Cheng-Chien Liu,
Kendall L. Carder,
Richard L. Miller,
and James E. Ivey
When this research was performed, C.-C. Liu, K. L. Carder, and J. E. Ivey were with the College of Marine Science, University of South Florida, 140 Seventh Avenue, South, St. Petersburg, Florida 33701.
C.-C. Liu (ccliu88@yahoo.com), as well as R. L. Miller, is now with the NASA Earth Science Applications Directorate, Stennis Space Center, Mississippi 39529.
Cheng-Chien Liu, Kendall L. Carder, Richard L. Miller, and James E. Ivey, "Fast and accurate model of underwater scalar irradiance," Appl. Opt. 41, 4962-4974 (2002)
A spectral model of scalar irradiance with depth is applied to calculations of photosynthetically available radiation for a vertically homogeneous water column. The model runs more than 14,000 times faster than the full Hydrolight code, while it limits the percentage error to 2.20% and the maximum error to less than 4.78%. The distribution of incident sky radiance and the effects of a wind-roughened surface are integrated into this model. It can be applied to case 1 waters as well as to case 2 waters that happen to be gelbstoff rich, and the volume-scattering phase function can be generated dynamically based on the backscatter fraction. This new model is both fast and accurate and is, therefore, suitable for use interactively in models of the oceanic system, such as biogeochemical models or the heat budget part of global circulation models. It can also be applied by use of remote-sensing data to improve light-field calculations as a function of depth, which is needed for the estimation of global ocean carbon production and the ocean heat budget.
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Example of Implementing Eqs. (3) and (5) to Calculate E0(z, 560)a
The computational conditions are Chl = 1.0 (mg m-3), θs = 60°, Cloud = 0.0. The values of atmospheric parameters are those default settings in Hydrolight without consideration of inelastic scattering. (a) Table of Ēd(0+; quadi), Ē0 (0-; Vwind; quadi), and Ē0(0-; Vwind′; quadi); (b) results of Hydrolight run; (c) table of Ē0 (z; quadi); (d) implementation of Eqs. (3)–(5).
Table 3
Part of the Chlorophyll-Based LUTc
BF
quadi
Chl
λ (nm)
B0
B1
P
B2
Q
PCC
0.012
0–5°
1.0
400
1.622968
0.041260
0.169544
0.392585
0.115434
0.999981
0.012
5–15°
1.0
400
1.621863
0.359990
0.120332
0.055450
0.187885
0.999984
0.012
15–25°
1.0
400
1.620264
0.079612
0.217664
0.298405
0.132753
0.999984
0.012
25–35°
1.0
400
1.620130
0.284019
0.176459
0.044905
0.280099
0.999986
0.012
35–45°
1.0
400
1.621477
-0.605406
0.166221
0.846156
0.206304
0.999988
0.012
45–55°
1.0
400
1.622204
-0.484247
0.157253
0.618031
0.261602
0.999944
0.012
55–65°
1.0
400
1.622143
-0.594980
0.150696
0.608611
0.312121
0.999965
0.012
65–75°
1.0
400
1.621727
-0.653226
0.147192
0.490206
0.451490
0.999996
0.012
75–85°
1.0
400
1.622437
-0.881774
0.157695
0.503827
0.588206
0.999995
0.012
