## 5 Examples of Forward-Inverse Modeling
### Atmospheric Input & Output
#### 5.4.3 Issues Specific to Geostationary Platforms  
_Low Solar Elevations_  
With geostationary platforms one has the opportunity to investigate diurnal variations in water properties. Hence, there is a desire to obtain useful observations throughout the day from sunrise to sunset. However, for solar zenith angles larger than about 75\({}^{\circ}\) (and large viewing angles) the _plane-parallel approximation_ (PPA) becomes invalid, and we need to take Earth curvature into account, as discussed in a recent paper [110]. An approximate way to deal with this problem is the so-called _pseudo-spherical approximation_ (PSA), in which the direct beam single scattering (solar pseudo-source) term is treated in spherical geometry: \(e^{-\tau/\mu_{0}}\to e^{-\tau\,Ch(\mu_{0})}\), where the Chapman function \(Ch(\mu_{0})\) takes curvature into account [14; 110], while the multiple scattering term is treated using the PPA. Hence, in the PSA the RTE becomes:  
\[u\frac{dI(\tau,u,\phi)}{d\tau} = I(\tau,u,\phi)-\overbrace{\frac{\omega(\tau)}{4\pi}\int_{0}^{2 \pi}d\phi^{\prime}\int_{-1}^{1}du^{\prime}p(\tau,u^{\prime},\phi^{\prime};u, \phi)I(\tau,u^{\prime},\phi^{\prime})}^{\text{single scattering}} \tag{157}\] \[-\overbrace{\frac{\omega(\tau)}{4\pi}p(\tau,-\mu_{0},\phi_{0};u, \phi)F_{0}e^{-\tau\,Ch(\mu_{0})}}^{\text{single scattering}}.\]  
It should be pointed out that low solar elevations is generally a problem at high latitudes also for polar-orbiting platforms.  
_Surface Roughness Considerations_  
As alluded to above, it would be worthwhile considering the advantage gained by using a 2D rather than a 1D slope distribution to deal with a wind-roughened surface (see Figure 4). Hence, one should consider using  
1. a 2D Gaussian surface slope distribution for singly scattered light, and
2. a 1D Gaussian surface slope distribution for multiply scattered light.  
Such an approach may be quite successful, as demonstrated in Figure 3, because a 2D BRDF simulates sunglint very well, while a 1D BRDF is sufficient to simulate the smoother (more directionally uniform) skylight.  
_Use of Vector (Polarized) RT Simulations_  
Analysis of polarization measurements require vector RT modeling [111], and retrieval results from radiance-only measurements are likely to be improved by employing a vector (polarized) forward RT model to compute the radiances and Jacobians used in the inversion step, particularly for modeling the radiance due to scattering by small aerosol particles at short wavelengths. Also, the increased information from polarization measurements can lead to significant improvements in atmospheric correction due to aerosols [112]. Hence, for ocean color retrievals from geostationary platforms, we should explore the advantage of using the pseudo-spherical approximation combined with  
* polarized (vector) radiative transfer simulations,
* a 2D Gaussian distribution of surface slopes,  
and neural networks and optimal estimation for  
* simultaneous retrieval of atmospheric and marine parameters from multi-spectral as well as hyperspectral measurements of total and polarized (if available) radiances, and
* assessments of retrieval accuracy and error budgets.