Optical closure for remote-sensing reflectance based on accurate radiative transfer approximations: the case of the Changjiang (Yangtze) River Estuary and its adjacent coastal area,

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Optical closure for remote-sensingreflectance based on accurate radiativetransfer approximations: the case ofthe Changjiang (Yangtze) River Estuaryand its adjacent coastal area, ChinaLeonid G. Sokoletskya & Fang Shena

a State Key Laboratory of Estuarine and Coastal Research, EastChina Normal University, Shanghai, 200062, ChinaPublished online: 05 Jun 2014.

To cite this article: Leonid G. Sokoletsky & Fang Shen (2014) Optical closure for remote-sensingreflectance based on accurate radiative transfer approximations: the case of the Changjiang(Yangtze) River Estuary and its adjacent coastal area, China, International Journal of RemoteSensing, 35:11-12, 4193-4224, DOI: 10.1080/01431161.2014.916048

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Optical closure for remote-sensing reflectance based on accurateradiative transfer approximations: the case of the Changjiang(Yangtze) River Estuary and its adjacent coastal area, China

Leonid G. Sokoletsky* and Fang Shen

State Key Laboratory of Estuarine and Coastal Research, East China Normal University,Shanghai, 200062, China

(Received 13 July 2013; accepted 25 October 2013)

Optical closure exercises are pivotal for evaluating the accuracy of water qualityremote-sensing techniques. The agreement between radiometrically derived and inher-ent optical property (IOP)-derived above-water spectral remote-sensing reflectanceRrs(λ) is necessary for resolving IOPs, the diffuse attenuation coefficient, and biogeo-chemical parameters from space. We combined spectral radiometric and IOP measure-ments to perform an optical closure exercise for two optically contrasting Chinesewaters – the Changjiang (Yangtze) River Estuary and its adjacent coastal area in theEast China Sea. The final aim of our investigation was to compare two derivations ofRrs(λ): Rrs(λ), derived from radiometric measurements; and Rrs(λ), derived from simul-taneous IOP measurements. Five subsequent steps have been taken to achieve thisgoal, including (1) estimation of the Rrs(λ) from radiometric measurements; (2) scatter-ing correction for the non-water spectral absorption coefficient apd(λ); (3) estimation ofthe below-water spectral remote-sensing reflectance rrs(λ) from IOPs measurements;(4) the estimation of the Rrs(λ) from the rrs(λ) values; and (5) the comparison betweenthe Rrs(λ) derived from radiometric and IOP measurements. All steps were realized byusing both direct measurements and different models based on radiative transfer theory.Results demonstrated that the impact of the errors caused by the scattering correctionprocedure and conversion of radiometric quantities into Rrs(λ) may be rather signifi-cant, especially in the long-wavelength spectrum range. Nevertheless, spectral featureswere similar between these Rrs(λ) sets for all waters – from relatively clear to veryturbid. Exploiting this fact allows use of the spectral reflectance ratios for remotesensing of the estuarine and coastal Chinese waters.

1. Introduction

An ‘optical closure’ between the radiometrically and inherent optical property (IOP)-derived upwelling radiance or reflectance is an important step towards the development ofgeographically localized remote-sensing algorithms (Zaneveld 1989; Barnard, Zaneveld,and Pegau 1999; Sokoletsky et al. 2003; Tzortziou et al. 2006; Gallegos, Davies-Colley,and Gall 2008; Shen, Verhoef, et al. 2010). However, this closure is difficult to realize inpractice due to several factors, including (1) difficulty in accounting an impact of sky andsolar glitter in calculating the above-water spectral remote-sensing reflectance Rrs(λ) = Lw(λ)/Ed(λ, 0+), where λ is the wavelength and Lw(λ) and Ed(λ, 0+) represent the water-leaving radiance, and the total downwelling (incoming) irradiance just above the surfacelevel, respectively (Mobley 1999); (2) difficulty in taking into account a scattering impacton the measured IOPs (McKee, Piskozub, and Brown 2008); (3) problems with

*Corresponding author. Email: [email protected]

International Journal of Remote Sensing, 2014Vol. 35, Nos. 11–12, 4193–4224, http://dx.doi.org/10.1080/01431161.2014.916048

© 2014 Taylor & Francis

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developing an adequate model for the underwater spectral remote-sensing reflectance rrs(λ) = Lu(λ, 0−)/Ed(λ, 0−), where Lu(λ, 0−) and Ed(λ, 0−) represent the upwelling radianceand the total downwelling irradiance just below the surface level, respectively (Sokoletskyet al. 2012); and (4) problems with the Rrs(λ) vs. rrs(λ) modelling (Morel and Gentili1996).

Thus, the current study used the following algorithm for solving the optical closuretask (Figure 1):

The extremely variable waters of the Changjiang (Yangtze) River Estuary (YRE) andadjacent coastal area (ACE) of the East China Sea, China, were used as an example of thedevelopment and validation of several Rrs(λ) algorithms. In parallel, with the aim ofclassifying Chinese waters, an algorithm for the spectral diffuse attenuation coefficientKd(λ) was developed and compared with the other known algorithms.

2. Data acquisition and preprocessing

2.1. Study area

A detailed description of the area under investigation is given in Shen, Verhoef, et al.(2010) and Shen, Salama, et al. (2010). The research stations are shown in Figure 2,where different symbols correspond to three types of waters: clear, intermediate, andturbid. This classification is based on using the diffuse attenuation coefficient at the solarzenith angle θ0 = 0°, wavelength λ = 550 nm, and the depth z just below the air–watersurface (see Section 3.1.1 for definitions).

Estimation of the ρsky(λ) and Rrs(λ)from the radiometric measurements

In situ measurements ofradiometrical quantities: Ed(λ, 0+),

Ls(λ), Ltot(λ, 0+)

In situ measurements of inherentoptical properties (IOPs) apd(λ, z),

cpd(λ, z), bbp(λ, z)

Scattering correction for the apd(λ, z)

Estimation of Kd(λ, 0–) and rrs(λ)from the IOPs

Kd(λ, 0–) based classification of YREand ACE waters the IOPs

Estimation of Rrs(λ) from the rrs(λ)

Validation of the radiative transfermodel by comparison between the

radiometrically and IOPs-based Rrs(λ)

Evaluation of the optimalwavelengths for remote sensing

Figure 1. Flow chart illustrating the steps of the optical closure algorithm developed.

4194 L.G. Sokoletsky and F. Shen

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2.2. In situ data sets

In situ optical and suspended sediment concentration (SSC) profile data were acquired atintervals of a few centimetres during the cruise campaign of 4–12 May 2011. Data werecollected mostly during daylight hours, but some recordings (at stations ‘c4’ and ‘c5’)were done during night hours (see Figure 2 for station locations). The weather was calm;however, on 4–5 May and 11 May, and in the morning hours of 12 May, the observationswere identified as cloudy, according to the criterion of Ruddick et al. (2006) (see Section3.3). The water samples were collected during the cruise simultaneously along with theradiometric and IOPs measurements, and then were used for SSC determination in thelaboratory. The concentrations were determined (Shen, Verhoef, et al. 2010) gravimetri-cally by filtering the water samples on 0.7 μm Whatman® GF/F glass fibre filters. Allfilters were rinsed with Milli-Q water to remove salts and then dried and reweighed on ahigh-precision balance in the laboratory.

