A simple model of single acoustic scattering is used to study the dependence of the shape of local earthquake coda on the anelastic and scattering structures of the lithosphere. The model is applied to the coda of earthquakes located near Stone Canyon, central California, and provides an explanation for the features observed in the data, which include an interesting temporal variation in the coda shape. A surficial layer with a Q of 50 and thickness of 10 or 25 km underlain by a zone with a Q of 1000 extending to the bottom of the lithosphere, together with a scattering scale length, a, that varies with depth z according to the relation a=0.3 exp[-(z/45)2] are found to constitute the simplest structure of the medium compatible with the coda data and with body and surface wave attenuation data. The profile of heterogeneity sizes implies that the scattering strength increases strongly with depth, a constraint required by the necessity to boost the energy of the later coda without forcing the intrinsic Q to be excessively high in the uppermost mantle. This constraint is viewed as an artifact of the single scattering model which overstimates the scattering coefficient due to the neglect of multiple scattering. The observed temporal variation of the signal is difficult to explain by a simple change of the intrinsic Q at some depth. Rather, it is suggested that the scattering properties at depth changed with time through a variation of the fractional rms velocity fluctuation on the order of one percent. ?? 1990 Birkha??user Verlag.