Algorithm Description Ver.2.1(Revision 2017.3.31)

PI No. 302 Takafumi Hirata (GCOM-C/SGLI Ocean Team)
Theoretical Description of the Euphotic Depth Algorithm
1. Physics of the problem
The euphotic depth z1% is defined as the depth z at which the photosynthetically available radiation(PAR) at the sea surface falls to 1% of that (i.e. PAR(z1%)= 0.01 PAR(z=0-), where 0- indicates “justbelow the sea surface”. The PAR is defined by PAR(z)=∫ E0(z, λ) dλ where 400 ≤ λ ≤ 700nm, andE0(z,λ) here represents the spectral scalar irradiance. PAR is attenuated with depth by:PAR(z)=PAR(z=0) exp(-∫ KPAR(z) ·dz) =PAR(z=0)exp(- K PAR ·z).

(1)

K PAR is the depth average of the diffuse attenuation coefficient for PAR:K PAR =∫ KPAR(z) dz / ∫1dz = ∫ KPAR(z) dz / z
and KPAR(z), which is a function of depth, is defined by:
KPAR(z)= d(ln PAR(z)) / dz.
From Eq. 1, we obtain

z1%= -ln[PAR (z=z1%)/PAR(z=0)]/ ∫KPAR (z)dz = -ln(0.01) / K PAR(2)

Equation (2) indicates that z1% is found when K PAR is known. Therefore, K PAR is a key factor toestimate z1%. In the following section, a derivation of K PAR(hence z1%) using a remotemeasurement of the radiance is shown. Essentially, the problem is composed of the four practicalproblems(i) How a conventional approximation of KPAR can be corrected to represent the exact KPAR

for a better estimation of z1%, (ii) How the spectrally-integrated quantity of KPAR is derived from aradiance measured remotely at a limited number of discrete wavelengths, (iii) How a depth average ofKPAR is derived from KPAR at the surface which a remote sensor would detect, (iv) How KPAR at thesurface can actually be derived from the remotely-sensed radiance. In the following, wavelength- anddepth dependencies of the variables such as PAR and KPAR are omitted unless otherwise specified.
2. Euphotic Depth Algorithm
2.1 How the spectrally-integrated property of KPAR is derived from a limited numbers of discrete
wavelengths?
The diffuse attenuation coefficient for PAR (KPAR(λ,z)) is related to that of the spectral scalar irradianceK0(λ,z) via

1K PAR (z) = − 700∫ E (λ, z)d λ# 700&d % ∫ E0 (λ, z)d λ ($ 400'dz

0400=−()

d 300E0 (λ, z)
1dz
300E0 (λ, z)= K 0 (λ ', z)(4)where λ’ is a wavelength at which E 0 (λ,z)=E 0 (λ',z) or PAR(z)=300 E0(λ’,z) is found. It shows that
KPAR(z) can be calculated from K0(λ’z) for any z, once if λ’ is known.

Fig. 1 The ratio of the spectral average of E0 to E0(λ) for the optical depth of 0 (left) and 5 (right).

Fig. 1 shows the ratio between E 0 (λ,z) and E 0 (λ,z) , derived from the radiative transfer
simulation (Appendix A). The wavelength at which this ratio equals unity is found as λ’. At the sea
surface (or the optical depth = 0), E 0 (λ,z) is often close to E 0 (λ,z) numerically at λ=560nm, sothat λ’ is often 560nm, although λ’ does vary. The same holds true for the optical depth of 5.0(which is close to the optical depth of 4.6 and corresponds to z1%). However, uncertaintyof the ratio at 560nm represented by the standard deviation (hence, uncertainty of λ’) isalso large at the optical depth of 5.0. Rather, the uncertainty is small between 412-490nm
at both the optical depth of 0.0 and 5.0 (and intermediate depths not shown), although
the ratio is no longer unity (i.e. E 0 (λ,z) ≠ E 0 (λ,z) ). Over the water column considered(i.e. the optical depth 0 to 5.0), we found that the uncertainty is smallest at thewavelength of 490 nm, with E 0 (λ,z)/E 0 (λ,z)=0.835 ± 0.04 among SGLI Visible andNear Infrared Radiometer wavelengths (i.e. 412, 443, 490, 530, 565, 673, 763, 868nm).
Although the ratio is not unity, E 0 (λ,z) can be estimated by using 490nm more
successfully than λ’ because, even if the ratio is 0.835, Eq. 4 can be re-written by

K PAR (z) = −=−()

d 300⋅E0 (λ, z)
1dz
300⋅E0 (λ, z)d ( 257.4 ⋅ E0 (490, z))1257.4 ⋅ E0 (490, z)
dz= K 0 (490, z)(4’)
to reach the same conclusion as Eq. 4 that KPAR can be found directly from K0 for any z. At a glance,
it then seems in Eq. 4’ that a choice of the wavelength doesn’t matter, as any value of the ratio between

