Algorithm Description Ver.0(2011.09.30)

Algorithm Description Ver.0 加筆修正 (2011.10.10)
PI No. 134 Takafumi Hirata (GCOM-C/SGLI Ocean Team)
CIs: Nick Hardman-Mountford, Tim Smyth, Shubha Sathyendranath, Trevor Platt,Yasuhiro Yamanaka
Theoretical Description of the Inherent Optical Property Algorithm
1. Derivation of the inherent optical properties (IOPs)
1.1 Physics of the problem
The IOP algorithm assumes that the remote sensing reflectance (Rrs) just above the sea

surface (denoted by z=0+, where z represents a depth), or the water-leaving reflectance(ρ), is obtained in prior to its implementation.
The Rrs for a wavelength λ is defined byRrs(θv, φv, z=0+, θs, φs, λ) = Lw(θv, φv, z=0+, θs, φs, λ) / Ed(z=0+, θs, φs,λ)(1)where Lw and Ed are the radiance and the downward plane irradiance at theobservation angle (zenith angle θv, azimuth angle, φv) and the solar angle (zenith angle
θs, azimuth angle φs). The water-leaving reflectance ρ can be obtained by ρ = π Rrs. Morel andGentili (1993, 1996) showed that the Eq. (1) can be related to the absorption coefficientat and the backscattering coefficient of the bulk water bbt by
Rrs (θv, φv, z=0+, θs, φs, λ)= R (W, θs, φs, λ) F(θv, φv, z=0-, θs, φs, λ) [bbt (z=0-, λ)/at(z=0-, λ)] (2)where W denotes the wind speed. For convenience, all dependencies of the variables in
Eq. 2 are omitted hereafter, unless otherwise specified. In addition, R F will be denoted
by F’ so that Eq.2 is simplified by
Rrs = F’ [bbt/at].

(3)

The absorption coefficient of the bulk seawater is decomposed into the absorption
coefficients of the pure seawater (aw) and any other materials (ao),at=aw+ao.

(4)

Similarly, the backscattering coefficient of the bulk seawater is decomposed into thebackscattering coefficients of the pure seawater (bbw) and any other particles (bbp)
bbt=bbw+bbp.

(5)

In the following sections, retrievals of the absorption and backscattering coefficientsand their components will be described.
2. IOP algorithm
2.1 Derivation of the IOPs of bulk seawater from the remote sensing reflectance
Substitutions of Eqs 4 and 5 into Eq. 3 give a set of equations for wavelengths λ1 and λ2:
Rrs (λ1) a0(λ1) - F’ (λ1) bbp(λ1) = F’ (λ1) bbw(λ1) - Rrs(λ1) aw(λ1)
Rrs (λ2) a0(λ2) - F’ (λ2) bbp(λ2) = F’ (λ2) bbw(λ2) - Rrs(λ2) aw(λ2),(6)

where Rrs is obtained by satellite observation and aw and bbw are assumed known foreach λ (i.e. they are approximated by those by Morel, 1974; Pope and Fry, 1997, althoughaw and bbw are not constant and varies with temperature and salinity). Therefore, Eq. 6 can besolved to derive a0 (λ1 and λ2) and bbp (λ1 and λ2), when (1) the wavelength dependencies of theinherent optical properties (bbp and a0) and (2) F’(λ) are known.
2.2 The wavelength dependencies of the inherent optical properties
The wavelength dependency of bbp (denoted by εbb) is defined by
εbb = bbp(λ1) /bbp(λ2) = (λ1/λ2)-m(7)where m is related to the particle size distribution and set as a parameter in the presentalgorithm. The wavelength dependency of a0 is given by
εa = a0(λ1) /a0(λ2)(8)

and it is another algorithm parameter. According to in situ observation, εa = a0(λ=490nm)/a0(λ=510nm) is least variable and therefore applicable for a wide range of seawaters (Fig. 1). SinceSGLI does not have 510nm, it is replaced by the SGLI 530nm. Substitutions of Eqs. 7 and 8 into Eq.
6, as well as a use of Rrs(λ=490) and Rrs(λ=530) observed by satellite, gives a0 (490), a0 (530),bbp(490) and bbp(530) for a known F’ (Determination of F’ will be described in the next subsection).
Once bbp(490) (or bbp(530) ) are obtained, substituting them into Eq. 7 gives bbp at any wavelength λfor an “m” assumed: (e.g.)bbp(λ)= bbp(490) (λ/490)m.

(9)

Eq. 9 allows us to derive a0 at any SGLI short wavelength λ, using Eqs. 3-5:a0(λ)=F’ (λ) [ bbw(λ) + bbp(λ) ] – aw(λ).