The following radiometric IOPs were measured during the campaign: the downwellingspectral irradiance Ed(λ, 0+), the incoming sea-surface spectral radiance Ls(λ), and thetotal upwelling radiance Ltot(λ). All three quantities were measured in the spectral rangebetween 349 and 850 nm (with a 1 nm resolution) near to the air–water surface by theHyperspectral Surface Acquisition System (HyperSAS) of Satlantic Inc.® The radiancesensor was pointed to the sea and sky at the same nadir and zenith angles, approximatelybetween 30° and 50°, with an optimum angle of 40° (Shen, Verhoef, et al. 2010). This wasdone to maximally avoid the wind speed impact (Mobley 1999). An azimuth angle of theradiance sensor was selected between 90° and 180°, with an optimum angle of 135° awayfrom the sun (Shen, Verhoef, et al. 2010). This angle was chosen due to the necessity tominimize solar glitter effects (Mobley 1999). Uncertainties of all radiometric

121.5° E 122.5° E 123.5° E 124° E

32° N

31.5° N

30.5° N

31° N

32° N

31.5° N

30.5° N

31° N

123° E122° E

121.5° E20 mi

50 km

122.5° E 123.5° E 124° E123° E122° E

Figure 2. Location of research stations in YRE and ACE. The blue, green, and brown symbolscorrespond to ‘clear’, ‘intermediate’, and ‘turbid’ waters as they are defined in Section 3.1.1.

International Journal of Remote Sensing 4195

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measurements may be accepted close to 5–10% (Mueller et al. 2003; Zibordi 2012). Noradiometric measurements for ‘c4’, ‘c5’, ‘e3’, and ‘f3’ stations were performed.

The IOPs were represented by the non-water (i.e. particulate and dissolved matter) spectralabsorption and attenuation coefficients apd(λ, z) and cpd(λ, z), respectively (z is the depth inmetres), and by the particulate angular spectral scattering coefficient βp(λ, z). The first twoproperties were measured in situ by the WETLabs® absorption and attenuation meter ac-s atnine wavelengths (412, 440, 488, 510, 532, 595, 650, 676, and 715 nm) during the downcastsand upcasts as water flowed through the ac-s meter. The value of βp(λ, z, θ) has been measuredsimultaneously by theWETLabs® ECO-VSF (at three wavelengths, λ − 470, 532, and 650 nm− and at three scattering angles, θ, of 100°, 125°, and 150° and by the ECO-BB9 (at values of λof 412, 440, 488, 510, 532, 595, 650, 676, and 715 nm, and at a scattering angle of 117°)backscattering sensors. A conversion from the βp(λ, z, θ) to the particulate backscattering bbp(λ,z) has been realized, following the recommendations of Moore, Twardowski, and Zaneveld(2000) and Boss et al. (2004). More definitively, βp(λ, z, θ) sin θ was fitted by a third-orderpolynomial at all θ in a backward direction (i.e. θ varied from 90° to 180°):

βpðλ;z;θÞsinθ ¼ a0ðλ;zÞ þ a1ðλ;zÞθ þ a2ðλ;zÞθ2 þ a3ðλ;zÞθ3: (1)

A solution of this equation at four angles (100°, 125°, 150°, and 180°) yields thecoefficients a0 … a3. Further, it is easy to find bbp(λ, z). By definition (Jerlov 1976, p.6; Boss et al. 2004):

bbp ¼ 2πZπ

π=2

βpðθÞsinθdθ: (2)

A substitution of Equation (1) into Equation (2) yields an analytical expression for bbp(λ, z):

bbpðλ; zÞ ¼ 2π2 ð1=2Þa0ðλ; zÞþð3=8Þπa1ðλ; zÞþð7=24Þπ2a2ðλ; zÞþð15=64Þπ3a3ðλ; zÞ� �

:

(3)

Differences between bbp(λ, z) determined by the ECO-VSF and ECO-BB9 were generallyinsignificant (see Figure 3 for near-surface depths). For example, at λ = 532 and 650 nm,the mean absolute percentage error (MAPE) for all depths was estimated as 14% and 9%,respectively; even smaller values of MAPE were found for the depths z < 1 m: 6.0% and4.8%, respectively. However, measurements performed by the ECO-BB9 at λ < 488 nmgave too low values in general (Figure 3). Therefore, a spectral power function fitting hasbeen conducted for the ECO-BB9 measurements based on the bbp(λ, z) values measured atλ ≥ 488 nm.

For the modelling of optical properties ‘just below the surface’ (i.e. at z = 0−), we usedarithmetically averaged data collected from the depths above 1 m. Although this depth waschosen arbitrarily, such a selection allowed us to compare our results with the classical water-type classification (Jerlov 1976). We used a relative standard error, RSE (mathematically it isexpressed by the standard deviation for the given statistical population being sampled,normalized by the mean value and the square root of the number of measurements) as ameasure of variability of the total absorption, a(λ, z) (Figure 4a) and total attenuation, c(λ, z)(Figure 4b) in the range of 0–1 m. These RSE values show that the variability of these IOPs isrelatively small (generally within 5%), especially in clear and intermediate waters.


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2.3. Scattering correction issue

The next step was converting the temperature and the salinity-corrected apd (we desig-nated it as að0Þpd ; i.e. a ‘null approximation’ to the finally obtained coefficient apd), cpd andbð0Þp ¼ cpd � að0Þpd into the scattering-corrected values of non-water absorption, apd andnon-water (particulate) scattering, bp. This step is necessary for optical instruments suchas ac-s because the absorption sensor of these instruments fails to collect all scatteredlight; thus, real absorption has to be less than measured (McKee, Piskozub, and Brown2008). However, different methods of the scattering correction deviate by up to 20% intheir estimate of absorption in the blue portion of the spectrum, while converging towardsthe red portion (Zaneveld et al. 1992, 1994; Boss et al. 2007). Two different scatteringcorrection algorithms were tested in the current study: (1) a standard ‘proportionalZaneveld’s method (PZM) (Zaneveld, Kitchen, and Moore 1994); and (2) a ‘modifiedBoss’s method (MBM)’ (Boss et al. 2013).

Figure 4. The relative standard error for the total absorption coefficient a(λ, z) (a) and the totalattenuation coefficient c(λ, z) (b) measured at z < 1 m. The symbols correspond to the different watertypes shown in Figure 2. All triangles were connected by the solid curves for the better visualizationof results.

Figure 3. Particulate backscattering coefficient spectra bbp(λ, 0−) measured by ECO-BB9 (curves)and ECO-VSF (symbols) instruments. The ECO-BB9 values corresponding to clear (C), intermedi-ate (I), and turbid (T) waters are shown by the dashed, dotted, and solid curves, respectively. Thecorresponding ECO-VSF values are shown by the squares, circles, and triangles.


