Curtis D. Mobley,
Lydia K. Sundman,
and Emmanuel Boss
C. D. Mobley (mobley@sequoiasci.com) and L. K. Sundman are with Sequoia Scientific, Incorporated, 15317 NE 90th Street, Redmond, Washington 98052-3562. USA
E. Boss is with Oregon State University, College of Ocean and Atmospheric Sciences, Corvallis, Oregon 97331. USA
Numerical simulations show that underwater radiances, irradiances, and reflectances are sensitive to the shape of the scattering phase function at intermediate and large scattering angles, although the exact shape of the phase function in the backscatter directions (for a given backscatter fraction) is not critical if errors of the order of 10% are acceptable. We present an algorithm for generating depth- and wavelength-dependent Fournier-Forand phase functions having any desired backscatter fraction. Modeling of a comprehensive data set of measured inherent optical properties and radiometric variables shows that use of phase functions with the correct backscatter fraction and overall shape is crucial to achieve model-data closure.
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Rms Percentage Differences in Phase Functions, as Defined by Eq. (6), for Intermediate Scattering Angles 5 ≤ ψ ≤ 90 dega
Phase Function
Phase Function
Petzold
Flat Back
FF
OTHG
Haltrin
Ad Hoc
Petzold
6.2
15.9
18.3
113.4
58.1
Flat back
6.2
17.3
19.9
112.9
58.8
FF
9.3
11.2
28.9
103.6
71.1
OTHG
7.1
11.0
14.4
120.9
49.0
Haltrin
84.8
89.9
79.4
88.6
146.0
Ad hoc
39.5
41.0
47.1
34.3
107.2
The lower-left triangle of numbers is Eq. (6) including the sin ψ factor in the integrand; the upper-right triangle of numbers is Eq. (6) without the sin ψ factor.
Table 2
Rms Percentage Differences in Phase Functions, as in Table
1, for Backscattering Angles 90 ≤ ψ ≤ 180 deg
Phase Function
Phase Function
Petzold
Flat Back
FF
OTHG
Haltrin
Ad Hoc
Petzold
15.5
6.4
26.1
88.8
83.6
Flatback
12.1
11.7
34.7
85.3
90.4
FF
3.4
10.2
27.6
87.9
85.1
OTHG
9.0
19.5
11.4
102.3
66.2
Haltrin
50.6
45.9
49.1
56.4
125.8
Ad hoc
31.4
40.7
33.7
24.1
68.2
Table 3
Rms Percentage Differences in Phase Functions, as in Table
1, for Intermediate and Backscattering Angles 5 ≤ ψ ≤ 180 deg
Phase Function
Phase Function
Petzold
Flat Back
FF
OTHG
Haltrin
Ad Hoc
Petzold
11.7
12.0
23.3
101.5
72.5
Flat back
9.4
14.5
28.4
99.8
76.7
FF
7.0
10.4
28.3
95.9
78.7
OTHG
8.1
15.6
12.9
111.7
58.6
Haltrin
69.2
67.1
65.4
73.6
136.0
Ad hoc
35.5
40.5
40.7
29.5
89.0
Table 4
Data Taken at the LEO-15 Site as Used to Model the In-Water Light Fielda
Quantity Measured
Instrument
Nominal Wavelength (nm)
Total a(z, λ), total c(z, λ)
Unfiltered ac-9
412, 440, 488, 532, 555, 650, 676, 715
Dissolved a(z, λ)
Filtered ac-9
412, 440, 488, 532, 555, 650, 676, 715
Backscatter b(z, λ) derived from VSF at ψ = 140 deg
HydroScat-6
442, 488, 532, 555, 620
Backscatter b(z, λ) derived from VSF at ψ = 100, 125, and 150 deg
ECO-VSF
530
VSF (ψ = 0.6–179.6 deg)
VSM
530
Ed(z, λ) and Lu(z, λ)
OCP
412, 443, 489, 533, 555, 591, 683
Sky Ed(λ)
Multichannel visible detector system
412, 443, 489, 533, 555, 591, 683
Sky Ed(λ) and Lu(z = 0.6 m, λ)
Hyper-TSRB
123 wavelengths between 396 and 798
Most instruments have a nominal 10-nm bandwidth centered on the listed wavelengths.
Table 5
Rms Percentage Differences in the Three Phase Functions of Fig.
13, as Defined by Eq. (6), for Intermediate Scattering Angles 5 ≤ ψ ≤ 90 dega
Phase Function
Phase Function
Petzold
VSM
FF
Petzold
103.7
113.0
VSM
86.5
26.9
FF
87.7
12.4
VSF is the measured phase function, and FF is the Fournier-Forand phase function with the same backscatter fraction. The lower-left triangle of numbers is Eq. (6) including the sin ψ factor in the integrand; the upper-right triangle of numbers is Eq. (6) without the sin ψ factor.
Table 6
Rms Percentage Differences in the Three Phase Functions of Fig.