85–90°
1.0
400
1.624094
-1.100926
0.171378
0.480971
0.848644
0.999983
0.012
0–5°
1.0
560
1.406156
0.024583
0.473172
0.296568
0.108950
0.999981
0.012
5–15°
1.0
560
1.408768
0.248220
0.151219
0.057948
0.056629
0.999899
0.012
15–25°
1.0
560
1.403002
0.252404
0.143545
0.019512
0.787108
0.999994
0.012
25–35°
1.0
560
1.404388
0.072719
0.335383
0.155298
0.172646
0.999961
0.012
35–45°
1.0
560
1.404837
-0.589266
0.165140
0.736479
0.210064
0.999850
0.012
45–55°
1.0
560
1.403158
-0.263195
0.121358
0.313748
0.320304
0.999807
0.012
55–65°
1.0
560
1.403785
-0.476039
0.135085
0.419990
0.342729
0.999942
0.012
65–75°
1.0
560
1.403959
-0.598409
0.140788
0.385590
0.474633
0.999993
0.012
75–85°
1.0
560
1.405005
-0.823804
0.154344
0.419845
0.595190
0.999992
0.012
85–90°
1.0
560
1.406644
-1.022498
0.168514
0.403268
0.861775
0.999983
0.035
0–5°
5.0
400
1.946314
0.042092
0.148112
0.549831
0.205600
0.999986
0.035
5–15°
5.0
400
1.945661
0.512749
0.219135
0.057444
0.160444
0.999986
0.035
15–25°
5.0
400
1.944717
0.088024
0.166086
0.438363
0.256887
0.999984
0.035
25–35°
5.0
400
1.945591
0.418034
0.308522
0.049296
0.155572
0.999987
0.035
35–45°
5.0
400
1.945348
-0.581844
0.278121
0.940185
0.336619
0.999987
0.035
45–55°
5.0
400
1.945740
-0.457318
0.257737
0.686127
0.422929
0.999968
0.035
55–65°
5.0
400
1.945499
-0.594447
0.246508
0.678134
0.500986
0.999950
0.035
65–75°
5.0
400
1.945279
-0.665562
0.240058
0.538762
0.721242
0.999993
0.035
75–85°
5.0
400
1.945685
-0.929317
0.259333
0.547040
0.932661
0.999993
0.035
85–90°
5.0
400
1.946528
-1.184181
0.282872
0.515979
1.329025
0.999985
0.035
0–5°
5.0
560
1.993911
0.043121
0.146628
0.577093
0.204160
0.999979
0.035
5–15°
5.0
560
1.993240
0.538552
0.217402
0.059420
0.158097
0.999983
0.035
15–25°
5.0
560
1.991906
0.091133
0.187666
0.460743
0.247576
0.999984
0.035
25–35°
5.0
560
1.991324
0.441338
0.292586
0.048417
0.221900
0.999980
0.035
35–45°
5.0
560
1.992105
-0.605649
0.275581
0.984666
0.331311
0.999984
0.035
45–55°
5.0
560
1.992702
-0.604170
0.274190
0.850269
0.403206
0.999971
0.035
55–65°
5.0
560
1.992645
-0.648478
0.254406
0.745793
0.487815
0.999948
0.035
65–75°
5.0
560
1.992339
-0.693967
0.244965
0.575341
0.704433
0.999990
0.035
75–85°
5.0
560
1.992622
-0.945963
0.260523
0.564988
0.928310
0.999993
0.035
85–90°
5.0
560
1.993437
-1.206370
0.283599
0.531964
1.320401
0.999985
Table 4
Part of the Albedo-Based LUTw
BF
quadi
ω0
B0
B1
P
B2
Q
PCC
0.012
0–5°
0.15
1.053646
0.038500
0.053935
0.007855
0.448326
0.999964
0.012
5–15°
0.15
1.055313
0.028511
0.085161
0.006237
0.588680
0.999972
0.012
15–25°
0.15
1.020224
-0.046376
0.004308
0.018152
0.277671
0.998879
0.012
25–35°
0.15
1.024035
-0.084536
0.015590
0.021749
0.272535
0.999165
0.012
35–45°
0.15
1.057579
-0.735063
0.086653
0.635782
0.107455
0.999920
0.012
45–55°
0.15
1.054412
-0.294799
0.068933
0.109806
0.189921
0.999949
0.012
55–65°
0.15
1.059850
-3.802800
0.126050
3.534720
0.136682
0.999896
0.012
65–75°
