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This is the last lecture in 12.510 2008. It includes a review of the moment tensor, focal mechanisms, radiation patter, seismic source, magnitude, magnitude saturation, and moment magnitude.
A quick review of the Moment Tensor and focal mechanisms
For a point source at ๐, ๐ the solution for the equation of motion is expressed with the Greenโs function ๐ฎ(๐, ๐) :
๐ข ๐ฅ, ๐ก ๐ท๐๐ ๐๐๐๐๐๐๐๐ก
๐๐๐ก๐๐๐๐ก๐ ๐๐ฃ๐๐ ๐ก๐๐ ๐ ๐๐๐๐๐๐ ๐ก๐๐๐ ๐๐ข๐๐๐ก๐๐๐
๐ ๐๐๐ก๐๐๐ ๐๐ฅ๐ก๐๐๐ก ๐๐ ๐ ๐๐ข๐๐๐
๐ ๐๐ข๐๐๐ ๐๐ข๐๐๐ก๐๐๐
The force is described by the following relation:
๐๐ = ๐ด๐ฟ ๐ฅ โ ๐ฅ โ ๐ฟ ๐ก โ ๐ก โ ๐ฟ๐๐ ; (2)
where๐ด is the amplitude, ๐, ๐ is the time, (๐, ๐) is the position, and n is the direction. Substituting equation (2) in the equation of motion and then solving for the displacement (u) resulting from the wave motion due to a point source, leads to the following relations:
๐ข๐ ๐ฅ, ๐ก = ๐บ๐๐ ๐ฅ, ๐ก; ๐ฅ, ๐ก ๐๐ ๐ฅ, ๐ก , ๐ข ๐ฅ, ๐ก ~๐บ(๐ฅ, ๐ก; ๐ฅ , ๐ก) ; ๐ข ๐ฅ, ๐ก = (^) ๐๐ฅ๐ ๐
where ๐ข๐ is the displacement, ๐๐ is the force vector. Greenโs function gives the displacement at
point x that results from the unit force function applied at point x. Internal forces ๐ must act in opposing directions โ๐, at a distance d so as to conserve momentum (force couple). For angular momentum conservation, a complementary couple balances the double couple forces. Figure 1 shows nine different force couples for the components of the moment tensor.
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Figure 1: Different force couples for the components of the moment tensor (Source: Shearer,
Single couple Double force couple
Figure 2: Single and double couple
Conserve momentum (angular) ๏จ force couple
Moment Tesnor
Starting with defining the moment tensor as:
โ๐
๐
๐
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We can diagonalize the moment matrix (equation 8) to find principal axes. In this case, the principal axes are at 45ยฐ to the original x 1 and x 2 axes, we get:
Principal axes become tension and pressure axis. The above matrix represents that 1 x โฒ coordinate is the tension axis, T , and 2xโฒ is the pressure axis, P. (see Figure 3)
Figure 3: The double-coupled forces and their rotation along the principal axes.(Source: Shearer,
Radiation Patterns
Figure 4: Radial componet
Flip goes to zero before it flips again
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The following figure shows the variation of the radiation patterns with the direction of the receiver. It assumed that the radiation field is in spherical coordinates (will be explained in this section), where ๐ is measured from the z axis and โ
is measured in the x-y plane.
Figure 4: The left graph shows body wave radiation patter for a double couple source. The right graph shows the radiation amplitude patterns of P and S waves in the x-z plane. (Source: Stein and Wysession, p221)
P-wave potential in spherical coordinate is given by dโAlembert
๐ ๐ฅ, ๐ก = ๐ ๐ก โ ๐ฅ ๐ผ + ๐(๐ก + (^) ๐ผ๐ฅ) ;
ฮฑ = P-wave speed, ๐ = ๐ผ^2 ๐ป^2 โ
;
ฮธ ฮธ
x 1 ฯ
x 1
x 1
x 1
x 3
x 2
Auxiliaryplane S waves T axis
Fault plane
x 2
x 2
x 3
x 3 x 3
x 1
x 1 x 1
x 3
x 1
x (^3) x 3
x 3
Receiver
+ +
-
+
+
**-
-**
Null axis
P waves
P waves
S waves
Auxiliary plane
Fault plane
(a)
(b)
(c)
The body wave radiation pattern for a double couple source has symmetry in the spherical coordinate system shown. ฮธ is measured from the x 3 axis, the normal to the fault ( x 1 - x 2 ) plane, and ฯ is measured in the fault plane. The P -wave radiation pattern has four lobes that go to zero at the nodal planes, which are the fault and auxiliary ( x 2 - x 3 ) planes. The S -wave radiation pattern describes a vector displacement that does not have nodal planes but is perpendicular to the P -wave nodal planes. S -wave motion converges toward the T axis, diverges from the P axis, and is zero on the null axis.
