## 3 Results
### Machine Learning Results  
Forward radiative transfer modeling from tools such as Hydrolight can be computationally inefficient due to the large number of complex non-linear physical equations. The current and future Earth observing satellites continue to provide measurements at higher spectral, spatial and temporal resolutions. As a result of the increased number of measurements, traditional physically based radiative transfer models will not be feasible in operational processing. Machine learning approaches have been proposed to overcome these computational challenges. Machine learning models such as neural networks (NNs) can learn how to represent the complex physical relationships in the radiative transfer simulations through statistical equations [55; 56].  
Here, we explore the feasibility of using an NN to reproduce the physics-based penetration depth simulations from Hydrolight so that the retrieval algorithm can be applied to an operational sensor such as OMI. The inputs used to estimate penetration depth are SZA, chlorophyll concentration, total column ozone and surface irradiance. For this work, we used a simple four-layer feed-forward neural network with an input layer, two hidden layers and an output layer that predicts planar and scalar penetration depth. The neural network uses a sigmoid activation function for each hidden layer and an adaptive moment estimation optimizer was used to minimize the error with a learning rate of 0.01. We trained the NN with 80% of the samples from two orbits from 7 January 2005 (orbits 2570 and 2572), which include a well-distributed sampling of Case I open waters (low chlorophyll concentration) as well as Case II coastal waters (higher chlorophyll concentration). The neural network was able to predict the penetration depth in approximately two seconds of the computer time for a single orbit as compared with a few minutes of the computer time needed for processing a single orbit by the physics-based algorithm.  
Figure 9 shows comparisons of the NN estimates of penetration depth compared to those from the Hydrolight simulations. This figure shows comparisons for orbits included in the training (left panel) as well as additional independent testing orbits not included in the NN training (right panel). There was excellent agreement between the NN results and the Hydrolight simulations with an \(R^{2}\) of nearly 1 and RMS error of 0.08-0.1 m. The comparisons were quite similar for the training orbits and independent orbits, suggesting that the NN works well for all conditions and is not overfitting to the training samples.  
Figure 10 shows maps of the planar penetration depth from Hydrolight (top) and the estimated planar penetration depth from the NN (middle). The NN does well in reproducing the planar penetration depth; differences with Hydrolight were generally  
Figure 9: Comparisons of Hydrolight penetration depth simulations and neural network estimate for 7 January 2005. Left panel shows comparisons for orbits included in neural network training while right panel shows orbits on this date not included in the training.  
\(<\)\(\pm\)2%. There were some larger differences off the immediate coast of Antarctica, which are likely due to misclassification of snow or sea ice.  
Figure 10: Map of planar penetration depth on 7 January 7 2005. Top panel shows planar penetration depth from Hydrolight, middle panel shows planar penetration depth from the neural network and the bottom shows the percent difference of Hydrolight versus the neural network estimate. The color bars represent the penetration depths in meters and the percent difference.