Fast and accurate model of underwater scalar irradiance for stratified Case 2 waters

Cheng-Chien Liu

doi:10.1364/OE.14.001703

Optics Express
Vol. 14,
Issue 5,
pp. 1703-1719
(2006)
•https://doi.org/10.1364/OE.14.001703

Fast and accurate model of underwater scalar irradiance for stratified Case 2 waters

Cheng-Chien Liu

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Cheng-Chien Liu, "Fast and accurate model of underwater scalar irradiance for stratified Case 2 waters," Opt. Express 14, 1703-1719 (2006)

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- Atmospheric and ocean optics
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About this Article
History
- Original Manuscript: December 15, 2005
- Revised Manuscript: February 19, 2006
- Manuscript Accepted: February 23, 2006
- Published: March 6, 2006
Virtual Issues
Virtual Journal for Biomedical Optics Vol. 1, Iss. 4

Abstract

This paper is devoted to the derivation of a fast and accurate model of scalar irradiance for stratified Case 2 waters. Five strategies are formulated and employed in the new model, including (1) reallocating the sky radiance, (2) approximating the influence of the air-water interface, (3) constructing a look-up table of average cosine based on the single-scattering albedo and the backscatter fraction, (4) calculating the phase function of surrogate particles in Case 2 waters, and (5) using the average cosine as an index to cope with stratified waters. A comprehensive model-to-model comparison shows that the new model runs more than 1,400 times faster than the commercially-available Hydrolight model, while it limits the percentage error to 2.03% and the maximum error to less than 6.81%. This new model can be used interactively in models of the oceanic system, such as biogeochemical models or the heat budget part of global circulation models.

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Fig. 1. Illustration of the validity of Eq. (18). The computational conditions are set as typical Case 1 waters with θ _s=30°, Cloudiness=0.0, V_wind =5.0 ms^-1, and (a) Chl=l.0 mg∙m^-3, BF_p =0.00915, (b) Chl=10.0 mg∙m^-3, BF_p =0.00915, (c) Chl=1.0 mg∙m^-3, BF_p =0.0183, (d) Chl=10.0 mg∙m^-3, BF_p =0.0183, (e) Chl=l.0 mg∙m^-3, BF_p =0.0366, (f) Chl=l0.0 mg∙m^-3, BF_p =0.0366. The values of the atmospheric parameters used are the default settings in Hydrolight. Both Hydrolight and the new model are employed to simulate μ̄(ζ).

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Fig. 2. (a) The analytical phase functions for ocean water proposed by Fournier and Forand [20]. A large range in β(ψ) can be obtained by varying the value of BF. (b) An example of applying the technique of optimization to calculate BF_p for simulating the phase function of surrogate particles in Case 2 waters.

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Fig. 3. Illustration of using average cosine as an index to cope with stratified waters. The computational conditions are θ_S = 30°, Cloud = 0.0, λ = 440 nm and V_wind = 5m∙s^-1. The atmospheric parateters are those default settings in Hydrolight without consideration of inelastic scattering. (a) A profile of chlorophyll with one meter intervals. (b) A set of μ̄(z) that are calculated by considering eight cases of homogeneous bodies of water, each with one of those chlorophyll concentrations under the same conditions of incident light. (c) Combining these “layer solutions” of μ̄(z) to approximate the full solution of μ̄(z) for the corresponding case of an inhomogeneous body of water under the same conditions of incident light. (d) Comparison of the approximated E ₀ to the Hydrolight simulated E ₀.

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Fig. 4. Twenty cases of stratified Case 2 waters with vertical profiles of (a) Chl, (b) a _g,440 and (c) M. The corresponding profiles of IOPs at wavelength 440 nm are illustrated as (d) a, (e) b and (f) BF_p .

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Fig. 5. Comparison of accuracy and speed in simulating E ^0,PAR(z) (W m^-2) between the new model and Hydrolight. A very high correlation (r = 0.999924) as well as a large CSR of 1402.8 was obtained for our model. The percentage error ε% is 2.03% and the maximum relative error ε _max is not more than 6.81%.

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Tables (5)

Table 1. A section of LUT_CW for quick reference to the corrective factor CW based on wind speed V_wind and albedo ω ₀.

