Earthquake magnitude, intensity, energy, and acceleration*

Abstract

Summary The paper investigates the principal physical elements of earthquakes: the magnitude M, energy E, intensity I, acceleration a, and their relation to the depth h and radius of perceptibility r. (r2 + h2 = R2. Subscript zero (0) refers to the epicenter.) Equations log ⁡ a = I 3 − 1 2 and A D 2 T 2 = constant (A = ground amplitude, T = period, D = hypocentral distance for a given shock) are established empirically for California shocks. Equation (9) holds very generally, and offers a basis for a more accurate definition of I, like that suggested by Cancani. Equation (4) is here used very generally at short distances; but it is approximate only, may differ regionally, and bridges over the probably discontinuous transition of the maximum acceleration from S̄ to some other transverse wave, with increasing distance. However, consequences derived from (4) nowhere conflict seriously with observation. The instrumental earthquake-magnitude scale has been extended to cover short distances. The results enter into an empirical relation M = 2.2 + 1.8 log ⁡ a 0 from which and (9) follows M = 1.3 + 0.6 I 0 These two equations are established and verified for the California region; they should also hold in other regions of similar structure for earthquakes originating at about the same depth (which is roughly 18 km.). The simplest possible assumptions (constant velocity, negligible absorption, sinusoidal waves) lead to the general equation log ⁡ E = 14.9 + 2 log ⁡ h + log ⁡ t 0 + 2 log ⁡ T 0 + 2 log ⁡ a 0 (t0 = duration, T0 = period, of sinusoidal wave train at the epicenter). Equations (27), (9), and (4) give the generally applicable results a 0 h 2 = a D 2 = a r R 2 I 1 − I 2 = 6 log ⁡ D 2 D 1 I 0 − 1.5 = 6 log ⁡ R h ar, the minimum perceptible acceleration, is approximately 1 gal. For shocks at the usual depth in California log ⁡ E = 11.3 + 1.8 M For other depths, and probably for other regions, log ⁡ E = 9.5 + 3.2 log ⁡ h + 1.1 I 0 log ⁡ E = 11.1 + 6.4 log ⁡ R − 3.2 log ⁡ h A summary of the physical elements for shocks in California is given in table 10. Equation (13) is used to calculate apparent depths for earthquakes in the United States and Europe. The results tend to confirm the relatively shallow origin of shocks on the Pacific Coast compared with those occurring elsewhere, particularly under the Canadian Shield, the central Mississippi Valley, and the southern Appalachians.