## 5 Examples of Forward-Inverse Modeling
### Bidirectional Reflectance of Water--Why Is It Important?
To obtain the water-leaving radiance, the upward radiance was computed just above the ocean surface twice using AccuRT. Assuming the ocean to be black, i.e., totally absorbing (no scattering), the upward radiance was first computed just above the ocean surface, \(I_{\text{black}}^{+}(0^{+},\lambda,\theta_{0},\theta,\Delta\phi)\), which includes the radiance due to Fresnel reflection of direct attenuated sunlight and skylight by the air-water interface, but no radiance from the water. The second time the ocean with water and its embedded constituents, was included and the radiance \(I^{+}(0^{+},\lambda,\theta_{0},\theta,\Delta\phi)\) was computed. Hence, \(I^{+}(0^{+},\lambda,\theta_{0},\theta,\Delta\phi)\) included the water-leaving radiance as well as the Fresnel reflected direct attenuated sunlight and skylight. Then the water-leaving radiance was obtained from the difference  
\[L_{\rm w}(0^{+},\lambda,\theta_{0},\theta,\Delta\phi)=I^{+}(0^{+},\lambda, \theta_{0},\theta,\Delta\phi)-I^{+}_{\rm black}(0^{+},\lambda,\theta_{0}, \theta,\Delta\phi). \tag{149}\]  
To work satisfactorily, a neural network must be properly trained. For this purpose AccuRT was used to generate a training dataset containing 30,000 data points of \(I^{+}(0^{+},\lambda,\theta_{0},\theta,\Delta\phi)\) and \(I^{+}_{\rm black}(0^{+},\lambda,\theta_{0},\theta,\Delta\phi)\) at seven wavelengths, 412, 443, 490, 510, 560, 620, and 665 nm, which are similar to the wavelengths used in the MAG02 algorithm. The synthetic dataset was generated by randomly selecting 5000 combinations of the aerosol optical depths at 865 nm (\(\tau_{a}(865)\)), the fraction of small aerosol particles (\(f_{a}\)), the relative humidity (RH), and the three ocean parameters: CHL, MIN, and CDOM. To cover a wide range of water and atmospheric IOPs, these six parameters were randomly sampled logarithmically from the following ranges: (i) \(\tau_{a}(865)\): 0.001-0.5, (ii) \(f_{a}\): 1-95 [%], (iii) RH: 30-95 [%], (iv) CHL: 0.01-100 [mg\(\cdot\)m\({}^{-3}\)], (v) MIN: 0.01-100 [g\(\cdot\)m\({}^{-3}\)], (vi) CDOM: 0.001-10 [m\({}^{-1}\)]. Then for each case in the 5000 combinations, six combinations of the Sun-sensor geometry were randomly selected in the following ranges: (i) \(\theta_{0}\): 0-80 [\({}^{\circ}\)], (ii) \(\theta\): 0-70 [\({}^{\circ}\)], (iii) \(\Delta\phi\): 0-180 [\({}^{\circ}\)]. The downward irradiance just above the ocean surface (\(F^{-}(\lambda,0^{+})\)) was also computed for each case as well as the remote sensing reflectance:  
\[R_{\rm rs}(\lambda,\theta_{0},\theta,\Delta\phi)=\frac{L_{\rm w}(0^{+},\lambda,\theta_{0},\theta,\Delta\phi)}{F^{-}(\lambda,0^{+})} \tag{150}\]  
where \(L_{\rm w}(0^{+},\lambda,\theta_{0},\theta,\Delta\phi)\) is given by Equation (149). The remote sensing reflectances were arranged into two groups: (i) one consisting of all the angle-dependent remote sensing reflectances \(R_{\rm rs}(\lambda_{i},\theta_{0},\theta,\Delta\phi)\), the other consisting of the the corresponding nadir remote sensing reflectances, \(R_{\rm rs}(\lambda_{i},\theta_{0})\). Then a neural network with two hidden layers was created. The first layer used RBFs as neurons, while the second layer used a linear function as neurons. The input to the neural network training was the three geometry angles \(\theta_{0},\theta,\Delta\phi\) plus the angle-dependent remote sensing reflectances \(R_{\rm rs}(\lambda_{i},\theta_{0},\theta,\Delta\phi)\), while the output consisted of the corresponding nadir remote sensing reflectances, \(R_{\rm rs}(\lambda_{i},\theta_{0})\). After the training, the nadir remote sensing reflectances can be derived from a single equation:  
\[R_{\rm rs}(\lambda_{i},\theta_{0})=\sum_{j=1}^{N}a_{ij}\exp[-b^{2}\sum_{k=1}^ {N_{\rm in}}(p_{k}-c_{jk})^{2}]+d_{i} \tag{151}\]  
where \(N\) is the number of neurons, \(b\) and \(c_{jk}\) are the bias and weight in the first layer, \(a_{ij}\) and \(d_{i}\) are the weight and bias in the second layer.