Calculation of attenuation and dispersion spectra for a general anelastic body may be posed as an inversion problem. We observe the time-dependent strain due to “instantaneous” changes in stress in order to characterize the anelastic response. This experimental technique reaches a range of frequencies which is lower than that used in resonance bar experiments, but we can make only indirect measurement of the viscoelastic properties. We employ the most general anelastic model which states that the observed compliance is due to the summed effect of an arbitrary spectrum of mechanisms. The analysis requires the solution of Fredholm integral equations of the first kind. It is well known that this problem is ill-conditioned so that any numerical scheme will have to involve some smoothing to obtain accurate solutions. The present work employs Butler’s method of constrained regularization which takes advantage of the fact that the solution is positive and uses data dependent smoothing. This work indicates that the imposition of the positivity constraint makes the computation of the solution much better conditioned. Computations with the method of constrained regularization employing near-optimal smoothing demonstrate its superiority over the method of Shapery for obtaining accurate solutions when the data are very noisy.