Hiroyuki Tan, Tomohiko Oishi, Akihiko Tanaka, and Roland Doerffer, "Accurate estimation of the backscattering coefficient by light scattering at two backward angles," Appl. Opt. 54, 7718-7733 (2015)

Backscattering coefficients are frequently estimated from light scattering at one backward angle multiplied by a conversion factor. We determined that the shapes of the volume scattering functions (VSFs), particularly for scattering angles larger than 170°, cause significant variations in the conversion factor at 120°. Our approach uses the ratio of scattering at 170° and at 120°, which is a good indicator of the shape differences of the VSFs for most oceanic waters and wavelengths in the visible range. The proposed method provides significant accuracy improvement in the determination of the backscattering coefficients with a prediction error of 3% of the mean.

Malik Chami, Eugeny B. Shybanov, Gueorgui A. Khomenko, Michael E.-G. Lee, Oleg V. Martynov, and Gennady K. Korotaev Appl. Opt. 45(15) 3605-3619 (2006)

References

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Average and ${\mathsf{\sigma}}^{*}$ of ${\mathsf{\chi}}_{\mathsf{p}}(\mathsf{\theta})$ for Individual Samples, Natural Waters, and Cultures at Selected Backward Angles^{a}

Water samples

$N$

90°

100°

110°

120°

130°

140°

150°

160°

170°

River Elbe

18

${\chi}_{p,\text{avg}}(\theta )$

0.81

0.99

1.11

1.16

1.21

1.17

1.08

0.88

0.54

${\sigma}^{*}$, %

4.6

5.7

4.8

3.7

2.7

2.5

4.2

8.9

23.6

Lake Schaalsee

16

${\chi}_{p,\text{avg}}(\theta )$

0.94

1.09

1.17

1.14

1.13

1.04

0.91

0.71

0.52

$\sigma *$, %

5.2

5.7

4.7

4.9

3.7

2.6

5.3

10.5

17.8

P. minimum

54

${\chi}_{p,\text{avg}(\theta )}$

0.99

1.19

1.31

1.30

1.27

1.10

0.88

0.65

0.34

${\sigma}^{*}$, %

9.9

8.5

7.1

5.9

5.3

5.7

7.5

7.9

12.0

R. baltica

54

${\chi}_{p,\text{avg}}(\theta )$

1.35

1.63

1.78

1.64

1.57

1.28

0.96

0.60

0.22

${\sigma}^{*}$, %

14.8

16.7

18.3

13.9

11.1

8.3

8.8

9.1

22.4

Synechococcus spp.

54

${\chi}_{p,\text{avg}}(\theta )$

1.15

1.39

1.48

1.44

1.35

1.12

0.84

0.53

0.25

${\sigma}^{*}$, %

25.0

13.9

11.3

10.8

11.1

9.7

8.7

9.0

12.4

Nannochloropsis spp.

54

${\chi}_{p,\text{avg}}(\theta )$

1.36

1.76

1.96

1.88

1.79

1.51

1.12

0.69

0.21

$\sigma *$, %

13.7

16.8

18.4

19.8

17.6

15.3

14.6

11.0

22.3

T. weissflogii

54

${\chi}_{p,\text{avg}}(\theta )$

1.18

1.37

1.39

1.30

1.23

1.06

0.85

0.62

0.32

${\sigma}^{*}$, %

8.9

9.6

9.0

8.1

4.9

4.4

6.5

9.4

19.2

Natural waters

34

${\chi}_{p,\text{avg}}(\theta )$

0.87

1.04

1.14

1.15

1.17

1.11

1.00

0.80

0.53

${\sigma}^{*}$, %

8.9

7.4

5.4

4.3

4.6

6.8

9.9

14.1

20.9

Cultures

270

${\chi}_{p,\text{avg}}(\theta )$

1.21

1.47

1.58

1.51

1.44

1.22

0.93

0.62

0.27

${\sigma}^{*}$, %

19.4

19.8

21.6

20.3

18.9

17.4

15.0

13.0

26.8

$N$ is the number of data points, and ${\sigma}^{*}$ is the standard deviation normalized by the mean.

Table 2.

Average of ${\mathsf{\phi}}_{\mathsf{p}}(\mathsf{120},\mathsf{\lambda})$ in Ascending Order, with Mean ${\mathsf{G}}_{\mathsf{p}}(\mathsf{\theta},\mathsf{\lambda})$ at 120° and 170° (%)

${\phi}_{p,\text{avg}}(120)$

${G}_{p,\text{avg}}(120)$

${G}_{p,\text{avg}}(170)$

Lake Schaalsee

1.14

1.34

0.61

River Elbe

1.16

1.31

0.60

P. minimum

1.30

1.17

0.91

T. weissflogii

1.30

1.18

1.00

Synechococcus spp.

