15-digit accuracy calculations of Ambartsumian-Chandrasekhar’s $H$-functions for four-term phase functions with the double-exponential formula

Abstract

We have established an iterative scheme to calculate with 15-digit accuracy the numerical values of Ambartsumian-Chandrasekhar’s $H$-functions for anisotropic scattering characterized by the four-term phase function: the method incorporates some advantageous features of the iterative procedure of Kawabata (Astrophys. Space Sci. 358:32, 2015) and the double-exponential integration formula (DE-formula) of Takahashi and Mori (Publ. Res. Inst. Math. Sci. Kyoto Univ. 9:721, 1974), which proved highly effective in Kawabata (Astrophys. Space Sci. 361:373, 2016). Actual calculations of the $H$-functions have been carried out employing 27 selected cases of the phase function, 56 values of the single scattering albedo $\varpi_{0}$, and 36 values of an angular variable $\mu(= \cos\theta)$, with $\theta$ being the zenith angle specifying the direction of incidence and/or emergence of radiation. Partial results obtained for conservative isotropic scattering, Rayleigh scattering, and anisotropic scattering due to a full four-term phase function are presented. They indicate that it is important to simultaneously verify accuracy of the numerical values of the $H$-functions for $\mu<0.05$, the domain often neglected in tabulation. As a sample application of the isotropic scattering $H$-function, an attempt is made in Appendix to simulate by iteratively solving the Ambartsumian equation the values of the plane and spherical albedos of a semi-infinite, homogeneous atmosphere calculated by Rogovtsov and Borovik (J. Quant. Spectrosc. Radiat. Transf. 183:128, 2016), who employed their analytical representations for these quantities and the single-term and two-term Henyey-Greenstein phase functions of appreciably high degrees of anisotropy. While our results are in satisfactory agreement with theirs, our procedure is in need of a faster algorithm to routinely deal with problems involving highly anisotropic phase functions giving rise to near-conservative scattering.

Appendix: Numerical calculations of plane and spherical albedos for semi-infinite, vertically homogeneous media

The Ambartsumian equation to determine the $m$-th order Fourier coefficient of the reflection function $R^{(m)}(\mu, \mu_{0})$ ($\mu, \mu_{0}\in[0,1]$) for a semi-infinite medium takes the following form (Sobolev 1975; Yanovitskij 1997; Mishchenko et al. 1999):

$$\begin{aligned} &R^{(m)}(\mu, \mu_{0}) \\ &\quad =\frac{1}{4(\mu+\mu_{0})} \biggl\{ P^{(m)}(- \mu, \mu_{0}) \\ &\qquad{} +2\mu\int_{0}^{1}R^{(m)}\bigl(\mu,\mu^{\prime}\bigr)P^{(m)}\bigl(\mu ^{\prime}, \mu_{0}\bigr)d\mu^{\prime} \\ &\qquad{} +2\mu_{0}\int_{0}^{1}P^{(m)}\bigl(\mu,\mu^{\prime}\bigr)R^{(m)}\bigl(\mu ^{\prime},\mu_{0}\bigr)d\mu^{\prime} \\ &\qquad{} +4\mu\mu_{0}\int_{0}^{1}R^{(m)}\bigl(\mu,\mu^{\prime}\bigr) \\ &\qquad{} \times\biggl[\int _{0}^{1} P^{(m)} \bigl(-\mu^{\prime},\mu^{\prime\prime}\bigr)R^{(m)}\bigl(\mu^{\prime\prime}, \mu_{0}\bigr)d\mu^{\prime\prime} \biggr] d \mu^{\prime} \biggr\} , \end{aligned}$$

(A.1)

where $P^{(m)}(\mu, \mu_{0})$ is the $m$-th order Fourier coefficient of the phase function of our interest (see Eq. (A.2) below). It must be stressed that $P^{(m)}$ includes the single scattering albedo $\varpi_{0}$ as a multiplicative factor (Eq. (8) of the main text). Equation (A.1) is usually discretized by approximating the integrals with respect $\mu^{\prime}$ and $\mu^{\prime\prime}$ by an $N_{\mu}$-th order quadrature as shown by Eq. (29) of Mishchenko et al. (1999), yielding a system of $N_{\mu}\times N_{\mu}$ simultaneous equations, which we intend to solve here by a successive approximation method to investigate how closely we can reproduce the results of Tables 1 and 2 as well as those in Figs. 5 and 6 of Rogovtsov and Borovik (2016).