(10)

Fig. 1 Wavelength dependencies of the absorption coefficient of optical components determinedfrom in situ observation (Smyth et al., 2006). εph: phytoplankton absorption, εd: detritus absorption,
εy: CDOM, εdy: detritus plus CDOM, εa: total absorption without pure sea water component.
2.3 Derivation of F’
F’ is initially assumed to be a certain value (e.g. a literature value). Firstly Eq. 6 is solved
using the initially assumed value of F’ (Fig. 2). The initial outputs at and bbt are then used to search
the new value of F’ from a Look-Up Table (LUT) of F’ which was pre-generated, by radiativetransfer simulations in prior to the algorithm implementation, for the assumed conditions of theobservation geometry (θv, φv), solar geometry (θs, φs) and inherent optical properties of seawater (at,bt) at SGLI-wavelengths. Eq. 6 is again solved, and at (λ) and bt (λ) are obtained from the second
value of F’ (λ), which are then used to search a third value, forth, … nth value of F’ (λ) from the LUT.

This procedure is repeated towards a convergence of at(λ) and bt (λ) values (or equivalently, F’ (λ))
obtained from Eq. 6. When either the IOP algorithm returns the outputs only within a tolerance rangeof at (λ) and bt(λ) or a pre-fixed number of iteration is implemented, the iteration is terminated and avalue of F’ (λ) at the termination is given as a F’ (λ) value wanted. At the same time, at and bt at thetermination are also given as the solutions for Eq. 6. In other words, F’ (λ) and the set of at(λ) andbt(λ) are simultaneously derived by the iteration.

Fig. 2 Flow chart of the algorithm implementation

2.1 Decomposition of the IOPs of optical components from the IOPs of bulk seawater
The total absorption coefficient without pure seawater contribution ao can further be decomposedby ao=aph+ady where the absorption coefficient of phytoplankton and of detritus plus
chromophoric dissolved organic matters (CDOM) are denoted as aph and ady, respectively.
The decomposition gives the simultaneous equations for two wavelengths λ1 and λ2.
ao(λ1)=aph(λ1)+ady(λ1)
ao(λ2)=aph(λ2)+ady(λ2).

(11)

In analogy to Eq. 8, Eq. 11 can be solved to obtain aph and ady at λ1 and λ2 when
wavelength dependencies of aph and ady defined by εph = aph(λ1) /aph(λ2) and εph = ady(λ1)

/ady(λ2) are known. Fig. 1 shows that variations in εph and εdy are least for the wavelength pair of
λ1=412nm and λ2=443nm so that their values may be applied to a wide range of water types.
Therefore, εph(412, 443) and εdy(412, 443) are used in the present algorithm.
Since the wavelength dependency of ady, or equivalently εdy, can be expressed by (Smythet al., 2006):

ady(λ)=ady(λ1)exp[-Y(λ-λ1)],(12)

ady for any SGLI short wavelength λ can be obtained using ady (413) and ady(443) derived
from Eq. 11. Given that ady(λ) exponentially decreases with wavelength, ady(λ) may be approximatedto ay(λ)(or aCDOM(λ)) at relatively longer wavelengths within SGLI bands to the first approximation:i.e. ay(490)~ ady(490). This however requires further investigations for a better estimation of ay(490).
Rearrangement of Eq. 11 givesaph(λ)=a0(λ) - ady(λ).

(13)

Since a0(λ) and ady(λ) can be obtained for any SGLI short wavelength, Eq. 13 completes thederivations of a0 , aph, ady and bbp at the SGLI wavelengths. Note that at and bbt can also
be derived from these IOPs using Eqs. 4 and 5. Fig. 3 summarizes the sequence of theIOPs retrieval.

Fig. 3 The sequence of IOP retrieval.

3. Algorithm evaluation
Figs. 4-6 show examples of evaluation results of the IOP algorithm, using the NOMAD
data set (Werdell and Bailey, 2005) from which simultaneous measurements of in situ
Rrs and IOPs can be obtained. The in situ Rrs is given as an input to the present IOP
algorithm, and the algorithm outputs were compared to the in situ IOPs in the data set.
Also Table 1-3 below summarizes statistical results of the evaluation.

Fig. 4 Comparison between ady (adg) measured in situ and derived from the IOP
algorithm.

Fig. 5 Comparison between bb measured in situ and derived from the IOP algorithm.

Fig. 6 Comparison between bb measured in situ and derived from the IOP algorithm.
Table 1. Statistical results of algorithm evaluation for ady against in situ data
Wavelength#

slope#

intercept#

r2#

RMSE*

412

0.939

-0.111

0.802

0.342

443

0.0.906-0.145

0.781

0.219

490

0.856

-0.258

0.741

0.118

510

0.836

-0.307

0.723

0.091

#log scale, *Liner scale

Table 2. Statistical results of algorithm evaluation for aph against in situ data
Wavelength#

slope#

intercept#

r2#

RMSE*

4120.815-0.292

0.724

0.484

443

0.796

-0.287

0.761

0.123

490

0.820

-0.180

0.722

0.089

510

0.748

-0.209

0.732

0.076

#log scale, *Liner scale

Table 3. Statistical results of algorithm evaluation for bb against in situ data
Wavelength

slope#

intercept#

r2#

RMSE*

412

0.811

-0.2380.411

0.0049443

0.784-0.2790.421

0.0049490

0.748-0.3480.430

0.0048510

0.729-0.3860.430

0.0047#log scale, *Liner scale

References
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