Multiwavelength lidar aerosol microphysical parameter retrieval represents a fundamental ill-posed inverse problem governed by Fredholm integral equations, where limited optical observations must constrain infinite-dimensional particle size distributions. Current regularization methods provide only point estimates without uncertainty quantification, severely limiting their applications in climate model validation and data assimilation. The IPCC sixth assessment report identifies aerosol-radiation interactions as the largest uncertainty source in climate forcing (-2.0~-0.4 W·m^-2), largely due to inadequate aerosol property characterization. Traditional approaches suffer from three critical limitations: subjective regularization parameter selection, absence of uncertainty bounds, and inability to systematically incorporate prior physical knowledge. This study develops a comprehensive Bayesian framework that transforms deterministic retrieval into probabilistic inference, providing rigorous uncertainty quantification essential for advancing atmospheric aerosol science and climate applications.

We establish a complete Bayesian inversion framework based on Mie scattering theory: g_p(λ)=∫Kp(λ, r, m)v(r)dr+ε, where g_p(λ) represents observed optical parameters, K_p(λ, r, m) is the Mie-derived kernel function, v(r) is volume size distribution, and ε denotes measurement noise. The continuous distribution is discretized using eight B-spline basis functions with weight coefficients w treated as random variables. The hierarchical Bayesian model specifies P(w|α)=N(0, α^-1I) and P(g|w, A, β)=N(Aw, β^-1I), where precision hyperparameters α and β follow Gamma priors and are automatically estimated through variational inference. This eliminates subjective parameter selection while providing complete posterior distributions P(w|g, A)=N(μ_w, Σ_w) with mean μ_w=(αI+βA^TA)^-1βA^Tg and covariance Σ_w=(αI+βA^TA)^-1. We rigorously establish mathematical equivalence between Tikhonov regularization and Bayesian maximum a posteriori (MAP) estimation, demonstrating that classical methods emerge as special cases of our probabilistic framework, thus providing unified theoretical guidance for method selection.

Comprehensive validation using 3β+2α lidar configurations (355, 532, 1064 nm backscatter; 355, 532 nm extinction) demonstrates exceptional performance across challenging scenarios. For bimodal distributions representing atmospheric fine and accumulation modes, geometric mean radius retrieval errors remain below 5% (fine mode: -0.6%, accumulation mode: +4.4%) with effective radius errors under 3.5% (Fig. 3). Under realistic noise conditions (10%?20%), the Bayesian method exhibits superior robustness through automatic regularization adaptation, while posterior variance analysis reveals physically meaningful uncertainty patterns: elevated uncertainty in modal boundaries (0.2?0.4 μm) reflecting observational constraints, and high uncertainty in large particle tails consistent with reduced lidar sensitivity (Fig. 4). Statistical validation through 1000 Monte Carlo experiments confirms 95% credible intervals, demonstrating accurate uncertainty quantification. The method correctly captures aerosol microphysics where fine particles dominate numerically (90.9%) while accumulation mode controls volume (93.5%) and optical properties. Comparative analysis reveals complementary characteristics: Tikhonov excels in computational efficiency and single-mode accuracy (1.24% mean error), while Bayesian provides comprehensive uncertainty information with comparable accuracy (2.29% mean error), making it optimal for scientific applications requiring rigorous uncertainty assessment.

This paper establishes a transformative Bayesian framework for multiwavelength lidar aerosol retrieval that addresses fundamental limitations of deterministic approaches. Key contributions include: 1) complete probabilistic formulation providing both accurate retrievals and comprehensive uncertainty quantification; 2) rigorous mathematical unification of regularization methods under Bayesian theory; 3) automatic hyperparameter estimation eliminating subjective parameter selection; 4) demonstrated superior robustness and statistically validated uncertainty bounds. The framework’s scientific impact extends to climate model evaluation through quantified parameter uncertainties, improved data assimilation via observation error covariances, and uncertainty-aware environmental monitoring. Experimental validation confirms geometric mean errors below 5% with reliable uncertainty propagation under observational noise. This probabilistic paradigm represents a significant methodological advancement for atmospheric remote sensing, providing essential tools for climate prediction, air quality assessment, and environmental policy applications requiring rigorous uncertainty quantification.