Page updated: April 21, 2020
Author: Curtis Mobley

The Henyey-Greenstein Phase Function

It is often convenient to have an analytic formula that approximates the shape of an actual phase function. The Henyey-Greenstein (HG) phase function has been widely used in the past for such purposes and is discussed here for historical completeness. The Henyey-Greenstein phase function (and many other simple analytical models) has now been supplanted in oceanography by the more complicated but more realistic Fournier-Forand phase function, which is discussed on the next page.

Henyey and Greenstein (1941) proposed the phase function

β̃HG(g,ψ) 1 4π 1 g2 1 + g2 2gcosψ32, (1)

for scattering by interstellar dust. The parameter g can be adjusted to control the relative amounts of forward and backward scattering in β̃HG; g = 0 corresponds to isotropic scattering, and g 1 gives highly peaked forward scattering. Note that β̃HG satisfies the normalization condition 2π11β̃HG(g,ψ)dcosψ = 1 for any g.

The physical interpretation of g comes from noting that

2π11β̃ HG(g,ψ)cosψdcosψ = g. (2)

Thus the Henyey-Greenstein parameter g is just the average of the cosine of the scattering angle for β̃HG.

Equation (1) can be integrated over ψ from π2 to π to obtain the backscatter fraction:

BHG = 1 g 2g 1 + g 1 + g2 1. (3)

Because of its mathematical simplicity, the HG phase function has been widely used in other fields, including oceanography. However, we can anticipate discrepancies between β̃HG and measured oceanic phase functions because oceanic particles are quite different in their physical properties from interstellar dust.

The average of cosψ for the Petzold average-particle phase function β̃p of the previous page is 0.924. Using g = 0.924 in Eq. (1) thus gives the Henyey-Greenstein phase function corresponding to the particle phase function β̃p, in the sense that each phase function then has the same average cosine. Figures 1 and 2 compare β̃HG(g = 0.924,ψ) and β̃p. β̃HG is also shown for g = 0.7 and 0.99. Note that the best-fit β̃HG differs noticeably from β̃p at scattering angles greater than ψ 150degrees and less than 20 degrees, and that β̃HG is much too small at angles of less than a few degrees. The small-angle behavior of Eq. (1) is inherently incompatible with β̃p because β̃HG always levels off as ψ 0, whereas β̃p continues to rise. Even for g = 0.99, β̃HG is nearly constant for ψ < 0.5degrees. For g = 0.924, β̃HG = 0.0170, compared to the backscatter fraction of 0.0183 for the Petzold phase function.


PIC

Figure 1: Fig. 1. Log-log Comparison of the Petzold average-particle phase function β̃p (black line) with β̃HG(g,ψ) of Eq. (1) for three values of g.


PIC

Figure 2: Fig. 2. Log-linear Comparison of the Petzold average-particle phase function β̃p (black line) with β̃HG(g,ψ) of Eq. (1) for three values of g.

Because of the poor fits of the HG phase function to measurements at small and large scattering angles, a linear combination of Henyey-Greenstein phase functions is sometimes used to improve the fit at small and large angles. The so-called two-term Henyey-Greenstein (TTHG) phase function is

β̃TTHG(α,g1,g2,ψ) αβ̃HG(g1,ψ) + (1 α)β̃HG(g2,ψ). (4)

Enhanced small-angle scattering is obtained by choosing g1 near one, and enhanced backscatter is obtained by making g2 negative; α is a weighting factor between zero and one. Kattawar (1975) shows how to determine best-fit values of α, g1, and g2 for a given phase function. Reasonable fits can be obtained with the TTHG function for phase functions that are not highly peaked, for example atmospheric haze phase functions (see Kattawar’s paper for example fits). However, the fit of β̃TTHG to highly peaked oceanic phase functions such as β̃p always remains unsatisfactory at very small angles, for the reason already noted.

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