For arbitrary phase functions, the Fourier coefficients $P^{(m)}$ involved in Eq. (A.1) can be numerically evaluated by

$$\begin{aligned} &P^{(m)}(u, \mu_{0}) \\ &\quad =\frac{1}{\pi}\int_{0}^{\pi} P(\varTheta)\cos m \phi^{\prime}d\phi^{\prime} \\ &\quad \simeq \frac{1}{\pi}\sum_{n=1}^{N_{\phi}}w_{n}P \Large \Bigl(\cos^{-1}\Bigl[u\mu_{0} +\sqrt{\bigl(1-u^{2}\bigr) \bigl(1-\mu_{0}^{2}\bigr)} \\ &\qquad {}\times\cos\phi_{n}\Bigr] \Large\Bigr)\cos(m\phi_{n}), \end{aligned}$$

(A.2)

where $u$ designates either $\mu$ or $-\mu$, while $\phi_{n}~( \in[0, \pi])$ and $w_{n}$ are the $n$-th division point and the associated integration weight of an $N_{\phi}$-th order numerical quadrature employed.

However, in the case of the (single-term) Henyey-Greenstein phase function having an anisotropy parameter $g\in(-1, 1)$:

$$ P(\varTheta)=P_{\mathrm{HG}}(\varTheta; g)=\frac{\varpi_{0}~(1-g^{2})}{(1+g ^{2}-2g\cos\varTheta)^{3/2}}, $$

(A.3)

the azimuth angle-averaged term $P^{(0)}(u, \mu_{0})$ can be expressed in the following manner (van de Hulst 1980, on p. 333):

$$\begin{aligned} P^{(0)}(u, \mu_{0}) =& \varpi_{0}\bigl[ \bigl(1-g^{2}\bigr)/\sqrt{\alpha+\beta}( \alpha-\beta)\bigr] \\ &{}\times(2/\pi)E\bigl(\pi/2, \sqrt{2\beta/(\alpha+\beta)}~\bigr), \end{aligned}$$

(A.4)

where $E(\pi/2, k)$ is the complete elliptic integral of the second kind, while $\alpha$ and $\beta$ are defined as

$$\begin{aligned} &\alpha = 1+g^{2}-2gu\mu_{0}, \end{aligned}$$

(A.5a)

$$\begin{aligned} &\beta = 2|g|\sqrt{\bigl(1-u^{2}\bigr) \bigl(1-\mu_{0}^{2}\bigr)}. \end{aligned}$$

(A.5b)

Because of the symmetry relations present in phase functions (Hansen and Travis 1974), we only need to calculate $P^{(m)}(-\mu, \mu _{0})$ and $P^{(m)}(\mu, \mu_{0})$ to solve Eq. (A.1). However, in applying a numerical quadrature to the integrals in Eq. (A.1), it is crucial to make sure the normalization condition (Hovenier et al. 2004)

$$\begin{aligned} &\Biggl\vert \frac{1}{2\varpi_{0}}\sum_{n=1}^{N_{\mu}} \bigl[ P^{(0)}(-\mu _{n},\mu_{k})+P^{(0)}( \mu_{n},\mu_{k}) \bigr] w_{n}-1 \Biggr\vert \\ &\quad\equiv B(\mu_{k})\le\varepsilon_{\mathrm{norm}}, \quad (k=1, 2, \ldots, N_{\mu}), \end{aligned}$$

(A.6)

is satisfied by the $N_{\mu}$-th order quadrature to avoid causing an artificial absorption, where $\varepsilon_{\mathrm{norm}}$ is a prescribed numerical value for error tolerance, while $\mu_{k}$ and $\mu_{n}$ signify the quadrature points. Whether or not a chosen value for $N_{\mu}$ is adequate can be assessed to a large extent by inspecting $\max B(\mu_{k})$ ($k=1,2,\ldots, N_{\mu}$). We iteratively achieve this renormalization with $\varepsilon_{\mathrm{norm}}=10^{-14}$ following the procedure of Hansen (1971).

To set up a starting approximation for $R^{(m)}(\mu, \mu_{0})$ on the right-hand side of Eq. (A.1), we treat both single scattering and second-order scattering rigorously, but approximate all the higher order scatterings by isotropic scattering. Then for $m=0$, we have

