# The Approximate Analytical Solution for the Top-of-Atmosphere Spectral Reflectance of Atmosphere—Underlying Snow System over Antarctica

## Abstract

**:**

## 1. Introduction

## 2. The Top-of-Atmosphere Reflectance Model

#### 2.1. The Reflection of Light from Atmosphere—Underlying Surface System

#### 2.2. The Atmospheric Path Reflectance

#### 2.3. The Atmospheric Spherical Albedo

#### 2.4. The Atmospheric Transmittance Outside Gaseous Absorption Bands

#### 2.5. The Atmospheric Gaseous Transmittance

#### 2.5.1. Ozone

#### 2.5.2. Water Vapor

#### 2.5.3. Oxygen

#### 2.6. Reflectance of Solar Light from a Snow Surface

## 3. The Intercomparison of Derived Parameterisation with Spaceborne Measurements of Spectral Top-of-Atmosphere Reflectance

## 4. Conclusions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**The intercomparison of different approximations for the calculation of the spherical albedo of atmosphere with molecular scattering with different values of atmospheric optical thickness. The exact RT calculations have been performed using the SORD code [26].

**Figure 2.**(

**a**) The intercomparison of modelled (dashed) and experimentally measured (solid line) ozone absorption cross section (at temperature t = 233 K). (

**b**) The error of the approximation for the ozone VOD.

**Figure 4.**Water vapor transmittance at ${N}_{{H}_{2}O}=0.33\mathrm{mm}$=,$\overline{t}=233\text{}\mathrm{K},\text{}\overline{P}=325\text{}\mathrm{hPa},$ ${m}_{{H}_{2}O}=3.32$.

**Figure 5.**The spectral function tabulated in [30]) (solid line) and derived using Equations (63) and (64) (dashed line).

**Figure 6.**The oxygen transmittance (solid line—the tabular data given in [30] are used, dashed line—Equations (63) and (64) for the spectral function are used).

**Figure 7.**The spectral TOA reflectance measured by OLCI (symbols) and also modelled using Equation (2) (solid red line). Dashed red line corresponds to the artificial atmosphere without gas. Blue dashed line corresponds to the case when atmospheric contribution is neglected. Solid blue line corresponds to the case when absorption by atmospheric gases (but not light scattering in atmosphere) is fully accounted for. The solar zenith angle is equal to 63.61 degrees, the VZA is 20.63 degrees and the relative azimuthal angle is 118.39 degrees. The satellite measurements have been performed on November 10, 2017, at Dome C (Antarctica) under geometry specified above.

**Figure 9.**OLCI reflectance at 761.25 nm for the case shown in Figure 8.

**Table 1.**The parameters of water vapor absorption model [28].

λ, nm | ${\mathit{B}}_{\mathit{j}}$$,\text{}{\mathbf{cm}}^{-1}$ | ${\mathit{w}}_{\mathit{j}}$$,\text{}{\mathbf{cm}}^{-1}$ | ${\Delta}_{\mathit{j}}$$,\text{}{\mathbf{cm}}^{-1}$ | Comment |
---|---|---|---|---|

910 (j = 1) | 0.744 | 11,099 | 23.4
$(w<{w}_{j})$ 73.8 $(w\ge {w}_{j})$ | Weak absorption band |

940 (j = 2) | 7.560 | 10,697 | 23.1
$(w<{w}_{j})$ 110.2 $(w\ge {w}_{j})$ | Strong absorption band |

**Table 2.**The parameters of the model used in Figure 7 for the case of clean snow underlying surface.

N | Parameter | Value | Comment |
---|---|---|---|

1 | α | 0.008 | atmospheric aerosol |

2 | β | 1.3 | atmospheric aerosol |

3 | ${N}_{{O}_{2}}$, cm-atm | $8.706853\times {10}^{4}$ | oxygen |

4 | ${N}_{{O}_{3}}$, DU | 250 | ozone |

5 | ${N}_{H{O}_{2}}$, mm | 0.33 | water vapor |

6 | $P$, hPa | 650 | surface pressure |

7 | $\overline{P}$, hPa | 325 | average pressure |

8 | $\overline{t}$, K | 233 | average temperature |

9 | L, mm | 2.24 | snow effective absorption length |

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**MDPI and ACS Style**

Kokhanovsky, A.
The Approximate Analytical Solution for the Top-of-Atmosphere Spectral Reflectance of Atmosphere—Underlying Snow System over Antarctica. *Remote Sens.* **2022**, *14*, 4778.
https://doi.org/10.3390/rs14194778

**AMA Style**

Kokhanovsky A.
The Approximate Analytical Solution for the Top-of-Atmosphere Spectral Reflectance of Atmosphere—Underlying Snow System over Antarctica. *Remote Sensing*. 2022; 14(19):4778.
https://doi.org/10.3390/rs14194778

**Chicago/Turabian Style**

Kokhanovsky, Alexander.
2022. "The Approximate Analytical Solution for the Top-of-Atmosphere Spectral Reflectance of Atmosphere—Underlying Snow System over Antarctica" *Remote Sensing* 14, no. 19: 4778.
https://doi.org/10.3390/rs14194778