Pd is the peak amplitude of displacement in the first three seconds after the arrival of the P wave. We investigated the attenuation of Pd with the hypocentral distance R in southern California as a function of magnitude M, and obtained the following relationship: log (Pd) = -3.463 + 0.729 ¿ M - 1.374 ¿ log (R) ¿ 0.305. Given an earthquake location determined by the P-wave arrival times at stations close to the epicenter, this relationship can be used to define a so-called "Pd magnitude" of earthquakes. Our result shows that for earthquakes in southern California the Pd magnitudes agree with the catalog magnitudes with a standard deviation of 0.18 for events less than magnitude 6.5. Therefore, Pd is a robust measurement for estimating the magnitudes of earthquakes and has practical application in earthquake early warning systems.