$$\begin{aligned} &R^{(0)}(\mu, \mu_{0}) \\ &\quad =\frac{1}{4(\mu+\mu_{0})} \biggl\{ P^{(0)}(- \mu,\mu_{0}) \\ &\qquad{} +\frac{\mu_{0}}{2}\int_{0}^{1}P^{(0)}\bigl(\mu,\mu^{\prime}\bigr)P ^{(0)}\bigl(-\mu^{\prime},\mu_{0}\bigr)d\mu^{\prime}/\bigl(\mu^{\prime}+\mu_{0}\bigr) \\ &\qquad{} +\frac{\mu}{2}\int_{0}^{1} P^{(0)}\bigl(-\mu,\mu^{\prime}\bigr)P^{(0)}\bigl(\mu^{\prime}, \mu_{0}\bigr)d\mu^{\prime}/\bigl(\mu+\mu^{\prime}\bigr) \\ &\qquad{} +\varpi_{0}H^{\mathrm{iso}}(\varpi_{0},\mu)H^{\mathrm{iso}}(\varpi_{0}, \mu_{0}) - \varpi_{0} \\ &\qquad{} -\frac{\varpi_{0}^{2}}{2} \bigl[\mu\log\bigl((1+ \mu)/\mu\bigr)+\mu_{0}\log\bigl((1+\mu_{0})/\mu_{0}\bigr) \bigr] \biggr\} , \end{aligned}$$

(A.7)

whereas for $m\ge1$, we substitute the Fourier coefficient that just precedes, viz.,

$$ R^{(m)}(\mu, \mu_{0})=R^{(m-1)}(\mu,\mu_{0}) \quad (m\ge1). $$

(A.8)

We evaluate $H^{\mathrm{iso}}(\varpi_{0}, \mu)$ and $H^{\mathrm{iso}}( \varpi_{0}, \mu_{0})$ in Eq. (A.7) using the approximation formula of Kawabata and Limaye (2011, 2013). The iteration for successive approximation for $R^{(m)}(\mu, \mu_{0})$ is terminated if the following condition is satisfied for all combinations of the division points of the quadrature employed for $\mu$ and $\mu_{0}$:

$$ \bigl\vert R^{(m)}(\mu, \mu_{0})^{\mathrm{new}}-R^{(m)}( \mu, \mu_{0})^{ \mathrm{old}} \bigr\vert \le10^{-7}. $$

(A.9)

The values for the plane albedo $A_{\mathrm{pl}}(\varpi_{0}, \mu)$ and the spherical albedo $A_{\mathrm{sp}}(\varpi_{0})$ are then calculated using those of $R^{(0)}(\mu, \mu_{0})$ produced on a square grid of the division points according to

$$\begin{aligned} &A_{\mathrm{pl}}(\mu, \varpi_{0}) = 2\int_{0}^{1} R^{(0)}\bigl(\mu,\mu^{\prime}\bigr)\mu^{\prime}d\mu^{\prime}, \end{aligned}$$

(A.10a)

$$\begin{aligned} &A_{\mathrm{sp}}(\varpi_{0}) = 2\int_{0}^{1} A_{\mathrm{pl}}\bigl(\mu^{\prime},\varpi_{0}\bigr)\mu^{\prime}d\mu^{\prime}. \end{aligned}$$

(A.10b)

For simplicity, the Gauss-Legendre quadrature is employed with $N_{\mu}=395$ for Eq. (A.1) and $N_{\phi}=300$ for Eq. (A.2).

Our computer code to solve Eq. (A.1) has been tested for the case of conservative isotropic scattering: the maximum relative deviation of the numerical values of reflection function obtained on a $N_{\mu} \times N_{\mu}$ square grid is $7.32\times10^{-7}$ in comparison with those given by the exact solution:

$$ R^{(0)}(\mu, \mu_{0})=\frac{\varpi_{0}}{4(\mu+\mu_{0})}H^{ \mathrm{iso}}( \mu)H^{\mathrm{iso}}(\mu_{0}), $$

(A.11)

where the values of $H^{\mathrm{iso}}(\mu)$ and $H^{\mathrm{iso}}(\mu _{0})$ are evaluated using the procedure discussed in the main text. In addition, the values of the Fourier coefficients $P^{(0)}(-\mu, \mu _{0})$ and $P^{(0)}(\mu, \mu_{0})$ of the Henyey-Greenstein phase function (Eq. (A.3)) calculated by using Eq. (A.2) on this grid have been checked against those generated by Eq. (A.4): for $g=0.989$, the maximum relative deviations from the latter results are $3.75\times10^{-12}$ and $3.23\times10^{-12}$ for $P^{(0)}(-\mu, \mu_{0})$ and $P^{(0)}(\mu, \mu_{0})$, respectively, whereas for $g=0.9965$, they are $2.71\times10^{-11}$ and $3.32\times10^{-11}$, respectively.

