A tutorial review is provided of forward and inverse radiative transfer in coupled atmosphere-snow/ice-water systems. The coupled system is assumed to consist of two adjacent horizontal slabs separated by an interface across which the refractive index changes abruptly from its value in air to that in ice/water. A comprehensive review is provided of the inherent optical properties of air and water (including snow and ice). The radiative transfer equation for unpolarized as well as polarized radiation is described and solutions are outlined. Several examples of how to formulate and solve inverse problems encountered in environmental optics involving coupled atmosphere-water systems are discussed in some detail to illustrate how the solutions to the radiative transfer equation can be used as a forward model to solve practical inverse problems.

Ocean color algorithms that rely on “atmospherically corrected” nadir water-leaving radiances to infer information about marine constituents such as the chlorophyll concentration depend on a reliable method to convert the angle-dependent measured radiances from the observation direction to the nadir direction. It is also important to convert the measured radiances to the nadir direction when comparing and merging products from different satellite missions. The standard correction method developed by Morel and coworkers requires knowledge of the chlorophyll concentration. Also, the standard method was developed based on the Case 1 (open ocean) assumption, which makes it unsuitable for Case 2 situations such as turbid coastal waters. We introduce a neural network method to convert the angle-dependent water-leaving radiance (or the corresponding remote sensing reflectance) from the observation direction to the nadir direction. This method relies on neither an “atmospheric correction” nor prior knowledge of the water constituents or the inherent optical properties. It directly converts the remote sensing reflectance from an arbitrary slanted viewing direction to the nadir direction by using a trained neural network. This method is fast and accurate, and it can be easily adapted to different remote sensing instruments. Validation using NuRADS measurements in different types of water shows that this method is suitable for both Case 1 and Case 2 waters. In Case 1 or chlorophyll-dominated waters, our neural network method produces corrections similar to those of the standard method. In Case 2 waters, especially sediment-dominated waters, a significant improvement was obtained compared to the standard method.

The treatment of strongly anisotropic scattering phase functions is still a challenge for accurate radiance computations. The new delta-M+ method resolves this problem by introducing a reliable, fast, accurate, and easy-to-use Legendre expansion of the scattering phase function with modified moments. Delta-M+ is an upgrade of the widely used delta-M method that truncates the forward scattering peak with a Dirac delta function, where the “+” symbol indicates that it essentially matches moments beyond the first M terms. Compared with the original delta-M method, delta-M+ has the same computational efficiency, but for radiance computations, the accuracy and stability have been increased dramatically.

The Ocean Color—Simultaneous Marine and Aerosol Retrieval Tool (OC-SMART) is a robust data processing platform utilizing scientific machine learning (SciML) in conjunction with comprehensive radiative transfer computations to provide accurate remote sensing reflectances ($R_{\mathrm{rs}}$ estimates), aerosol optical depths, and inherent optical properties. This paper expands the capability of OC-SMART by quantifying uncertainties in ocean color retrievals. Bayesian inversion is used to relate measured top of atmosphere radiances and a priori data to estimate posterior probability density functions and associated uncertainties. A framework of the methodology and implementation strategy is presented and uncertainty estimates for $R_{\mathrm{rs}}$ retrievals are provided to demonstrate the approach by applying it to MODIS, OLCI Sentinel-3, and VIIRS sensor data.