Abstract
<jats:p>Angle-resolved X-ray photoelectron spectroscopy (ARXPS) provides non-destructive information about composition within the first few nanometres of a surface, but reconstruction of a free-form concentration depth profile is a severely ill-posed inverse Laplace-transform problem. An earlier analysis showed that the experimentally meaningful depth resolution is governed principally by signal-to-noise ratio and is a substantial fraction of the depth itself. It also raised an unresolved question: can nonlinear constraints such as non-negativity, bounded concentration and finite support materially improve this depth resolution, or do they mainly suppress unphysical oscillations and make a poorly resolved answer appear sharper? Here the question is reformulated using the modern language of restricted stability and admissible-set geometry. Three different quantities are separated: free-form depth resolution, constrained free-form resolution and conditional precision within a low-dimensional parametric model. Work by Sabatier, Bertero, Hansen, Donoho and others shows that weak constraints can improve feasibility and numerical stability, whereas large super-resolution gains require much stronger structure such as sparsity, separation or a narrowly specified model class. The finitesupport calculation reported previously for ARXPS is consistent with this picture: a useful but moderate support restriction reduced the depth-resolution interval by approximately 40%, while improvements approaching a factor of two required extremely strong localisation of the unknown profile. Illustrative calculations for a five-angle ARXPS experiment show how a sharp interface model can yield a fitted interface-position standard deviation of approximately 0.075 nm, even when the corresponding free-form depth-resolution estimate is approximately 2.3 nm. This is not a contradiction: the former is conditional parameter precision, while the latter is the ability to discriminate arbitrary admissible profiles. We propose an algorithm-independent definition of constrained ARXPS resolution based on the diameter of the set of profiles compatible with the data and noise. This provides a route to quantitative limits on what weak constraints can achieve, and a transparent basis for assessing claims of exceptionally sharp model-fitted depth profiles. The treatment is extended to multicomponent reconstruction, where concentration closure and shared-interface constraints reduce the admissible dimension but uncertainties in number density, attenuation lengths, sensitivity factors, angular asymmetry and elastic scattering act as nuisance parameters. We propose separate reporting of free-form resolution, constrained resolution, conditional parameter precision, predictive reproducibility and absolute depth-scale uncertainty, together with a minimum interleaved-angle cross-validation test.</jats:p>