Table I
Phase Functions (×102)
θ(deg) | KA | KB | KC | A | B | C | PA | PB |
---|
1 | 4916 | 4577 | 4285 | 2630 | 2490 | 2360 | 3972 | 1662 |
5 | 574 | 534 | 500 | 710 | 678 | 634 | 407 | 271 |
10 | 169 | 158 | 148 | 201 | 182 | 168 | 130 | 131 |
20 | 29.5 | 29.4 | 29.2 | 52.3 | 46.2 | 42.1 | 34.4 | 54.9 |
30 | 12.6 | 12.0 | 11.4 | 15.8 | 15.6 | 15.4 | 12.2 | 13.3 |
45 | 3.06 | 3.66 | 4.19 | 4.20 | 4.82 | 5.51 | 4.17 | 5.30 |
60 | 1.09 | 1.58 | 2.00 | 1.47 | 1.94 | 2.36 | 2.37 | 3.09 |
75 | 0.546 | 0.915 | 1.19 | 0.655 | 1.02 | 1.32 | 1.81 | 2.37 |
90 | 0.344 | 0.661 | 0.952 | 0.383 | 0.708 | 0.998 | 1.56 | 1.95 |
105 | 0.311 | 0.641 | 0.928 | 0.318 | 0.648 | 0.933 | 1.57 | 1.90 |
120 | 0.317 | 0.732 | 1.09 | 0.324 | 0.717 | 1.05 | 1.76 | 2.08 |
135 | 0.410 | 0.829 | 1.31 | 0.395 | 0.871 | 1.31 | 2.02 | 2.53 |
150 | 0.492 | 1.02 | 1.62 | 0.542 | 1.16 | 1.71 | 2.29 | 2.97 |
165 | 0.579 | 1.26 | 1.86 | 0.735 | 1.60 | 2.36 | 3.04 | 3.56 |
170 | 0.595 | 1.31 | 1.93 | 0.795 | 1.91 | 2.71 | 3.43 | 4.15 |
175 | 0.610 | 1.34 | 1.97 | 1.17 | 2.61 | 4.00 | 3.55 | 4.30 |
180 | 0.617 | 1.36 | 2.00 | 2.04 | 4.30 | 6.31 | 3.66 | 4.45 |
B | 0.0236 | 0.0506 | 0.0746 | 0.0257 | 0.0544 | 0.0799 | 0.1193 | 0.1462 |
Table II
Comparison Between Matrix Operator (MO) and Monte Carlo (MC) Calculations of R(τ,−) for a Rayleigh Scattering Atmosphere
τ | ω0 = 0.1
| ω0 = 0.5
|
---|
MO | MC | MO | MC |
---|
(×102) | (×10) |
---|
0 | 1.777 | 1.783 | 1.207 | 1.215 |
0.4795 | 1.785 | 1.799 | 1.241 | 1.251 |
0.9795 | 1.790 | 1.799 | 1.258 | 1.263 |
2.98 | 1.797 | 1.792 | 1.289 | 1.289 |
8.98 | 1.802 | 1.364 | 1.311 | 1.491 |
Table III
Variation in K(τ,−)/cD0 with the Incident Radiance Distribution for τ = 1.50 and Phase Function B
θ0w | ω0 = 0.2 | ω0 = 0.4 | ω0 = 0.6 | ω0 = 0.8 |
---|
0° | 0.8307 | 0.6551 | 0.4704 | 0.2754 |
10° | 0.8303 | 0.6549 | 0.4707 | 0.2762 |
20° | 0.8323 | 0.6555 | 0.4707 | 0.2767 |
30° | 0.8304 | 0.6538 | 0.4772 | 0.2803 |
40° | 0.8330 | 0.6543 | 0.4779 | 0.2841 |
Sky | 0.8253 | 0.6555 | 0.4784 | 0.2853 |
Table IV
Expansion Coefficients for Eqs. (19) and (19′)
τ | k0 | k1 | k2 | k3 | k0′ | k1′ | k2′ | k3′ |
---|
0 | 1.0016 | −0.9959 | 0.1089 | −0.1527 | 0.9588 | −0.7408 | −0.3935 | 0.1776 |
1 | 0.9999 | −0.9221 | 0.0522 | −0.1584 | 0.9690 | −0.7124 | −0.3870 | 0.1303 |
2 | 0.9999 | −0.9259 | 0.1749 | −0.2780 | 0.9673 | −0.6148 | −0.5520 | 0.1988 |
3 | 0.9999 | −0.8956 | 0.1526 | −0.2838 | 0.9683 | −0.5926 | −0.5536 | 0.1772 |