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APZMmay bewritten as follows (upper indices indicate the subsequent algorithmic steps):

apdðλ; zÞ ¼ að1Þpd ðλ; zÞ � ɛðzÞbð1Þpd ðλ; zÞ; (4)

where

að1Þpd ðλ; zÞ ¼ að0Þpd ðλ; zÞ; if að0Þpd ðλ; zÞ � 00; otherwise

;

�(5)

bð1ÞP ðλ; zÞ ¼ cpdðλ; zÞ � að1Þpd ðλ; zÞ; (6)

ɛðzÞ ¼ að1Þpd ð715; zÞbð1Þp ð715; zÞ

: (7)

A new method (MBM) is presented as the following algorithmic sequence:

að1Þpd ðλ; zÞ ¼ að1Þpd ð412; zÞ exp SpdðzÞ 412� λð Þ� �; (8)

where non-water absorption coefficient að1Þpd ð412; zÞ and exponential slope SpdðzÞ areestimated from conditions:

Xλ

að1Þpd ðλ; zÞ � að0Þpd ðλ; zÞh i2

¼ min; λ ¼ ½412 nm; 715 nm� (9)

by the least-squares method, and

að2Þpd ðλ; zÞ ¼að0Þpd ðλ; zÞ � ɛðzÞcpdðλ; zÞ

1� ɛðzÞ ; where ɛðzÞ ¼ að0Þpd ð412; zÞ � að1Þpd ð412; zÞcpdð412; zÞ � að1Þpd ð412; zÞ

;

(10)

apdðλ; zÞ ¼ að2Þpd ðλ; zÞ; if að2Þpd ðλ; zÞ � 00; otherwise

:

�(11)

Four important features of the MBM method should be noted here: (1) apdð715; zÞ is notnecessarily equal to 0 (as it is in the PZM method), but cannot be negative; (2) apdðλ; zÞ isnot necessarily a decreasing exponential function, though it is close to it; (3) the quantityɛðzÞ is expressed explicitly (that differentiates it from the ɛðzÞ in the Boss et al. 2013method); and (4) the ɛðzÞ is a wavelength-independent quantity. The physical sense of ɛðzÞis the contribution of scattering into measured non-water absorption at the depth z; i.e. thefollowing equation holds at any wavelength:

að0Þpd ðλ; zÞ ¼ að2Þpd ðλ; zÞ þ ɛðzÞbð2Þp ðλ; zÞ; where bð2Þp ðλ; zÞ ¼ cpdðλ; zÞ � að2Þpd ðλ; zÞ:(12)


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Numerical values of ɛð0�Þ varied from 0 to 0.025, much less than was derived in previousstudies (Kirk 1991; Boss et al. 2013). However, this is similar to the ɛð0�Þ values derivedfrom applying the PZM method.

The results of calculating apdðλ; 0�Þ by both of the methods for the samples collectedin the area (shown in Figure 5) provide a visual impression of a good correlation betweenthe initially measured að0Þpd and scattering corrected values (see Figure 5a, b). However,differences between the non-water absorption (and, hence, total absorption) derived by thetwo methods may be rather large. Figure 5c yields a relative difference between the totalabsorption coefficient a(λ, 0−) (which includes a pure water absorption coefficient aw(λ)in addition to apd(λ, 0−)) obtained by the PZM method and the coefficient obtained by theMBM method. Figure 5c also demonstrates that the differences between the scatteringcorrection methods strongly depend on water turbidity. The best convergence (with thedifference < 4%) was found for the clear and intermediate samples, i.e. those at which thediffuse attenuation coefficient Kd(550 nm, z) was less than 5 m−1 (see Section 3.1.1).However, for the more turbid samples, the relative differences achieved 70% atλ = 650 nm.

The values of aw(λ) were interpolated from Buiteveld et al. (1994), Pope and Fry(1997), and Fry, Lu, and Qu (2006), while the pure water scattering coefficients bw(λ)were derived from Zhang, Hu, and He (2009) at measured temperatures and salinities(note that the bw(λ) values almost coincide with the values derived from Morel’s (1974)observations and model at salinities of less than 30 practical salinity units (psu)). For thespectral attenuation cw(λ) and backscattering bbw(λ) of pure water, the well-known

Figure 5. Scattering correction for surface spectral non-water absorption coefficient. Correctedvalues vs. measured shown for the PZM (a) and MBM (b) scattering correction methods. (c)illustrates the relative difference for the total spectral absorption coefficient derived by the PZMmethod compared with the MBM method. The symbols correspond to the different water typesshown in Figure 2.


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relationships (Smith and Baker 1981; Morel and Maritorena 2001) were applied, namely,cw(λ) = aw(λ) + bw(λ) and bbw(λ) = 0.5bw(λ).

We did not perform a scattering correction for attenuation, assuming that this wouldmerely improve attenuation estimates by an insignificant degree (Zaneveld et al. 1992;Wet Labs ac Meter Protocol Document 2011), especially for turbid case 2 waters.

The exponential slope at the near-surface level Spdð0�Þwas estimated to be in therange from 0.0080 to 0.0116 nm−1, which is in a full correspondence with the slope valuesderived for the non-algal particulate absorption in the same geographical area duringMarch, July, and August 2009 (Shen, Zhou, and Hong 2012).

3. Apparent optical properties (AOPs) modelling

3.1. Estimation of Kd(λ, 0−) and rrs(λ) from the IOPs

3.1.1. The review of existing models for Kd(λ)

We consider a computation of the spectral diffuse attenuation coefficient Kd(λ) andunderwater remote-sensing reflectance rrs(λ) in the same subsection to stress that thesetwo quantities are related to one another. Although reliable models for reflectance aremore important for remote-sensing retrieval of IOPs and water components, Kd(λ) modelsget a better impression about water quality and allow the classification of natural watersby their turbidity (Jerlov 1976; Prieur and Sathyendranath 1981; Baker and Smith 1982;Morel and Berthon 1989; Kaczmarek and Woźniak 1995; Pelevin and Rutkovskaya1997).

For the current study, Kd at the zenith angle of the sun (θ0 = 0°), the middle of thevisible spectrum (λ = 550 nm), and just below the surface (z = 0−) was accepted as anindex for the water turbidity. We accepted (although rather arbitrarily) that the waterspossessing (Kd(0°, 550 nm, 0−) ≤ 1 m−1, 1 m−1 < Kd(0°, 550 nm, 0−) ≤ 5 m−1, and Kd

(0°, 550 nm, 0−) > 5 m−1) were considered ‘clear’ (C), ‘intermediate’ (I), and ‘turbid’(T) waters, respectively. Note that such classification sharply contrasts withthe classical Jerlov’s classification system (Jerlov 1976, 135, 135). According tothat classification, the clearest coastal type 1 waters correspond to Kd(0°, 550 nm,0−) ≈ 0.12 m−1 and the most turbid coastal type 9 correspond to Kd(0°, 550 nm,0−) ≈ 0.63 m−1. However, such a range of values is much narrower than the realmeasurements and models yielded in the YRE area. Thus, all water types (bothoceanic and coastal) presented in Jerlov’s classification system should be referred toas ‘clear’-type waters according to our classification. A similar problem arises at useof the other known classifications based on the spectral Kd(λ) – all of them dealingwith the too small values of Kd(λ).

Below (Table 1), we present several different computational models for the average(from just below the surface to depth z; i.e. 0− → z) diffuse attenuation coefficient of thespectral solar downward irradiance Kdðθ1; λ; 0� ! zÞ omitting designations of angle,wavelength, and depth:

Here the following designations were used: Tðθ1; λ; 0� ! zÞ is the transmittance oflight at wavelength λ and the solar zenith angle just after refraction, θ1 = arcos μ1; a, b,and bb are the absorption, scattering, and backscattering coefficients, respectively; g is theasymmetry parameter; ω0 is the single-scattering albedo and ω0 = b/(a + b) = b/c; τef is theeffective optical depth, defined as τef = cz/μ1.

To calculate the asymmetry parameter, g, we first calculated the particulate asymmetryfactor gp from the measured particulate backscattering ratio, Bp = bbp/bp, based on an


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Table

1.Com

putatio

nalmod

elsforthespectral

diffuseattenu

ationcoefficient.

