Zachary K. Erickson, P. Jeremy Werdell, and Ivona Cetinić, "Bayesian retrieval of optically relevant properties from hyperspectral water-leaving reflectances," Appl. Opt. 59, 6902-6917 (2020)
Current methods to retrieve optically relevant properties from ocean color observations do not explicitly make use of prior knowledge about property distributions. Here we implement a simplified Bayesian approach that takes into account prior probability distributions on two sets of five optically relevant parameters, and conduct a retrieval of these parameters using hyperspectral simulated water-leaving reflectances. We focus specifically on the ability of the model to distinguish between two optically similar phytoplankton taxa, diatoms and Noctiluca scintillans. The inversion retrieval gives most-likely concentrations and uncertainty estimates, and we find that the model is able to probabilistically predict the occurrence of Noctiluca scintillans blooms using these metrics. We discuss how this method can be expanded to include a priori covariances between different parameters, and show the effect of varying measurement uncertainty and spectral resolution on Noctiluca scintillans bloom predictions.
Amir Ibrahim, Bryan A. Franz, Andrew M. Sayer, Kirk Knobelspiesse, Minwei Zhang, Sean W. Bailey, Lachlan I. W. McKinna, Meng Gao, and P. Jeremy Werdell Appl. Opt. 61(22) 6453-6475 (2022)
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Priors and Their Standard Deviations (in Parentheses) for Each Retrieval
Retrieval #
Forward Model
mg Chla
mg Chla
(555 nm)
(443 nm)
(443 nm)
(443 nm)
Range
0.02, 0.1, 0.5, 1.0
0–5
0.1, 0.2
0.002, 0.02
0.005, 0.05
0.007, 0.025, 0.05, 0.07
0.012, 0.016, 0.018
1
1.0 (1.0)
1.0 (1.0)
Truth (0.01)
Truth (0.002)
Truth (0.005)
2
1.0 (1.0)
1.0 (1.0)
0.15 (0.05)
0.01 (0.01)
0.025 (0.025)
3
1.0 (1.0)
1.0 (1.0)
0.15 (0.05)
0.035 (0.035)
0.015 (0.003)
Table 2.
Retrieved Variables and Their Standard Deviations (in Parentheses) for the Examples in Figs. 3 and 4a
Retrieval
mg Chla
mg Chla
(555 nm)
(443 nm)
(443 nm)
(443 nm)
DoF
Truth
1.00
0.34
0.1
0.002
0.005
0.007
0.016
27
1
1.04 (0.12)
0.41 (0.11)
0.098 (0.007)
0.002 (0.002)
0.004 (0.002)
3.5
18.5
2
0.95 (0.18)
0.45 (0.14)
0.108 (0.015)
0.009 (0.009)
0.000 (0.005)
4.0
20.7
3
1.03 (0.14)
0.41 (0.12)
0.099 (0.011)
0.006 (0.003)
0.015 (0.003)
3.9
20.1
Total number of degrees of freedom (DoFs) is also indicated, as is the goodness of fit ${\chi ^2}$. Note that for a good fit, ${\chi ^2} \sim m$. ${\chi ^2}$ calculated for the truth is from the forward model results.
Table 3.
Median Absolute Percent Error and Median Degrees of Freedom for Each Variable, from the Full Suite of Simulated (Considering Only Those Retrievals with )a
Retr.
(555 nm)
(443 nm)
(443 nm)
(443 nm)
1
16%, 1.0
47%, 1.0
2%, 0.5
5%, 0.1
14%, 0.8
12%
4%
3%
84%
2
26%, 1.0
51%, 1.0
9%, 0.9
60%, 0.1
24%, 1.0
18%
4%
3%
71%
3
25%, 0.9
46%, 1.0
11%, 0.7
13%, 0.9
8%, 0.4
16%
4%
2%
72%
Total chlorophyll $C = {\textbf{C}_\textbf{D}} + {\textbf{C}_\textbf{N}}$, total absorption at 443 nm $a(443)$, and total backscattering at 555 nm $b(555)$ are derived, and therefore, only the median absolute percent error is shown.
Table 4.
Symbols Used in This Paper and Their Units and Definitions
Priors and Their Standard Deviations (in Parentheses) for Each Retrieval
Retrieval #
Forward Model
mg Chla
mg Chla
(555 nm)
(443 nm)
(443 nm)
(443 nm)
Range
0.02, 0.1, 0.5, 1.0
0–5
0.1, 0.2
0.002, 0.02
0.005, 0.05
0.007, 0.025, 0.05, 0.07
0.012, 0.016, 0.018
1
1.0 (1.0)
1.0 (1.0)
Truth (0.01)
Truth (0.002)
Truth (0.005)
2
1.0 (1.0)
1.0 (1.0)
0.15 (0.05)
0.01 (0.01)
0.025 (0.025)
3
1.0 (1.0)
1.0 (1.0)
0.15 (0.05)
0.035 (0.035)
0.015 (0.003)
Table 2.
Retrieved Variables and Their Standard Deviations (in Parentheses) for the Examples in Figs. 3 and 4a
Retrieval
mg Chla
mg Chla
(555 nm)
(443 nm)
(443 nm)
(443 nm)
DoF
Truth
1.00
0.34
0.1
0.002
0.005
0.007
0.016
27
1
1.04 (0.12)
0.41 (0.11)
0.098 (0.007)
0.002 (0.002)
0.004 (0.002)
3.5
18.5
2
0.95 (0.18)
0.45 (0.14)
0.108 (0.015)
0.009 (0.009)
0.000 (0.005)
4.0
20.7
3
1.03 (0.14)
0.41 (0.12)
0.099 (0.011)
0.006 (0.003)
0.015 (0.003)
3.9
20.1
Total number of degrees of freedom (DoFs) is also indicated, as is the goodness of fit ${\chi ^2}$. Note that for a good fit, ${\chi ^2} \sim m$. ${\chi ^2}$ calculated for the truth is from the forward model results.
Table 3.
Median Absolute Percent Error and Median Degrees of Freedom for Each Variable, from the Full Suite of Simulated (Considering Only Those Retrievals with )a
Retr.
(555 nm)
(443 nm)
(443 nm)
(443 nm)
1
16%, 1.0
47%, 1.0
2%, 0.5
5%, 0.1
14%, 0.8
12%
4%
3%
84%
2
26%, 1.0
51%, 1.0
9%, 0.9
60%, 0.1
24%, 1.0
18%
4%
3%
71%
3
25%, 0.9
46%, 1.0
11%, 0.7
13%, 0.9
8%, 0.4
16%
4%
2%
72%
Total chlorophyll $C = {\textbf{C}_\textbf{D}} + {\textbf{C}_\textbf{N}}$, total absorption at 443 nm $a(443)$, and total backscattering at 555 nm $b(555)$ are derived, and therefore, only the median absolute percent error is shown.
Table 4.
Symbols Used in This Paper and Their Units and Definitions
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