Table 1
Symbols Used in This Paper, with Their Definitions and Units
Symbol Definition Unit a , b Absorption and scattering coefficients m−1 a u , d d Diffuse absorption coefficient for upwelling and downwelling light m−1 b u , d d Diffuse backscattering coefficient for upwelling and downwelling light m−1 b b Backscattering coefficient m−1 B Backscatter fraction β Volume scattering function sr−1 m−1 c Beam attenuation coefficient (sum of a and b ) m−1 E d , E u Downwelling and upwelling irradiance Wm−2 E od , E ou Downwelling and upwelling scalar irradiance Wm−2 ζ Optical depth, ζ = cz θ Zenith scattering angle θ0 Zenith angle of the Sun ϑ Vector specified by a zenith and an azimuth angle L Radiance Wm−2 sr−1
μ
¯
d
,
μ
¯
u Downwelling and upwelling mean cosines of the zenith angle of downwelling and upwelling traveling photons, respectively
μ
¯
s Mean cosine of the scattering phase function R Irradiance reflectance r d , r u Downwelling and upwelling shape factors z Depth m
Table 2
Petzold’s Scattering Phase Functions Used in This Paper
Source B
μ
¯
s Station 11 (Offshore California) 0.013 0.947 Station 5 (Offshore California) 0.014 0.947 Sea water 0.015 0.934 Sea water filtered for 40 min 0.017 0.931 Filtered fresh water + scattering agent 0.018 0.926 Average of San Diego Harbor measurements 0.019 0.922 (stations 2040, 2200, 2240) Sea water filtered for 100 min 0.025 0.919 Station 7 (Bahamas Islands) 0.025 0.915 Station 9 (Bahamas Islands) 0.038 0.885 Station 8 (Bahamas Islands) 0.044 0.872 Filtered fresh water 0.119 0.727 Sea water filtered for 18 h 0.146 0.660
Table 3
Values of r d Averaged over Four Solar Zenith Angles for Various Values of b /a and
μ
¯
s for E d = E d (z = 0)
μ
¯
s r d at Depth Where E d = E d (z = 0)
0.1 0.2 0.3 0.5 1 5 10 15 20 25 30 0.66 1.0801 1.0860 1.0904 1.1006 1.1238 1.2354 1.3009 1.3381 1.3625 1.3796 1.3925 0.727 1.0691 1.0738 1.0772 1.0860 1.1059 1.2072 1.2721 1.3106 1.3364 1.3550 1.3691 0.872 1.1588 1.1643 1.1719 1.1843 1.2139 1.3991 1.5531 1.6603 1.7393 1.8005 1.8492 0.885 1.1767 1.1839 1.1896 1.2031 1.2330 1.4244 1.5886 1.7057 1.7933 1.8620 1.9173 0.915 1.2330 1.2369 1.2430 1.2569 1.2874 1.4874 1.6700 1.8067 1.9137 2.0003 2.0711 0.919 1.2002 1.2040 1.2096 1.2205 1.2510 1.4476 1.6308 1.7697 1.8792 1.9683 2.0421 0.922 1.3518 1.3586 1.3727 1.3892 1.4384 1.7453 2.0287 2.2438 2.4125 2.5500 2.6638 0.926 1.2935 1.2994 1.3130 1.3315 1.3805 1.7211 2.0495 2.3051 2.5094 2.6780 2.8191 0.931 1.3196 1.3205 1.3342 1.3510 1.3927 1.6764 1.9495 2.1608 2.3316 2.4716 2.5887 0.934 1.3746 1.3781 1.3933 1.4104 1.4586 1.7864 2.1014 2.3496 2.5482 2.7137 2.8525 0.947a 1.2784 1.2848 1.2938 1.3096 1.3391 1.5702 1.8068 2.0020 2.1663 2.3059 2.4272 0.947b 1.2797 1.2879 1.2933 1.3039 1.3341 1.5494 1.7689 1.9512 2.1031 2.2332 2.3447
a Petzold’s SPF for Station 5.
