A formalism is developed for simultaneous inversion of surface wave phase velocity and attenuation to determine shear wave velocity and Q-1 in the earth. A simultaneous inversion takes full account of the dependence of surface wave velocity and attneuation on both the elastic and the dissipative part of an earth structure and permits inclusion of the physical relationship between anelasticity and intrinsic dispersion that arises from linearity. The procedure is mathematically more complete than the approach of correcting phase velocity data for the intrinsic dispersion due to anelasticity and inverting the corrected phase velocity data for the intrinsic dispersion due to anelasticity and inverting the corrected velocity data alone, and it gives different results. The proposed fomalism, including resolution analysis, weighted least squares inversion, and extremal inversion, is applied to Love waves in western North America. Various intrinsic dispersion-attenuation relations are tested, including Q independent of frequency, Q varying as a power of frequency, and Q specified by a sum of relaxation mechanisms. The results of the inversions confirm the coincidence of the low-velocity and low Q zones beneath western North America for frequencies in the surface wave band. Compared with previous inversion of Q-1 data alone, the simultaneous inversion results in improved depth resolution of Qs-1 and the elimination of an apparent incompatibility of low-attenuation data at 20- to 25-s periods. The Love wave data do not discriminiate among the various dispersion-attenuation relations, though a constant Q leads to the removal of the requirement for a low-velocity zone at frequencies above 1 Hz. The predicted intrinsic dispersion within the low-velocity zone at frequencies above 1 Hz. The predicted intrinsic ispersion within the low-velocity zone varies from 1% to 10% between 0.01 and 1 Hz for the various models; broadband measurements of body wave dispersion offer the greatest promise for choosing among the models.