Mod

elname

Abb

reviation

Mod

els

References

Bou

guer–L

ambert–

Beerlaw

for

absorptio

n

BLB-a

Kd¼

�lnT z¼

a μ 1Stavn

(198

1,19

88)

Quasi-single-

scattering

approx

imation

QSSA

Kd¼

aþb b

μ 1Gordo

nandBrown(197

4);

Gordo

n,Brown,

andJacobs

(197

5)

Kirk’smod

elKM

Kd¼

azffiffiffiffiffiffiffiffi

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi



1þðg

1μ 1

�g 2Þω

0=ð1

�ω0Þ

pμ 1

;

g 1¼

2:63

6=g�2:44

7;g 2

¼0:84

9=g�0:73

9

Kirk(199

1)

Ben–D

avid’smod

elBDM

Kd¼

lncoshðyτ

efÞþ

ðx=yÞsinhðyτ

efÞ

½�

z;

x¼

1�0:5ð1þgÞω0;y

¼ffiffiffiffiffiffiffiffi


ffiffiffiffiffiffiffiffiffiffiffiffiffi

ð1�ω

0Þð1

�gω

0Þ

pBen–D

avid

(199

5,19

97)

Lee’smod

elLM

Kd¼

ð1þ0:00

5θ0Þa

þ4:18

1�0:52e�

10:8a

� b b

;

θ 0in

degrees

Lee,Du,

andArnon

e(200

5)


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equation derived by us for the scattering phase function, called the Fournier–Forand–Mobley (FFM), which is very popular in marine optics (Mobley, Sundman, and Boss2002; Sokoletsky, Nikolaeva, et al. 2009):

gp ¼ 1� 3:587Bp þ 9:475B2p � 22:40B3

p þ 19:88B4p: (13)

The error MAPE of this approximation was estimated at 0.8%, and the coefficient ofdetermination, R2, was 0.9998 at 31 selected points (these points corresponded to thevalues for the real part of the relative refractive index of 1.01, 1.02,…, 1.31). The furtherconversion from gp values to the total asymmetry factor g was realized by the equation(see Morel and Loisel 1998; Videen et al. 1998)

g ¼ bpbgp: (14)

A geometrical thickness of the layer z was chosen for each of the different models basedon our desire to calculate Kd close to the surface. Calculations showed that at z < 10−4, allcalculations yielded the same results within 0.2%. Somewhat arbitrarily, we selected thevalue z = 10−6. Note, however, that all IOP values needed for our calculations wereobtained by averaging within the 0–1 m layer.

Note that the QSSA model was obtained initially by analytical reasoning,while Kirk’s model used a wide set of experimental data; the BDM model isactually a modified van de Hulst (1980, Chapter 14) two-stream radiativemodel; and Lee’s model was derived from the numerical radiative transfer computa-tions by the Hydrolight (Mobley and Sundman 2001) code. Note that bb is given alarger weighting than a in Lee’s model. This may be explained by the multiplebackscattering processes that are not taken into account in BLB-a and QSSAapproximations.

3.1.2. Modified quasi-single-scattering approximation (MQSSA) for Kd(λ)

Below, we give a concise derivation of a newly MQSSA for the diffuse attenuationcoefficient. Let us assume that a QSSA is a model that truly describes a transmittance(we designated it as T1) after the first light traverses a given layer of thickness z asfollows:

T1 ¼ exp �ð1� ω0FÞτeff g; (15)

where F is a forward-scattering ratio (F = 1–B = 1–bb/b), and τef = cz/μ1 or τef = cz/μef (μefis the effective cosine of scattering in the layer), which depends on the incomingirradiance – collimated or diffuse – used in modelling (Sathyendranath and Platt 1988;Sokoletsky et al. 2014). For classification purposes, we will assume here that μ1 = μef = 1.Then τef = cz is independent of a type of incoming irradiance.

Taking into account the reflectances from the top of the layer, Rt(z), and the bottom,Rb(z), a downwelling irradiance at the bottom Ed(z) after the first layer passage can bewritten as

EdðzÞ ¼ Edð0�Þ 1� RtðzÞ½ � 1� RbðzÞ½ � T1ðzÞ: (16)


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To differentiate between the diffuse (sky) and collimated (solar) transmittances, wewill use, instead of T1, designations t1 (‘spherical transmittance’) and Tp1 (‘planetransmittance’) for the diffuse and collimated cases, respectively. Similarly, we willdesignate the irradiance reflectance before the first light traverses the layer of thick-ness z as r1 in the case of the incoming diffuse irradiance (called the ‘sphericalalbedo’; see van de Hulst 1980; Hapke 1993; Kokhanovsky and Sokoletsky 2006a)and Rp1 in the case of the incoming collimated irradiance (called the ‘plane albedo’;see van de Hulst 1980; Hapke 1993; Kokhanovsky and Sokoletsky 2006a, 2006b). Itappears obvious that beginning from the first reflectance from the bottom of the layerthat extends to a depth z, all subsequent transmittances and reflectances are just thesame as the first.

Therefore, for the incoming diffuse light, all light traversing cycles lead to the totaldownwelling irradiance at the bottom of the layer:

EdðzÞ ¼ Edð0�Þ 1� r1ðzÞ½ � 2t1ðzÞ 1þ r21ðzÞt21ðzÞ þ r41ðzÞt41ðzÞ þ :::� �

¼ Edð0�Þ 1� r1ðzÞ½ � 2t1ðzÞ1� r21ðzÞt21ðzÞ

;(17)

and the total spherical transmittance:

tðzÞ ¼ 1� r1ðzÞ½ � 2t1ðzÞ1� r21ðzÞt21ðzÞ

: (18)

The same equation as the last one has been derived, for instance, by Stokes (2009),Tuckerman (1947), and by Bohren and Huffman (2004). Similarly, for the case ofincoming collimated irradiance, we can write

EdðzÞ ¼ Edð0�Þ 1� Rp1ðzÞ� �

1� r1ðzÞf gTp1ðzÞ 1þ r21ðzÞt21ðzÞ þ r41ðzÞt41ðzÞ þ…� �

¼ Edð0�Þ 1� Rp1ðzÞ� �

1� r1ðzÞ½ � Tp1ðzÞ1� r21ðzÞt21ðzÞ

;

(19)

and the total plane transmittance

TpðzÞ ¼1� Rp1ðzÞ

� �1� r1ðzÞ½ � Tp1ðzÞ

1� r21ðzÞt21ðzÞ: (20)

For combined (direct and diffuse) illumination, it is easy to obtain the transmittance,Tc(z):

TcðzÞ ¼ EdðzÞEdð0�Þ ¼

Ed;dirð0�ÞTpðzÞ þ Ed;dif ð0�ÞtðzÞEdð0�Þ ¼ 1� dEð ÞTpðzÞ þ dEtðzÞ; (21)

where dE is a contribution of the incoming diffuse irradiance Ed, dif to the total incomingirradiance, and Ed,dir is the incoming direct (collimated) irradiance. We used the expres-sion by Højerslev (2001) for calculating dE:


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dEðλ; θ0Þ ¼ 0:3735 þ 4:15� 10�3ðθ0 � 40Þ þ 1:095� 10�4ðθ0 � 40Þ2

þ 1:1� 10�6ðθ0 � 40Þ3 þ ½�8:0� 10�4ðλ� 500Þþ 2:0� 10�6ðλ� 500Þ2� � ½1þ 3:0� 10�3ðθ0 � 40Þ�;

(22)

where θ0 and λ are expressed in degrees and nanometres, respectively. However, dE(λ) wasassigned the value 1 (i.e. downwelling irradiance was considered as completely diffuse)for the observations from two stations (‛c4’ and ‘c5’), which were conducted during nighthours.

