The mean upward-scattering coefficient of the downward-traveling photons and the mean downward-scattering coefficient of the upward-traveling photons are two factors needed for the two-stream approximation to the radiative-transfer equation. Numerical values of each shape factor just beneath the surface and at asymptotic depths give an indication of the range of values at intermediate depths in spatially uniform waters with no sources and are used to obtain an approximate depth-dependent model for each shape factor. The shape factors are computed for different surface-illumination conditions, wavelengths, and chlorophyll concentrations.

Ammar H. Hakim and Norman J. McCormick Appl. Opt. 42(6) 931-938 (2003)

References

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Inherent Optical Properties c, b, and b_{b} (in Inverse Meters) at Various Wavelengths λ (in Nanometers) and Chlorophyll Concentrations C (in Milligrams per Cubic Meter)

λ

C

c

b

b_{b}/b

430

0.01

0.0456

0.0276

0.3980

430

10

1.8530

1.6051

0.0498

500

0.01

0.0501

0.0219

0.3533

500

10

1.5646

1.3786

0.0401

570

0.01

0.0993

0.0184

0.0394

570

10

1.3546

1.2084

0.0349

700

0.01

0.6651

0.0143

0.2329

700

10

1.6849

0.9833

0.0295

Table 2

Mean Cosines in the Downward and Upward Directions in Deep Water at Various Wavelengths λ (in Nanometers) for Two Chlorophyll Concentrations C (in Milligrams per Cubic Meter)

λ

${\overline{\mu}}_{\pm}(\infty )$

C

0.01

10.0

430

${\overline{\mu}}_{+}(\infty )$

0.714

0.680

${\overline{\mu}}_{-}(\infty )$

−0.471

−0.409

500

${\overline{\mu}}_{+}(\infty )$

0.799

0.673

${\overline{\mu}}_{-}(\infty )$

−0.483

−0.400

570

${\overline{\mu}}_{+}(\infty )$

0.915

0.667

${\overline{\mu}}_{-}(\infty )$

−0.508

−0.395

700

${\overline{\mu}}_{+}(\infty )$

0.992

0.820

${\overline{\mu}}_{-}(\infty )$

−0.525

−0.374

Table 3

Two-Stream Shape Factors r_{±}(∞) in Deep Water at Various Wavelengths λ (in Nanometers) for Two Chlorophyll Concentrations C (in Milligrams per Cubic Meter)

λ

r_{±}(∞)

C

0.01

10.0

430

r_{+}(∞)

1.031

1.978

r_{−}(∞)

1.405

14.898

500

r_{+}(∞)

1.032

2.269

r_{−}(∞)

1.647

18.525

570

r_{+}(∞)

1.017

2.519

r_{−}(∞)

1.949

21.344

700

r_{+}(∞)

1.002

1.554

r_{−}(∞)

2.754

23.976

Table 4

Two-Stream Shape Factors r_{±}(0^{+}) just beneath the Surface at Various Wavelengths λ (in Nanometers) for Two Chlorophyll Concentrations C (in Milligrams per Cubic Meter)

The surface-illumination conditions from Eq. (15) are isotropic illumination given by case 1, normal monodirectional illumination given by case 2 with μ_{0} = 1, and a boundary condition given by case 2 with μ_{0} = 0.7071.

Tables (4)

Table 1

Inherent Optical Properties c, b, and b_{b} (in Inverse Meters) at Various Wavelengths λ (in Nanometers) and Chlorophyll Concentrations C (in Milligrams per Cubic Meter)

λ

C

c

b

b_{b}/b

430

0.01

0.0456

0.0276

0.3980

430

10

1.8530

1.6051

0.0498

500

0.01

0.0501

0.0219

0.3533

500

10

1.5646

1.3786

0.0401

570

0.01

0.0993

0.0184

0.0394

570

10

1.3546

1.2084

0.0349

700

0.01

0.6651

0.0143

0.2329

700

10

1.6849

0.9833

0.0295

Table 2

Mean Cosines in the Downward and Upward Directions in Deep Water at Various Wavelengths λ (in Nanometers) for Two Chlorophyll Concentrations C (in Milligrams per Cubic Meter)

λ

${\overline{\mu}}_{\pm}(\infty )$

C

0.01

10.0

430

${\overline{\mu}}_{+}(\infty )$

0.714

0.680

${\overline{\mu}}_{-}(\infty )$

−0.471

−0.409

500

${\overline{\mu}}_{+}(\infty )$

0.799

0.673

${\overline{\mu}}_{-}(\infty )$

−0.483

−0.400

570

${\overline{\mu}}_{+}(\infty )$

0.915

0.667

${\overline{\mu}}_{-}(\infty )$

−0.508

−0.395

700

${\overline{\mu}}_{+}(\infty )$

0.992

0.820

${\overline{\mu}}_{-}(\infty )$

−0.525

−0.374

Table 3

Two-Stream Shape Factors r_{±}(∞) in Deep Water at Various Wavelengths λ (in Nanometers) for Two Chlorophyll Concentrations C (in Milligrams per Cubic Meter)

λ

r_{±}(∞)

C

0.01

10.0

430

r_{+}(∞)

1.031

1.978

r_{−}(∞)

1.405

14.898

500

r_{+}(∞)

1.032

2.269

r_{−}(∞)

1.647

18.525

570

r_{+}(∞)

1.017

2.519

r_{−}(∞)

1.949

21.344

700

r_{+}(∞)

1.002

1.554

r_{−}(∞)

2.754

23.976

Table 4

Two-Stream Shape Factors r_{±}(0^{+}) just beneath the Surface at Various Wavelengths λ (in Nanometers) for Two Chlorophyll Concentrations C (in Milligrams per Cubic Meter)

The surface-illumination conditions from Eq. (15) are isotropic illumination given by case 1, normal monodirectional illumination given by case 2 with μ_{0} = 1, and a boundary condition given by case 2 with μ_{0} = 0.7071.