Comparisons with Other Models

The polarized radiative transfer model described in this chapter has been compared against a number of other models. A summary of the results is given in Table 4.4. All comparisons are made for plane-parallel, homogeneous atmospheres consisting entirely of either Rayleigh or Mie scatterers. Both the total optical depth and single-scattering albedo are prescribed. For Mie atmospheres the effective radius, $r_{\rm eff}$, and the effective variance, $v_{\rm eff}$, are given for the standard-Gamma size distribution. All surfaces are Lambertian of albedo $\Lambda$ and depolarizing. Full references are given in the bibliography and D&A refers to the doubling and adding solution method. For the orders of scattering convergence, q=4 was used. The overall agreement was found to be quite good with maximum errors of less than 0.5% for a simulation of the thick Venusian atmosphere and less than 0.1% for optical depths of unity. In general, the Mie atmospheres seem to have slightly better agreement than the Rayleigh. This is believed to be a result of the fact that in the Mie atmospheres the energy is distributed over a greater number of Fourier harmonic frequencies. As a result, the Rayleigh m=0 term does not converge as fast as the Mie m=0 term.

 

Table 4.4: Comparison of model results with various sources of tabulated model output (D&A=doubling and adding).

 

No.	Ref.	Solution	Atmosphere and	Comparison	Percent
		Method	Geometry		Difference
1	1	D&A	Rayleigh	Reflected	avg=0.04
			$\tau$=1, $\tilde{\omega}$=1.0	I,Q,U	max=0.08
			$\Lambda$=0.25, $\mu_o$=0.8
			$\phi-\phi_o$=90o
2	1	D&A	Mie ( $r_{\rm eff}$=0.2 $\mu$m,	Reflected	avg=0.03
			$v_{\rm eff}$=0.07)	I,Q,U,V	max=0.06
			$\tau$=1, $\tilde{\omega}$=0.99	expansion
			$\Lambda$=0.1, $\mu_o$=0.2	coefficients
3	2/3	Spherical	Mie ( $r_{\rm eff}$=0.2 $\mu$m,	Internal	avg=0.03
		Harmonics	$v_{\rm eff}$=0.07)	I,Q,U,V	max=0.07
			$\tau$=1, $\tilde{\omega}$=0.99
			$\Lambda$=0.1, $\mu_o$=0.2,
			$\phi-\phi_o$=0, 90, 180o
4	4	D&A	Rayleigh	Internal	avg=0.1
			$\tau$=10, $\tilde{\omega}$=1.0	I	max=0.4
			$\Lambda$=0.1, $\mu_o$=0.6,
			$\phi-\phi_o$=0, 90, 180o
5	4a	D&A	Mie ( $r_{\rm eff}$=1.05 $\mu$m,	Internal	avg=0.08
			$v_{\rm eff}$=0.07)	I	max=0.35
			$\tau$=10, $\tilde{\omega}$=1.0
			$\Lambda$=0.1, $\mu_o$=0.6,
			$\phi-\phi_o$=0, 90, 180o
6	5/6	Matrix	Rayleigh and	plots of I,	identical
		Operator	Mie (haze-H)	polarization	(to within
				and neutral	width of
				points	lines)

^a Simulates Venusian atmosphere
1: Evans and Stephens (1991)
2: Garcia and Siewert (1986)
3: Garcia and Siewert (1989)
4: Stamnes et al. (1989)
5: Kattawar et al. (1976)
6: Hitzfelder et al. (1976)

Table 4.5 shows some numbers used in comparison 2 (cf. Table 4.4). The first three (of eight) expansion I, Q, U, and V coefficients are compared at four zenith angles. A number are identical to three significant figures and off by one or two in the fourth significant digit. This amounts to an average difference of about 0.03%. Also, the fractional difference does not increase for the smaller magnitude numbers as is evident by comparing V.

 

Table 4.5: Summary of comparison 2 as described in Table 4.4.

 

	Evans and Stephens (1991)			this model
$\mu$	m=0	m=1	m=2	m=0	m=1	m=2
	Reflected $I^{cm}(0;\mu)$
0.09501	3.16625(-1)a	2.99206(-1)	1.41050(-1)	3.1657(-1)	2.9928(-1)	1.4109(-1)
0.45802	1.52211(-1)	1.02308(-1)	4.46799(-2)	1.5216(-1)	1.0235(-1)	4.4693(-2)
0.75540	8.76554(-2)	3.83168(-2)	1.38881(-2)	8.7614(-1)	3.8336(-2)	1.3894(-2)
0.98940	5.58402(-2)	4.81263(-3)	4.54656(-4)	5.5810(-2)	4.8162(-3)	4.5469(-4)
	Reflected $Q^{cm}(0;\mu)$
0.09501	6.35745(-2)	1.35402(-2)	-4.85223(-2)	6.3537(-2)	1.3539(-2)	-4.8515(-2)
0.45802	2.52572(-2)	-7.97154(-3)	-3.09958(-2)	2.5234(-2)	-7.9729(-3)	-3.1005(-2)
0.75540	8.27355(-3)	-1.10523(-2)	-2.54604(-2)	8.2636(-3)	-1.0948(-2)	-2.5469(-2)
0.98940	2.76513(-4)	-2.87733(-3)	-2.17265(-2)	2.7611(-4)	-2.8777(-3)	-2.1735(-2)
	Reflected $U^{sm}(0;\mu)$
0.09501		4.65680(-2)	2.96372(-2)		4.6567(-2)	2.9648(-2)
0.45802		2.83557(-2)	2.87641(-2)		2.8357(-2)	2.8774(-2)
0.75540		1.59988(-2)	2.53115(-2)		1.6000(-2)	2.5320(-2)
0.98940		2.91467(-2)	2.17265(-2)		2.9150(-2)	2.1735(-2)
	Reflected $V^{sm}(0;\mu)$
0.09501		-6.77792(-5)	4.81852(-5)		-6.7764(-5)	4.8163(-5)
0.45802		2.13234(-5)	-1.14599(-4)		2.1317(-5)	-1.1451(-4)
0.75540		6.18229(-5)	-7.27901(-5)		6.1794(-5)	-7.2747(-5)
0.98940		2.04401(-5)	-3.61377(-6)		2.0433(-5)	-3.6123(-6)

^a Read 3.16625(-1) as 3.16625 $\times 10^{-1}$.

In addition to the comparisons carried out against published model results, tests of the spherical corrections were done through a comparison against a single-scattering model. This model, developed at the Meteorological Institute of Stockholm University (MISU) and it uses the same spherical shell geometry to calculate unpolarized, single-scattered radiances. Using the same atmosphere and cross-sections, calculated internal radiances were nearly identical to that of the MISU code.