13, as in Table
5, for Backscattering Angles 90 ≤ ψ ≤ 180 deg
Phase Function
Phase Function
Petzold
VSM
FF
Petzold
118.4
127.4
VSM
90.1
34.4
FF
88.3
7.0
Table 7
Rms Percentage Differences in the Three Phase Functions of Fig.
13, as in Table
5, for Intermediate and Backscattering Angles 5 ≤ ψ ≤ 180 deg
Phase Function
Phase Function
Petzold
VSM
FF
Petzold
111.4
120.4
VSM
88.1
31.0
FF
87.8
10.0
Tables (7)
Table 1
Rms Percentage Differences in Phase Functions, as Defined by Eq. (6), for Intermediate Scattering Angles 5 ≤ ψ ≤ 90 dega
Phase Function
Phase Function
Petzold
Flat Back
FF
OTHG
Haltrin
Ad Hoc
Petzold
6.2
15.9
18.3
113.4
58.1
Flat back
6.2
17.3
19.9
112.9
58.8
FF
9.3
11.2
28.9
103.6
71.1
OTHG
7.1
11.0
14.4
120.9
49.0
Haltrin
84.8
89.9
79.4
88.6
146.0
Ad hoc
39.5
41.0
47.1
34.3
107.2
The lower-left triangle of numbers is Eq. (6) including the sin ψ factor in the integrand; the upper-right triangle of numbers is Eq. (6) without the sin ψ factor.
Table 2
Rms Percentage Differences in Phase Functions, as in Table
1, for Backscattering Angles 90 ≤ ψ ≤ 180 deg
Phase Function
Phase Function
Petzold
Flat Back
FF
OTHG
Haltrin
Ad Hoc
Petzold
15.5
6.4
26.1
88.8
83.6
Flatback
12.1
11.7
34.7
85.3
90.4
FF
3.4
10.2
27.6
87.9
85.1
OTHG
9.0
19.5
11.4
102.3
66.2
Haltrin
50.6
45.9
49.1
56.4
125.8
Ad hoc
31.4
40.7
33.7
24.1
68.2
Table 3
Rms Percentage Differences in Phase Functions, as in Table
1, for Intermediate and Backscattering Angles 5 ≤ ψ ≤ 180 deg
Phase Function
Phase Function
Petzold
Flat Back
FF
OTHG
Haltrin
Ad Hoc
Petzold
11.7
12.0
23.3
101.5
72.5
Flat back
9.4
14.5
28.4
99.8
76.7
FF
7.0
10.4
28.3
95.9
78.7
OTHG
8.1
15.6
12.9
111.7
58.6
Haltrin
69.2
67.1
65.4
73.6
136.0
Ad hoc
35.5
40.5
40.7
29.5
89.0
Table 4
Data Taken at the LEO-15 Site as Used to Model the In-Water Light Fielda
Quantity Measured
Instrument
Nominal Wavelength (nm)
Total a(z, λ), total c(z, λ)
Unfiltered ac-9
412, 440, 488, 532, 555, 650, 676, 715
Dissolved a(z, λ)
Filtered ac-9
412, 440, 488, 532, 555, 650, 676, 715
Backscatter b(z, λ) derived from VSF at ψ = 140 deg
HydroScat-6
442, 488, 532, 555, 620
Backscatter b(z, λ) derived from VSF at ψ = 100, 125, and 150 deg
ECO-VSF
530
VSF (ψ = 0.6–179.6 deg)
VSM
530
Ed(z, λ) and Lu(z, λ)
OCP
412, 443, 489, 533, 555, 591, 683
Sky Ed(λ)
Multichannel visible detector system
412, 443, 489, 533, 555, 591, 683
Sky Ed(λ) and Lu(z = 0.6 m, λ)
Hyper-TSRB
123 wavelengths between 396 and 798
Most instruments have a nominal 10-nm bandwidth centered on the listed wavelengths.
Table 5
Rms Percentage Differences in the Three Phase Functions of Fig.
13, as Defined by Eq. (6), for Intermediate Scattering Angles 5 ≤ ψ ≤ 90 dega
Phase Function
Phase Function
Petzold
VSM
FF
Petzold
103.7
113.0
VSM
86.5
26.9
FF
87.7
12.4
VSF is the measured phase function, and FF is the Fournier-Forand phase function with the same backscatter fraction. The lower-left triangle of numbers is Eq. (6) including the sin ψ factor in the integrand; the upper-right triangle of numbers is Eq. (6) without the sin ψ factor.
Table 6
Rms Percentage Differences in the Three Phase Functions of Fig.
13, as in Table
5, for Backscattering Angles 90 ≤ ψ ≤ 180 deg
Phase Function
Phase Function
Petzold
VSM
FF
Petzold
118.4
127.4
VSM
90.1
34.4
FF
88.3
7.0
Table 7
Rms Percentage Differences in the Three Phase Functions of Fig.
13, as in Table
5, for Intermediate and Backscattering Angles 5 ≤ ψ ≤ 180 deg