  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.
