Absorption correction and phase function shape effects on the closure of apparent optical properties

J. Pitarch; G. Volpe; S. Colella; R. Santoleri; V. Brando

doi:10.1364/AO.55.008618

Applied Optics
Vol. 55,
Issue 30,
pp. 8618-8636
(2016)
•https://doi.org/10.1364/AO.55.008618

Absorption correction and phase function shape effects on the closure of apparent optical properties

J. Pitarch, G. Volpe, S. Colella, R. Santoleri, and V. Brando

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J. Pitarch, G. Volpe, S. Colella, R. Santoleri, and V. Brando, "Absorption correction and phase function shape effects on the closure of apparent optical properties," Appl. Opt. 55, 8618-8636 (2016)

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- Atmospheric and Oceanic Optics
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About this Article
History
- Original Manuscript: July 7, 2016
- Revised Manuscript: September 19, 2016
- Manuscript Accepted: September 19, 2016
- Published: October 20, 2016
Virtual Issues
Virtual Journal for Biomedical Optics Vol. 11, Iss. 11

Abstract

We present a closure experiment between new inherent optical properties (IOPs: absorption a, scattering $b$ , backscattering $b_{b}$ ) and apparent optical properties (AOPs: remote-sensing reflectance $R_{rs}$ , irradiance reflectance $R$ , and anisotropic factor at nadir $Q_{n}$ ) data of Ionian and Adriatic seawaters, from very clear to turbid waters, ranging across one order of magnitude in $R_{rs}$ . The internal consistency of the IOP–AOP matchups was investigated though radiative transfer closure. Using the in situ IOPs, we predicted the AOPs with the commercial radiative transfer solver Hydrolight. Closure was limited by two unresolved issues, one regarding processing of in situ data and the other related to radiative transfer modeling. First, different correction methods of the absorption data measured by the Wetlabs ac-s produced high variations in simulated reflectances, reaching 40% for the highest reflectances in our dataset. Second, the lack of detailed volume scattering function measurements forces us to adopt analytical functions that are consistent with a given particle backscattering ratio. The analytical phase functions named Fournier-Forand and two-term Kopelevich presented reasonable angular shapes with respect to measurements at a few backward angles. Between these phase functions, induced changes were within 4% for $R_{rs}$ , within 11% for $R$ , and within 10% for $Q_{n}$ . Additionally, closure of $Q_{n}$ was generally not successful considering radiometric uncertainties. Simulated $Q_{n}$ overestimated low values and underestimated high values, especially at 665 nm, where Hydrolight was unable to predict measured $Q_{n}$ values greater than 6 sr. The physical nature of $Q_{n}$ makes this mismatch almost independent of the measured IOPs, thus precluding $Q_{n}$ tuning by varying the former. The non-closure of $Q_{n}$ might be caused by an inaccurate phase function and, to a lesser extent, by the modeling of the incoming radiance. For the future, this remains the task of accurate absorption and phase function measurements, especially at red wavelengths.

