# Analysis of the Discrete Theory of Radiative Transfer in the Coupled “Ocean–Atmosphere” System: Current Status, Problems and Development Prospects

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## Abstract

**:**

## 1. Introduction

## 2. Boundary Value Problems of the Radiation Transfer Theory

## 3. Architecture of the Boundary Value Problems in Radiative Transfer

**r**) is the reflection coefficient of the surface. The inelastic scattering processes changing the wavelength of the light are not included. The BVP (1) is defined in the Cartesian coordinate system OXYZ, in which the OZ axis is directed downward perpendicular to the layer boundary, $\hat{z}$ is a unit vector along OZ. The upper boundary of the layer is located at $z=0$. Throughout the paper, the unit vectors are denoted with a “^” sign, while column vectors, row vectors and matrices are marked with the right arrows, left arrows and double arrows respectively.

**r**and defines the radiance of the surface, which is due to the multiple reflection events from the surface $\overline{\mathsf{\rho}}$ as well as multiple scattering events in the medium.

## 4. The Boundary Value Problem for the Radiative Transfer Equation for a Slab

## 5. Discretization of the Radiative Transfer Equation

_{j}are the Gaussian quadrature points of the N/2 order. For the sake of simplicity, index m is further omitted. Such discretization allows substituting the integral with a finite sum. The BVP is transformed into the matrix nonlinear differential equation of the first order with constant coefficients:

_{j}are the Gaussian quadrature weights, ${\mathrm{P}}_{l}^{n}(\mathsf{\mu})$ are the associated Legendre polynomials, and ${\mathrm{P}}_{l}^{0}(\mathsf{\mu})\equiv {\mathrm{P}}_{l}(\mathsf{\mu})$ are the Legendre polynomials, $\overleftrightarrow{\mathrm{A}}\equiv {\displaystyle \sum _{k=0}^{K}(2k+1){\mathrm{P}}_{k}^{m}({\mathsf{\mu}}_{i}^{\pm}){x}_{k}{\mathrm{P}}_{k}^{m}({\mathsf{\mu}}_{j}^{\pm})},$ $\overleftrightarrow{\mathrm{W}}\equiv \frac{\Lambda}{4}\left[\begin{array}{cc}{w}_{i}& 0\\ 0& {w}_{i}\end{array}\right]$, $\overleftrightarrow{\mathrm{M}}\equiv \left[\begin{array}{cc}{\mathsf{\mu}}_{i}^{-}& 0\\ 0& {\mathsf{\mu}}_{i}^{+}\end{array}\right]$, $x(\gamma )={\displaystyle \sum _{k=0}^{\infty}\frac{2k+1}{4\mathsf{\pi}}{x}_{k}{\mathrm{P}}_{k}(\mathrm{cos}\gamma )}$.

## 6. Propagators and Scatters

## 7. Invariance Property of the Solution

## 8. The Anisotropic Part of the Solution

_{0}= 0° and 70° for different sight angles is shown in Figure 1. The observation scheme is adopted, in which the direction to the Nadir ϑ = 180°. As one can see, the error does not exceed 10

^{−5}in almost the entire range of sight angles.

## 9. Reflection and Refraction at the Ocean–Atmosphere Interface

_{o}> 1. The discrete ordinates in the atmosphere are related to those in the ocean through Snell’s law as follows:

_{t}points per hemisphere. Consequently, we obtain the two column-vectors ${\overrightarrow{\mathrm{C}}}_{+}^{t},{\overrightarrow{\mathrm{C}}}_{-}^{t}$ of downwelling and upwelling radiance values, respectively, in the Fourier space along discrete ordinate directions. Note that in the case of the flat oceanic surface ${\overrightarrow{\mathrm{C}}}_{+}^{t}$ and ${\overrightarrow{\mathrm{C}}}_{-}^{t}$ are related to each other by the law of specular reflection. For the first and last integrals in Equation (47) we perform the change of variables according to Equation (46):

## 10. Synthetic Iteration Method

_{i}are the roots of P

_{N}

_{+1}(μ). Substituting Equation (52) into Equation (51) gives:

_{m}(τ,μ). These relations can be regarded as analogous to the Nyquist-Shannon-Kotelnikov sampling theorem. The following comments are in order:

- 1)
- Since the separation of the anisotropic part of the solution provides the most accurate discrete representation of the scattering integral, the MDOM determines the mean convergence;
- 2)
- The convergence of the solution in the uniform metric is determined by the features of the angular distribution of the scattering phase function, then the convergence here of any method for isolating anisotropy will be equivalent;
- 3)
- To achieve good convergence in a uniform metric, the sampling interval must correspond to the angular size of the finest detail of the radiance distribution necessary to solve a practical problem.

## 11. Polarized Radiation Transfer

## 12. Three-Dimensional Radiative Transfer Models

## 13. Inverse Models

## 14. Numerical Aspects

^{th}digit has been obtained. The computations have been performed for the coupled model with the flat oceanic surface. The ocean has been modeled as a two-layer system with the dissolved organic matter in the upper oceanic layer. The atmosphere was modeled as a 14-layer system containing the trace gases (O

_{3}, NO

_{2}) as well as the urban aerosol.

## 15. Conclusions

- 1)
- The synthetic iteration techniques equipped with the small-angle approximation, DOM or SHM or;
- 2)
- MOM with various degrees of assumptions.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**The relative error of the MDOM to the exact RTE solution for a semi-infinite layer of a cloudy medium with Rayleigh scattering.

**Figure 2.**Validation of the coupled MDOM model against the coupled LIDORT model. The solar zenith angle is 35 degrees.

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**MDPI and ACS Style**

Afanas’ev, V.P.; Basov, A.Y.; Budak, V.P.; Efremenko, D.S.; Kokhanovsky, A.A.
Analysis of the Discrete Theory of Radiative Transfer in the Coupled “Ocean–Atmosphere” System: Current Status, Problems and Development Prospects. *J. Mar. Sci. Eng.* **2020**, *8*, 202.
https://doi.org/10.3390/jmse8030202

**AMA Style**

Afanas’ev VP, Basov AY, Budak VP, Efremenko DS, Kokhanovsky AA.
Analysis of the Discrete Theory of Radiative Transfer in the Coupled “Ocean–Atmosphere” System: Current Status, Problems and Development Prospects. *Journal of Marine Science and Engineering*. 2020; 8(3):202.
https://doi.org/10.3390/jmse8030202

**Chicago/Turabian Style**

Afanas’ev, Viktor P., Alexander Yu. Basov, Vladimir P. Budak, Dmitry S. Efremenko, and Alexander A. Kokhanovsky.
2020. "Analysis of the Discrete Theory of Radiative Transfer in the Coupled “Ocean–Atmosphere” System: Current Status, Problems and Development Prospects" *Journal of Marine Science and Engineering* 8, no. 3: 202.
https://doi.org/10.3390/jmse8030202