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Scattering

The Henyey-Greenstein phase function

Page updated: Apr 9, 2013
Principal author: Curtis Mobley
 

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

$\displaystyle \tilde{\beta}_{{\rm HG} } (g, \psi )~ \equiv{\frac{~1}{4 \pi }} \,{\frac{1 - g ^{2} }{\left (1 + g^{2} - 2 g \cos \psi \right ) ^{3/2} }} \,\, ,$ (1)

for scattering by interstellar dust. The parameter $ g$ can be adjusted to control the relative amounts of forward and backward scattering in $ \tilde{\beta}_{\rm HG}$; $ g = 0$ corresponds to isotropic scattering, and $ g \to 1$ gives highly peaked forward scattering. Note that $ \tilde{\beta}_{\rm HG}$ satisfies the normalization condition $ 2 \pi \int_{-1}^{1} \, \tilde{\beta} _{{\rm HG} } (g, \psi )\, d \cos \psi = 1$ for any $ g$.

The physical interpretation of $ g$ comes from noting that

$\displaystyle 2 \pi \int_{-1}^{1} \,\, \tilde{\beta}_{{\rm HG} } (g, \psi )\, \cos \psi \, d \cos \psi ~=~g\,\, .$ (2)

Thus the Henyey-Greenstein parameter $ g$ is just the average of the cosine of the scattering angle for $ \tilde{\beta}_{\rm HG}$.

Equation (1) can be integrated over $ \psi$ from $ \pi/2$ to $ \pi$ to obtain the backscatter fraction:

$\displaystyle B _{\rm HG} ~=~{\frac{1 - g}{2g}} \left [{\frac{1 + g}{\sqrt{1 + g ^{2} } }} - 1 \right ]\,\, .$ (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 $ \tilde{\beta}_{\rm HG}$ and measured oceanic phase functions because oceanic particles are quite different in their physical properties from interstellar dust.

The average of $ \cos \psi$ for the Petzold average-particle phase function $ \tilde{\beta}_p$ 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 $ \tilde{\beta}_p$, in the sense that each phase function then has the same average cosine. Figures 1 and 2 compare $ \tilde{\beta}_{\rm HG}(g = 0.924, \psi)$ and $ \tilde{\beta}_p$. $ \tilde{\beta}_{\rm HG}$ is also shown for $ g = 0.7$ and 0.99. Note that the best-fit $ \tilde{\beta}_{\rm HG}$ differs noticeably from $ \tilde{\beta}_p$ at scattering angles greater than $ \psi \approx 150~\rm {degrees}$ and less than 20 degrees, and that $ \tilde{\beta}_{\rm HG}$ is much too small at angles of less than a few degrees. The small-angle behavior of Eq. (1) is inherently incompatible with $ \tilde{\beta}_p$ because $ \tilde{\beta}_{\rm HG}$ always levels off as $ \psi \to 0$, whereas $ \tilde{\beta}_p$ continues to rise. Even for $ g = 0.99$, $ \tilde{\beta}_{\rm HG}$ is nearly constant for $ \psi < 0.5~\rm {degrees}$. For $ g = 0.924$, $ \tilde{\beta}_{\rm HG} = 0.0170$, compared to the backscatter fraction of 0.0183 for the Petzold phase function.

Figure: Fig. 1. Log-log Comparison of the Petzold average-particle phase function $ \tilde{\beta}_p$ (black line) with $ \tilde{\beta}_{\rm HG}(g,\psi)$ of Eq. (1) for three values of $ g$.
Image 3baea9dffdacc03549e23e62176fb014

Figure: Fig. 2. Log-linear Comparison of the Petzold average-particle phase function $ \tilde{\beta}_p$ (black line) with $ \tilde{\beta}_{\rm HG}(g,\psi)$ of Eq. (1) for three values of $ g$.
Image e2bbba58597e78f48b848c2058a07b0b

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

$\displaystyle \tilde{\beta}_{{\rm TTHG} } (\alpha , g _{1} , g _{2}, \psi )~ \equiv ~\alpha \, \tilde{\beta}_{{\rm HG} } (g _{1} , \psi ) + (1 - \alpha ) \tilde{\beta}_{{\rm HG} } (g _{2} , \psi )\,\, .$ (4)

Enhanced small-angle scattering is obtained by choosing $ g_1$ near one, and enhanced backscatter is obtained by making $ g_2$ negative; $ \alpha$ is a weighting factor between zero and one. Kattawar (1975) shows how to determine best-fit values of $ \alpha$, $ g_1$, and $ g_2$ 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 $ \tilde{\beta}_{\rm TTHG}$ to highly peaked oceanic phase functions such as $ \tilde{\beta}_p$ always remains unsatisfactory at very small angles, for the reason already noted.