Responds to IE Bulletin 79-07 re seismic stress analyses ...
Source parameters II Stress drop determination Energy balance Seismic energy and seismic efficiency...
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Transcript of Source parameters II Stress drop determination Energy balance Seismic energy and seismic efficiency...
![Page 1: Source parameters II Stress drop determination Energy balance Seismic energy and seismic efficiency The heat flow paradox Apparent stress drop.](https://reader035.fdocuments.in/reader035/viewer/2022062515/56649d215503460f949f7000/html5/thumbnails/1.jpg)
Source parameters II
• Stress drop determination• Energy balance• Seismic energy and seismic efficiency• The heat flow paradox• Apparent stress drop
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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!
![Page 3: Source parameters II Stress drop determination Energy balance Seismic energy and seismic efficiency The heat flow paradox Apparent stress drop.](https://reader035.fdocuments.in/reader035/viewer/2022062515/56649d215503460f949f7000/html5/thumbnails/3.jpg)
Source parameters II: stress drop
Seismic moment, stress drop and rupture dimensions are related.
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Source parameters II: stress drop
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.
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Source parameters II: stress drop
Use of aftershock distribution for stress drop determination
Figures from Lay and Wallace
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Source parameters II: stress drop
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
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Source parameters II: stress drop
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.
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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.
Source parameters II: stress drop
![Page 9: Source parameters II Stress drop determination Energy balance Seismic energy and seismic efficiency The heat flow paradox Apparent stress drop.](https://reader035.fdocuments.in/reader035/viewer/2022062515/56649d215503460f949f7000/html5/thumbnails/9.jpg)
Source parameters II: stress drop
Despite this uncertainty, the result that earthquake stress drops are typically 10-100 bars over a very wide range of seismic moments is convincing.
![Page 10: Source parameters II Stress drop determination Energy balance Seismic energy and seismic efficiency The heat flow paradox Apparent stress drop.](https://reader035.fdocuments.in/reader035/viewer/2022062515/56649d215503460f949f7000/html5/thumbnails/10.jpg)
Source parameters II: stress drop
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:
![Page 11: Source parameters II Stress drop determination Energy balance Seismic energy and seismic efficiency The heat flow paradox Apparent stress drop.](https://reader035.fdocuments.in/reader035/viewer/2022062515/56649d215503460f949f7000/html5/thumbnails/11.jpg)
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.
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Source parameters II: seismic energy
• The ground velocity is then:
• The kinetic energy is just:
• The average of this over one cycle gives the kinetic energy density:
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Source parameters II: seismic energy
• 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.
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Source parameters II: seismic energy
• 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!
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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?
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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:
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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.
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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 .
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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.
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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:
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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).
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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:
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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.
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Source parameters II: seismic efficiency
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.
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Source parameters II: seismic efficiency
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.
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Source parameters II: seismic efficiency
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
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Source parameters II: seismic efficiency
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:
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Source parameters II: seismic efficiency
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.
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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).
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Source parameters II: the heat flow paradox
A section parallel to the SAF plane:
Figure from: Scholz, 1990
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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.
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Source parameters II:
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.
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Source parameters II:
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.
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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
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Source parameters II: radiated energy versus seismic moment and the apparent stress drop
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
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Source parameters II: radiated energy versus seismic moment and the apparent stress drop
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)