1.44

1.07

1.24

R. baltica

1.64

0.94

1.48

Nannochloropsis spp.

1.88

0.84

1.54

Table 3.

Summary of Quantified Errors in the Fixed-Angle Approach by the Use of Different Conversion Factors for ${\mathsf{\beta}}_{\mathsf{p}}(\mathsf{\theta},\mathsf{\lambda})$ Based on the Lorenz–Mie Calculations^{a}

Conversion Factor Used for ${b}_{bp}(\lambda )$ Estimation

Types of Waters

$N$

${\nu}_{p}[{R}_{p}(120:170,\lambda )]$

${\chi}_{p,\mathrm{ave}}(120)$

Junge PSD

450

RMSE relative to ${b}_{bp,\text{in}\text{tegral}}(\lambda )$

0.033

0.031

Average of $|\mathrm{\Delta}{b}_{bp,\nu p}|$ or $|\mathrm{\Delta}{b}_{bp,\chi p}|$

1.9%

1.7%

Frequency within $\pm 5\%$ of $\mathrm{\Delta}{b}_{bp,\nu p}$ or $\mathrm{\Delta}{b}_{bp,\chi p}$

90%

93%

Gauss PSD

972

RMSE relative to ${b}_{bp,\text{in}\text{tegral}}(\lambda )$

0.134

0.129

Average of $|\mathrm{\Delta}{b}_{bp,\nu p}|$ or $|\mathrm{\Delta}{b}_{bp,\chi p}|$

9.6%

9.3%

Frequency within $\pm 5\%$ of $\mathrm{\Delta}{b}_{bp,\nu p}$ or $\mathrm{\Delta}{b}_{bp,\chi p}$

39%

22%

Note that ${\chi}_{p,\mathrm{ave}}(120)=1.08$ for Junge PSD and 1.18 for Gauss PSD.

Table 4.

Summary of Quantified Errors in the Fixed-Angle Approach by the Use of Different Conversion Factors for Measured ${\mathsf{\beta}}_{\mathsf{p}}(\mathsf{\theta},\mathsf{\lambda})$

Conversion Factor Used for ${b}_{bp}(\lambda )$ Estimation

Types of Waters

N

${\nu}_{p}[{R}_{p}(120:170,\lambda )]$

${\chi}_{p,\text{mean}(120)}$

Natural waters with the historical VSFs

141

RMSE relative to ${b}_{bp,\text{in}\text{tegral}}(\lambda )$

0.047

0.080

Average of $|\mathrm{\Delta}{b}_{bp,\nu p}|$ or $|\mathrm{\Delta}{b}_{bp,\chi p}|$

3.4%

6.1%

Frequency within $\pm 5\%$ of $\mathrm{\Delta}{b}_{bp,\nu p}$ or $\mathrm{\Delta}{b}_{bp,\chi p}$

74%

52%

Cultures

270

RMSE relative to ${b}_{bp,\text{in}\text{tegral}}(\lambda )$

0.040

0.238

Average of $|\mathrm{\Delta}{b}_{bp,\nu p}|$ or $|\mathrm{\Delta}{b}_{bp,\chi p}|$

3.2%

19.7%

Frequency within $\pm 5\%$ of $\mathrm{\Delta}{b}_{bp,\nu p}$ or $\mathrm{\Delta}{b}_{bp,\chi p}$

79%

13%

Table 5.

Summary of Quantified Errors in the Fixed-Angle Approach by the Use of Different Conversion Factors for Measured $\mathsf{\beta}(\mathsf{\theta},\mathsf{\lambda})$

Conversion Factor Used for ${b}_{b}(\lambda )$ Estimation

Types of Waters

N

${\nu}_{t}[R(120:170,\lambda )]$

${\chi}_{t}(120)$

Natural waters with the historical VSFs

154

RMSE relative to ${b}_{b,\text{in}\text{tegral}}(\lambda )$

0.039

0.080

Average of $|\mathrm{\Delta}{b}_{b,\nu t}|$ or $|\mathrm{\Delta}{b}_{b,\chi t}|$

3.1%

6.9%

Frequency within $\pm 5\%$ of $\mathrm{\Delta}{b}_{b,\nu t}$ or $\mathrm{\Delta}{b}_{b,\chi t}$

84%

33%

Cultures

270

RMSE relative to ${b}_{b,\text{in}\text{tegral}}(\lambda )$

0.032

0.239

Average of $|\mathrm{\Delta}{b}_{b,\nu t}|$ or $|\mathrm{\Delta}{b}_{b,\chi t}|$

2.5%

21.1%

Frequency within $\pm 5\%$ of $\mathrm{\Delta}{b}_{b,\nu t}$ or $\mathrm{\Delta}{b}_{b,\chi t}$

88%

4%

Tables (5)

Table 1.

