  The analysis of Tikhonov regularization for nonlinear ill-posed equations
with smoothness promoting penalties is an important topic in inverse problem
theory. With focus on Hilbert scale models, the case of oversmoothing
penalties, i.e., when the penalty takes an infinite value at the true solution
gained increasing interest. The considered nonlinearity structure is as in the
study B. Hofmann and P. Math\'{e}. Tikhonov regularization with oversmoothing
penalty for non-linear ill-posed problems in Hilbert scales. Inverse Problems,
2018. Such analysis can address two fundamental questions. When is it possible
to achieve order optimal reconstruction? How to select the regularization
parameter? The present study complements previous ones by two main facets.
First, an error decomposition into a smoothness dependent and a (smoothness
independent) noise propagation term is derived, covering a large range of
smoothness conditions. Secondly, parameter selection by balancing principles is
presented. A detailed discussion, covering some history and variations of the
parameter choice by balancing shows under which conditions such balancing
principles yield order optimal reconstruction. A numerical case study, based on
some exponential growth model, provides additional insights.