14
b Petzold’s SPF for Station 11.
14
Table 4
Values of r d Averaged over Four Solar Zenith Angles for Various Values of b /a and
μ
¯
s for E d = 50% E d (z = 0)
μ
¯
s r d at Depth Where E d = 50% E d (z = 0)
0.1 0.2 0.3 0.5 1 5 10 15 20 25 30 0.66 1.0797 1.0917 1.1003 1.1163 1.1518 1.2908 1.3527 1.3844 1.4008 1.4135 1.4221 0.727 1.0684 1.0782 1.0848 1.0982 1.1285 1.2583 1.3246 1.3578 1.3790 1.3918 1.4023 0.872 1.1571 1.1726 1.1881 1.2128 1.2737 1.5772 1.7637 1.8667 1.9315 1.9791 2.0127 0.885 1.1761 1.1939 1.2072 1.2312 1.2954 1.6236 1.8278 1.9436 2.0162 2.0745 2.1093 0.915 1.2299 1.2440 1.2589 1.2885 1.3553 1.7240 1.9750 2.1284 2.2271 2.2936 2.3533 0.919 1.1962 1.2077 1.2235 1.2515 1.3152 1.6818 1.9416 2.0965 2.2012 2.2791 2.3357 0.922 1.3532 1.3773 1.3993 1.4444 1.5531 2.1534 2.5546 2.7900 2.9386 3.0525 3.1328 0.926 1.2984 1.3131 1.3468 1.3948 1.5183 2.2182 2.7076 2.9895 3.1717 3.3148 3.4101 0.931 1.3208 1.3346 1.3598 1.4002 1.5015 2.0846 2.4852 2.7272 2.8866 3.0163 3.0974 0.934 1.3740 1.3914 1.4218 1.4714 1.5877 2.2715 2.7392 3.0241 3.2174 3.3629 3.4710 0.947a 1.2822 1.2958 1.3078 1.3432 1.4230 1.9336 2.3269 2.5903 2.7705 2.9086 3.0132 0.947b 1.2755 1.2879 1.3069 1.3313 1.4073 1.8756 2.2439 2.4893 2.6580 2.7877 2.8847
a Petzold’s SPF for Station 5.
14
b Petzold’s SPF for Station 11.
14
Table 5
Values of r d Averaged over Four Solar Zenith Angles for Various Values of b /a and
μ
¯
s for E d = 10% E d (z = 0)
μ
¯
s r d at Depth Where E d = 10% E d (z = 0)
0.1 0.2 0.3 0.5 1 5 10 15 20 25 30 0.66 1.0787 1.0924 1.0991 1.1203 1.1598 1.3041 1.3620 1.3887 1.4047 1.4156 1.4233 0.727 1.0664 1.0749 1.0868 1.0995 1.1351 1.2718 1.3338 1.3645 1.3825 1.3956 1.4050 0.872 1.1537 1.1680 1.1848 1.2194 1.2918 1.6235 1.8005 1.8933 1.9527 1.9942 2.0266 0.885 1.1644 1.1836 1.2008 1.2375 1.3148 1.6748 1.8689 1.9753 2.0432 2.0917 2.1273 0.915 1.2045 1.2354 1.2548 1.2859 1.3721 1.7883 2.0374 2.1747 2.2630 2.3303 2.3793 0.919 1.1697 1.1988 1.2200 1.2534 1.3338 1.7473 1.9992 2.1449 2.2411 2.3105 2.3637 0.922 1.3368 1.3561 1.3870 1.4408 1.5852 2.2614 2.6408 2.8578 2.9928 3.0935 3.1699 0.926 1.2553 1.3086 1.3307 1.4005 1.5597 2.3480 2.8049 3.0683 3.2381 3.3593 3.4517 0.931 1.2730 1.3180 1.3396 1.3977 1.5283 2.1864 2.5859 2.8052 2.9550 3.0614 3.1428 0.934 1.3396 1.3747 1.4052 1.4753 1.6196 2.3917 2.8576 3.1200 3.2930 3.4237 3.5157 0.947a 1.2556 1.2730 1.2977 1.3404 1.4437 2.0320 2.4366 2.6810 2.8496 2.9748 3.0679 0.947b 1.2530 1.2765 1.3025 1.3309 1.4262 1.9710 2.3458 2.5749 2.733 2.8499 2.9391
a Petzold’s SPF for Station 5.