Average and ${\mathsf{\sigma}}^{*}$ of ${\mathsf{\chi}}_{\mathsf{p}}(\mathsf{\theta})$ for Individual Samples, Natural Waters, and Cultures at Selected Backward Angles^{a}

Water samples

$N$

90°

100°

110°

120°

130°

140°

150°

160°

170°

River Elbe

18

${\chi}_{p,\text{avg}}(\theta )$

0.81

0.99

1.11

1.16

1.21

1.17

1.08

0.88

0.54

${\sigma}^{*}$, %

4.6

5.7

4.8

3.7

2.7

2.5

4.2

8.9

23.6

Lake Schaalsee

16

${\chi}_{p,\text{avg}}(\theta )$

0.94

1.09

1.17

1.14

1.13

1.04

0.91

0.71

0.52

$\sigma *$, %

5.2

5.7

4.7

4.9

3.7

2.6

5.3

10.5

17.8

P. minimum

54

${\chi}_{p,\text{avg}(\theta )}$

0.99

1.19

1.31

1.30

1.27

1.10

0.88

0.65

0.34

${\sigma}^{*}$, %

9.9

8.5

7.1

5.9

5.3

5.7

7.5

7.9

12.0

R. baltica

54

${\chi}_{p,\text{avg}}(\theta )$

1.35

1.63

1.78

1.64

1.57

1.28

0.96

0.60

0.22

${\sigma}^{*}$, %

14.8

16.7

18.3

13.9

11.1

8.3

8.8

9.1

22.4

Synechococcus spp.

54

${\chi}_{p,\text{avg}}(\theta )$

1.15

1.39

1.48

1.44

1.35

1.12

0.84

0.53

0.25

${\sigma}^{*}$, %

25.0

13.9

11.3

10.8

11.1

9.7

8.7

9.0

12.4

Nannochloropsis spp.

54

${\chi}_{p,\text{avg}}(\theta )$

1.36

1.76

1.96

1.88

1.79

1.51

1.12

0.69

0.21

$\sigma *$, %

13.7

16.8

18.4

19.8

17.6

15.3

14.6

11.0

22.3

T. weissflogii

54

${\chi}_{p,\text{avg}}(\theta )$

1.18

1.37

1.39

1.30

1.23

1.06

0.85

0.62

0.32

${\sigma}^{*}$, %

8.9

9.6

9.0

8.1

4.9

4.4

6.5

9.4

19.2

Natural waters

34

${\chi}_{p,\text{avg}}(\theta )$

0.87

1.04

1.14

1.15

1.17

1.11

1.00

0.80

0.53

${\sigma}^{*}$, %

8.9

7.4

5.4

4.3

4.6

6.8

9.9

14.1

20.9

Cultures

270

${\chi}_{p,\text{avg}}(\theta )$

1.21

1.47

1.58

1.51

1.44

1.22

0.93

0.62

0.27

${\sigma}^{*}$, %

19.4

19.8

21.6

20.3

18.9

17.4

15.0

13.0

26.8

$N$ is the number of data points, and ${\sigma}^{*}$ is the standard deviation normalized by the mean.

Table 2.

Average of ${\mathsf{\phi}}_{\mathsf{p}}(\mathsf{120},\mathsf{\lambda})$ in Ascending Order, with Mean ${\mathsf{G}}_{\mathsf{p}}(\mathsf{\theta},\mathsf{\lambda})$ at 120° and 170° (%)

${\phi}_{p,\text{avg}}(120)$

${G}_{p,\text{avg}}(120)$

${G}_{p,\text{avg}}(170)$

Lake Schaalsee

1.14

1.34

0.61

River Elbe

1.16

1.31

0.60

P. minimum

1.30

1.17

0.91

T. weissflogii

1.30

1.18

1.00

Synechococcus spp.

1.44

1.07

1.24

R. baltica

1.64

0.94

1.48

Nannochloropsis spp.

1.88

0.84

1.54

Table 3.

