A Note on the Ratio of the Moment Magnitude Scale to Other Magnitude Scales: Theory and Applications

The relationship between the moment magnitude scale and other magnitude scales is a subject of continuing research ever since the moment magnitude ( M w ) was first proposed. Empirical results show that in western North America, the moment magnitude is greater than, equal to, and less than the Richter ( M L ) and Nuttli magnitudes ( m b Lg ), whereas over 97% of the earthquakes in eastern North America have an M w value that is less than M L , m b Lg , or M e . To explain the large differences that exist between the two regions, first it is shown theoretically that the magnitudes that are based on a peak amplitude measurement are approximately equal to the energy magnitude ( M e ) that is based on the square root of the observed seismic trace Lg coda energy. If we define p as the ratio M w / M e and α as the seismic moment scaling constant, then it is shown theoretically that p =(4/9) α . Most of the major fault‐parameter relations such as stress drop, corner frequency, fault area, etc., can be obtained from the observed ratio of M w / M e . The value of this ratio is not unique but depends on the nature of faulting and the tectonic environment. For self‐similar earthquakes α =3, M w =(4/3) M e . For most earthquakes in western North America, α varies from 2 to 3, leading to a range of values for the ratio M w / M e , whereas for most earthquakes in eastern North America ( α ≈2 and M w ≈(8/9) M e ). This type of scaling for small‐to‐intermediate earthquakes may be due to earthquake fault zones being shaped by the geometry of sloping rock layers within the crust.