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Name and Units	Symbol (and definition or comments if applicable)
Radiances, irradiances, and apparent optical properties
Underwater radiance ( $W m^{- 2} {nm}^{- 1} {sr}^{- 1}$ )	$L (Ω, λ, z)$
Water-leaving radiance toward zenith ( $W m^{- 2} {nm}^{- 1} {sr}^{- 1}$ )	$L_{w} (λ, Ω_{zenith})$ or simply $L_{w} (λ)$
Downwelling irradiance ( $W m^{- 2} {nm}^{- 1}$ )	$E_{d} (λ, z) = \int_{Ξ_{d}} L (Ω, λ, z) \cos θ d Ω$ or $E_{s} (λ, 0^{+})$ for above-water measurements
Upwelling irradiance ( $W m^{- 2} {nm}^{- 1}$ )	$E_{u} (λ, z) = \int_{Ξ_{u}} L (Ω, λ, z) \cos θ d Ω$
Upwelling radiance toward zenith ( $W m^{- 2} {nm}^{- 1} {sr}^{- 1}$ )	$L_{u} (λ, z) = L (Ξ_{zenith}, λ, z)$
Irradiance reflectance (adimensional)	$R (λ, z) = \frac{E_{u} (λ, z)}{E_{d} (λ, z)}$ . Commonly evaluated just below the surface $z = 0^{-}$ , becomes $R (λ, 0^{-})$ or simply $R (λ)$
Remote-sensing reflectance ( ${sr}^{- 1}$ )	$R_{rs} (λ) = \frac{L_{w} (λ)}{E_{d} (λ, 0^{+})}$
Anisotropy factor of the upwelling radiance, or “ $Q$ -factor” toward zenith	$Q_{n} (λ, z) = \frac{E_{u} (λ, z)}{L_{u} (λ, z)}$
Inherent optical properties
Volume scattering function (VSF) of pure seawater [1]	$β_{w} (θ, λ)$
VSF of particles	$β_{p} (θ, λ, z)$
Total VSF	$β (θ, λ, z) = β_{w} (θ, λ) + β_{p} (θ, λ, z)$
Scattering coefficients: of water ( $b_{w}$ ), of particles ( $b_{p}$ ), and total ( $b$ )	$b (λ, z) = b_{w} (λ) + b_{p} (λ, z) b_{x} (λ, z) = 2 π \int_{0}^{π} β_{x} (θ, λ, z) \sin θ d θ$ , $x = w, p$
Backscattering coefficients: of water ( $b_{bw}$ ), of particles ( $b_{bp}$ ), and total ( $b_{b}$ )	$b_{b} (λ, z) = b_{bw} (λ) + b_{bp} (λ, z) b_{b x} (λ, z) = 2 π \int_{\frac{π}{2}}^{π} β_{x} (θ, λ, z) \sin θ d θ$
Water backscattering ratio	$B_{w} (λ) = b_{bw} (λ) / b_{w} (λ) = 0.5$
Particle backscattering ratio	$B_{p} (λ, z) = b_{bp} (λ, z) / b_{p} (λ, z)$
Particle phase function (PF)	${\tilde{β}}_{p} (θ, λ, z) = β_{p} (θ, λ, z) / b_{p} (θ, λ, z)$
Total absorption	$a (λ, z)$
Pure seawater absorption	$a_{w} (λ)$
Pure water absorption	$a_{pw} (λ, z)$
Drift-corrected absorption, but not for scatter yet	$a_{m} (λ, z)$
“Final” non-water absorption	$a_{t - w} (λ, z)$
Various attenuations	$c_{x} (λ, z)$ same notation as the absorptions
Single-scattering albedo	$ω_{0} (λ, z) = b (λ, z) / c (λ, z)$
Quasi-single-scattering albedo, or backscattering albedo, or “Gordon parameter”	$ω_{b} (λ, z) = b_{b} (λ, z) / (a (λ, z) + b_{b} (λ, z))$

Acronym	Full Name	References
FF	Mobley’s Fournier-Forand	[14,16]
FFTw	Twardowski’s Fournier-Forand	[16,17]
OTHG	One-Term Henyey-Greenstein	[18]
TTHG	Two-Term Henyey-Greenstein	[19]
NTTHG	New Two-Term Henyey-Greenstein	[20]
TTKO	Two-Term Kopelevich	[21]

	Radiometry
Article	Instruments (hyperspectral?)	Above or Below Surface	Derived AOPs	IOP Measurements	Nr. of Stations	Model	Statistics
[30]	Satlantic HyperTSRB (yes) Satlantic OCP-100 (no)	Below	$R_{rs}$	ac-9 Hyperangular VSF measurements at 532 nm	4	Hydrolight	Relative bias $\sim 17 %$
[27]	Satlantic OCR-200 and OCI-200	Below	$Q_{n}$ . Also, $E_{u}$ and $L_{u}$ , forcing the measured $E_{d}$ as input	ac-9 Hydroscat-6	3	Finite-element code	Relative bias for $L_{u}$ and $E_{u} \sim 30 %$ . For $Q_{n}$ : from $- 4 %$ to 12%
[23]	Satlantic MicroPro (no)	Below	$R_{rs}$	ac-9 Lab. absorption ECO-VSF3	16	Hydrolight	Relative bias $\sim 8.5 %$
[28]	TriOs-RAMSES (yes)	Below	$L_{w}$ (forcing the measured $E_{d}$ as input)	ac-9 Lab. absorption ECO-VSF3	4	Hydrolight	Variable bias (see paper). Relative RMS $\sim 12.8 %$
[10]	Various instruments (see paper)	Above and below		Lab. absorption Hydroscat-6	20	$R_{rs} \sim ω_{b}$	Relative bias from $\sim - 15.0 %$ to $\sim - 27.9 %$
[31]	Satlantic HyperOCR (yes)	Above	$R_{rs}$	ac-s Hydroscat-6, BB-9	23	$R_{rs} \sim ω_{b}$	Not reported, but very high $\sim \pm 100 %$ (see paper)
[29]	TriOs-RAMSES (yes)	Above	$R_{rs}$	ac-s Hydroscat-6	31	Hydrolight	Depending on the ac-s data correction, $\sim \pm 25 %$

