Several widely implemented and tested earthquake early warning algorithms employ empirical equations that relate earthquake magnitudes with ground-motion peak amplitudes and hypocentral distances. This approach is effective to the extent that the offline dataset available for setting the fitting coefficients in those equations is of sufficient quality and quantity. However, to address the problem of having a limited dataset, it is instructive to gain physical understanding of the main factors controlling the P-wave attenuation. In this study, theoretical expressions are derived that relate the root mean square (rms) of the P-wave displacement drms and velocity vrms to the seismic moment, stress drop, and hypocentral distance. The theoretical attenuation laws are then validated against observed attenuation, using earthquake data from southern California and Japan. Good agreement is found between observed and predicted ground motions. The similar ground-motion attenuation in California and Japan suggests that the attenuation laws are similarly applicable for the two regions and implies that they may also be implemented in other regions without having to go through a lengthy calibration phase. Because drms is more strongly dependent on the seismic moment than vrms, use of the attenuation law for drms yields better magnitude prediction than that of vrms. It is shown that the drms-to-vrms ratio is proportional to the characteristic length of the rupture and that the stress drop is a function of the seismic moment and the cube of drms/vrms. This result paves the way for a new stress-drop determination scheme that is totally independent of previously used approaches. Finally, it is shown that the rms of the ground motions are proportional to their peak values.
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