Inverse modeling of circular lattices via orbit response measurements in the presence of degeneracy

The number and relative placement of BPMs and steerers with respect to the quadrupoles in a circular lattice can lead to degeneracy in the context of inverse modeling of accelerator optics. Further, the measurement uncertainties introduced by beam position monitors can propagate by the inverse modeling process in ways that prohibit the successful estimation of model errors. In this contribution, the influence of BPM and steerer placement on the conditioning of the inverse problem is studied. An analytical version of the Jacobian, linking the quadrupole gradient errors along with BPM and steerer gain errors with the orbit response matrix, is derived. It is demonstrated that this analytical version of the Jacobian can be used in place of the numerically obtained Jacobian during the fitting procedure. The approach is first tested with simulations and the findings are verified by measurement data taken on SIS18 synchrotron at GSI. The results are crosschecked with the standard numerical Jacobian approach. The quadrupole errors causing tune discrepancies observed at SIS18 are identified.