  A main drawback of classical Tikhonov regularization is that often the
parameters required to apply theoretical results, e.g., the smoothness of the
sought-after solution and the noise level, are unknown in practice. In this
paper we investigate in new detail the residuals in Tikhonov regularization
viewed as functions of the regularization parameter. We show that the residual
carries, with some restrictions, the information on both the unknown solution
and the noise level. By calculating approximate solutions for a large range of
regularization parameters, we can extract both parameters from the residual
given only one set of noisy data and the forward operator. The smoothness in
the residual allows to revisit parameter choice rules and relate a-priori,
a-posteriori, and heuristic rules in a novel way that blurs the lines between
the classical division of the parameter choice rules. All results are
accompanied by numerical experiments.