E 0 (λ,z) and E 0 (λ,z) will be canceled out in Eq. 4’ to always reach the same conclusion ofKPAR(z)=K0(z). Given that λ’ is more or less variable and cannot be predicted easily (see Fig. 1), it is
the stability and uncertainty of the ratio in various environmental conditions, not the ratio value itself,that affects the robustness of the replacement of E 0 (λ,z) with E 0 (λ,z) , hence the validity of theconclusion from Eq. 4(4’). Since a choice of wavelength affects the stability and uncertainty, the choicemust carefully be made, unless one can always know and change in advance the variable wavelengththat gives the best estimate of E 0 (λ,z) and can find the wavelengths within a constrain of satellite
sensor bands (i.e. a limited numbers of sensor wavelengths). From Eq. 4’, it is obvious that KPAR=K0(490) for z=0.
2.1 How a depth average of KPAR is derived from KPAR at the surface?
Fig. 2a shows vertical profiles of K0(490,z) (=KPAR(z), see Eq. 4’). Depending on the solar zenith
angle θs, K0(490,z) varies with depth over 9%, even if the inherent optical properties of water bodyremain uniform over the water column. Thus, the approximation of K0(490,0)=K0(490,z) (hence,KPAR(0)= KPAR(z)) would lead to an inneligible error and must be corrected.

Fig.2 K0 at 490 nm as a function of the optical depth (left) and the ratio of the depth average ofK0(490) to the surface value of K0(490).

Fig.2b shows the ratio between K 0 (490, z) and K 0 (490, z) at θs=0°, as a function of the
optical properties (i.e. the absorption and the scattering coefficients) of bulk water. This result was
similar to those at other θs, hence not shown. At realistic combinations of the absorption and scattering
coefficients obtained from in situ measurements (NOMAD, Werdell et al. 2005), variability in the ratio
is rather confined to a narrow range, namely 1.35±0.05, so that

K PAR (z) = K 0 (490, z) = (1.3± 0.05)⋅ K 0 (490, 0) .

(5)

The next step is to find K 0 (490, 0) from a remote measurement of the radiance. This may involve

two scientific challenges in our strategy; (1)How K 0 (490, 0) can be related to a routinely evaluatedvariable such as Kd(490,0) and (2)How Kd(490,0) can be found from the remotely-sensed radiance.
To proceed, we define the ratio between the scalar irradiance E0 to the downward scalarirradiance Eod so that E0(λ,z)= ε1(λ,z) E0d(λ,z). We also define the average cosine for the downward
radiance µd(λ,z)= Ed(λ,z)/E0d(λ,z). For simplicity, the wavelength- and depth dependency of thesevariables are omitted hereafter unless otherwise specified. As in the case for K0 (λ,z)(and KPAR (λ,z),etc), the diffuse attenuation coefficient for ε1 and µd(λ,z) can be defined as Kε1(λ,z)=-(1/ε1(λ,z))(dε1(λ,z)/dz) and Kµd (λ,z) =-(1/µd(λ,z)) (dµd (λ,z)/dz), respectively. As a result, we get

K 0 (490, 0) = −=−1d(ε1(490, 0)⋅ Eod (490, 0))ε1(z, λ )⋅ Eod (z, λ )dzµ d (490, 0)d(ε1(490, 0)⋅ Ed (490, 0) / µ d (490, 0)ε1(490, 0)⋅ Eod (490, 0)dz= K d (490, 0) + Kε1 (490, 0) − K µd (490, 0)= K d (490, 0) + ΔK(490, 0)(6)where ΔK(490,0)=Kε1 (490,0)- Kµd(490,0). At a glance, K0 (490,0) is linearly related to Kd (490,0) in
this expression (although ΔK(490,0) may or may not be a function of Kd(490,0), which we do notknow at this point). In order to find a relationship between Kd (490,0) and Kd, (490,0), Fig. 3 is plottedto show their ratio at z=0 as a function of the absorption and scattering coefficients of bulk water
bodies for different solar angles θs =0°(Fig. 3b) and 50°(Fig. 3c). For realistic combinations of theabsorption and scattering represented by in situ data, the ratio tends to lie on a contour of 0.80 and0.98, respectively. This suggests that K0(490,0) co-varies with Kd(490,0) for different water bodies(i.e. for different combination of the absorption and scattering), although K0 (490,0) ≠ Kd (490,0)
necessarily. Also shown in Fig. 3c is the ratio at 490nm varying with θs, which can mathematically beformulated by

K 0 (490, 0) = K d (490, 0) − [ K d (490, 0)⋅ 0.201⋅ exp(−0.043θ s )]= K d (490, 0) + ΔK '(490, 0)(7)

Fig. 3 The ratio of K0 (λ=490nm, z=0m) to Kd(490,0) for θs=0°(left) and θs=50°(center), and thesolar dependency of the ratio (right).

