Using P wave amplitudes from 52 deep earthquakes, recorded by the Global Seismograph Network (GSN) between 1991 and 1996, we empirically determined depth-distance corrections for body wave magnitude mb and moment magnitude Mw. After correcting vertical, broadband records from stations of the GSN for instrument response and source radiation factors, P wave amplitudes from velocity records were read for the case of mb and the surface area under the displacement pulse of the P wave for Mw. Amplitudes were then normalized to a scalar moment M0=1018 Nm, or mb=5.7, Mw=5.9. Using the generic logarithmic relationship for each magnitude, correction factors Q(Δ,h) were determined. A total of 973 observed values of Q in the plane of distance Δ vs. depth h were smoothed by local averaging, as well as fitted by a parameterized smooth surface. These empirical Q(Δ,h) surfaces differ in important detail from those historically established by Gutenberg and Richter, and markedly from the theoretical curves predicted by Veith and Clawson (1971). We present a first empirical correction factor for the determination of Mw from P wave displacements. ¿ 1998 American Geophysical Union