To estimate r1(z) and Rp1(z), Stokes invariants (St_dif and St_dir), which are inde-pendent of the layer depth, were used (Stokes 2009; Tuckerman 1947; Shifrin 2001):

St dif ¼ 1þ r2ðzÞ � t2ðzÞrðzÞ ; St dir ¼ 1þ R2

pðzÞ � T2p ðzÞ

RpðzÞ ; (23)

from which follows that

St dif ¼ 1þ r2ð1Þrð1Þ ; St dir ¼ 1þ R2

pð1ÞRpð1Þ ; (24)

and

r1ðzÞ ¼ 0:5 ðSt dif Þ �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðSt difÞ 2 � 4 1� t21ðzÞ

� �q� ; (25)

Rp1ðzÞ ¼ 0:5 ðSt dirÞ �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðSt dirÞ2 � 4 1� T2

p1ðzÞh ir�

: (26)

Assuming again μ1 = μef = 1, we obtain from Equation (15)

t1(z) = Tp1(z) = a(z) + bb(z). (27)

The following accurate approximations for rð1Þ and Rpð1Þ were used (van de Hulst1974; Kokhanovsky and Sokoletsky 2006b; Sokoletsky, Kokhanovsky, and Shen 2013):

rð1Þ ¼ ð1� 0:139sÞð1� sÞ1þ 1:170s

; (28)

where s is Hulst’s similarity parameter, defined as

s ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffi1�ω0

1�gω0

s; (29)


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Rpð1Þ ¼Φðζ Þ 1� s

1þ 2μ1s; (30)

where ζ¼μ1 � 0:5

andΦðζ Þ ¼ exp A1ζ þ A2ζ

2� �

sþ A3ζ þ A4ζ2

� �s2

� �; (31)

where Ai ¼P3j¼1

αijgj�1 and each αij corresponds to the ijth element of the following matrix:

α̂ ¼�0:991 3:139 �1:8741:435 �4:294 2:0890:719 �5:801 2:117�0:509 0:418 3:360

0BB@

1CCA: (32)

Note that this approximation has been created on the base of a numerical solution of theradiative transfer equation by the Mishchenko et al.’s (1999) invariant imbedding method(IIM); the validation of this approximation may be found in Kokhanovsky and Sokoletsky(2006a, 2006b), Sokoletsky, Nikolaeva et al. (2009), and Sokoletsky, Kokhanovsky, andShen (2013). An asymmetry factor g has been estimated by Equations (13) and (14).

3.1.3. Comparison results for Kd(λ)

Figure 6 shows spectral Kd computed by the MQSSA method for μ1 = μef = 1 just belowthe surface, while Figure 7 yields comparisons of the other above-mentioned methodswith MQSSA. It seems that the BLB-a, QSSA, and BDM systematically underestimateKd values when compared with MQSSA with the largest differences occurring forBLB-a (giving a MAPE of 49%) and the least differences in the case of the BDM(23%). In contrast, the Lee model overestimates Kd at values of Kd(0°) > 1 m−1;however, differences between this model and MQSSA are small (MAPE = 16%).Overall, the KM model yields the maximal closeness to the MQSSA model, with anMAPE of 13%. These results may be considered excellent, taking into account the widerange of Kd(λ) values, which span three orders of magnitude. Moreover, all consideredmodels demonstrate an excellent correlation with the MQSSA method with the values ofR2 being 0.948, 0.986, 0.949, 0.997, and 0.999 for the BLB-a, QSSA, KM, BDM, andLM models, respectively.

An important feature of Kd(λ), as obtained for the YRE, is not only that there weresome extremely large values − reaching 100 m−1 − in the blue region of the spectrum,but also its unusual spectral form. For relatively clear and intermediate waters, with Kd

(0°, 550) < 5 m−1, this form corresponds with a U-type or reversed J-type, asdescribed by Prieur and Sathyendranath (1981). However, for extremely turbid waters− Kd(0°, 550) > 5 m−1 − the spectra become monotonically decreasing within thevisual and near infrared (NIR) regions of the spectrum. This occurred due to theenormous impact of absorption and backscattering caused by sediments, much largerthan the impact caused by pure water and phytoplankton, which have non-monotonicspectral features.


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Figure 7. A comparison of Kd(0°, λ, 0−) computed by the MQSSA method, Kd, MQSSA, with thesame quantity computed by BLB-a (a), QSSA (b), KM (c), BDM (d), and LM (e) models.

Figure 6. The spectral diffuse attenuation coefficient Kd(0°, λ, 0−) computed at μ1 = μef = 1 for justbelow the surface level by the MQSSA method from the station’s IOP values. The symbolscorrespond to the different water types shown in Figure 2.


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3.1.4. The review of existing models for the IOPs-derived rrs(λ)

Table 2 contains a set of several different computational models for the IOP-derivedspectral remote-sensing reflectance:

Actually, all the models from Table 2 have been derived from the numerical radiativetransfer computations using the following designations: G = bb/(a + bb) is Gordon’sparameter; and μ2 is the cosine of the viewing nadir angle θ2 before refraction throughthe sea–air interface. Note that at a viewing angle equal to 40° (as was the case for our

observations) and following Snell’s law, μ2ðλÞ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� sin 40�=nwðλÞ½ �2

q� 0:876� 0:879

(where nw is the real part of the water refractive index) in the spectral range of 412−715 nm.

3.1.5. Comparison results for rrs(λ)

The rrs(λ) spectra computed by the PAA method from the station’s measured IOP valuesare shown in Figure 8. The spectral behaviour of rrs(λ) is highly predictable and incomplete accordance with previous studies of clear oceanic and highly sediment-loadedturbid waters (Bukata et al. 1995; Forget et al. 1999; Lahet, Forget, and Ouillon 2001;Doxaran, Froidefond, and Castaing 2002; Karnieli and Figueras 2004; Shen, Verhoef,et al. 2010; Shen, Salama, et al. 2010). Even though quantitative spectral relationshipsvery much depend on the place (and sometimes, the season) under investigation, thequalitative content is similar to the literature data. For example, the main rrs(λ) peak in theblue spectral range and the negative slope between 532 and 650 nm are obvious forrelatively clear waters (with Kd(0°, 550 nm, 0−) ≤ 1 m−1). For more turbid waters (with1 m−1 < Kd(0°, 550 nm, 0−) ≤ 5 m−1), the blue peak becomes flatter and the 532–650 nmslope is smaller than for clear waters. The 532–650 nm slope becomes positive and themain peak shifts towards the red−NIR spectral range in the murkiest waters, which haveKd(0°, 550 nm, 0−) > 5 m−1. This effect is also in full correspondence with the shift of theKd(0°, λ, 0−) minimum with turbidity growth shown in Figure 6. Thus, we observe a veryimportant result for remote-sensing applications: the obvious spectral reflectance featuresfor different natural waters. However, these results should be taken with some cautionbecause our location had a relatively small impact from phytoplankton and dissolvedorganic matter. In the waters where their impact is large, the spectral features may bedifferent from ours.

Figure 9 presents comparisons between the values of rrs(λ) as derived by five differentalgorithms (described above). A comparison was made between each of the Gordon, Lee,Albert and Gege, and SAG models and the PAA model. This yielded MAPE values of15%, 16%, 17%, and 20%, respectively, and values for R2 of 0.979, 0.981, 0.982, and0.985, respectively. These results are very similar to those for the Kd(λ) and similarly maybe considered excellent, taking into account that the rrs(λ) values spanned more than twoand a half orders of magnitude.

