  The goal of this paper is to further develop an approach to inverse problems
with imperfect forward operators that is based on partially ordered spaces.
Studying the dual problem yields useful insights into the convergence of the
regularised solutions and allow us to obtain convergence rates in terms of
Bregman distances - as usual in inverse problems, under an additional
assumption on the exact solution called the source condition. These results are
obtained for general absolutely one-homogeneous functionals. In the special
case of TV-based regularisation we also study the structure of regularised
solutions and prove convergence of their level sets to those of an exact
solution. Finally, using the developed theory, we adapt the concept of
debiasing to inverse problems with imperfect operators and propose an approach
to pointwise error estimation in TV-based regularisation.
