How much do helioseismological inferences depend on the assumed reference model?

We investigate systematic uncertainties in determining the profiles of the solar sound speed, density, and adiabatic index using helioseismological te chniques. We find that rms uncertainties (averaged over the Sun) of similar to 0.02%-0.04% are contributed to the sound-speed profile by each of three sources: (1) the choice of assumed reference model, (2) the width of the i nversion kernel, and (3) the measurement errors. The rms agreement between the standard solar model sound speeds and the best helioseismological deter minations is about 0.07%. The profile of the adiabatic index, Gamma(1), is determined to an accuracy of about 0.02% with the Michelson Doppler Imager (MDI) data set. The density profile is about an order of magnitude less wel l determined by the helioseismological measurements. Five state-of-the-art models, each with a significant difference in the input physics or a parame ter choice, all give comparably good agreement with global helioseismologic al measurements. We consider four deficient solar models that are construct ed either using old input data, assuming the He-3 + He-4 fusion reaction do es not occur, neglecting element diffusion, or artificially mixing the inte rior of the Sun. When used as reference models in the inversion process, th ese deficient models yield sound speeds for the Sun that differ only by 0.1 % from the sound speeds obtained using the standard model. We conclude that even relatively crude reference models yield reasonably accurate solar par ameters. Although acceptable for most purposes as reference models, nonstan dard solar models in which the core is artificially mixed or in which eleme nt diffusion is neglected are strongly disfavored by the p-mode oscillation data. These nonstandard models produce sound-speed profiles with respect t o the Sun that are 4.5 and 18 times worse, respectively, than the agreement obtained with the standard solar model.