# Physically Based Thermal Infrared Snow/Ice Surface Emissivity for Fast Radiative Transfer Models

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## Abstract

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## 1. Introduction

#### 1.1. Satellite Observing System Experiments over Snow-Covered Regions

#### 1.2. Surface Emissivity Considerations and Modeling Approaches

## 2. Methodology

#### 2.1. Wiscombe–Warren (WW80) Model

#### 2.2. Hybrid Physical Model

## 3. Results and Discussion

#### 3.1. Preliminary OSE Using WW80 Model

`FV3GFS_V16`over the time period of 28 July to 30 September 2021, with obs obtained from the SNPP/NOAA-20 CrIS and the Metop-B/C IASI for a longwave TIR microwindow, $\nu \approx 962.5$ cm${}^{-1}$). All the satellite bias corrections were reset and a 5-week spin-up was conducted. The baseline (control) OSE was based on the existing CRTM v2.3.0 snow/ice emissivity a priori [4]. The test run was conducted using a stripped-down LUT from the v1 (WW80) snow/ice emissivity physical model that was hard-coded into the GSI for two zenith observing angles (${\theta}_{o}={10}^{\circ},\phantom{\rule{0.166667em}{0ex}}{60}^{\circ}$) and a nominal particle size of $r=200$ $\mathsf{\mu}$m. The LUT-interpolated coarse-resolution model emissivity spectra were then adjusted based on a linear fit to the surface channel data.

#### 3.2. Comparison of Models against Published Laboratory and Field Measurements

## 4. Conclusions and Future Work

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Radiative Transfer within a Mie-Scattering Layer

## Appendix B. Delta-Eddington (D-E) Approximation

#### Appendix B.1. Phase Function

#### Appendix B.2. Scaled RTE

#### Appendix B.3. Simplified RTE

## Appendix C. Solutions for the Surface Fluxes

## Appendix D. Determination of Spectral Albedo

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**Figure 2.**Stacked histograms of longwave TIR microwindow (962.5 cm${}^{-1}$) obs − calc obtained from a global snow surface OSE over the time period of 28 July to 30 September 2021, where the “calc” were computed from the existing release version of CRTM (v2.3.0), and “obs” were observations from TIR hyperspectral sounders assimilated within the FV3GFS GSI (

`FV3GFS_V16`) from: (

**top**) the SNPP (orange-red) and NOAA-20 (blue) CrIS in the “afternoon” (01:30, 13:30 local equator crossing time LEXT) ascending orbit, and (

**bottom**) the Metop-B/C (orange-red/blue) IASI in the “morning” (09:30, 21:30 LEXT) descending orbit.

**Figure 3.**Snow surface emissivity spectra calculated based on the WW80 snow albedo model, Equations (5) and (7), for snow particle radius $r=200$ $\mathsf{\mu}$m and zenith observing angles ${\theta}_{o}={0}^{\circ},\phantom{\rule{0.166667em}{0ex}}{30}^{\circ},\phantom{\rule{0.166667em}{0ex}}{45}^{\circ},\phantom{\rule{0.166667em}{0ex}}{60}^{\circ},\phantom{\rule{0.166667em}{0ex}}{75}^{\circ}$. The figure aspect ratio, x and y axes, and choice of ${\theta}_{o}$ and r are meant to duplicate visually the results published by Ref. [35] (Figure 1b, op. cit.).

**Figure 4.**Emissivity particle size dependence at ${\theta}_{o}={10}^{\circ}$. The (

**top**) and (

**bottom**) rows show the particle size dependencies in terms of log-r (radius) and log-$\chi $ (size parameter), respectively, with these defined by the mean of the radii ranges for snow types reported in Salisbury et al. [51], namely, frost, fresh, or newly fallen snow, and medium and coarse granular snow. The

**left**column shows the results from the WW80 model and the

**right**column shows field-measurements from ECOSTRESS [29,51], with the circles plotted at $r=1000$ $\mathsf{\mu}$m denoting the flat surface emissivities defined as ${\u03f5}_{\nu}\equiv 1-{\rho}_{\nu}$, where ${\rho}_{\nu}$ is the unpolarized Fresnel reflectance.

**Figure 5.**Fractional non-Fresnel reflective surface areas, ${\eta}_{s}$ or $1-{\eta}_{\rho}$. The blue line/circles depict the values derived by Hori et al. [53] in their “semi-empirical” model, $1-{\eta}_{\rho}$; the red line/asterisks depict the values interpreted as surface area reflecting as a multiple-scattering layer, ${\eta}_{s}$, extrapolated in log-r space to smaller snow grain sizes where emissivity is observed to behave in this manner according to the WW80 Mie-scattering model.

**Figure 6.**As Figure 2, except calc, which are derived from CRTM v2.3.0 using a subset (${\theta}_{o}={10}^{\circ},\phantom{\rule{0.166667em}{0ex}}{60}^{\circ}$, and $r=200$ $\mathsf{\mu}$m) of the physical snow emissivity model LUT (v1, based only on the WW80 albedo model), hard-coded into the GSI.