14
b Petzold’s SPF for Station 11.
14
Table 6
Values of r d Averaged over Four Solar Zenith Angles for Various Values of b /a and
μ
¯
s for E d = 1% E d (z = 0)
μ
¯
s r d at Depth Where E d = 1% E d (z = 0)
0.1 0.2 0.3 0.5 1 5 10 15 20 25 30 0.66 1.0746 1.0803 1.0968 1.1214 1.1595 1.3045 1.3614 1.3877 1.4055 1.4139 1.4232 0.727 1.0568 1.0813 1.0791 1.0998 1.1360 1.2709 1.3345 1.3653 1.3826 1.3956 1.4054 0.872 1.1415 1.1518 1.1645 1.1994 1.2782 1.6273 1.8014 1.8938 1.9520 1.9946 2.0273 0.885 1.1495 1.1743 1.1976 1.2194 1.3092 1.6733 1.8728 1.9779 2.0443 2.0910 2.1273 0.915 1.2011 1.2222 1.2349 1.2531 1.3657 1.7912 2.0444 2.1772 2.2670 2.3236 2.3794 0.919 1.1759 1.1958 1.2116 1.2384 1.3252 1.7453 2.0071 2.1468 2.2481 2.3097 2.3655 0.922 1.2846 1.3284 1.3309 1.4186 1.5529 2.2579 2.6483 2.8583 2.9986 3.1007 3.1682 0.926 1.2648 1.2832 1.3334 1.3701 1.5285 2.3408 2.8178 3.0802 3.2327 3.3604 3.4585 0.931 1.2822 1.2647 1.3139 1.3771 1.5055 2.1767 2.5941 2.8112 2.9663 3.0572 3.1428 0.934 1.3837 1.3258 1.3779 1.4218 1.5829 2.3951 2.8579 3.1224 3.2942 3.4188 3.5176 0.947a 1.2232 1.2273 1.2934 1.3290 1.4240 2.0283 2.4337 2.7015 2.8539 2.9793 3.0694 0.947b 1.2426 1.2691 1.2521 1.3013 1.4107 1.9621 2.3531 2.5820 2.7347 2.8579 2.9435
a Petzold’s SPF for Station 5.
14
b Petzold’s SPF for Station 11.