0.15
1.061696
-2.980420
0.139046
2.578965
0.156566
0.999979
0.012
75–85°
0.15
1.065108
-5.053614
0.179318
4.500936
0.199396
0.999877
0.012
85–90°
0.15
1.066206
-7.376005
0.209640
6.640651
0.228794
0.999859
0.012
0–5°
0.41
1.134323
0.104837
0.091273
0.013352
0.650900
0.999993
0.012
5–15°
0.41
1.134048
0.090558
0.107042
0.013761
0.665505
0.999989
0.012
15–25°
0.41
1.134348
0.067074
0.171292
0.009859
0.983724
0.999993
0.012
25–35°
0.41
1.130415
0.067198
0.289831
-0.034495
0.042748
0.999375
0.012
35–45°
0.41
1.133539
-0.185658
0.093456
0.150322
0.253500
0.999603
0.012
45–55°
0.41
1.133855
-0.269064
0.101965
0.150173
0.329097
0.999918
0.012
55–65°
0.41
1.135699
-0.530234
0.128551
0.319484
0.288401
0.999975
0.012
65–75°
0.41
1.136023
-0.574711
0.132875
0.227846
0.439203
0.999992
0.012
75–85°
0.41
1.138137
-0.839223
0.157968
0.329256
0.475856
0.999974
0.012
85–90°
0.41
1.139139
-0.974873
0.171682
0.277810
0.715086
0.999973
0.035
0–5°
0.61
1.364718
0.238690
0.155216
0.041475
0.851003
0.999993
0.035
5–15°
0.61
1.360770
-0.034832
0.208278
0.287954
0.208390
0.999123
0.035
15–25°
0.61
1.361444
-0.079530
0.253336
0.302621
0.253685
0.999090
0.035
25–35°
0.61
1.365871
0.154675
0.302159
0.035968
0.969081
0.999934
0.035
35–45°
0.61
1.363445
-0.088155
0.113320
0.194225
0.473085
0.999529
0.035
45–55°
0.61
1.363856
-0.207092
0.146423
0.218831
0.556780
0.999367
0.035
55–65°
0.61
1.364457
-0.400895
0.176294
0.306439
0.553353
0.999848
0.035
65–75°
0.61
1.364966
-0.545908
0.194507
0.297485
0.745182
0.999972
0.035
75–85°
0.61
1.366048
-0.915217
0.240828
0.472261
0.664322
0.999881
0.035
85–90°
0.61
1.366992
-0.986013
0.248573
0.338646
1.251294
0.999975
0.035
0–5°
0.77
1.613194
0.416072
0.190456
0.017797
1.452924
0.999980
0.035
5–15°
0.77
1.611246
-0.080315
0.214176
0.488650
0.214279
0.999804
0.035
15–25°
0.77
1.610765
-0.115537
0.250267
0.487512
0.248249
0.999751
0.035
25–35°
0.77
1.613106
0.310606
0.294654
0.019077
1.446102
0.999981
0.035
35–45°
0.77
1.620358
0.225593
0.554577
0.021778
0.554565
0.998564
0.035
45–55°
0.77
1.614895
-0.295328
0.212123
0.422794
0.465301
0.999680
0.035
55–65°
0.77
1.614162
-0.482955
0.217270
0.487635
0.526686
0.999805
0.035
65–75°
0.77
1.613853
-0.584198
0.217586
0.411343
0.756011
0.999981
0.035
75–85°
0.77
1.617718
-1.446369
0.310555
1.044260
0.568603
0.999599
0.035
85–90°
0.77
1.615448
-1.066770
0.267118
0.433326
1.320442
0.999983
Table 5
Arbitrary Composition of the Concentration of Chlorophyll and Extra CDOM at λ = 670 nma
Example of Implementing Eqs. (3) and (5) to Calculate E0(z, 560)a
The computational conditions are Chl = 1.0 (mg m-3), θs = 60°, Cloud = 0.0. The values of atmospheric parameters are those default settings in Hydrolight without consideration of inelastic scattering. (a) Table of Ēd(0+; quadi), Ē0 (0-; Vwind; quadi), and Ē0(0-; Vwind′; quadi); (b) results of Hydrolight run; (c) table of Ē0 (z; quadi); (d) implementation of Eqs. (3)–(5).