	$R_{rs} (665) ({sr}^{- 1})$	$a_{t - w} (443) (m^{- 1})$	$b_{bp} (556) (m^{- 1})$
Min.	0.0001723	0.0238	0.00153
Max.	0.0048329	0.4332	0.05964

Acron.	Abs. at NIR	Extrap. Toward Blue	Attenuation
M1	Zero	Flat	Unchanged
M2	Zero	Proportional	Unchanged
M3	$a_{t - w} (λ_{NIR}) = 0.212 a_{m} {(λ_{NIR})}^{1.135}$	Proportional	$c_{t - w} (λ_{NIR}) = 1.78 c_{m} (λ_{NIR})$
M4	$a_{t - w} (λ_{NIR}) = 0.06 a_{m} {(λ_{NIR})}^{0.7}$	Proportional	$c_{t - w} (λ_{NIR}) = 1.30 c_{m} (λ_{NIR})$

Clear	412	443	489	510	556	665
M2 versus M1	−1.8	−2.9	−4.0	−4.4	−3.5	0.5
	−4.9	−9.0	−26.5	−41.9	−71.6	19.3
M3 versus M1	−3.1	−2.9	−2.3	−2.2	−0.6	2.3
	−8.6	−10.8	−14.9	−21.1	−12.7	83.9
M4 versus M1	2.5	1.1	0.2	0.2	1.1	3.9
	7.0	4.0	1.4	1.5	22.9	144.5
Turbid	412	443	489	510	556	665
M2 versus M1	−119	−108	−99	−94	−83	−12
	−21.6	−25.5	−36.0	−41.4	−59.9	−17.1
M3 versus M1	−52	−32	−9	−0.1	18	72
	−9.4	−7.6	−3.3	−0.3	12.9	95.6
M4 versus M1	−103	−86	−67	−59	−42	23
	−18.7	−20.3	−24.2	−26.1	−30.0	31.2

M1	412	443	489	510	556	665	all
$r^{2}$	0.82	0.81	0.85	0.90	0.93	0.96	0.91
$m$	0.75	0.68	0.62	0.68	0.74	0.98	0.77
$n (\times 10^{3})$	1.04	1.74	2.48	1.54	1.01	0.12	0.95
bias	−1.80	2.01	2.79	−3.93	−1.79	17.00	2.38
rmse	12.34	12.78	14.31	12.73	12.83	24.07	15.42
M2	412	443	489	510	556	665	all
$r^{2}$	0.86	0.90	0.92	0.94	0.95	0.96	0.95
$m$	1.18	1.26	1.26	1.22	1.15	1.01	1.21
$n (\times 10^{3})$	−0.32	−0.41	−0.01	−0.29	0.05	0.11	−0.13
bias	10.55	16.75	24.89	15.39	13.14	17.89	16.43
rmse	18.99	22.42	29.38	21.97	20.24	24.73	23.20
M3	412	443	489	510	556	665	all
$r^{2}$	0.82	0.82	0.83	0.88	0.91	0.95	0.90
$m$	0.97	0.91	0.78	0.79	0.75	0.89	0.87
$n (\times 10^{3})$	0.52	1.06	2.05	1.30	1.04	0.14	0.86
bias	8.47	11.20	10.89	1.28	−0.88	13.52	7.41
rmse	16.82	18.01	18.72	13.68	13.95	21.38	17.30
M4	412	443	489	510	556	665	all
$r^{2}$	0.85	0.89	0.91	0.93	0.94	0.94	0.94
$m$	1.04	1.06	1.00	0.97	0.91	0.93	0.99
$n (\times 10^{3})$	0.07	0.16	0.64	0.30	0.44	0.13	0.30
bias	5.26	8.39	9.23	1.31	0.34	15.26	6.63
rmse	15.36	15.32	15.06	12.12	12.40	24.02	16.20