14
Table 7
Values of r u Averaged over Four Solar Zenith Angles for Various Values of b /a and
μ
¯
s for E d = E d (z = 0)
μ
¯
s r u at Depth Where E d = E d (z = 0)
0.1 0.2 0.3 0.5 1 5 10 15 20 25 30 0.66 1.5879 1.5867 1.6041 1.6046 1.5965 1.5816 1.5731 1.5672 1.5631 1.5597 1.5569 0.727 1.6077 1.6023 1.6054 1.5998 1.5961 1.5814 1.5729 1.5672 1.5631 1.5595 1.5568 0.872 2.8757 2.8204 2.8666 2.8464 2.8175 2.7087 2.6537 2.6244 2.6062 2.5924 2.5815 0.885 3.1206 3.1058 3.1293 3.0957 3.0619 2.9295 2.8626 2.8281 2.8061 2.7893 2.7764 0.915 3.9515 3.9219 3.9009 3.8794 3.8228 3.5914 3.4710 3.4069 3.3649 3.3365 3.3132 0.919 3.8965 3.8625 3.9134 3.9047 3.8361 3.6239 3.5059 3.4432 3.4032 3.3771 3.3543 0.922 5.9156 6.2127 6.1407 6.0840 5.9229 5.3470 5.0617 4.9204 4.8327 4.7743 4.7286 0.926 6.7762 6.8868 6.8586 6.8002 6.6929 6.0347 5.7227 5.5667 5.4746 5.4136 5.3680 0.931 5.9927 6.0746 6.1022 6.1859 6.0222 5.4585 5.1790 5.0269 4.9380 4.8762 4.8280 0.934 7.5502 7.2947 7.3642 7.3356 7.1586 6.3784 5.9997 5.8127 5.6939 5.6139 5.5533 0.947a 6.4272 6.1879 6.1739 6.1856 6.0787 5.6395 5.3761 5.2379 5.1487 5.0860 5.0386 0.947b 5.7655 6.0074 5.8956 5.8173 5.7496 5.3436 5.1023 4.9703 4.8873 4.8279 4.7823
a Petzold’s SPF for Station 5.
14
b Petzold’s SPF for Station 11.
14
Table 8
Values of r u Averaged over Four Solar Zenith Angles for Various Values of b /a and
μ
¯
s for E d = 50% E d (z = 0)
μ
¯
s r u at Depth Where E d = 50% E d (z = 0)
0.1 0.2 0.3 0.5 1 5 10 15 20 25 30 0.66 1.6393 1.6080 1.6090 1.6136 1.6195 1.6141 1.6000 1.5908 1.5824 1.5768 1.5719 0.727 1.6187 1.6247 1.6059 1.6083 1.6178 1.6165 1.6046 1.5950 1.5873 1.5805 1.5756 0.872 2.8334 2.8799 2.8759 2.8904 2.9217 2.8714 2.8113 2.7661 2.7300 2.7049 2.6817 0.885 3.1090 3.1169 3.1869 3.1608 3.1706 3.1259 3.0512 2.9970 2.9539 2.9259 2.8969 0.915 4.0813 3.9226 3.9247 3.9336 3.9477 3.8445 3.7236 3.6442 3.5817 3.5302 3.4943 0.919 4.0370 3.9533 3.9030 3.9495 3.9696 3.9174 3.8051 3.7212 3.6579 3.6088 3.5682 0.922 6.3389 6.1638 6.2274 6.2171 6.1639 5.8069 5.5302 5.3527 5.2203 5.1283 5.0490 0.926 7.1246 7.0856 6.9480 7.0908 7.0790 6.7954 6.4674 6.2427 6.0725 5.9590 5.8524 0.931 6.2404 6.5357 6.3435 6.3082 6.3048 5.9839 5.6993 5.5103 5.3786 5.2855 5.1985 0.934 7.3766 7.5562 7.6813 7.4843 7.4445 7.0285 6.6514 6.4110 6.2464 6.1224 6.0217 0.947a 5.7012 6.3710 6.2704 6.4025 6.4061 6.3004 6.0704 5.8929 5.7557 5.6460 5.5595 0.947b 5.8290 5.6943 5.9204 5.8507 5.9712 5.9125 5.7119 5.5588 5.4315 5.3348 5.2531
a Petzold’s SPF for Station 5.
14
b Petzold’s SPF for Station 11.
