  We study the choice of the regularisation parameter for linear ill-posed
problems in the presence of data noise and operator perturbations, for which a
bound on the operator error is known but the data noise-level is unknown. We
introduce a new family of semi-heuristic parameter choice rules that can be
used in the stated scenario. We prove convergence of the new rules and provide
numerical experiments that indicate an improvement compared to standard
heuristic rules.
