Estimates of the composition of the lower mantle are commonly obtained by a process of adiabatic decompression, in which zero-pressure mantle parameters are determined with a least squares fit of an equation of state to mantle densities. We show that this procedure may lead to biased estimates of the zero-pressure density and compressibility of the mantle. The bias can be eliminated or minimized by subjecting the equation of state fit to a thermodynamic constraint at zero pressure. Such inversions provide equations of state that are compatible with the lower mantle as well as the thermal properties of selected mineral models. We used this method to evaluate assemblages containing (Mg, Fe)SiO3 perovskite and (Mg, Fe)O, with Fe/(Fe+Mg) ratios as high as 0.2. Thermodynamic systematics were used to place bounds on the Anderson-Gruneisen parameter ΔS for perovskite. We find a fairly large range of acceptable models for the composition of the lower mantle. They range from upper-mantle-like compositions to assemblages that are enriched in both Si and Fe relative to the upper mantle. However, a pure perovskite lower mantle, whether enriched in Fe or not, seems improbable. Small Fe enrichments of the order of 1%, given density contrasts across the 670-km discontinuity that are probably insufficient to prevent convective mixing of the upper and lower mantles. We therefore suggest that unless the mantle is homogeneous in its major oxide components, the lower mantle is likely to be enriched in Fe by at least several percent. ¿ American Geophysical Union 1990