## 5 Examples of Forward-Inverse Modeling
### Bidirectional Reflectance of Water--Why Is It Important? The **B**idirectional **R**eflectance **D**istribution **F**unction (BRDF) is defined as the ratio of the reflected radiance to the incident power per unit surface area:  
\[\rho(\mu,\phi;-\mu^{\prime},\phi^{\prime})=\frac{dI_{\text{refl}}(\tau^{*},\mu, \phi)}{I(\tau^{*},-\mu^{\prime},\phi^{\prime})\ \mu^{\prime}\ d\mu^{\prime}d\phi^{\prime}}. \tag{148}\]  
Here \(dI_{\text{refl}}(\tau^{*},\mu,\phi)\) is the reflected radiance in direction \((\mu,\phi)\), while \(I(\tau^{*},-\mu^{\prime},\phi^{\prime})\) is the incident radiance in direction \((-\mu^{\prime},\phi^{\prime})\). Understanding bidirectional effects including sunlight is important for several reasons [63; 94]:  
1. correct interpretation of ocean color data;
2. comparing consistency of spectral radiance data derived from space observations with a single instrument for a variety of illumination and viewing conditions;
3. merging data collected by different instruments operating simultaneously. The BRDF defined in Equation (148) has unit per steradian [sr\({}^{-1}\)]. The remote sensing reflectance defined as \(R_{\text{rs}}=I(0^{+},\mu,\phi)/F^{-}(0^{+})\), where \(0^{+}\) refers the level just above the air-water interface, and \(F^{-}(0^{+})\) is the downward irradiance, also has unit [sr\({}^{-1}\)]. It should be noted that the frequently used bidirectional reflectance factor defined as BRF = \(\pi R_{\text{rs}}\) is dimensionless, because \(\pi\) has unit [sr]. A BRDF correction algorithm [63] (denoted as MAG02) was developed for application to open ocean water based on the following expression for the normalized water-leaving radiance \(nL_{w}\)  
\[nL_{w}=L_{w}\times\frac{\mathfrak{R}_{\otimes}}{\mathfrak{R}}\times\frac{f_{0} (\tau_{a},W,\text{IOP})}{Q_{0}(\tau_{a},W,\text{IOP})}\times\left[\frac{f( \theta_{0},\theta,\Delta\phi,\tau_{a},W,\text{IOP})}{Q(\theta_{0},\theta, \Delta\phi,\tau_{a},W,\text{IOP})}\right]^{-1}\]  
where \(W\) is the wind speed, and the function \(\mathfrak{R}\) accounts for refraction and reflection effects when radiances propagate through the air-water interface. The function \(f\) relates the irradiance reflectance (\(R=F^{+}/F^{-}\)) to the IOPs, and the function \(Q\) is a bidirectional function, defined as \(Q(\theta_{0},\theta^{\prime},\Delta\phi)=F^{+}(0^{-})/I^{+}(0^{-},\theta_{0}, \theta^{\prime},\Delta\phi)\). The subscripts \"0\" on \(\mathfrak{R}_{0}\), \(f_{0}\) and \(Q_{0}\) are the values of the three functions evaluated in the nadir direction. However, the MAG02 algorithm requires knowledge of CHL to derive the \(f/Q\) correction factor, and it does not work well in turbid (coastal) water. To remedy these shortcomings, [PERSON] et al. [95] developed a neural network method to correct for bidirectional effects in water-leaving radiances for both clear (open ocean) and turbid (coastal) water. This neural network algorithm directly derives the entire spectral nadir remote sensing reflectances \(R_{\text{rs}}(\lambda_{i},\theta_{0})\) from the angular values \(R_{\text{rs}}(\lambda_{i},\theta_{0},\theta,\Delta\phi)\), without any prior knowledge of the water IOPs. Based on AccuRT simulations, [PERSON] et al. [95] showed that differences in spectral \(R_{\text{rs}}\) values are significant between clear (open ocean) and turbid (coastal) water, but relatively small between nadir- and slant-viewing directions for a given water type. Consequently, a trained Radial Basis Function Neural Network (RBF-NN) can be used to convert the spectral \(R_{\text{rs}}\) values from the slant- to the nadir-viewing direction. To this end, AccuRT was used to simulate \(R_{\text{rs}}\) values at both nadir- and slant-viewing directions for a 13-layer atmosphere with aerosols added in the bottom 0-2 km layer, by randomly selecting aerosol models based on fraction of small-mode aerosol particles (\(f_{a}\)) and relative humidity (RH). The CCR bio-optical model, parameterized in terms of CHL, CDOM, and MIN as described above, was used to represent the water IOPs.