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Table 2. A section of the LUT_μ for quick reference to a set of parameters (B ₀, B ₁, P, B ₂, Q) used by the McCormick five-parameter model [21] of μ̄(ζ). This new LUT is based on two non-dimensional variables BF (0.0001 – 0.5) and ω ₀(0.01 – 0.99).

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Table 3. A section of the LUT_BF for quick reference to the value of backscattering fraction BF_p simulating the phase function of surrogate particles in Case 2 waters. The LUT_BF is based on three variables: ratio_l (0.0 – 1.0), the backscattering fraction for large particles BF_l (0.0001 – 0.0 the backscattering fraction for small particles BF_s (0.018 – 0.3).

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Table 4. The randomly specified values of parameters for the 20 cases that are used in the model-to-model comparison to examine the accuracy, flexibility and applicability of the new model. The parameters include the solar zenith angle θ_S , cloudiness, surface wind speed V_wind , the backscattering fraction for large particles BF_l , the backscattering fraction for mineral particles BF_S , and the exponential coefficient of CDOM absorption γ.

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Table 5. The randomly specified values of parameters in Eq. (23) for describing the vertical profiles of chlorophyll concentration Chl(z) and the mineral particle concentration M(z), respectively. The CDOM absorption at a reference wavelength a _g,440(z) is set to be varied linearly from the surface value $a_{g, 440}^{Surface}$ to the bottom value $a_{g, 440}^{Bottom}$ , which are specified randomly as well.

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Equations (30)

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\frac{DL (\vec{x}; \hat{ξ}; λ)}{Dr} = - c (\vec{x}; \hat{ξ}; λ) L (\vec{x}; \hat{ξ}; λ) + L_{*} + L_{*}^{S},

\frac{D}{Dr} = \frac{1}{v} \frac{D}{Dt} = \frac{1}{v} (\frac{\partial}{\partial t} + \vec{v} ∙ \nabla) = \frac{1}{v} \frac{\partial}{\partial t} + \hat{ξ} ∙ \nabla = \hat{ξ} ∙ \nabla,

\hat{ξ} ∙ \nabla L (\vec{x}; \hat{ξ}; λ) = - c (\vec{x}; λ) L (\vec{x}; \hat{ξ}; λ) + \int_{\hat{ξ′} \inΞ} L (\vec{x}; \hat{ξ′}; λ) β (\vec{x}; \hat{ξ'} \to \hat{ξ}; λ) d Ω (\hat{ξ'})

\cos θ \frac{dL (z; θ; ϕ; λ)}{dz} = - c (z; λ) L (z; θ; ϕ; λ) + \int_{ϕ′ = 0}^{2 π} \int_{θ′ = 0}^{π} L (z; θ'; ϕ'; λ) β (z; θ' \to θ; ϕ' \to ϕ; λ) \sin θ' dθ' dϕ' .

E_{0} (z) = \int_{Ξ} L (z; \hat{ξ}) d Ω = \int_{ϕ = 0}^{2 π} \int_{θ = 0}^{π} L (z; θ; ϕ) \sin θd θdθ .

E_{d} (0^{+}) = \int_{ϕ = 0}^{2 π} \int_{θ = 0}^{\frac{π}{2}} L (0^{+}; θ; ϕ) \cos θ \sin θd θdϕ = \int_{θ = 0}^{\frac{π}{2}} \bar{L} (0^{+}; θ; ϕ_{0}) \cos θ \sin θd θ,

\overset{̅}{L} (0^{+}; θ; ϕ_{0}) = \int_{ϕ = 0}^{2 π} L (0^{+}; θ; ϕ) dϕ .

E_{0} (z) = \int_{θ = 0}^{\frac{π}{2}} {\overset{̅}{E}}_{0} (z; θ; ϕ_{0}) \bar{L} (0^{+}; θ; ϕ_{0}) \cos θ \sin θd θ .