The results for $A_{\mathrm{sp}}(\varpi_{0})$ obtained using the Henyey-Greenstein phase function with $g=0.99$ and 0.9965 are shown respectively in columns 2 and 6 of Table A.1 as functions of $\varpi_{0}$. The $\Delta$-values given in columns 3 and 7 indicate the excess of the last digit figures over those of Rogovtsov and Borovik (2016) (RB for short):

$$ \Delta\equiv\mbox{last digit}~(\mbox{present})-\mbox{last digit}~(\mbox{RB}). $$

(A.12)

Deviations by one unit in the last digit are seen at four locations in the case of $g=0.9965$ in contrast to just one location for $g=0.99$. The columns 4 and 8 designated by ‘Iter’ give the number of iterations required to solve Eq. (A.1) for $R^{(0)}(\mu, \mu _{0})$ with a given value of $\varpi_{0}$ respectively for $g=0.99$ and 0.9965 under the convergence criterion shown by Eq. (A.9). A rapid increase in Iter is clearly seen when we move from $\varpi_{0}=0.9995$ to 0.9999 especially in the case of $g=0.9965$, where the number of iterations exceeds $10^{4}$. For reference purpose, the resulting values for $R^{(0)}(1, 1)$ are given in columns 5 and 9.

Table A.1 The spherical albedos $A_{\mathrm{sp}}(\varpi_{0})$ and the 0-th order Fourier coefficient $R^{(0)}(1,1)$ of reflection function calculated for two values of anisotropy parameter $g$ of the Henyey-Greenstein phase function

Table A.2 shows the values of plane albedo $A_{\mathrm{pl}}(\mu,\varpi_{0})$ obtained for 6 values of $\varpi_{0}$ and 14 values of $\mu$ using the Henyey-Greenstein phase function with $g=0.989$. The differences $\Delta$ by one unit in the last digit of $A_{\mathrm{pl}}( \mu)$ are found at 14 locations in comparison with the values given in Table 2 of Rogovtsov and Borovik (2016). Also shown in the bottom row are the corresponding values of the spherical albedo $A_{\mathrm{sp}}(\varpi _{0})$. They also are in good agreement with those of Rogovtsov and Borovik (2016) with a one-unit difference in the fourth decimal place found only for $\varpi_{0}=0.9995$.

Table A.2 The plane albedos $A_{\mathrm{pl}}(\mu , \varpi_{0})$ and spherical albedos $A_{\mathrm{sp}}(\varpi_{0})$ calculated for the Henyey-Greenstein phase function with anisotropy parameter $g=0.989$

In order to make a further check of the reliability of our procedure to solve Eq. (A.1) with double peaked phase functions $P(\varTheta)$, we have also tried the cases with the two-term Henyey-Greenstein phase function of the form:

$$\begin{aligned} &P(\varTheta)=f\,P_{\mathrm{HG}}(g_{1}; \varTheta)+(1-f)\,P_{\mathrm{HG}}(g_{2};\varTheta), \\ &\quad \bigl(g_{1}\ge0, g_{2}\le0, f\in[0, 1]\bigr). \end{aligned}$$

(A.13)

Following Rogovtsov and Borovik (2016), we have employed $g_{1}=0.995$, $g_{2}=-0.995$, and $f=0.99$. With $N_{\mu}=395$ and $N_{\phi}=300$, the maximum relative deviations of the values of $P^{(0)}(-\mu, \mu _{0})$ and $P^{(0)}(\mu, \mu_{0})$ obtained by Eq. (A.2) from those given by Eq. (A.4) with $f=0.99$ are found to be $1.87\times10^{-11}$ and $1.35\times10^{-11}$, respectively.

The resulting values for $A_{\mathrm{pl}}(\mu)$, $A_{\mathrm{sp}}$, and $R^{(0)}(\mu, 1)$ obtained for four values of $\varpi_{0}$, viz., 0.993, 0.997, 0.999, and 0.9995, are shown in Table A.3. Also given in the bottom row of Table A.3 are the number of iterations required to get the solution $R^{(0)}(\mu, \mu_{0})$ for each value of $\varpi_{0}$. Graphical comparisons of our results for $A_{\mathrm{pl}}(\mu)$ and $R^{(0)}(\mu, 1)$ with the plots displayed in Fig. 5 and Fig. 6 of Rogovtsov and Borovik (2016) indicate that the two data sets are in close agreement.

Table A.3 The plane albedos $A_{\mathrm{pl}}(\mu , \varpi_{0})$, spherical albedos $A_{\mathrm{sp}}(\varpi_{0})$, and the 0-th order Fourier coefficient $R^{(0)}(\mu , 1)$ of reflection function for the two-term Henyey-Greenstein phase function defined by Eq. (A.13) with $g_{1}=0.995$, $g_{2}=-0.995$, and $f=0.99$

A significant improvement in execution speed is nevertheless requisite for our procedure to be of practical use for applications for which azimuth-angle dependent quantities such as intensity distributions over a planetary disk must be calculated extensively using highly anisotropic phase functions giving rise to near-conservative scattering (see, e.g., Yanovitskij 1997).