# Optimized Wavelength Sampling for Thermal Radiative Transfer in Numerical Weather Prediction Models

^{1}

^{2}

^{*}

## Abstract

**:**

_{2}, and other trace gases. Radiative transfer calculations of wavelength-integrated quantities, such as irradiance and heating rate, are computationally expensive, requiring a high spectral resolution for accurate numerical weather prediction and climate modeling. This paper introduces a method that could highly reduce the cost of integration in the thermal spectrum by employing an optimized wavelength sampling method. Absorption optical thicknesses for various trace gases were calculated from the HITRAN 2012 spectroscopic dataset using the ARTS line-by-line model as input to a fast Schwarzschild radiative transfer model. Using a simulated annealing algorithm, different optimized sets of wavelengths and corresponding weights were identified, which allowed for accurate integrated quantities to be computed as a weighted sum, reducing the computational time by several orders of magnitude. For each set of wavelengths, a lookup table, including the corresponding weights and absorption cross-sections, is created and can be applied to any atmospheric setups for which it was trained. We applied the lookup table to calculate irradiances and heating rates for a large set of atmospheric profiles from the ECMWF 91-level short-range forecast. Ten wavelength nodes are sufficient to obtain irradiances within an average root mean square error (RMSE) of upward and downward radiation at any height below 1 Wm

^{−2}while 100 wavelengths allowed for an RSME of below 0.05 Wm

^{−2}. The applicability of this method was confirmed for irradiances and heating rates in clear conditions and for an exemplary cloud at 3.2 km height. Representative spectral gridpoints for integrated quantities in the thermal spectrum (REPINT) is available as absorption parameterization in the libRadtran radiative transfer package, where it can be used as an efficient molecular absorption parameterization for a variety of radiative transfer solvers.

## 1. Introduction

_{2}O, CO

_{2}, and O

_{3}at a time.

## 2. Thermal Radiative Transfer Model

_{2}O (called perturbations in ARTS) to encompass all atmospheric variations in the datasets of [1,11].

_{2}O with its own VMR, an interpolation method was developed, which optimally considers these dependencies. For a general trace gas, the method first identifies the position of the desired absorption cross-section ${\sigma}_{ij}({\overline{p}}_{i},{\overline{T}}_{i})$ in the grid of the stored absorption cross-sections, producing the following four values:

^{−2}. This discrepancy, however, did not affect the optimization process, since both the to be approximated values and the approximations use the same interpolation.

## 3. Simulated Annealing Method

#### 3.1. Training Dataset

_{2}O, O

_{3}, CO

_{2}, N

_{2}O, CO, and CH

_{4}. For applications to different climatic scenarios, these atmospheres were additionally modified in their concentrations of CO

_{2}and CH

_{4}. For the data considered in this case, each atmosphere in the Garand set of atmospheres was considered for 0, 1 and 5 times the original concentration in CO

_{2}and CH

_{4}. Hence, 378 atmospheric scenarios were defined as a training set, with a total of 32,508 integrated values for upward and downward irradiance at the respective atmospheric layer interfaces, as well as the corresponding spectral values for 100,000 wavelengths each.

#### 3.2. Simulated Annealing Algorithm

#### 3.3. Absorption Parameterization for the Thermal Spectral Region

## 4. Performance of the Parameterization

- The linear regression method produces few negative weights. These did not produce any unphysical results during testing.
- Sampling nodes concentrate on the part of the spectrum in which most of the absorption of different trace gases takes place. This can be seen by comparing to the impact of trace gas absorption (CO
_{2}, H_{2}O and O_{3}) in Figure 2. - Comparatively large weights at longer wavelengths, where little absorption takes place, account for the bulk of the integrated value, while sampling nodes at shorter wavelengths with high absorption determine the fine-tuned values with respect to specific atmospheric scenarios.

**Figure 4.**Positions of sampling nodes and chosen weights from reduced lookup tables for 10 (

**a**), 30 (

**b**), 50 (

**c**) and 100 (

**d**) sampling nodes.

${\mathit{n}}_{\mathbf{wvl}}$ | $4\phantom{\rule{3.33333pt}{0ex}}\mathsf{\mu}\mathbf{m}<\mathit{\lambda}<20\phantom{\rule{3.33333pt}{0ex}}\mathsf{\mu}\mathbf{m}$ | $4\phantom{\rule{3.33333pt}{0ex}}\mathsf{\mu}\mathbf{m}<\mathit{\lambda}<40\phantom{\rule{3.33333pt}{0ex}}\mathsf{\mu}\mathbf{m}$ | $\left|\mathit{w}\right|<1000$ | $\left|\mathit{w}\right|<5000$ | $\mathit{w}<0$ |
---|---|---|---|---|---|

10 | 0.70 | 1.00 | 0.20 | 0.70 | 0.00 |

30 | 0.73 | 0.87 | 0.63 | 0.90 | 0.00 |

50 | 0.80 | 0.90 | 0.70 | 0.88 | 0.04 |

100 | 0.82 | 0.92 | 0.77 | 0.94 | 0.06 |

#### 4.1. Test Dataset

#### 4.2. Accuracy of Irradiances

#### 4.3. Accuracy of Heating Rates

#### 4.4. Cloudy Atmospheres

## 5. Conclusions

^{−2}. With 100 wavelength nodes, an average RSME below 0.05 Wm

^{−2}can be achieved. The method was verified for a large variety of atmospheric profiles taken from the ECMWF model. This performed equally well for atmospheres with or without clouds.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Spectral upward irradiance at the ground and at the top of the atmosphere, considering emission and absorption for an exemplary atmosphere in the dataset of [1].