M1	412	443	489	510	556	665	all
$r^{2}$	0.78	0.76	0.73	0.68	0.68	0.68	0.64
$m$	0.38	0.38	0.42	0.38	0.41	0.27	0.32
$n$	2.57	2.69	2.60	2.86	2.78	3.21	3.02
bias	−3.11	0.35	2.95	6.56	7.12	−10.95	0.49
rmse	9.28	8.68	8.08	11.44	11.38	15.62	11.04
M2	412	443	489	510	556	665	all
$r^{2}$	0.76	0.70	0.53	0.48	0.53	0.67	0.61
$m$	0.37	0.35	0.37	0.32	0.37	0.27	0.31
$n$	2.61	2.77	2.75	3.02	2.89	3.20	3.00
bias	−3.65	−0.55	1.38	5.16	6.17	−10.96	−0.41
rmse	9.60	9.07	8.52	11.56	11.44	15.60	11.22
M3	412	443	489	510	556	665	all
$r^{2}$	0.76	0.72	0.63	0.59	0.63	0.70	0.65
$m$	0.40	0.38	0.40	0.35	0.39	0.31	0.34
$n$	2.63	2.79	2.77	3.05	2.96	3.17	3.04
bias	−0.34	2.78	4.93	8.97	10.00	−7.98	3.06
rmse	9.07	9.58	9.59	13.55	13.80	13.61	11.73
M4	412	443	489	510	556	665	all
$r^{2}$	0.77	0.72	0.62	0.56	0.60	0.68	0.63
$m$	0.39	0.37	0.39	0.35	0.40	0.29	0.33
$n$	2.60	2.76	2.72	2.99	2.88	3.18	3.02
bias	−2.17	1.14	3.46	7.31	8.30	−9.56	1.41
rmse	9.13	9.06	8.81	12.41	12.53	14.64	11.32

AOP	$R_{rs} (665)$	$R (665)$	$Q_{n} (665)$
$r^{2}$	0.9998	0.99991	0.975
$m$	1.014	1.0095	1.044
$n$	$3.3 \cdot 10^{- 5}$	$2.2 \cdot 10^{- 4}$	−0.355
bias	9.52	6.59	−2.26
rmse	11.13	7.73	2.98

