Come analyze HEASARC, IRSA, and MAST data in the cloud! The Fornax Initiative is now welcoming all interested beta users.
Next: recorn: change correction norm Up: projct: project 3-D ellipsoidal Previous: projct: project 3-D ellipsoidal Contents
Failure modes and regularization
The shell-to-annulus mixing that projct performs is upper-triangular:
the emission in annulus 
is a sum of contributions from all shells 
,
weighted by the geometric path length of shell 
through annulus .
Recovering the shell parameters is the inverse of that mixing — each
inner shell is obtained by subtracting the fitted outer-shell contributions from
the observed annulus — and that subtraction is an ill-conditioned operation.
Two consequences follow, and neither is flagged by the fit output:
- Noise amplifies inward. Outer-shell uncertainties propagate into every inner shell they overlap, so the innermost shells — built on the most subtractions — are the noisiest, and adjacent shells become strongly anti-correlated (an upward fluctuation in one shell forces a compensating downward one in its neighbour).
- The conditioning is set by the annular partition. Thin shells give nearly-degenerate mixing rows and a poorly-conditioned inverse; the same data, re-binned, can yield a qualitatively different deprojected profile. There is no warning and no built-in conditioning diagnostic.
Typical symptoms and their mitigations:
| Symptom | Cause | Mitigation | |||
| Innermost shell kT/norm wild or railed | noise amplified through repeated subtraction | regularize; widen the inner bin | |||
| Adjacent shells anti-correlated; oscillating profile | ill-conditioned inverse, over-fit to noise | smoothness prior; coarser partition | |||
| Profile shape changes on re-binning | conditioning set by the partition | re-bin-and-refit sensitivity scan | |||
| Error bars implausibly small after a good fit | degeneracy hidden by the point estimate | sample the posterior under a smoothness prior | |||
| Outermost shell unconstrained | few/no annuli beyond it | fix or tie the outer shell |
The robust deprojection tools outside XSPEC all replace the raw inverse with some form of regularization or physical prior: dsdeproj (Sanders & Fabian 2007) propagates Monte-Carlo errors through a model-independent geometric deprojection; MBProj2 (Sanders et al. 2018) forward-fits the projected data with a smooth parameterised 3-D model; and mass-profile codes such as clmass impose a physical profile. Within XSPEC the same effect is available through the bayes smooth command, which couples a chosen per-datagroup parameter (e.g. norm or kT) across the shells with a Tikhonov smoothness prior built from the shell radii (taken automatically from the XFLT major keywords). It leaves projct itself unchanged, is off by default, regularises both the Levenberg–Marquardt fit and the posterior samplers, and can select its strength automatically by the L-curve or discrepancy criteria. See the bayes command description and the Bayesian Methods chapter for details.