Table 3
Part of the Chlorophyll-Based LUTc
BF
quadi
Chl
λ (nm)
B0
B1
P
B2
Q
PCC
0.012
0–5°
1.0
400
1.622968
0.041260
0.169544
0.392585
0.115434
0.999981
0.012
5–15°
1.0
400
1.621863
0.359990
0.120332
0.055450
0.187885
0.999984
0.012
15–25°
1.0
400
1.620264
0.079612
0.217664
0.298405
0.132753
0.999984
0.012
25–35°
1.0
400
1.620130
0.284019
0.176459
0.044905
0.280099
0.999986
0.012
35–45°
1.0
400
1.621477
-0.605406
0.166221
0.846156
0.206304
0.999988
0.012
45–55°
1.0
400
1.622204
-0.484247
0.157253
0.618031
0.261602
0.999944
0.012
55–65°
1.0
400
1.622143
-0.594980
0.150696
0.608611
0.312121
0.999965
0.012
65–75°
1.0
400
1.621727
-0.653226
0.147192
0.490206
0.451490
0.999996
0.012
75–85°
1.0
400
1.622437
-0.881774
0.157695
0.503827
0.588206
0.999995
0.012
85–90°
1.0
400
1.624094
-1.100926
0.171378
0.480971
0.848644
0.999983
0.012
0–5°
1.0
560
1.406156
0.024583
0.473172
0.296568
0.108950
0.999981
0.012
5–15°
1.0
560
1.408768
0.248220
0.151219
0.057948
0.056629
0.999899
0.012
15–25°
1.0
560
1.403002
0.252404
0.143545
0.019512
0.787108
0.999994
0.012
25–35°
1.0
560
1.404388
0.072719
0.335383
0.155298
0.172646
0.999961
0.012
35–45°
1.0
560
1.404837
-0.589266
0.165140
0.736479
0.210064
0.999850
0.012
45–55°
1.0
560
1.403158
-0.263195
0.121358
0.313748
0.320304
0.999807
0.012
55–65°
1.0
560
1.403785
-0.476039
0.135085
0.419990
0.342729
0.999942
0.012
65–75°
1.0
560
1.403959
-0.598409
0.140788
0.385590
0.474633
0.999993
0.012
75–85°
1.0
560
1.405005
-0.823804
0.154344
0.419845
0.595190
0.999992
0.012
85–90°
1.0
560
1.406644
-1.022498
0.168514
0.403268
0.861775
0.999983
0.035
0–5°
5.0
400
1.946314
0.042092
0.148112
0.549831
0.205600
0.999986
0.035
5–15°
5.0
400
1.945661
0.512749
0.219135
0.057444
0.160444
0.999986
0.035
15–25°
5.0
400
1.944717
0.088024
0.166086
0.438363
0.256887
0.999984
0.035
25–35°
5.0
400
1.945591
0.418034
0.308522
0.049296
0.155572
0.999987
0.035
35–45°
5.0
400
1.945348
-0.581844
0.278121
0.940185
0.336619
0.999987
0.035
45–55°
5.0
400
1.945740
-0.457318
0.257737
0.686127
0.422929
0.999968
0.035
55–65°
5.0
400
1.945499
-0.594447
0.246508
0.678134
0.500986
0.999950
0.035
65–75°
5.0
400
1.945279
-0.665562
0.240058
0.538762
0.721242
0.999993
0.035
75–85°
5.0
400
1.945685
-0.929317
0.259333
0.547040
0.932661
0.999993
0.035
85–90°
5.0
400
1.946528
-1.184181
0.282872
0.515979
1.329025
0.999985
0.035
0–5°
5.0
560
1.993911
0.043121
0.146628
0.577093
0.204160
0.999979
0.035
5–15°
5.0
560
1.993240
0.538552
0.217402
0.059420
0.158097
0.999983
0.035
15–25°
5.0
560
1.991906
0.091133
0.187666
0.460743
0.247576
0.999984
0.035
25–35°
5.0
560
1.991324
0.441338
0.292586
0.048417
0.221900
0.999980
0.035
35–45°
5.0
560
1.992105
-0.605649
0.275581
0.984666
0.331311
0.999984
0.035
45–55°
5.0
560
1.992702
-0.604170
0.274190
0.850269
0.403206
0.999971
0.035
55–65°
5.0
560
1.992645
-0.648478
0.254406
0.745793
0.487815
0.999948
0.035
65–75°
5.0
560
1.992339
-0.693967
0.244965
0.575341
0.704433
0.999990
0.035
75–85°
5.0
560
1.992622
-0.945963
0.260523
0.564988
0.928310
0.999993
0.035
85–90°
5.0
560
1.993437
-1.206370
0.283599
0.531964