FF	412	443	489	510	556	665	all
$r^{2}$	0.79	0.79	0.85	0.90	0.93	0.95	0.89
$m$	0.63	0.59	0.57	0.61	0.69	0.79	0.71
$n$	0.013	0.019	0.025	0.018	0.012	0.001	0.011
bias	−1.80	4.85	7.13	3.86	6.53	6.25	4.47
rmse	16.24	18.55	20.18	18.38	18.91	19.00	18.58
FFTw	412	443	489	510	556	665	all
$r^{2}$	0.76	0.77	0.84	0.90	0.93	0.95	0.88
$m$	0.58	0.54	0.51	0.56	0.63	0.73	0.65
$n$	0.014	0.020	0.024	0.018	0.011	0.002	0.011
bias	−5.24	0.39	1.48	−1.57	1.10	3.55	−0.05
rmse	17.36	18.08	18.63	17.91	17.77	18.46	18.05
OTHG	412	443	489	510	556	665	all
$r^{2}$	0.78	0.79	0.85	0.90	0.93	0.95	0.89
$m$	0.72	0.69	0.66	0.71	0.80	0.88	0.81
$n$	0.013	0.019	0.025	0.017	0.011	0.001	0.010
bias	6.26	13.85	16.69	11.81	13.86	8.44	11.82
rmse	22.97	25.65	26.72	21.75	22.61	19.46	23.31
TTHG	412	443	489	510	556	665	all
$r^{2}$	0.73	0.74	0.83	0.88	0.92	0.93	0.85
$m$	0.49	0.44	0.41	0.46	0.53	0.66	0.56
$n$	0.017	0.022	0.027	0.020	0.014	0.002	0.013
bias	−6.66	−2.18	−2.69	−4.90	−1.16	9.00	−1.43
rmse	18.11	18.88	20.03	20.32	20.75	23.82	20.40
NTTHG	412	443	489	510	556	665	all
$r^{2}$	0.78	0.79	0.83	0.88	0.91	0.93	0.86
$m$	0.74	0.66	0.60	0.65	0.73	1.00	0.79
$n$	0.020	0.028	0.035	0.028	0.020	0.003	0.018
bias	25.72	32.60	32.31	31.15	39.10	63.36	37.37
rmse	33.60	39.94	40.90	40.63	48.74	71.21	47.42
TTKO	412	443	489	510	556	665	all
$r^{2}$	0.79	0.79	0.85	0.90	0.93	0.95	0.89
$m$	0.72	0.68	0.66	0.71	0.79	0.87	0.80
$n$	0.013	0.019	0.024	0.017	0.011	0.001	0.01
bias	5.79	13.30	15.13	11.20	13.33	8.77	11.25
rmse	22.33	25.09	24.57	21.46	22.33	19.71	22.66

FF	412	443	489	510	556	665	all
$r^{2}$	0.79	0.79	0.85	0.90	0.93	0.95	0.89
$m$	0.63	0.59	0.57	0.61	0.69	0.79	0.71
$n$	0.013	0.019	0.025	0.018	0.012	0.001	0.011
bias	−1.80	4.85	7.13	3.86	6.53	6.25	4.47
rmse	16.24	18.55	20.18	18.38	18.91	19.00	18.58
FFTw	412	443	489	510	556	665	all
$r^{2}$	0.76	0.77	0.84	0.90	0.93	0.95	0.88
$m$	0.58	0.54	0.51	0.56	0.63	0.73	0.65
$n$	0.014	0.020	0.024	0.018	0.011	0.002	0.011
bias	−5.24	0.39	1.48	−1.57	1.10	3.55	−0.05
rmse	17.36	18.08	18.63	17.91	17.77	18.46	18.05
OTHG	412	443	489	510	556	665	all
$r^{2}$	0.78	0.79	0.85	0.90	0.93	0.95	0.89
$m$	0.72	0.69	0.66	0.71	0.80	0.88	0.81
$n$	0.013	0.019	0.025	0.017	0.011	0.001	0.010
bias	6.26	13.85	16.69	11.81	13.86	8.44	11.82
rmse	22.97	25.65	26.72	21.75	22.61	19.46	23.31
TTHG	412	443	489	510	556	665	all
$r^{2}$	0.73	0.74	0.83	0.88	0.92	0.93	0.85
$m$	0.49	0.44	0.41	0.46	0.53	0.66	0.56
$n$	0.017	0.022	0.027	0.020	0.014	0.002	0.013
bias	−6.66	−2.18	−2.69	−4.90	−1.16	9.00	−1.43
rmse	18.11	18.88	20.03	20.32	20.75	23.82	20.40
NTTHG	412	443	489	510	556	665	all
$r^{2}$	0.78	0.79	0.83	0.88	0.91	0.93	0.86
$m$	0.74	0.66	0.60	0.65	0.73	1.00	0.79
$n$	0.020	0.028	0.035	0.028	0.020	0.003	0.018
bias	25.72	32.60	32.31	31.15	39.10	63.36	37.37
rmse	33.60	39.94	40.90	40.63	48.74	71.21	47.42
TTKO	412	443	489	510	556	665	all
$r^{2}$	0.79	0.79	0.85	0.90	0.93	0.95	0.89
$m$	0.72	0.68	0.66	0.71	0.79	0.87	0.80
$n$	0.013	0.019	0.024	0.017	0.011	0.001	0.01
bias	5.79	13.30	15.13	11.20	13.33	8.77	11.25
rmse	22.33	25.09	24.57	21.46	22.33	19.71	22.66

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