Normal mode structure coefficients provide important constraints on the long-wavelength component of 3-D mantle density (&rgr;) structure, but inversions for independent models of vs, vp, and &rgr; using normal mode data alone are ill-posed even at long wavelengths. Ill-posed inversions typically are regularized by imposing a priori assumptions on the set of estimated models, but such regularization can introduce important uncertainties in the models. We characterize these uncertainties for &rgr; models estimated from current normal mode data using a set of 512 different regularization schemes. These schemes sample a variety of plausible a priori assumptions about the nature and distribution of mantle heterogeneity by specifying allowable vs, vp, &rgr;, and boundary topography structures. The estimated &rgr; models are fairly robust with respect to prior constraints on vs, vp, and topography. However, the character and amplitude of the estimated &rgr; models depend strongly on how &rgr; is allowed to decorrelate from vs, and we display several models in which &rgr; and vs decorrelate in very different depth intervals. Because these models all result from plausible prior constraints and fit the data equally and acceptably well, inversions of current normal mode data cannot robustly locate the decorrelation of &rgr; from vs. It remains possible that reliable &rgr; models may be obtained in the future as more normal mode measurements are introduced to break the strong tradeoffs between upper and lower mantle &rgr; structures that characterize current inversions. ¿ 1999 American Geophysical Union