14
Table 9
Values of r u Averaged over Four Solar Zenith Angles for Various Values of b /a and
μ
¯
s for E d = 10% E d (z = 0)
μ
¯
s r u at Depth Where E d = 10% E d (z = 0)
0.1 0.2 0.3 0.5 1 5 10 15 20 25 30 0.66 1.6069 1.6095 1.6113 1.6208 1.6303 1.6215 1.6061 1.5938 1.5846 1.5782 1.5727 0.727 1.6455 1.5846 1.6026 1.6173 1.6303 1.6264 1.6119 1.5990 1.5900 1.5833 1.5775 0.872 2.8228 2.8268 2.8861 2.9009 2.9238 2.9173 2.8381 2.7845 2.7449 2.7131 2.6898 0.885 2.8862 3.0072 3.2103 3.1589 3.2152 3.1729 3.0817 3.0175 2.9724 2.9364 2.9084 0.915 3.5981 3.8719 3.9713 3.8964 3.9717 3.8985 3.7717 3.6781 3.6075 3.5559 3.5133 0.919 3.1638 3.9192 3.9741 3.9760 4.0250 3.9894 3.8635 3.7653 3.6902 3.6326 3.5863 0.922 6.2344 6.2522 6.1870 6.1858 6.1849 5.9201 5.6004 5.4014 5.2580 5.1537 5.0753 0.926 7.1407 6.7614 7.2974 7.1035 7.1552 6.9728 6.5695 6.3258 6.1324 5.9922 5.8818 0.931 5.4425 6.3119 6.4198 6.3164 6.2236 6.0816 5.7867 5.5848 5.4258 5.3116 5.2295 0.934 7.9650 7.2656 7.3673 7.6344 7.3930 7.1833 6.7676 6.5004 6.3105 6.1689 6.0575 0.947a 6.5651 7.0223 6.2543 6.2431 6.4534 6.4771 6.1926 5.9826 5.8271 5.7061 5.6078 0.947b 6.3453 6.0751 6.0794 5.9779 6.1362 6.0759 5.8241 5.6488 5.4992 5.3928 5.2993
a Petzold’s SPF for Station 5.
14
b Petzold’s SPF for Station 11.
14
Table 10
Values of r u Averaged over Four Solar Zenith Angles for Various Values of b /a and
μ
¯
s for E d = 1 % Ed (z = 0)
μ
¯
s r u at Depth Where E d = 1% E d (z = 0)
0.1 0.2 0.3 0.5 1 5 10 15 20 25 30 0.66 1.6185 1.5291 1.6171 1.6166 1.6280 1.6233 1.6089 1.5947 1.5846 1.5773 1.5726 0.727 1.4828 1.5631 1.6744 1.5827 1.6379 1.6255 1.6125 1.5984 1.5899 1.5829 1.5757 0.872 2.3766 2.8209 2.8815 2.9845 2.9089 2.9200 2.8380 2.7838 2.7407 2.7092 2.6889 0.885 3.4174 2.8773 3.1711 3.1118 3.2176 3.1710 3.0835 3.0196 2.9720 2.9352 2.9050 0.915 2.7345 4.3381 4.0629 3.6915 4.0786 3.9134 3.7775 3.6830 3.6080 3.5521 3.5143 0.919 6.5086 3.8669 3.7187 3.7309 4.0271 3.9473 3.8758 3.7651 3.6955 3.6268 3.5929 0.922 5.2944 6.3842 5.9976 6.1493 5.8994 5.8880 5.5931 5.3899 5.2731 5.1600 5.0730 0.926 4.5542 6.8715 7.5339 7.0769 7.2332 6.9364 6.6071 6.3205 6.1239 5.9838 5.8972 0.931 3.7619 3.5290 7.0930 6.9480 6.5340 6.1089 5.8010 5.6036 5.4438 5.3106 5.2188 0.934 6.0178 7.3894 5.4892 7.8775 7.4017 7.2355 6.7606 6.5073 6.3060 6.1743 6.0490 0.947a 3.5764 6.4152 5.7277 5.6224 6.2528 6.4215 6.1670 6.0085 5.8334 5.6917 5.5996 0.947b 3.6821 6.2724 5.8746 5.6193 6.3723 6.0198 5.8328 5.6535 5.5027 5.3910 5.2966
a Petzold’s SPF for Station 5.
14
b Petzold’s SPF for Station 11.
14