**Figure 7.**TIR snow/ice model spectral emissivity calculations versus ECOSTRESS laboratory measurements [29,51] for an observer zenith angle ${\theta}_{o}={10}^{\circ}$. The (

**top**) row shows the snow grain size dependencies in log-r space (x-axes) for a sample of 6 monochromatic channels within the TIR spectral windows ($\nu =$ 740, 840, 900, 980, 1160, 2620 cm${}^{-1}$), and the (

**bottom**) row shows the corresponding spectral dependencies for a sample of 6 snow grain sizes corresponding to those reported in Ref. [51] ($r=$ 5, 10, 30, 212.5, 750, and 1000 $\mathsf{\mu}$m). The four columns from (

**left**) to (

**right**) show the results of the (1) H15 “semi-empirical” model [53]; (2) the WW80 albedo model [31]; (3) the hybrid physical model, Equation (13); and (4) laboratory measurements from the ECOSTRESS/ASTER library [29,51].

**Figure 8.**Similar to the top row of Figure 7 but showing TIR snow/ice model spectral emissivity calculations (for a different set of monochromatic channels spanning the lonwave TIR) as a function of linear-r ($\mathsf{\mu}$m) versus multi-angular field measurements of snow cover with median particle sizes $r\ge 35$ $\mathsf{\mu}$m taken from Hori et al. [52]. The columns are arranged as in Figure 7, with emissivity models H13, WW80, and hybrid physical model, shown in the

**left three**columns, and the field measurements in the

**rightmost**column; the rows are arranged from top to bottom, according to zenith observing angles ${\theta}_{o}={30}^{\circ},\phantom{\rule{0.166667em}{0ex}}{45}^{\circ},\phantom{\rule{0.166667em}{0ex}}{60}^{\circ},\phantom{\rule{0.166667em}{0ex}}{75}^{\circ}$.

**Table 1.**Measured Mean Sizes of Snow Grains ($\mathsf{\mu}$m) During Metamorphosis (From Yosida, 1962 [50]).

# of Days after Snow Deposition | Mean Length $\overline{\mathit{\ell}}$ ($\mathbf{\mu}$m) | Mean Radius $\overline{\mathit{r}}$ ($\mathbf{\mu}$m) | Mean Constriction $\overline{\mathit{w}}$ ($\mathbf{\mu}$m) |
---|---|---|---|

1 | 200 | 45 | 50 |

5 | 190 | 45 | 50 |

9 | 260 | 60 | 80 |

15 | 430 | 80 | 100 |

24 | 600 | 110 | 140 |

31 | 570 | 130 | 160 |

Particle Size, r ($\mathsf{\mu}$m) | ${\mathit{\eta}}_{\mathit{\rho}}$ | Snow Morphology | |
---|---|---|---|

Median | Range | ||

35 | 20–50 | 0.22 | fine dendrite snow |

300 | 150–550 | 0.29 | medium granular snow |

400 | 25–500 | 0.41 | coarse-grained snow |

550 | 400–750 | 0.53 | sun crust |

≳1000 (flat) | 0.95 | bare ice |

Model | ${\mathit{\theta}}_{\mathit{o}}$ | $\mathit{\nu}$ (cm${}^{-1}$) | Grain Size, r | T (K) |
---|---|---|---|---|

Original CRTM a priori | N/A | 666–3333 | “fresh” and “aged” | N/A |

(CRTM release versions v1.0 to v2.3.0) | ||||

WW80 physical model | 0–75${}^{\circ}$ | 600–3000 | 5–1000 $\mathsf{\mu}$m | 230–270 |

(CRTM v3, snow emissivity v1.0) | ||||

Hybrid physical model | 0–75${}^{\circ}$ | 50–3000 | 1–1000 $\mathsf{\mu}$m | 230–270 |

(CRTM v3, snow/ice emissivity v1.1) |

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**MDPI and ACS Style**

Nalli, N.R.; Dang, C.; Jung, J.A.; Knuteson, R.O.; Borbas, E.E.; Johnson, B.T.; Pryor, K.; Zhou, L.
Physically Based Thermal Infrared Snow/Ice Surface Emissivity for Fast Radiative Transfer Models. *Remote Sens.* **2023**, *15*, 5509.
https://doi.org/10.3390/rs15235509

**AMA Style**

Nalli NR, Dang C, Jung JA, Knuteson RO, Borbas EE, Johnson BT, Pryor K, Zhou L.
Physically Based Thermal Infrared Snow/Ice Surface Emissivity for Fast Radiative Transfer Models. *Remote Sensing*. 2023; 15(23):5509.
https://doi.org/10.3390/rs15235509

**Chicago/Turabian Style**

Nalli, Nicholas R., Cheng Dang, James A. Jung, Robert O. Knuteson, E. Eva Borbas, Benjamin T. Johnson, Ken Pryor, and Lihang Zhou.
2023. "Physically Based Thermal Infrared Snow/Ice Surface Emissivity for Fast Radiative Transfer Models" *Remote Sensing* 15, no. 23: 5509.
https://doi.org/10.3390/rs15235509