1.320401
0.999985
Table 4
Part of the Albedo-Based LUTw
BF
quadi
ω0
B0
B1
P
B2
Q
PCC
0.012
0–5°
0.15
1.053646
0.038500
0.053935
0.007855
0.448326
0.999964
0.012
5–15°
0.15
1.055313
0.028511
0.085161
0.006237
0.588680
0.999972
0.012
15–25°
0.15
1.020224
-0.046376
0.004308
0.018152
0.277671
0.998879
0.012
25–35°
0.15
1.024035
-0.084536
0.015590
0.021749
0.272535
0.999165
0.012
35–45°
0.15
1.057579
-0.735063
0.086653
0.635782
0.107455
0.999920
0.012
45–55°
0.15
1.054412
-0.294799
0.068933
0.109806
0.189921
0.999949
0.012
55–65°
0.15
1.059850
-3.802800
0.126050
3.534720
0.136682
0.999896
0.012
65–75°
0.15
1.061696
-2.980420
0.139046
2.578965
0.156566
0.999979
0.012
75–85°
0.15
1.065108
-5.053614
0.179318
4.500936
0.199396
0.999877
0.012
85–90°
0.15
1.066206
-7.376005
0.209640
6.640651
0.228794
0.999859
0.012
0–5°
0.41
1.134323
0.104837
0.091273
0.013352
0.650900
0.999993
0.012
5–15°
0.41
1.134048
0.090558
0.107042
0.013761
0.665505
0.999989
0.012
15–25°
0.41
1.134348
0.067074
0.171292
0.009859
0.983724
0.999993
0.012
25–35°
0.41
1.130415
0.067198
0.289831
-0.034495
0.042748
0.999375
0.012
35–45°
0.41
1.133539
-0.185658
0.093456
0.150322
0.253500
0.999603
0.012
45–55°
0.41
1.133855
-0.269064
0.101965
0.150173
0.329097
0.999918
0.012
55–65°
0.41
1.135699
-0.530234
0.128551
0.319484
0.288401
0.999975
0.012
65–75°
0.41
1.136023
-0.574711
0.132875
0.227846
0.439203
0.999992
0.012
75–85°
0.41
1.138137
-0.839223
0.157968
0.329256
0.475856
0.999974
0.012
85–90°
0.41
1.139139
-0.974873
0.171682
0.277810
0.715086
0.999973
0.035
0–5°
0.61
1.364718
0.238690
0.155216
0.041475
0.851003
0.999993
0.035
5–15°
0.61
1.360770
-0.034832
0.208278
0.287954
0.208390
0.999123
0.035
15–25°
0.61
1.361444
-0.079530
0.253336
0.302621
0.253685
0.999090
0.035
25–35°
0.61
1.365871
0.154675
0.302159
0.035968
0.969081
0.999934
0.035
35–45°
0.61
1.363445
-0.088155
0.113320
0.194225
0.473085
0.999529
0.035
45–55°
0.61
1.363856
-0.207092
0.146423
0.218831
0.556780
0.999367
0.035
55–65°
0.61
1.364457
-0.400895
0.176294
0.306439
0.553353
0.999848
0.035
65–75°
0.61
1.364966
-0.545908
0.194507
0.297485
0.745182
0.999972
0.035
75–85°
0.61
1.366048
-0.915217
0.240828
0.472261
0.664322
0.999881
0.035
85–90°
0.61
1.366992
-0.986013
0.248573
0.338646
1.251294
0.999975
0.035
0–5°
0.77
1.613194
0.416072
0.190456
0.017797
1.452924
0.999980
0.035
5–15°
0.77
1.611246
-0.080315
0.214176
0.488650
0.214279
0.999804
0.035
15–25°
0.77
1.610765
-0.115537
0.250267
0.487512
0.248249
0.999751
0.035
25–35°
0.77
1.613106
0.310606
0.294654
0.019077
1.446102
0.999981
0.035
35–45°
0.77
1.620358
0.225593
0.554577
0.021778
0.554565
0.998564
0.035
45–55°
0.77
1.614895
-0.295328
0.212123
0.422794
0.465301
0.999680
0.035
55–65°
0.77
1.614162
-0.482955
0.217270
0.487635
0.526686
0.999805
0.035
65–75°
0.77
1.613853
-0.584198
0.217586
0.411343
0.756011
0.999981
0.035
75–85°
0.77
1.617718
-1.446369
0.310555
1.044260
0.568603
0.999599
0.035
85–90°
0.77
1.615448
-1.066770
0.267118
0.433326
1.320442
0.999983
Table 5
Arbitrary Composition of the Concentration of Chlorophyll and Extra CDOM at λ = 670 nma