4 | 0.9999 | −0.8653 | 0.1305 | −0.2897 | 0.9695 | −0.5737 | −0.5465 | 0.1501 |
Table V
Variation in R(τ,−) with the Incident Radiance Distribution for τ = 1.50 and Phase Function B
θ0 | ω0 = 0.2 | ω0 = 0.4 | ω0 = 0.6 | ω0 = 0.8 |
---|
0° | 0.00458 | 0.01191 | 0.02655 | 0.06910 |
10° | 0.00455 | 0.01169 | 0.02675 | 0.06962 |
20° | 0.00461 | 0.01227 | 0.02736 | 0.07211 |
30° | 0.00467 | 0.01306 | 0.02972 | 0.07673 |
40° | 0.00520 | 0.01429 | 0.0327 | 0.08264 |
Sky | 0.00485 | 0.01322 | 0.02987 | 0.07575 |
Table VI
Expansion Coefficients for Eqs. (20) and (20′) for the Sun and Sky Cases
τ | r0 | r1 | r2 | r3 | r0′ | r1′ | r2′ | r3′ |
---|
0 | 0.0001 | 0.3244 | 0.1425 | 0.1308 | −0.0003 | 3.0770 | −4.2158 | 3.5012 |
| 0.0003 | 0.3687 | 0.1802 | 0.0740 | −0.0008 | 2.6987 | −3.2310 | 2.8947 |
1 | 0.0000 | 0.3355 | 0.1564 | 0.1233 | 0.0002 | 2.9592 | −3.9051 | 3.1737 |
| 0.0005 | 0.3741 | 0.2353 | −0.0037 | −0.0008 | 2.6204 | −3.2048 | 3.2993 |
2 | −0.0002 | 0.3502 | 0.1202 | 0.1730 | 0.0004 | 2.8809 | −3.5590 | 2.5424 |
| 0.0004 | 0.3826 | 0.2229 | 0.0081 | −0.0005 | 2.5650 | −2.9298 | 2.8087 |
3 | −0.0003 | 0.3589 | 0.1184 | 0.1796 | 0.0007 | 2.8131 | −3.3566 | 2.2948 |
| 0.0004 | 0.3874 | 0.2274 | 0.0017 | −0.0003 | 2.5262 | −2.8074 | 2.6296 |
4 | −0.0004 | 0.3668 | 0.1177 | 0.1820 | 0.0010 | 2.7524 | −3.1727 | 2.0866 |
| 0.0004 | 0.3892 | 0.2382 | −0.0096 | −0.0003 | 2.5054 | −2.7901 | 2.6772 |
Table VII
Variation in D(τ,−) − D0(τ,−) with the Incident Radiance Distribution for τ = 1.50 and Phase Function B
θ0w | ω0 = 0.2 | ω0 = 0.4 | ω0 = 0.6 | ω0 = 0.8 |
---|
0° | 0.0218 | 0.0504 | 0.0879 | 0.1548 |
10° | 0.0208 | 0.0497 | 0.0875 | 0.1483 |
20° | 0.0215 | 0.0545 | 0.0952 | 0.1677 |
30° | 0.0228 | 0.0582 | 0.1014 | 0.1870 |
40° | 0.0241 | 0.0599 | 0.1117 | 0.2008 |
Sky | 0.0274 | 0.0589 | 0.1105 | 0.1927 |
Table VIII
Expansion Coefficients in Eqs. (21) for the Sun and Sky Cases
τ | d0 | d1 | d2 | d3 |
---|
0 | 0.0002 | 0.2991 | 0.1492 | 0.3115 |
| −0.0037 | 0.3664 | 0.1689 | 0.0131 |
1 | 0.0058 | 0.8432 | −1.1932 | 1.1765 |
| 0.0112 | 1.2491 | −2.4523 | 2.4076 |
2 | 0.0123 | 1.0381 | −1.3613 | 1.2780 |
| 0.0171 | 1.5103 | −2.9670 | 2.7573 |
3 | 0.0174 | 1.2921 | −1.7987 | 1.5510 |
| 0.0238 | 1.8187 | −3.6898 | 3.3283 |
4 | 0.0220 | 1.5151 | −2.1851 | 1.7876 |
| −0.003 | 2.5054 | −2.7901 | 2.6772 |
Table IX
Test of Scheme for Finding ω0 and B
τ | ω0 | B |
---|
0 | 0.499 | 0.0235 |
1 | 0.512 | 0.0197 |
2 | 0.520 | 0.0209 |
3 | 0.523 | 0.0206 |
4 | 0.526 | .0.0207 |
True | 0.500 | 0.0253 |