CW (θ; ϕ_{0}) ∣_{V_{wind} \to {V'}_{wind}} \equiv \frac{{\bar{E}}_{0} (0^{-}; θ; ϕ_{0}) ∣_{{V'}_{wind}}}{{\bar{E}}_{0} (0^{-}; θ; ϕ_{0}) ∣_{V_{wind}}},

E_{0} (z) ∣_{{V'}_{wind}} = \int_{θ = 0}^{\frac{π}{2}} {\bar{E}}_{0} (z; θ; ϕ_{0}) \bar{L} (0^{+}; θ; ϕ_{0}) CW (θ; ϕ_{0}) ∣_{V_{wind} \to {V'}_{wind}} \cos θ \sin θd θ .

\frac{1}{\bar{μ} (ζ)} = B_{0} + B_{1} \exp (- Pζ) + B_{2} \exp (- Qζ) .

\bar{μ} (z) \equiv \frac{E (z)}{E_{0} (z)} .

K_{NET} (z) = \frac{a (z)}{\bar{μ} (z)} .

K_{NET} (z) \equiv - \frac{d In E (z)}{dz} = - \frac{d In [E_{d} (z) - E_{u} (z)]}{dz},

BF (λ) \equiv \frac{b_{w} (λ)}{b (λ)} \times {BF}_{w} + \frac{b_{p} (λ)}{b (λ)} {BF}_{p},

Δ (0^{-}) = (1 - \frac{{BF}_{p}}{BF}) [{\bar{μ}}_{p} (0^{-}) - \bar{μ'} (0^{-})],

Δ (ζ_{\infty}) = \frac{1}{2} [{\bar{μ}}_{p} (ζ_{\infty}) - {\bar{μ'}}_{p} (ζ_{\infty})] .

\bar{μ} (ζ) = {\bar{μ}}_{p} (ζ) + Δ (ζ) .

Δ (ζ) = [Δ (0^{-}) - Δ (ζ_{\infty})] [\frac{{\bar{μ}}_{p} (ζ) - {\bar{μ}}_{p} (ζ_{\infty})}{{\bar{μ}}_{p} (0^{-}) - {\bar{μ}}_{p} (ζ_{\infty})}] .

\tilde{β} (ψ; λ) \equiv \frac{b_{w} (λ)}{b (λ)} {\tilde{β}}_{w} (ψ) + \frac{b_{l} (λ)}{b (λ)} {\tilde{β}}_{l} (ψ; λ) + \frac{b_{s} (λ)}{b (λ)} {\tilde{β}}_{s} (ψ; λ) .

{\tilde{β}}_{p} (ψ; λ) = \frac{b_{l} (λ)}{b_{l} (λ) + b_{s} (λ)} {\tilde{β}}_{l} (ψ; λ) + \frac{b_{s} (λ)}{b_{l} (λ) + b_{s} (λ)} {\tilde{β}}_{s} (ψ; λ) .

{ratio}_{l} \equiv \frac{b_{l} (λ)}{b_{l} (λ) + b_{s} (λ)} .

C (z) = C_{o} + \frac{h}{s \sqrt{2 π}} \exp [- \frac{1}{2} {(\frac{z - z_{max}}{s})}^{2}] .

a_{g} (z, λ) = a_{g, 440} (z) \exp [- γ (λ - 440)],

a_{c} (z; λ) = 0.06 a_{c}^{*} (λ) Chl {(z)}^{0.65},

b_{c} (z; λ) = 0.30 (\frac{550}{λ}) Chl {(z)}^{0.62},

a_{m} (z; λ) = M (z) a_{m}^{*} (λ),

b_{m} (z; λ) = M (z) b_{m}^{*} (λ) .

ε_{%} = 10^{{RMSE}_{\log 10}} - 1,

{RMSE}_{\log 10} = \sqrt{\frac{\sum_{n = 1}^{N} {(\log_{10} E_{0, PAR}^{Model} - \log_{10} E_{0, PAR}^{Hydrolight})}^{2}}{N}} .