3.2. Estimation of Rrs(λ) from the rrs(λ) values

3.2.1. The review of existing models for deriving Rrs(λ) from rrs(λ)

Surface optical effects play a significant role in the change in the upwelling spectralradiance Lu(θ2, λ, 0−) when it passes through the water–air surface. Hence, rrs(θ2, λ, 0−)also undergoes radical changes. Below, we consider several current solutions (Table 3) to


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Table

2.Com

putatio

nalmod

elsfortheIO

P-derived

spectral

remote-sensingreflectance.

Mod

elname

Abb

reviation

Mod

els

References

Gordo

n’smod

elGM

r rs¼

Gð0:094

9þ0:07

94GÞ

Gordo

net

al.(198

8)Lee’smod

elLM

r rs¼

Gð0:084

þ0:17GÞ

Lee

etal.(199

9)

AlbertandGegemod

elAGM

r rs¼

0:05

12Gð1

þ4:66

59G�7:83

87G

2þ5:45

71G

3Þ

1þ0:10

98=μ 1

ðÞ1þ0:40

21=μ 2

ðÞ

AlbertandGege(200

6)

Com

binedSok

oletskyand

Albert–Gegemod

elSAG

r rsðμ

2Þ¼

r rsðμ

2¼

1Þ�

1þ0:40

21=μ 2

1:40

21

r rsðμ

2¼

1Þ¼

0:28

74Gð1

þ0:28

21G

�1:01

9μ1þ0:45

61μ2 1Þ

Sok

oletskyet

al.(201

2)andAlbertandGege(200

6)

Plane

albedo

approx

imation

PAA

r rs¼

Rpð1

Þ=π,

Equ

ations

(29)–(32

)vande

Hulst(197

4),Kok

hano

vsky

andSok

oletsky(200

6b)

andSok

oletsky,

Bud

ak,Lun

etta

(200

9)


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this problem, while the Saunderson−Aas model will be considered separately. We willassume a plane-parallel layer, which may be considered a good approach at wind speeds,W, of less than or equal to 5 m s−1, and at angles of incidence, θ0, and viewing angles ofmore than 20° (Mobley 1999). These conditions were always fulfilled throughout ourobservations.

Figure 8. The spectral underwater remote-sensing coefficient rrs(λ) computed by the PAA methodfrom the station’s IOP values. The blue, green, and brown symbols correspond to ‘clear’, ‘inter-mediate’, and ‘turbid’ waters. All symbols are connected by solid curves for better visualizationof results.

Figure 9. A comparison of rrs(λ) computed by the PAA method, rrs,PAA(λ), with the rrs(λ) computedby the Gordon (a), Lee (b), Albert and Gege (c), and SAG (d) models.


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Here �ρ is the air–water Fresnel reflection at the interface for the whole (sun + sky)downwelling irradiance; ρ is the internal Fresnel reflectance of upwelling irradiance; �r isthe water–air Fresnel reflection for the whole diffuse upwelling irradiance; and Q(λ, 0−) isthe ratio of upwelling irradiance Eu(λ, 0−) to the upwelling radiance Lu(λ, 0+). The valuesof �ρ and �r were assumed to be 0.043 and 0.489, respectively (Morel and Gentili 1996),while the ρ was calculated by the Fresnel law and Q(λ, 0−) was estimated from radiativetransfer computations carried out by Mishchenko et al.’s (1999) IIM. Our computationshave shown that for the FFM scattering phase function with θ0 and ω0 varying within theranges of 0°–80° and 0.70–0.95, respectively, Q(λ, 0−) may be reasonably expressed bythe following cubic polynomial in θ0 (in degrees):

Qðθ0; λ; 0�Þ ¼ a0ðλÞ þ a1ðλÞθ0 þ a2ðλÞθ20 þ a3ðλÞθ30 ; (33)

where coefficients a0(λ)−a3(λ) are as listed in Table 4:Note that our model for Q(λ, 0−) yields values ranging from 3.0 to 3.8 at any values of

θ0 in the range 0°−90° and any values of λ in the range 412−715 nm. Similar values werederived in some other studies, both empirically (Siegel 1984; Aas and Højerslev 1999)and theoretically (Hirata et al. 2009).

The real part of the refractive index of pure water nw(λ) was estimated as a function ofλ, temperature, and salinity according to Quan and Fry’s (1995) approximation.

3.2.2. Saunderson (1942)–Aas (2010) model (SAM)

Saunderson was one of the first to propose a physical model of surface optical effects(Saunderson 1942); however, he did not promote a clear physical derivation of hisequation. Further elaboration of this equation for different observational conditions hasbeen given by Mudgett and Richards (1971), Judd and Wyszecki (1975), and García–Valenzuela, Cuppo, and Olivares (2011). For the air−water system, Saunderson’s equationfor the total reflectance at the surface, Rtotal, may be written in the form

Table 3. Computational models for the Rrs(λ).

Model name Abbreviation Models References

Morel and Gentilimodel

MGM Rrs ¼ 1� �ρð Þ 1� ρð Þ1� �rQð 0�Þrrs½ �n2w

rrsMorel and Gentili(1996)

Mobley model MM Rrs ¼ 0:54þ0:03�0:04

� rrs Mobley (1999)

Lee model LM Rrs ¼ 0:52rrs1� 1:17rrs

Lee, Carder, andArnone (2002)

Loisel model LoM Rrs ¼ αðθ0Þrrs1� βðθ0Þrrs , where α and β are

found from the table:

Loisel (2008)

θ0 (°) α β

0 0.5236 2.194130 0.5169 2.300160 0.4933 2.6796


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Rtotalðλ; 0þÞ ¼ rskyðλ; 0þÞ þ 1� rwaterðλ; 0�Þf g 1� rskyðλ; 0þÞ� �1� rwaterðλ; 0�Þrðλ; 0�Þ rðλ; 0�Þ; (34)

where rwater is the internal reflectance of upwelling irradiance from the water–air surface;and rsky is the external reflectance of downwelling diffuse (sky) irradiance from theair−water surface. r is the subsurface irradiance ratio (reflectance) of upwelling irradianceEu(λ, 0−) to the downwelling irradiance Ed(λ, 0−); similarly, the total surface reflectanceRtotal = Eu(λ, 0+)/Ed(λ, 0+).

Then the above-water spectral reflectance Rwaterðλ; 0þÞ is

Rwaterðλ; 0þÞ ¼ Rtotalðλ; 0þÞ � rskyðλ; 0þÞ

¼ 1� rwaterðλ; 0�Þf g 1� rskyðλ; 0þÞ� �1� rwaterðλ; 0�Þrðλ; 0�Þ rðλ; 0�Þ:

(35)

To convert Rwaterðλ; 0þÞ and rðλ; 0�Þ to Rrs(λ) and rrs(λ), respectively, let us assume thatthe upwelling radiance at the interface just above the water−air surface is completelydiffuse (Jerlov 1976, p. 79), while at the interface just below the air−water surface it isnot. Then,

Rwaterðλ; 0þÞ ¼ πRrsðλ; 0þÞ; rðλ; 0�Þ ¼ Qðλ; 0�Þrrsðλ; 0�Þ; (36)

where Q(λ, 0−) was calculated by Equation (33).From Equations (35) and (36) it follows that

RrsðλÞ ¼ 1

π

1� rwaterðλ; 0�Þf g 1� rskyðλ; 0þÞ� �1� rwaterðλ; 0�ÞQðλ; 0�ÞrrsðλÞ Qðλ; 0�ÞrrsðλÞ: (37)

Accepting that the reflected surface light is completely diffuse, rskyðλ; 0þ Þ can beexpressed as

rskyðλÞ � πLr; skyðλÞEd; skyðλÞ

� �: (38)

Table 4. Coefficients of Equation (33) for different wavelengths.