**Figure 2.**Spectral upward irradiance at the ground and at the top of the atmosphere, considering the emission and absorption of different trace gases separately for the atmosphere from Figure 1.

**Figure 3.**Performance of the simulated annealing algorithm on a set of 6000 irradiance datapoints of ${E}_{\mathrm{up}}/{E}_{\mathrm{up}}$ for different profiles at different heights. The average RSME of 30 runs for each number of annealing iterations is depicted on the graph for 10 resp. 100 sampling nodes.

**Figure 5.**Absolute RSME of irradiance, as determined by weighted sum and high spectral resolution calculations, for 500 randomly chosen atmospheres in the dataset of [11].

**Figure 6.**Heating rates of the U.S. Standard Atmosphere, calculated using different numbers of sampling nodes.

**Figure 7.**Heating rates of the U.S. Standard Atmosphere with a cloud calculated using different numbers of sampling nodes.

**Table 1.**Parameters used for the generation of a lookup table using ARTS. The non-linear species (NLS) perturbations enable a calculation of the absorption of cross-sections of H

_{2}O depending on their own concentration. Each perturbation corresponds to the factor by which the water vapor concentration of the reference atmosphere is altered.

Parameter | Amount | Scope |
---|---|---|

T Perturbation | 9 | (−120, 120) °C |

NLS Perturbation | 5 | (0, 10) |

Pressure grid | 41 | (110,000, 0.0006892) Pa |

Gas species | 9 | H_{2}O, CO_{2}, O_{3}, N_{2}O, CO, |

CH_{4}, O_{2}, HNO_{3}, N_{2} |

Annealing Parameters | Values |
---|---|

Total annealing iterations ${n}_{\mathrm{tot}}$ | 10,000 |

${S}_{\mathrm{start}}$ | 0.5 |

${S}_{\mathrm{end}}$ | 1 × 10${}^{-7}$ |

Annealing datapoints | 15,000 |

Sampling nodes | 10, 30, 50, 100 |

**Table 4.**Average $\mu $ and standard deviation $\sigma $ of the absolute RSME of radiative upward and downward irradiance ${E}_{\mathrm{up}},{E}_{\mathrm{dn}}$ for 500 atmospheric dataset profiles by [11], for four different reduced lookup tables. Each calculation was made for the original atmosphere, as well as the same atmosphere with an added cloud. The unit for values in the table is [Wm

^{−2}].

Clear | Cloudy | |||
---|---|---|---|---|

${\mathbf{n}}_{\mathrm{wvl}}$ | $\mathbf{\mu}$ | $\mathbf{\sigma}$ | $\mathbf{\mu}$ | $\mathbf{\sigma}$ |

10 | 0.874 | 0.163 | 0.723 | 0.165 |

30 | 0.256 | 0.067 | 0.236 | 0.054 |

50 | 0.118 | 0.027 | 0.207 | 0.079 |

100 | 0.046 | 0.012 | 0.060 | 0.020 |

**Table 5.**Absolute RSME $\u03f5$ and maximum error ${\u03f5}_{\mathrm{max}}$ for heating rates of the U.S. Standard Atmosphere calculated with different numbers of sampling nodes. The unit for the values in the table is [Kd

^{−1}].

${\mathit{n}}_{\mathbf{wvl}}$ | $\mathit{\u03f5}$ | ${\mathit{\u03f5}}_{\mathbf{max}}$ |
---|---|---|

10 | 0.1817 | 0.3555 |

30 | 0.0323 | 0.0806 |

50 | 0.0178 | 0.0491 |

100 | 0.0049 | 0.0128 |

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**MDPI and ACS Style**

de Mourgues, M.; Emde, C.; Mayer, B.
Optimized Wavelength Sampling for Thermal Radiative Transfer in Numerical Weather Prediction Models. *Atmosphere* **2023**, *14*, 332.
https://doi.org/10.3390/atmos14020332

**AMA Style**

de Mourgues M, Emde C, Mayer B.
Optimized Wavelength Sampling for Thermal Radiative Transfer in Numerical Weather Prediction Models. *Atmosphere*. 2023; 14(2):332.
https://doi.org/10.3390/atmos14020332

**Chicago/Turabian Style**

de Mourgues, Michael, Claudia Emde, and Bernhard Mayer.
2023. "Optimized Wavelength Sampling for Thermal Radiative Transfer in Numerical Weather Prediction Models" *Atmosphere* 14, no. 2: 332.
https://doi.org/10.3390/atmos14020332