An analytical formulation of the optical inversion of aerosol mixing state and characterization of solution space

Accurate determination of the aerosol mixing state is indispensable to assess aerosol direct and indirect effects. However, the characterization of the mixing state is often limited by the scarcity of direct, in situ measurements of chemical composition and single-particle morphology. Consequently, the aerosol community has largely relied on optical closure techniques to infer the aerosol mixing states from optical measurements, which were generally deemed only as probable mixing states. The heuristic nature of these techniques restricts the quantification of inherent uncertainties in the inferred mixing states. To address this gap, this study presents an analytical formulation of the optical inversion problem as a linear system using the Python aerosol optical model, AeroMix. This formulation explicitly characterizes the problem as both ill-posed and ill-conditioned, while offering a scalable, modular framework that remains agnostic to the specific forward model and measurement techniques. To mitigate mathematical instabilities, system dimensionality is reduced by eliminating physically infeasible core-shell components and grouping spectrally indistinguishable core-shell components. Establishing that a unique solution is mathematically impossible, the solution space is characterized as a high-dimensional convex polytope bounded by linear inequalities defined by the range of measured optical properties and physical component constraints. Finally, this study proposes retrieving physically meaningful, sparse solutions by using Markov Chain Monte Carlo (MCMC) techniques to sample the polytope boundaries lying on coordinate hyperplanes. This stochastic approach transforms optical inversion from a heuristic estimation into a probabilistic characterization of the valid solution space, enabling robust uncertainty quantification in the inferred aerosol mixing states.