Laboratory evidence shows that the attenuating properties of geological materials depend nonlinearly on strain amplitude for strains greater than 10-6. Such a dependence is predicted by the amplitude-dependent frictional attenuation models of Mavko (1979) and Stewart et al. (1983). At low enough strain these inelastic mechanisms become secondary to anelastic, linear mechanisms and are masked. The nonlinear regime may be encountered in the very near field around a seismic source. We study the case of a spherically symmetric source (explosion) embedded in an attenuating medium and examine the decay of outgoing compressional waves. Comparison with observations from the Cowboy series of explosions in salt shows that it is possible to explain simultaneously the radial decay of peak displacement and peak velocity by either one of two general models: (1) a linear, anelastic model with low Q (~20) or (2) an amplitude-dependent Q model in which the anelastic contribution is small in comparison with the inelastic contribution. Laboratory evidence tends to favor the latter. Strictly speaking, neither model preserves the observed Y-1/3 scaling of the wave field with yield of the explosion, but departures from perfect scaling are small in comparison with the scatter of observations. We conclude that near-source rheology may have an important influence on the effective seismic source associated with far-field observations.