Comparison between Eqs. 6 and 7 suggests ΔK=ΔK’, so that ΔK can be expressed as a function of Kd:

ΔK(490, 0) = ΔK '(490, 0) = K d (490, 0)⋅ 0.201⋅ exp(−0.043θ s ) .

(8)

From Eq. 7, K0(490,0) can be expressed by only Kd(490) for a given θs (i.e. for a given
observation condition). From the two-flow radiative transfer model (Hirata et al., 2008) for ahomogenous water (i.e. inherent optical properties of water body is uniform over the water column),
Kd(490,0) may be expressed byK d (490, 0) =c − rdd b f rud bb R−µdµu= ε2a + bbµd(9)where the beam attenuation coefficient is denoted by c, the forward- and backscattering coefficientsby bf and bb, respectively. The irradiance ratio is denoted by R, and the shape factors for the incident
and reflected photon streams by rdd and rud, respectively. Finally, ε2=(c-rddbf)/(a+bb). Gordon (1989)
showed from Monte Carlo radiative transfer simulation that ε2=1.035. For the clear sky condition inwhich the satellite remote sensing is feasible, photons just below the sea surface can be dominated bythose from the direct Sun beam incident on the sea surface. In such a case µd ≈ cos(θsw) where θsw andµu represent the zenith angle of the apparent solar zenith angle just below the sea surface and theaverage cosine for the upward riddance field, respectively. When cos(θsw) just below the sea surface

sin 2 θ sis calculated from θs by 1−using the Snell’s law, substitution of Eq. 9 in to 7 and then2
resultant equation into Eq. 5 gives:

K PAR (z) = 1.339 ⋅1− 0.201⋅ exp(−0.043⋅ θ s )(1− sin 2 θ s ) / n 2[a(490) + bb (490)](10)

where the index of refraction n of seawater relative to air may be assumed 1.33. Eq. 10 indicates that

K PAR is essentially proportional to a(490)+b(490), with the proportionality factor being a function of
the solar zenith angle.
Substituting Eq. 10 into 2 finally gives z1%:

z1 % =k0 (1− sin 2 θ s ) / n 21⋅
[1− k1 exp(k2 ⋅ θ s )] [a(490) + bb (490)](11)

where k0=3.003, k1=0.201, k2=-0.043 in this study. Thus, z1% is obtained once a(490), bb(490) and θsis known. The former two (a(490) and bb(490)) are derived from the Inherent Optical Property
algorithm (see ATBD for GCOM-C/SGLI

IOP algorithm, otherwise Smyth et al., 2006), while thelast can be obtained once time (year, month,date) and observation location is known (the code isembedded within the GCOM-C/SGLI IOP algorithm).
3. Algorithm evaluation
Fig.4 shows an example of z1% derived from Eq. 9, using the NOMAD data set (Werdell and Bailey,2005) from which simultaneous measurements of in situ z1% and IOPs can be obtained. Also plotted
are z1% derived from other models of Kd(λ) using z1% ~ 4.605 / Kd(490). Statistics are summarized inTable 1. Eq. 9 performs relatively well with RMSE=23.7m, although some scatters are foundcompared to other models. The regression slope of Eq. 9 (and of Lee et al, 2005) is relatively close to1.0, too.

Fig. 4 Comparison between z1% observed(x) and modeled (y), as well as between Eq.9 (this study)
and other models.

Table 1. Statistical results of algorithm evaluation for ady against in situ data
Slope#Intercept#RMSE*

Eq.9 (This study)

0.899

0.079

23.7

Lee et al., 2005

0.901

0.117

20.3

Mitchel & Kahru, 1998

1.205

-0.26035.7

Werdell, 2005

1.164

-0.10655.6

Morel et al., 2007A

0.672

0.487

24.7

Morel et al., 2007B

0.759

0.413

18.0

#log scale, *Liner scale
5. Algorithm calibration

When the statistical coefficients (the slope and intercept in Table 1) obtained for Eq. 9 is used tocalibrate the algorithm, z1%calib=(z1%-0.079)/0.899. When the SGLI is actually launched andthe algorithm uses the SGLI satellite signal, the calibration coefficients should be re-determined.
References
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bio-optical data set. Calif. Coop. Oceanic Fish. Invest. Rep. 39, 133-147, 1998
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irradiance, J. Geophys. Res., 110, C02016, doi:10.1029/2004JC002275, 2005.
Morel, A., Antoine, D., and B. Gentili, Bidirectional reflectance of oceanic waters: Accounting for
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8116-8131, 2006.
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