A formalism for the simultaneous inversion of surface wave phase velocity and attenuation, previously developed for Love waves, is extended in this paper to Rayleigh waves. The simultaneous inversion technique permits the specification of the intrinsic dispersion-attenuation relation that arises from linearity and causality, and takes full account of the dependence of surface wave phase velocity and Q to the -1 on the real and imaginary parts of an anelastic earth structure. The formalism, including resolution analysis and extremal inversion, is applied to combined Love and Rayleigh wave data sets for a tectonically active and a stable continental path and to Rayleigh wave data for a stable oceanic path. The depth to the low Q zone is 60 + or - 20 km for the central Pacific, 80 + or - 20 km for western North America, and 130 + or - 30 km for the east-central North America. Q to the -1 within the low-Q, low- velocity zone however, is greater beneath western North America than beneath the central Pacific; the low-Q zone beneath east-central North America need not be a low velocity zone at frequencies above 1 Hz. The surface wave data cannot distinguish among several possible intrinsic dispersion-attenuation relations for the upper mantle.