### Abstract

Four types of approximations of the Mie phase function were studied in calculating multiple scattering by snow particles with the doubling method. These involve the two renormalizations of Hansen and Grant, the delta-M method and direct truncation. These four approximations were compared for snow surface albedo with effective grain radii of 50, 200 and 1000μ m in a wavelength region from 0.3 to 3.0μ m with the delta-Eddington approximation as a reference. In the Hansen's renormalization, the maximum albedo error exceeds 0.1 for snow with an effective radius of 1000 μ m at small solar zenith angles. The delta-M method overestimates snow albedos at all solar zenith angles in a wavelength region smaller than 1.4 μ m for snow with effective radius of 1000 μ m. This is due to insufficient angle resolution (0.1° in a scattering angle region less than 2°) in the forward peak region of the look-up table of the Mie phase function. It has been shown that even with ten times higher resolution in the scattering angle region less than 10° a sufficiet accuracy could not be obtained for an effective radius of 1000μ m in a wavelength region smaller than 0.6μ m. Reasonable results were obtained by the Grant's renormalization and direct truncation approximation for all cases of effective grain radii studied. It was also found that these methods save computation time and memory because sufficient accuracy is obtained even with a low angle resolution of 0.1° in the forward peak region of phase function. In direct truncation, the result was not sensitive to the choice of a truncation angle between 5° and 20°.

Original language | English |
---|---|

Pages (from-to) | 141-156 |

Number of pages | 16 |

Journal | Papers in Meteorology and Geophysics |

Volume | 47 |

Issue number | 3-4 |

Publication status | Published - Mar 1997 |

Externally published | Yes |

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### ASJC Scopus subject areas

- Atmospheric Science
- Geophysics

### Cite this

*Papers in Meteorology and Geophysics*,

*47*(3-4), 141-156.

**Approximations of phase function in calculating the spectral albedo of snow surface with multiple scattering.** / Aoki, Teruo; Aoki, Tadao; Fukabori, Masashi.

Research output: Contribution to journal › Article

*Papers in Meteorology and Geophysics*, vol. 47, no. 3-4, pp. 141-156.

}

TY - JOUR

T1 - Approximations of phase function in calculating the spectral albedo of snow surface with multiple scattering

AU - Aoki, Teruo

AU - Aoki, Tadao

AU - Fukabori, Masashi

PY - 1997/3

Y1 - 1997/3

N2 - Four types of approximations of the Mie phase function were studied in calculating multiple scattering by snow particles with the doubling method. These involve the two renormalizations of Hansen and Grant, the delta-M method and direct truncation. These four approximations were compared for snow surface albedo with effective grain radii of 50, 200 and 1000μ m in a wavelength region from 0.3 to 3.0μ m with the delta-Eddington approximation as a reference. In the Hansen's renormalization, the maximum albedo error exceeds 0.1 for snow with an effective radius of 1000 μ m at small solar zenith angles. The delta-M method overestimates snow albedos at all solar zenith angles in a wavelength region smaller than 1.4 μ m for snow with effective radius of 1000 μ m. This is due to insufficient angle resolution (0.1° in a scattering angle region less than 2°) in the forward peak region of the look-up table of the Mie phase function. It has been shown that even with ten times higher resolution in the scattering angle region less than 10° a sufficiet accuracy could not be obtained for an effective radius of 1000μ m in a wavelength region smaller than 0.6μ m. Reasonable results were obtained by the Grant's renormalization and direct truncation approximation for all cases of effective grain radii studied. It was also found that these methods save computation time and memory because sufficient accuracy is obtained even with a low angle resolution of 0.1° in the forward peak region of phase function. In direct truncation, the result was not sensitive to the choice of a truncation angle between 5° and 20°.

AB - Four types of approximations of the Mie phase function were studied in calculating multiple scattering by snow particles with the doubling method. These involve the two renormalizations of Hansen and Grant, the delta-M method and direct truncation. These four approximations were compared for snow surface albedo with effective grain radii of 50, 200 and 1000μ m in a wavelength region from 0.3 to 3.0μ m with the delta-Eddington approximation as a reference. In the Hansen's renormalization, the maximum albedo error exceeds 0.1 for snow with an effective radius of 1000 μ m at small solar zenith angles. The delta-M method overestimates snow albedos at all solar zenith angles in a wavelength region smaller than 1.4 μ m for snow with effective radius of 1000 μ m. This is due to insufficient angle resolution (0.1° in a scattering angle region less than 2°) in the forward peak region of the look-up table of the Mie phase function. It has been shown that even with ten times higher resolution in the scattering angle region less than 10° a sufficiet accuracy could not be obtained for an effective radius of 1000μ m in a wavelength region smaller than 0.6μ m. Reasonable results were obtained by the Grant's renormalization and direct truncation approximation for all cases of effective grain radii studied. It was also found that these methods save computation time and memory because sufficient accuracy is obtained even with a low angle resolution of 0.1° in the forward peak region of phase function. In direct truncation, the result was not sensitive to the choice of a truncation angle between 5° and 20°.

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M3 - Article

AN - SCOPUS:0008987323

VL - 47

SP - 141

EP - 156

JO - Papers in Meteorology and Geophysics

JF - Papers in Meteorology and Geophysics

SN - 0031-126X

IS - 3-4

ER -