λ (nm) a0 a1 a2 a3

412 3.316 −1.870 × 10−3 3.027 × 10−4 −3.581 × 10−6

440 3.329 −1.919 × 10−3 3.221 × 10−4 −3.759 × 10−6

470 3.341 −1.962 × 10−3 3.415 × 10−4 −3.939 × 10−6

488 3.348 −1.983 × 10−3 3.523 × 10−4 −4.041 × 10−6

510 3.356 −2.005 × 10−3 3.647 × 10−4 −4.157 × 10−6

532 3.363 −2.022 × 10−3 3.760 × 10−4 −4.264 × 10−6

595 3.380 −2.052 × 10−3 4.016 × 10−4 −4.509 × 10−6

650 3.388 −2.062 × 10−3 4.148 × 10−4 −4.637 × 10−6

676 3.390 −2.063 × 10−3 4.178 × 10−4 −4.666 × 10−6

715 3.391 −2.064 × 10−3 4.184 × 10−4 −4.671 × 10−6


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The ratio of reflected diffuse (sky) radiance Lr; sky to the incoming diffuse irradiance Ed;skywas computed as a function of W, λ, and θ0 according to polynomial approximations takenfrom Table 5 by Aas (2010), for W = 5 m s−1, as follows:

Lr; skyðλÞEd; skyðλÞ ¼

A0ðλÞ þ A1ðλÞθ0 þ A2ðλÞθ20B0ðλÞ þ B1ðλÞθ0 þ B2ðλÞθ20

; (39)

where θ0 is in degrees and the coefficients are as listed in Table 5:The linear interpolations and extrapolations for θ0 and λ were used because Aas

(2010) provided his coefficients only for θ0 = 37°−76° and separate values of λ.rwaterðλ; 0� Þ has been expressed as (Brockes 1960; Molenaar, ten Bosch, and Zijp

1999)

rwaterðλÞ ¼ 1� 1� rskyðλÞn2wðλÞ

: (40)

3.2.3. Comparison of results for Rrs(λ) and rrs(λ)

The results of the calculations of Rrs(λ) and rrs(λ) using the SAM are shown in Figure10. For comparison, the approximated curves by Mobley (1999), Lee, Carder, andArnone (2002), and Loisel (2008) (at θ0 = 30°) along with the adjusted curve for theSAM are also shown in this figure. It seems that at rrs(λ) < 0.03 sr−1 all models yieldclose results − that is within 15% of each other; however, differences between modelsincreased with increasing rrs(λ). Nevertheless, even at 0.03 sr−1 < rrs(λ) < 0.14 sr−1,Loisel and Saunderson’s models are close − within approximately 10%. Figure 11 showsthe comparisons between different models and the SAM model. Numerical estimatesgive MAPE = 10.0%, 12.0%, 9.4%, and 9.1% for the MGM, and Mobley, Lee, andLoisel’s models, respectively; the values of R2 are 0.996, 0.977, 0.993, and 0.993,respectively.

3.3. Estimation of ρsky(λ) and Rrs(λ) from the radiometric measurements

3.3.1. The review of existing models for the ρsky(λ)

The above-surface spectral remote-sensing reflectance Rrs(λ) may be expressed via radio-metric measurements [Ltot(λ), Ls(λ), and Ed(λ)] as follows:

RrsðλÞ ¼ LtotðλÞ � Lr; skyðλÞEdðλÞ ¼ LtotðλÞ � ρskyðλÞLs; skyðλÞ

EdðλÞ ; (41)

Table 5. Coefficients of Equation (39) for different wavelengths.

λ (nm) A0 A1 A2 B0 B1 B2

405 2.08 × 10−2 3.36 × 10−5 6.85 × 10−8 −0.886 0.165 −1.03 × 10−3

450 2.03 × 10−2 7.57 × 10−5 −36.3 × 10−8 −4.20 0.286 −1.95 × 10−3

520 2.55 × 10−2 −3.81 × 10−5 15.6 × 10−8 −3.09 0.266 −1.80 × 10−3

550 1.43 × 10−2 29.3 × 10−5 −212 × 10−8 −7.49 0.388 −2.44 × 10−3

650 0.879 × 10−2 50.0 × 10−5 −390 × 10−8 −6.44 0.354 −2.04 × 10−3


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where Ls; skyðλÞ and Lr; skyðλÞ are the sky incoming spectral radiance and reflected sky spectralradiance, respectively; ρskyðλÞ is the sky radiance spectral reflectance, expressed as

ρskyðλÞ ¼Lr; skyðλÞLs; skyðλÞ ¼

Lr; skyðλÞEd; skyðλÞ �

Ed; skyðλÞLs; skyðλÞ �

Lr; skyðλÞEd; skyðλÞ �

EdðλÞLsðλÞ : (42)

Figure 10. The spectral above-water remote-sensing coefficient Rrs(λ) computed by the SAMmodel from the station’s rrs(λ) values. Several other models are also presented for comparison.

Figure 11. A comparison of Rrs(λ) computed by the SAM model with the Rrs(λ) computed byMGM (a), Mobley (b), Lee (c), and Loisel (d) models.


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Equation (41) does not contain solar reflectance (glitter) because at the viewing angle of40°, the impact of glitter is considered negligible (Mobley 1999; Aas 2010). In the firstapproximation, Ls; skyðλÞ can be expressed as

Ls; skyðλÞ ¼ LsðλÞdEðλÞ; (43)

where dE(λ) was estimated by Equation (22). Thus, a problem of the Rrs(λ) estimation isreduced to either a reasonable estimate of sky radiance reflectance ρsky(λ) or theLr; skyðλÞ=Ed; skyðλÞ ratio. Table 6 contains a set of several different computational modelsfor ρsky(λ):

We derived the equation for ρsky in the SOHM model from 3000 pairs of ρsky andπLs, sky(400)/Ed(400) values measured by Simis and Olsson (2013) in the Baltic Sea. Notethat only the AHM model yields a spectral dependence for ρsky; recent observational andmodelling efforts by Lee et al. (2010) proved to be a very strong evidence of suchdependence.

3.3.2. Results for ρsky(λ)

According to our radiometric measurements and equation by Ruddick et al. (2006) for ρsky(see Table 6), May 4 (stations ‘e1, e2, e4, e5’), May 5 (‘f1−f3’), May 11 (‘c1−c3’) and themorning hours of May 12 (‘b1, b2’) were characterized as cloudy time periods, while May7 (stations ‘d1−d5’) and the afternoon of May 12 (‘b3−b5’) were identified as clear skytime periods. For stations ‘c4’, ‘c5’, and ‘e3’ radiometric measurements were not con-ducted and sky conditions were not determined.

We show results (Figure 12) for the AHM model only because all the other modelsyield non-spectral values of ρsky. The calculations demonstrated that ρsky is almostspectrally neutral (with the values of 0.01–0.02) during cloudy days and/or hours.However, in clear days and/or hours, spectral features with the ρsky(400) = 0.018–0.047and ρsky(800) = 0.027–0.137 were obvious. The maximal values of the ratio ρsky(800)/ρsky(400) − 2.2 to 2.9 − were found for the ‘d1−d5’ stations, which had the clearest skyconditions. Conversely, the minimal ratios − 0.64 to 0.86 − were found for stations‘c1−c3’ and ‘e1, e2, e4, e5’, which had the cloudiest sky conditions. These results are

Table 6. Computational models for ρsky(λ) and Rrs(λ).

