In 1983 teleseismic short-period P waves were recorded in the southwest United States along a linear array which stretched 1000 km across the Rio Grande rift. In 1985 a similar experiment was conducted across the East African rift, in which stations were installed along a 600 km array which ran east-west across Kenya. We have developed a spectral comparison technique to calculate the attenuation at each station. For a given event we simultaneously solve for the effective source spectrum, which includes mantle path effects as well as instrument response, an amplitude factor, and an attenuation operator t* at each station. We use downward projection of P wave travel time residuals to define the lithosphere-asthenosphere boundary, which we infer to upwarp beneath both rifts. The velocity structure so determined is then used to determine the Q of the asthenosphere relative to that of the lower lithosphere. When we assume a lower lithosphere Q of 100, we obtain asthenospheric Q values of 57 beneath the Rio Grande rift and 26 beneath the East African rift, with corresponding velocity contrasts of -8% and -12%, respectively. The low Q values as well as low velocity and density contrasts support the interpretation that the upwarped asthenosphere contains partial melt and that beneath the East African rift the concentration is higher than beneath the Rio Grande. Comparison of the results with experiments on peridotite (Sato et al., 1989a,b) suggests that the asthenosphere is at homologous temperatures of 1.01 and 1.03 with corresponding partial melt fractions of 1% and 3% beneath the Rio Grande and East African rifts, respectively. ¿ American Geophysical Union 1991