We apply the theory of radiative transfer to invert for the small-scale structure of the mantle, by fitting mean amplitude envelopes of precursors to PKP in the time domain. The data set previously used by Hedlin and Shearer <2000> have been augmented with new deep events from the 1997--2000 period, thereby doubling the number of high-quality records selected for this study. Regional and global average amplitudes are measured in six bins covering the 124--142¿ epicentral distance range. The inversion focuses on stacks for strongly scattering wave paths only and aims to determine the depth extent and power spectrum of the mantle heterogeneities that cause the scattered signals. We find that the restriction of scatterers to the D″ layer is incompatible with observations. Instead, whole mantle scattering models reproduce with reasonable accuracy the time dependence of precursors, in agreement with previous findings. Exponential correlations predict a spatial rate of decay of the precursor amplitude which is much faster than observed. Thus we are led to propose a new model of mantle heterogeneity, richer in small-scale than exponential media, that fits uniformly the data. An analytic form of the correlation function and power spectrum of our new model is given explicitly. It is shown that the precise length scale of heterogeneities is fundamentally unresolvable because of the limited range of observations. The perturbations in P wave velocities required to fit the data are only of order 0.1--0.2%, which is more than 1 order of magnitude less than found in previous studies.