Summary of Quantified Errors in the Fixed-Angle Approach by the Use of Different Conversion Factors for ${\mathsf{\beta}}_{\mathsf{p}}(\mathsf{\theta},\mathsf{\lambda})$ Based on the Lorenz–Mie Calculations^{a}

Conversion Factor Used for ${b}_{bp}(\lambda )$ Estimation

Types of Waters

$N$

${\nu}_{p}[{R}_{p}(120:170,\lambda )]$

${\chi}_{p,\mathrm{ave}}(120)$

Junge PSD

450

RMSE relative to ${b}_{bp,\text{in}\text{tegral}}(\lambda )$

0.033

0.031

Average of $|\mathrm{\Delta}{b}_{bp,\nu p}|$ or $|\mathrm{\Delta}{b}_{bp,\chi p}|$

1.9%

1.7%

Frequency within $\pm 5\%$ of $\mathrm{\Delta}{b}_{bp,\nu p}$ or $\mathrm{\Delta}{b}_{bp,\chi p}$

90%

93%

Gauss PSD

972

RMSE relative to ${b}_{bp,\text{in}\text{tegral}}(\lambda )$

0.134

0.129

Average of $|\mathrm{\Delta}{b}_{bp,\nu p}|$ or $|\mathrm{\Delta}{b}_{bp,\chi p}|$

9.6%

9.3%

Frequency within $\pm 5\%$ of $\mathrm{\Delta}{b}_{bp,\nu p}$ or $\mathrm{\Delta}{b}_{bp,\chi p}$

39%

22%

Note that ${\chi}_{p,\mathrm{ave}}(120)=1.08$ for Junge PSD and 1.18 for Gauss PSD.

Table 4.

Summary of Quantified Errors in the Fixed-Angle Approach by the Use of Different Conversion Factors for Measured ${\mathsf{\beta}}_{\mathsf{p}}(\mathsf{\theta},\mathsf{\lambda})$

Conversion Factor Used for ${b}_{bp}(\lambda )$ Estimation

Types of Waters

N

${\nu}_{p}[{R}_{p}(120:170,\lambda )]$

${\chi}_{p,\text{mean}(120)}$

Natural waters with the historical VSFs

141

RMSE relative to ${b}_{bp,\text{in}\text{tegral}}(\lambda )$

0.047

0.080

Average of $|\mathrm{\Delta}{b}_{bp,\nu p}|$ or $|\mathrm{\Delta}{b}_{bp,\chi p}|$

3.4%

6.1%

Frequency within $\pm 5\%$ of $\mathrm{\Delta}{b}_{bp,\nu p}$ or $\mathrm{\Delta}{b}_{bp,\chi p}$

74%

52%

Cultures

270

RMSE relative to ${b}_{bp,\text{in}\text{tegral}}(\lambda )$

0.040

0.238

Average of $|\mathrm{\Delta}{b}_{bp,\nu p}|$ or $|\mathrm{\Delta}{b}_{bp,\chi p}|$

3.2%

19.7%

Frequency within $\pm 5\%$ of $\mathrm{\Delta}{b}_{bp,\nu p}$ or $\mathrm{\Delta}{b}_{bp,\chi p}$

79%

13%

Table 5.

Summary of Quantified Errors in the Fixed-Angle Approach by the Use of Different Conversion Factors for Measured $\mathsf{\beta}(\mathsf{\theta},\mathsf{\lambda})$

Conversion Factor Used for ${b}_{b}(\lambda )$ Estimation

Types of Waters

N

${\nu}_{t}[R(120:170,\lambda )]$

${\chi}_{t}(120)$

Natural waters with the historical VSFs

154

RMSE relative to ${b}_{b,\text{in}\text{tegral}}(\lambda )$

0.039

0.080

Average of $|\mathrm{\Delta}{b}_{b,\nu t}|$ or $|\mathrm{\Delta}{b}_{b,\chi t}|$

3.1%

6.9%

Frequency within $\pm 5\%$ of $\mathrm{\Delta}{b}_{b,\nu t}$ or $\mathrm{\Delta}{b}_{b,\chi t}$

84%

33%

Cultures

270

RMSE relative to ${b}_{b,\text{in}\text{tegral}}(\lambda )$

0.032

0.239

Average of $|\mathrm{\Delta}{b}_{b,\nu t}|$ or $|\mathrm{\Delta}{b}_{b,\chi t}|$

2.5%

21.1%

Frequency within $\pm 5\%$ of $\mathrm{\Delta}{b}_{b,\nu t}$ or $\mathrm{\Delta}{b}_{b,\chi t}$