## 5 Examples of Forward-Inverse Modeling
### Atmospheric Input & Output
#### Retrieval Surface Roughness and Aerosol Parameters  
Although the variances in the crosswind and upwind directions, \(\sigma_{\text{c}}^{2}\) and \(\sigma_{\text{u}}^{2}\), can be parameterized in terms of wind speed and direction [75], they may instead be considered to be model input parameters describing the 2D surface slope distribution. For a 1D surface, the slope variance is then automatically given as \(\sigma^{2}=\sigma_{\text{u}}^{2}+\sigma_{\text{c}}^{2}\). This approach has the advantage that the slope variances do not depend on parameterizations in terms of wind speed, which provides more freedom to reproduce the measured 2D glint pattern by varying \(\sigma_{\text{c}}\) and \(\sigma_{\text{u}}\). In addition to reflection from Gaussian surface slope facets, multiple surface reflections, shadowing [see Equation (130)], and polarization effects also influence the glint signal [105; 106]. Since multiple reflections and shadowing become important only for very low solar elevations and polarization effects are relatively unimportant at 1035 nm due to weak molecular (and aerosol) scattering, these effects can be ignored [80]. To determine the optimum match between model-simulated and CAR-measured reflectances, we need to invert the measurements to find the best estimate of the model parameters described by the state vector \(\mathbf{x}=[\sigma_{\text{c}}^{2},\sigma_{\text{u}}^{2},\phi_{\text{wind}}, \tau_{\text{mix}}]^{T}\) including the four retrieval parameters: (i) the slope variance in the crosswind direction, \(\sigma_{\text{c}}^{2}\), (ii) the slope variance in the upwind direction, \(\sigma_{\text{u}}^{2}\), (iii) the wind direction, \(\phi_{\text{wind}}\), and (iv) the optical thickness \(\tau_{\text{mix}}\) of the layer with a mixed population of aerosols and molecules. To this end, [80] used a Gauss-Newton/Levenberg-Marquardt non-linear inversionalgorithm, in which the residual between reflectances produced by the RT forward model \(\mathbf{F}(\mathbf{x})\) and the CAR measurements, stored in the vector \(\mathbf{y}\), is minimized. The \(k^{\text{th}}\) iteration of \(\mathbf{x}\) yields:  
\[\mathbf{x}_{k+1}=\mathbf{x}_{k}+\left[\mathbf{J}_{k}^{T}\mathbf{J}_{k}+\gamma_ {k}\mathbf{I}\right]\mathbf{J}_{k}^{T}(\mathbf{F}(\mathbf{x}_{k})-\mathbf{y}_ {k}) \tag{154}\]  
where the vector \(\mathbf{J}\) contains the Jacobians of the forward model, \(\mathbf{I}\) is identity matrix, and the parameter \(\gamma_{k}\) (\(0\leq\gamma_{k}\leq\infty\)) is chosen at each step of the iteration to minimize the residual. If \(\gamma_{k}=0\) we have a classic Gauss-Newton method, while if \(\gamma_{k}\) is large we have a steepest descent method. Figure 3 shows a comparison of model-simulated and measured reflectances at 1036 nm on 20 July 2001. Clearly, model-simulated results agree very well with the measurements, reproducing the main characteristics of the glint pattern. Both the shape of the simulated glint ellipse and its tilt are generally well matched with the measurements. The shape of the elliptical glint pattern in Figure 3 is determined by the crosswind and upwind slope variances (\(\sigma_{\text{c}}^{2}\) and \(\sigma_{\text{u}}^{2}\)), while the wind direction is determined from the tilt angle. Such a titled ellipse indicates that use of the 2D asymmetric Gaussian BRDF is needed to fit the angular distribution of the reflectance measurements, and that a 1D Gaussian is insufficient because it averages those slopes (\(\sigma^{2}=\sigma_{\text{c}}^{2}+\sigma_{\text{u}}^{2}\)) to give only a circular glint pattern. Figure 4 shows a comparison of 1D and 2D results for the case shown in Figure 3. It is clear that model simulations based on a 1D Gaussian BRDF are unable to match the measured tilted elliptical glint pattern. Hence, employing a 2D BRDF in the data analysis is essential. Figure 3: Comparison between model-simulated and measured reflectances. The measurements are in solid blue and the simulations in dashed red lines.