Radiation amplitude patterns of P and S waves in the x 1 - x 3 plane. a: Fault geometry, showing the symmetry of the double couple about the x 2 axis. b: Radiation pattern for P waves, showing the amplitude (left) and direction (right). c: Same as (b), but for S waves.
Figure by MIT OpenCourseWare.
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Figure 5: The relationships between near-field and far-field displacement and velocity (Source: Shearer, 1999).
Figure 6: Radial component
In spherical coordinates, the far field displacement is given by:
๐ข๐ ๐ฅ, ๐ก = (^4) ๐๐^1 ๐ผ (^2) ๐ ๐(๐ก โ ๐ ๐ผ) ๐ ๐๐ 2 ๐ ๐๐๐ ๐ ๐๐๐๐๐๐ก๐๐๐ ๐๐๐ก๐ก๐๐๐
The first amplitude term decays as (^) ๐^1. The second term reflects the pulse radiated from the fault,
๐ ๐ก , which propagates away with the P-wave speed ฮฑ ๏ and arrives at a distance ๐ at time ๐ก โ ๐๐ผ.
๐ ๐ก is called the seismic moment rate function or source time function. Its integration form in
๐ข๐
๐ข๐ง ๐ข๐ก
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terms of time is given by equation (9). The final term describes the P-wave radiation pattern depedig on the two nodal planes. The first term describes P-wave radiation pattern depending on the two angles (๐, ๐). At ๐ = ๐ = 90ยฐ^ , the siplacement is zero on the two nodal planes. The maximum amplitudes are between the two nodal planes. Figure 8 shows the far-field radiation pattern for P-waves and S-waves for a double-couple source.
Figure 7: The far-field radiation pattern for P-waves (top) and S-waves (bottom) for a double- couple source (Source: Shearer, 1999).
Magnitude
Seismic Source ๏จwe have to find the location
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The body wave magnitude mb : measured from the early portion of the body wave train:
Mb = log
A
T
Measurements of mb depend on the seismometer used and the portion of the wave train measured. Common practice uses a period of ~1sec for the P and ~4s for the S.
The surface wave magnitude Ms : measured using the largest amplitude (zero to peak) of the surface waves
where the first form is general and the second uses the amplitudes of Rayleigh waves with a period of 20 sec, which often have the largest amplitudes.
Limitations
These relations are empirical and thus no direct connection to the physics of earthquakes. Additionally, body and surface wave magnitudes do not correctly reflect the size of large earthquakes.
Magnitude saturation Itโs a general phenomenon for Mb above about 6.2 and Ms above about 8.3.
Figure 10 shows the theoretical source spectra of surface and body waves. The two are identical below the ๏ท-2^ corner frequency. As the fault length increases, the seismic moment increases and the corner frequency moves to the left, to lower frequencies. The moment M 0 determining the zero-frequency level becomes larger. However, Ms, measured at 20 s, depends on the spectral amplitude at this period. For earthquakes with moments less than 10^26 dyn-cm, a 20s period corresponds to the flat part of the spectrum, so Ms increases with moment. But for larger moments, 20s is to the right of the first corner frequency, so Ms does not increase as the same rate as the moment. Once the moment exceeds 5.10^27 dyn-cm, 20 s is to the right of the second corner. Thus Ms saturates at about 8.2 even if the moment increases. It is similar for body wave magnitude, which depends on the amplitude at a period of 1s. Because this period is shorter that 20s, mb saturates at a lower moment (~10^25 dyn-cm), and remains at about 6 even for larger earthquakes.
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Figure 10: Saturated body and surface wave magnitudes (Source: Stein and Wysession, p221).
Moment magnitude
A simple solution by Kanamori (equation 21) defines the magnitude scale based on the seismic moment. The moment magnitude:
๐๐ = ๐~ 1 ๐ ๐๐ ๐๐ = ๐~ 20 ๐ ๐๐
This expression gives a magnitude directly tied to earthquake source processes that does not saturate. Mw is the common measure for large earthquakes. Estimation of M0 requires more analysis than for mb or Ms. However, semi-automated programs like the Harvard CMT project
