# Optimal Estimation Retrieval of Aerosol Fine-Mode Fraction from Ground-Based Sky Light Measurements
## 2 Model and Methods
### Methodology  
#### 2.3.1 OE inversion Method  
Based on the optimal estimation theory, the forward model can be described as the following formula [29]:  
\[\mathbf{y}\ =\ \mathbf{F}(\mathbf{x},\mathbf{b})+\mathbf{e}, \tag{7}\]  
where \(\mathbf{y}\) is the observation vector with \(N_{\rm y}\) measurement elements, \(\mathbf{x}\) represents a state vector that contains \(N_{\rm a}\) parameters to be retrieved, \(\mathbf{b}\) means the non-state vector, including all model parameters that we do not seek to optimize, \(\mathbf{F}\) denotes the physical process from state vector to observation vector, and \(\mathbf{e}\) is an error term, including uncertainties contained in measurements and forward modeling. In the presence of \(\mathbf{e}\), the inversion result is a statistical estimate other than the exact value. Thus, under the assumption of a Gaussian distribution of errors, an inversion problem can be converted to solve the minimum value of cost function \(J(\mathbf{x})\), which is composed of two terms, expressed as the following:  
\[J(\mathbf{x}^{\rm p})\ =\ \frac{1}{2}[\mathbf{y}-\mathbf{F}(\mathbf{x}^{\rm p})]^{\rm T}S_{ \rm y}^{-1}[\mathbf{y}-\mathbf{F}(\mathbf{x}^{\rm p})]+\frac{1}{2}\gamma_{\rm a}(\mathbf{x}^{ \rm p}-\mathbf{x}_{\rm a})^{\rm T}S_{\rm a}^{-1}(\mathbf{x}^{\rm p}-\mathbf{x}_{\rm a}), \tag{8}\]  
where the superscript \"\(-1\)\" represents an inversion operation and \"\(T\)\" represents a transpose operation; \(\mathbf{x}_{\rm a}\) describes an estimate of state vector, and \(\mathbf{x}^{\rm p}\) represents the state vector of the p-th iteration; \(S_{\rm a}\) denotes the error covariance matrix for a priori estimate, and \(S_{\rm y}\) is the error covariance matrix representing the uncertainty occurring in the measuring process. We use zero off-diagonal elements for \(S_{\rm y}\) by assuming that errors are independent between measurements. Besides, the a priori estimate errors are also assumed to be noncorrelated, which gives a diagonal matrix of \(S_{\rm a}\). Two terms in the right side of Equation (8) describe the contributions of observation and a priori estimate into the solution separately. The Lagrange multiplier \(\gamma_{\rm a}\) is a regularization parameter, and is defined following the work of [PERSON] et al. [31]:  
\[\gamma_{\rm a}\ =\ N_{\rm y}/N_{\rm a}. \tag{9}\]  
Usually, solving the minimum value of the \(J(\mathbf{x})\) is a nonlinear problem and always needs multiple iterations. Quasi-Newton methods are widely used for finding the local maximum and/or minimum of the nonlinear function [44; 45],  
\[\mathbf{x}^{\rm p+1}\ =\ \mathbf{x}^{\rm p}-\alpha_{\rm p}\mathbf{H}_{\rm p}\
abla J(\mathbf{x}^{ \rm p})^{\rm T}, \tag{10}\]  
where \(\mathbf{H}\) is the inverse matrix of the Hessian matrix constructed with successive gradient vectors in the quasi-Newton method, and \(\alpha_{\rm p}\) is the iterated step length which is chosen to minimize \(J\big{(}\mathbf{x}^{\rm p+1}\big{)}\). The gradient of \(J(\mathbf{x})\) can be defined as,  
\[\
abla J(\mathbf{x}^{\rm p})\ =\ \frac{\partial J(\mathbf{x}^{\rm p})}{\partial\mathbf{x}}\ =\ \mathbf{K}_{\rm p}^{\rm T}\mathbf{\xi}^{-1}[\mathbf{F}(\mathbf{x}^{\rm p})-\mathbf{y}]+ \gamma_{\rm a}\mathbf{S}_{\rm a}^{-1}(\mathbf{x}^{\rm p}-\mathbf{x}_{\rm a}), \tag{11}\]  
where \(\mathbf{K}\) is the Jacobians matrix of \(\mathbf{F}(\mathbf{x})\) with respect to \(\mathbf{x}\), and \(\mathbf{K}\ =\ \frac{\partial\mathbf{F}(\mathbf{x})}{\partial\mathbf{x}}\). The sensitivity of radiance with respect to the aerosol microphysical parameters can be calculated by UNL-VRTM. The second term in the right side of Equation (11) describes the convergence direction and step length of the state vector from the p-th iteration and (p + 1)-th iteration. We perform an iterative Quasi-Newton approach using the L-BFGS-B algorithm to find the minimization of \(J(\mathbf{x})\)[46; 47; 48].