Quad		15°~25°			35°~45°
ω ₀	V_wind	1.0	3.0	10.0	1.0	3.0	10.0
0.01		1.000477	1.000185	1.000058	1.005423	1.002095	0.998207
0.03		1.000433	1.000163	1.000038	1.005430	1.002088	0.998149
0.05		1.000399	1.000142	1.000038	1.005444	1.002089	0.998093
0.07		1.000366	1.000122	1.000028	1.005447	1.002090	0.998039
0.09		1.000324	1.000102	1.000009	1.005457	1.002090	0.997987
0.11		1.000284	1.000073	0.999991	1.005455	1.002080	0.997920
0.13		1.000253	1.000063	0.999982	1.005460	1.002087	0.997872
0.15		1.000205	1.000036	0.999955	1.005478	1.002084	0.997794
0.17		1.000158	1.000018	0.999947	1.005470	1.002080	0.997743
0.19		1.000113	0.999991	0.999922	1.005467	1.002076	0.997678
0.21		1.000068	0.999966	0.999897	1.005462	1.002071	0.997609
0.23		1.000025	0.999941	0.999874	1.005454	1.002064	0.997543
0.25		0.999959	0.999917	0.999851	1.005458	1.002065	0.997480
0.27		0.999918	0.999894	0.999820	1.005437	1.002064	0.997420
0.29		0.999871	0.999871	0.999791	1.005421	1.002055	0.997348
0.31		0.999818	0.999850	0.999762	1.005409	1.002045	0.997281
0.33		0.999758	0.999821	0.999727	1.005387	1.002034	0.997210
0.35		0.999701	0.999801	0.999701	1.005356	1.002022	0.997136
0.37		0.999639	0.999767	0.999669	1.005328	1.002015	0.997074
0.39		0.999571	0.999741	0.999630	1.005297	1.002001	0.997002
0.41		0.999499	0.999709	0.999600	1.005257	1.001986	0.996935
0.43		0.999423	0.999679	0.999565	1.005207	1.001969	0.996867
0.45		0.999343	0.999650	0.999538	1.005154	1.001951	0.996797
0.47		0.999253	0.999609	0.999500	1.005099	1.001933	0.996727
0.49		0.999167	0.999577	0.999463	1.005034	1.001906	0.996662

quad	BF	ω ₀	B ₀	B ₁	P	B ₂	Q	PCC
0°~5°	0.018	0.31	1.119133	0.086067	0.087585	0.017142	0.589276	0.999981
5°~15°	0.018	0.31	1.119114	0.073322	0.107362	0.016351	0.647022	0.999977
15°~25°	0.018	0.31	1.119486	0.062006	0.236405	-0.001879	0.236448	0.999166
25°~35°	0.018	0.31	1.114392	0.057128	0.324473	-0.040133	0.039546	0.998720
35°~45°	0.018	0.31	1.118637	-0.179313	0.089442	0.129340	0.269664	0.999598
45°~55°	0.018	0.31	1.119143	-0.261702	0.100093	0.128522	0.350085	0.999912
55°~65°	0.018	0.31	1.121492	-0.561937	0.134360	0.337626	0.281429	0.999969
65°~75°	0.018	0.31	1.121764	-0.582089	0.138694	0.221943	0.434598	0.999989
75°~85°	0.018	0.31	1.124118	-0.885466	0.170924	0.363883	0.454816	0.999959
85°~90°	0.018	0.31	1.125022	-1.013489	0.187065	0.304746	0.660470	0.999960

ratio_i	BF_t	BF_S	BF_p	*PCC*
0.31	0.001	0.018	0.012388	0.999982
0.31	0.001	0.020	0.013687	0.999975
0.31	0.001	0.025	0.016998	0.999952
0.31	0.001	0.030	0.020336	0.999919
0.31	0.001	0.035	0.023545	0.999878
0.31	0.001	0.040	0.026882	0.999821
0.31	0.001	0.050	0.033383	0.999671
0.31	0.001	0.060	0.039866	0.999464
0.31	0.001	0.080	0.053316	0.998891
0.31	0.001	0.100	0.066865	0.998132
0.31	0.001	0.150	0.103043	0.994607
0.31	0.001	0.200	0.137565	0.990413
0.74	0.018	0.018	0.018000	1.000000
0.74	0.018	0.020	0.018520	1.000000
0.74	0.018	0.025	0.019801	1.000000
0.74	0.018	0.030	0.021077	1.000000
0.74	0.018	0.035	0.022331	0.999999
0.74	0.018	0.040	0.023562	0.999997
0.74	0.018	0.050	0.026020	0.999989
0.74	0.018	0.060	0.028401	0.999976
0.74	0.018	0.080	0.033082	0.999931
0.74	0.018	0.100	0.037621	0.999862
0.74	0.018	0.150	0.048219	0.999605
0.74	0.018	0.200	0.057609	0.999279

