The Richter Magnitude Scale


The Richter magnitude is based on the formula:
		ML = log(A/Azero)
which can also be written:
                A = Azero 10ML

where A is the zero-to-peak amplitude measured in mm on a Wood-Anderson photographic instrument and Azero is the predicted amplitude for a magnitude zero earthquake. Azero is, of course, a function of distance. The table given by Richter can be approximated, out to 200 km distance, by the formula
          
		log(Azero) = 0.15 - 1.6 log(distance in km) 

and between 200 and 600 km distance by
		log(Azero) = 3.38 - 3.0 log(distance in km) 

By the approximation above, log(Azero) is equal to minus 3.05 at 100 km. In Richter's table (Richter, Charles F., 1958, Elementary Seismology, W.H. Greeman and Company, San Francisco, page 342) the value is exactly minus 3.0, so the zero-to-peak amplitude of a magnitude zero event measured on a Wood-Anderson seismogram would be equal to .001 mm at 100 km distance.

To compute the ground motion of a magnitude zero event at 100 km distance, one must know the sensitivity of the Wood-Anderson instrument. For frequencies well above the natural frequency (1.25 Hz) of the Wood-Anderson, the magnification is be 2080 (see Urhammer and Collins, 1990, Bulletin of the Seismological Society of America, v. 80, p. 702-716). This means that the zero-to-peak ground motion for a magnitude zero earthquake measured 100 km away is
        mag 0        .001/2080 mm = 0.00000048 mm = .00048 microns
For larger magnitudes, the ground motion would be:
        mag 1        .0048 microns
        mag 2        .048 microns
        mag 3        .48 microns
        mag 4       4.8 microns          .0048 mm
        mag 5      48 microns            .048 mm
        mag 6     480 microns            .48 mm
        mag 7   4,800 microns           4.8 mm

How could a magnitude be less than zero? Remember the formula:
		ML = log(A/Azero)

If A is less than Azero, then ML will equal the log of a number less than one, and will be negative. To continue the table above for smaller events:
        mag 0    	.00048 microns
	mag -1		.000048 microns
	mag -2		.0000048 microns

For comparison, the thickness of a human hair is about 50 microns (the range is 30 to 100 microns), and the thickness of a cotton fiber is about 25 microns.

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