Model name Abbreviation Models References

Whitlock–Højerslevmodel

WHM ρsky = 0.02; Equations (22), (41)–(43) Whitlock et al.(1981),Højerslev (2001)

Ruddick–Højerslevmodel

RHM ρsky ¼ 0:0256þ 0:00039W þ 0:000034W 2, ifLs; skyð750Þ=Edð750Þ<0:05 (clear-skyconditions); otherwise, ρsky ¼ 0:0256;Equations (22), (41)–(43)

Højerslev (2001),Ruddick et al.(2006)

Aas–Højerslevmodel

AHM Equations (22), (39), (41)–(43) Højerslev (2001),Aas (2010)

Simis–Olsson–Højerslevmodel

SOHM ρsky ¼ 0:0272 πLs; skyð400ÞEdð400Þ

h i�0:239; Equations

(22), (41)–(43)

Højerslev (2001),Simis and Olsson(2013)


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in agreement with Lee et al.’s (2010) findings. Also, similar to their results, we did notdiscover any significant relation between the water quality and ρsky(λ). Note that theresults for station ‘f3’ are not shown here due to the large errors caused by measurementproblems encountered during the observations made at this station.

3.3.3. Comparative results for radiometric determination of Rrs(λ)

The computation results for Rrs(λ) derived from the radiometry of the WHM, RHM, andSOHM models, when compared with the AHM-derived Rrs(λ) (Figure 13), showed anoverall good agreement. The error index MAPE equalled 11%, 13%, and 14% for theWHM, RHM, and SOHM models, respectively, and the values of the coefficient ofdetermination, R2, were 0.995, 0.985, and 0.987, respectively. Thus, even the verysimplified WHM method, which had ρsky = 0.02, demonstrates overall excellent results.However, the accuracy of radiometrically derived Rrs(λ) very much depends on thecontribution of Lw(λ) to the measured Ltot(λ), and this accuracy generally deterioratestowards longer wavelengths (Mobley 1999). Our results fully confirm the last statement.For example, we show (Figure 14) a plot of relative errors in Rrs(λ) for the WHM methodrelative to AHM for the stations, revealing the maximal discrepancies in the red and NIRspectral range (‘b5’, ‘d4’, and ‘d5’). It appears that in the blue-green range (where water-leaving radiance contribution into the total radiance is large and almost spectrally inde-pendent) errors are relatively small (<30%); however, these errors then sharply grow,revealing large and increasing errors at longer wavelengths.

3.4. Validation of the radiative transfer model

The preceding analysis showed that the impact of uncertainties in Rrs(λ) modellingcaused by potential errors in the rrs(λ) vs. IOPs(λ) and Rrs(λ) vs. rrs(λ) models is quite

Figure 12. The spectral sky radiance reflectance ρsky(λ) computed by the Aas (2010) method fromthe station’s radiometric measurements. Note that the observations on May 7 and May 12 (after-noon) were performed during clear sky conditions, while the other observations were identified ashaving been performed during cloudy sky conditions.


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insignificant, at least for our geographical and temporal situation. However, the impactof the errors caused by the scattering correction procedure and conversion of radio-metric quantities into Rrs(λ) may be rather significant. Therefore, we decided tocompare the radiometrically and IOP-derived Rrs(λ) by including different scatteringcorrection methods (more specifically, PZM and MBM methods) and radiometrical

Figure 14. Examples of calculated relative errors in Rrs(λ) obtained using the WHM method(δ(Rrs, WHM(λ)) as compared with the values obtained using the AHM method. The errors aregiven for selected stations that yielded the largest discrepancies between these two methods.

Figure 13. The Rrs(λ) computed by the WHM (a), RHM (b), and SOHM (c) methods from theradiometric measurements and plotted against the values obtained using the AHM method.


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conversion (more specifically, AHM and WHM) procedures into the analysis.Figure 15 provides such an analysis for each station. A comparison reveals thefollowing conclusions of the study:

Figure 15. Individual station comparisons between two radiometric conversion (AHM and WHM)and two scattering correction (PZM and MBM) methods for research stations b1–b4 (a), b5–c3 (b),c4–d2 (c), d3–e1 (d), e2–e5 (e), and f1–f3 (f).


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– Even though errors may be relatively large (up to 30%), the spectral forms of Rrs(λ)computed by using PZM and MBM scattering correction methods seemed to bevery close for all stations.

Figure 15. Continued.


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– A similar conclusion may be made concerning the AHM and WHM comparison;the maximal relative error was 199%, but the spectral forms of Rrs(λ) appear to bevery close.

Figure 15. Continued.


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– Discrepancies between radiometrically and IOPs-derived Rrs(λ) generally increasedtowards longer wavelengths; however, the spectral features for different modelsmay be considered close to one another at λ < 600 nm.

4. Discussion and conclusion

This study carried out a detailed analysis of experimental in situ data and different opticalmodels with the aim of verifying Rrs(λ) models for the Changjiang (Yangtze) RiverEstuary, and its adjacent coastal area in the East China Sea. This analysis showed thatthe main sources of potential errors are inaccurate procedures of scattering correction forabsorption and conversion of radiometric quantities into Rrs(λ). For example, differencesbetween spectral absorption values, derived by the proportional Zaneveld’s method and amodified Boss’s method, may achieve from −40% to 70% for very turbid waters at the redportion of the spectrum (Figure 5c). Similarly, differences between the Rrs(λ) values,derived by the Whitlock–Højerslev model and Aas–Højerslev model, in the red andNIR spectral range may be much more than 100% (Figure 14).

However, possible errors introduced by the rrs(λ) vs. IOPs(λ) (Figure 9) and Rrs(λ) vs. rrs(λ) (Figures 10 and 11) models seemed to be quite insignificant in the whole visible andNIR spectral range. Also, possible experimental IOPs errors hardly can introduce seriouscontributions to differences between radiometrically and IOPs-derived Rrs(λ). More specifi-cally, error of the measured attenuation coefficient (even without any scattering correction)is expected to be anywhere between 5% and 20% underestimation for the ac-s instrument(Wet Labs ac Meter Protocol Document 2011). The relative standard errors of averaged(from 0 to 1 m) values of total absorption and attenuation were generally less than 5%(Figure 4), reaching a maxima of 11% and 14% for absorption and attenuation, correspond-ingly. A comparison of backscattering coefficient measured by two optical instruments(ECO-VSF and ECO-BB9) at λ = 532 and 650 nm also yielded encouraging results(Figure 3) with the mean errors of 6.0% and 4.8% for the depths <1 m.

Unfortunately, expected good correspondence in the blue–orange range was not alwaysthe case, and discrepancies between the radiometrically and IOP-derived Rrs(λ) were large forsome of the stations (Figure 15). The possible reasons for such discrepancies include spatial(both horizontal and vertical) and temporal variability of water components and, hence, theiroptical responses. Another possible reason for these discrepancies is inaccurate accounting ofthe contribution of Lw(λ) into measured Ltot(λ). Nevertheless, the spectral features for differentradiometric and IOPs models were rather close to one another at λ < 600 nm, allowing the useof spectral ratios for further analysis. Thus, we hope that the radiative transfer (optical) closureapproach presented in this study may be useful for developing remote-sensing algorithms forregular monitoring and assessment of estuarine and coastal waters.

FundingThe research leading to these results received funding from the National Science Foundation ofChina (grant number 41271375) and the Doctoral Fund of Ministry of Education of China(grant number 20120076110009).

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Documents

Optical closure for remote-sensing reflectance based on accurate radiative transfer approximations: the case of the Changjiang (Yangtze) River Estuary and its adjacent coastal area,