Improved δ-Eddington approximation for optically thin clouds

Tong Ren, Ping Yang, Guanglin Tang, Xianglei Huang, Eli Mlawer

Research output: Contribution to journalArticlepeer-review

3 Scopus citations


The δ-Eddington simulations of broadband shortwave net radiation fluxes at the top of the atmosphere (FTOA) and the surface (FSURF) are evaluated with different parameterizations of the forward fraction of scattering (f), including the square of the asymmetry factor (f = g2), the fraction of the forward single-scattered intensity over the total single-scattered intensity (f = fp), and the cube of the asymmetry factor (f = g3). g2 and g3 are respectively the 2nd and 3rd moments of the Henyey–Greenstein (HG) phase function and hence approximate measures of the variance and skewness of the phase function. The factor fp for spherical droplets is estimated using a truncation angle, which separates the forward peak and diffusive portions of a highly anisotropic phase function. The results show that the simulations of FTOA and FSURF are not improved, if the conventional approach f = g2 is replaced by f = fp in the δ-Eddington approximation for an atmosphere in the presence of liquid clouds. For the optically thick conditions, multiple scattering plays a dominant role in determining the reflectance (R) and transmittance (T) of the cloudy layer; the conventional parameterization f = g2 is most accurate among the three parameterizations. For the optically thin conditions, single scattering dominates over multiple scattering and thus f = g2 results in biased FTOA and FSURF calculations, particularly with low solar elevations. For such cases, f = g3 shows most accurate FTOA and FSURF results for both liquid and ice clouds, though f = g3 also results in smaller cloud layer heating rates in general.

Original languageEnglish (US)
Article number106694
JournalJournal of Quantitative Spectroscopy and Radiative Transfer
StatePublished - Jan 2020


  • Forward fraction of scattering
  • The asymmetry factor
  • δ-Eddington approximation

ASJC Scopus subject areas

  • Radiation
  • Atomic and Molecular Physics, and Optics
  • Spectroscopy


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