanks, 1977

Stress drops vary between 0.1 and 10 MPa over a range of seismic moments between 1018 and 1027 dyn cm.


constraints on absolute stresses: In a hydrostatic state of stress, the friction stress increases with depth according to:

where is the coefficient of friction, g is the acceleration of gravity, and c and w are the densities of crustal rocks and water, respectively.

Laboratory experiments show:

Byerlee, 1978

Friction measured at maximum stress


Using:

, the coefficient of friction = 0.6c, rock density = 2600 Kg m-3

w, water density = 1000 Kg m-3

g, the acceleration of gravity = 9.8 m s-2

D, the depth of the seismogenic zone, say 12x103 m

We get an average friction of:

and the inferred seismic efficiency is:


So, the radiated energy makes only a small fraction of the energy that is available for faulting.

Based on this conclusion a strong heat-flow anomaly is expected at the surface right above seismic faults.

Source parameters II: the heat flow paradox

At least in the case of the San-Andreas fault in California, the expected heat anomaly is not observed.

The disagreement between the expected and observed heat-flow profiles is often referred to as the HEAT FLOW PARADOX.

Heat flow as a function of distance from the San Andreas fault in the Mojave segment. Theoretical anomaly is for a slip velocity of 25 mm/yr and average friction of 50 MPa (Lachenbruch and Sass, 1988).

Source parameters II: the heat flow paradox

A section parallel to the SAF plane:

Figure from: Scholz, 1990

Source parameters II:

The basis for the heat flow paradox is a simplified model, according to which the slip is occurring along a single plane. In practice, however, the deformation is distributed among several faults.


Other conceptual models:constant friction slip weakening quasi-static

• The simple model.• The slip-weakening model. Significant amount of energy is dissipated in the process of fracturing the contact surface. In the literature this energy is interchangeably referred to as the break-up energy, fracture energy or surface energy.• A silent (or slow) earthquake - no energy is radiated.


In reality, things are probably more complex than that.

We now know that the distribution of slip and stresses is highly heterogeneous, and that the source time function is complex.

Source parameters II: radiated energy versus seismic moment and the apparent stress drop

Radiated energy and seismic moment of a large number of earthquakes have been independently estimated. It is interesting to examine the radiated energy and seismic moment ratio.

Figure from: Kanamori, Annu. Rev. Earth Planet. Sci., 1994


Remarkably, the ratio of radiated energy to seismic moment is fairly constant over a wide range of earthquake magnitudes.

Figure from: Figure from Kanamori and Brodsky, Rep. Prog. Phys., 2004


What is the physical interpretation of the ratio ES to M0? Recall that the seismic moment is:

and the radiated energy for constant friction (i.e., F = 2):

Thus, ES/M0 multiplied by the shear modulus, G, is simply:

This is often referred to as the 'apparent stress drop’(apparent, because it is based on a highly simplified model)

Documents

Source parameters II Stress drop determination Energy balance Seismic energy and seismic efficiency The heat flow paradox Apparent stress drop