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Magnitude
After learning to locate earthquakes, we now need to quantify their size. The first measure is the magnitude, which is based on the amplitude of the waves recorded on a seismogram. Concept: the wave amplitude reflects the earthquake size once the amplitudes are corrected for the decrease with distance due to geometric spreading and attenuation. Magnitude scales have the general form:
M = log
⎝⎜^
A ⎞
where A: amplitude of the signal T: its dominant period f: correction for the variation of amplitude with the earthquake’s depth h and distance ∆ from the seismometer C: regional scale factor
For global studies, the primary magnitude scales are:
- The body wave magnitude m (^) b : measured from the early portion of the body wave train:
mb = log ⎝⎜
⎛ A^ ⎞ (^) + Q ( h , ∆ ) 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 M (^) s : measured using the largest amplitude (zero to peak) of the surface waves
M = log ⎛ ⎝⎜^
A ⎞
s +^ 1.66 log^ ∆ +^ 3. T ⎠⎟ M (^) s = log A 20 + 1.66 l og ∆ + 2.
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.
- Body and surface wave magnitudes do not correctly reflect the size of large earthquakes.
Magnitude saturation It’s a general phenomenon for m (^) b above about 6.2 and Ms above about 8.3.
May 9, 2005
The figure below shows the theoretical source spectra of surface and body waves. The two are identical below the ω-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.
0
(^
)
( )
8
L ( )
7
5 4
3
2
0 2
M (^) s
M (^) s m^ b
m (^) b
0 2
Log M
dyn-cm
Log f Hz
76
km
43 24
10
18
= 6
determined
20
22
24
26
28
Explanation for the saturation of body and surface wave magnitudes.
= 5.
SURFACE WAVES BODY WAVES
Figure by MIT OCW. (Adapted from Stein and Wysession textbook)
2
May 9, 2005
Tsunami
(Stein & Wysession 2.8.4)
Dispersion is observed for tsunamis, the water-waves generated by earthquakes. They involve gravitational potential energy stored by vertical displacements of the water. Tsunami dispersion is similar to that of Rayleigh and Love waves, in that the waves with longer periods travel faster and thus arrive earlier. At long periods, where the wavelengths are much greater than the ocean depth d, the phase velocities are essentially non-dispersive and are given by:
c = gd
where g is the acceleration gravity. Tsunami velocities depend on ocean depth. However, at shorter periods, where the wavelengths are much less than the ocean depth and so do not feel the ocean floor, the tsunami velocities depend on wavelength as
c = λ g / 2 π
so shorter waves travel more slowly.
Tsunami warning system: An example of detection system is represented below. These stations give detailed information about tsunamis while they are still far off shore. Each station consists of a sea-bed bottom pressure recorder (at a depth of about 6000 m) which detects the passage of a tsunami and transmits the data to a surface buoy via sonar. The surface buoy then radios the information to the Pacific Tsunami Warning Center (PTWC) via the GOES satellite system. The bottom pressure recorder lasts for two years while the surface buoy is replaced every year. The system has considerably improved the forecasting and warning of tsunamis in the Pacific.
May 9, 2005
More on that: http://en.wikipedia.org/wiki/Tsunami_warning_system