No.	θ_S	Cloudiness	V_wind	BF_l	BF_S	γ
1	36.4	0.59	7.2	0.0056	0.0325	0.0100
2	7.5	0.99	6.7	0.0074	0.0291	0.0105
3	27.3	0.78	12.0	0.0076	0.0225	0.0127
4	38.0	1.00	15.0	0.0100	0.0754	0.0152
5	12.3	0.59	10.4	0.0039	0.0385	0.0101
6	27.4	0.57	5.4	0.0072	0.0721	0.0136
7	22.7	0.15	14.2	0.0146	0.0695	0.0130
8	8.0	0.82	7.1	0.0055	0.0323	0.0149
9	1.6	0.52	9.9	0.0074	0.0794	0.0107
10	13.4	0.74	8.5	0.0088	0.0903	0.0127
11	28.2	0.66	3.0	0.0103	0.0800	0.0132
12	33.6	0.25	2.2	0.0048	0.0622	0.0150
13	3.3	0.44	3.0	0.0049	0.0581	0.0145
14	15.1	0.18	14.9	0.0039	0.0322	0.0142
15	9.1	0.63	9.1	0.0045	0.0484	0.0114
16	40.7	0.24	2.8	0.0046	0.0589	0.0111
17	39.0	0.91	9.2	0.0115	0.0605	0.0148
18	14.7	0.08	1.1	0.0046	0.0296	0.0151
19	43.2	0.87	10.9	0.0081	0.0788	0.0144
20	22.7	0.23	4.4	0.0080	0.0260	0.0139

No.	Chlorophyll				CDOM		Mineral particle
No.	C_o	h	s	z _max	$a_{g, 440}^{Surface}$	$a_{g, 440}^{Bottom}$	C_o	h	s	z _max
1	0.70	35.84	24.68	29.86	0.05	0.26	0.21	5.14	9.12	0.60
2	0.24	0.19	0.27	15.12	0.16	0.17	0.18	6.07	4.99	26.52
3	1.04	12.08	26.28	29.07	0.29	0.28	0.16	1.42	13.86	9.41
4	1.22	15.70	7.99	11.89	0.25	0.01	0.11	0.93	20.32	2.25
5	1.68	29.06	14.55	8.22	0.22	0.14	0.14	9.49	22.33	4.33
6	0.30	9.00	12.75	32.11	0.16	0.30	0.23	3.46	5.07	26.29
7	0.28	36.20	20.79	12.12	0.13	0.02	0.29	6.83	4.60	35.09
8	0.31	20.16	21.96	16.22	0.08	0.17	0.20	7.56	21.66	19.01
9	0.85	4.19	28.48	36.85	0.16	0.10	0.14	3.75	25.41	12.68
10	0.39	30.45	25.18	15.91	0.15	0.27	0.01	9.95	17.18	2.02
11	1.68	4.93	3.30	29.72	0.09	0.28	0.09	3.36	4.21	29.32
12	0.00	2.44	24.19	34.10	0.06	0.03	0.17	0.14	3.41	18.18
13	1.39	11.61	13.10	9.30	0.17	0.16	0.19	1.60	15.12	38.52
14	0.91	39.92	2.93	25.01	0.03	0.13	0.28	0.48	26.84	11.60
15	0.90	18.65	17.93	25.39	0.26	0.25	0.19	7.21	16.97	15.01
16	1.21	27.94	17.54	14.05	0.15	0.02	0.22	6.12	18.61	27.64
17	1.46	1.73	20.03	39.06	0.09	0.17	0.09	1.74	3.26	34.76
18	1.28	32.80	16.35	17.93	0.12	0.09	0.14	5.01	4.58	12.92
19	0.60	37.76	3.82	2.63	0.24	0.16	0.18	9.56	2.17	35.02
20	1.84	22.05	19.88	4.58	0.15	0.11	0.15	7.93	15.28	15.29

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