A fundamental relationship between the remote sensing reflectance (ρrs) and IOPs was formulated as follows (Gordon, Brown, and Jacobs 1975):
ρ λ λ
λ λ
rs b
b
( ) ( )
( ) ( )
∝ +
b
a b (18�1)
where a(λ) is the absorption coefficient and bb(λ) is the backscattering coefficient�
Recently, a conceptual model based on Equation 18�1 was developed and used to esti- mate pigment concentration in terrestrial vegetation at leaf and canopy levels (Gitelson, Gritz, and Merzlyak 2003; Gitelson et al� 2005):
Pigment content∝[ρ λ−1( )1 −ρ λ−1( )]2 ×ρ λ( )3 (18�2) where ρ(λ1), ρ(λ2), and ρ(λ3) are the reflectance values at wavelengths λ1, λ2, and λ3 respec- tively� λ1 is in a spectral region such that ρ(λ1) is maximally sensitive to absorption by the pigment of interest, although it is still affected by absorption by other pigments and scat- tering by all particulates� λ2 is in a spectral region such that ρ(λ2) is minimally sensitive to absorption by the pigment of interest and its sensitivity to absorption by other constituents is comparable to that of ρ(λ1)� Thus, the difference [ρ−1(λ1) − ρ−1(λ2)] is related to the concentra- tion of the pigment of interest� However, the difference is still potentially affected by varia- tions in scattering by particles� Consequently, information on λ3 is required� Wavelength λ3
is located in a spectral region where the reflectance ρ(λ3) is minimally affected by absorp- tion due to any constituent and is therefore used to account for the variability in scattering between samples�
Dall’Olmo, Gitelson, and Rundquist (2003) suggested the use of this conceptual model (Equation 18�2) for estimating chl-a concentration in turbid productive waters� In Equation 18�1, the absorption coefficient, a(λ), is the sum of the absorption coefficients of water, aw(λ), phytoplankton, aϕ(λ), nonalgal particles, aNAP(λ), and CDOM, aCDOM(λ)� Follow- ing Gordon’s concept, the model presented in Equation 18�2 (called henceforth the three- band NIR–red model) was designed by choosing three optimal wavelengths, such that the contributions due to absorption by constituents other than chl-a and backscattering by particles are kept to a negligible minimum, and the model output is maximally sensitive to chl-a concentration� The red region around 670 nm, where chl-a absorption is maximal (but the reflectance may be affected also by other constituents), was chosen as λ1� λ2 is longer than λ1, where absorption by chl-a, aϕ(λ), is minimal and the absorption by other constitu- ents, aNAP(λ) and aCDOM(λ), is about the same as at λ1� Thus, ρ−1(λ1) is a measure of absorp- tion by chl-a and other constituents, and ρ−1(λ2) is a measure of absorption by constituents other than chl-a� λ3 is at a wavelength beyond λ2 in the NIR region, where the absorption
by all particles and dissolved constituents is null� The backscattering coefficient is consid- ered spectrally uniform across the range of wavelengths considered (from λ1 through λ3; Dall’Olmo and Gitelson 2005), which is a fundamental assumption in the model�
The subtraction of ρ−1(λ2) from ρ−1(λ1) isolates the absorption by chl-a as follows:
ρ λ−1 −ρ λ− λ + φλ + λ + λ +
1 1
( ) ( )2 ∝aw 1 a 1 aNAP 1 aCDOM 1 bb( ) ( )
( ) (
λ λ
λ φλ λ λ λ
b
a a a a b
b b
w NAP CDOM b
b
2 2 2 2
− + + + +
λλ)
ρ λ ρ λ
λ
φ λ λ
−1 − − + −
1 1
( ) ( )2
∝a a ( )a b
w w
b
1 2 (18�3)
Another assumption is that the absorption by water at λ3 is much greater than the total backscattering, such that aw(λ3) >>bb(λ) and a(λ) ≅ aw(λ3)�
ρ λ λ
λ
( ) ( )
3 ∝b a
b w 3
(18�4) Considering the fact that the absorption by water, aw(λ), is independent of the constituent concentrations and ignoring its dependence on temperature, the model has the following form:
[ρ λ−1( )1 −ρ λ−1( )]2 ×ρ λ( )3 ∝aφ( )λ (18�5) Absorption by phytoplankton is related to chl-a concentration as follows:
aφ( )λ =aφ*( )λ ×Cchl-a (18�6)
where a*ϕ(λ) is the chl-a specific absorption coefficient and Cchl-a is the concentration of chl-a�
Thus, the three-band NIR–red model was finally formulated as [ρ λ−1( )1 −ρ λ−1( )]×ρ λ( )∝
2 3 chl-a (18�7)
Dall’Olmo and Gitelson (2005) have found that the optimal wavelengths for the accurate esti- mation of chl-a concentrations in the range of 2 to 180 mg ⋅ m−3 were as follows: λ1 = 670 nm, λ2 = 710 nm, and λ3 = 740 nm� Testing the model for several data sets collected in inland and estuarine waters, Gitelson, Schalles, and Hladik (2007) and Gitelson et al� (2008) found relatively wide optimal spectral bands of wavelengths of λ1 = 660–670 nm, λ2 = 700–720 nm, and λ3 = 730–760 nm, which provided accurate estimations of chl-a concentration with the three-band NIR–red model�
For waters that do not have significant concentrations of nonalgal particles and colored dissolved organic matter, the subtraction of ρ−1(λ2) in the model may be omitted (Dall’Olmo and Gitelson 2005), which leads to the special case of a two-band NIR–red model (Stumpf and Tyler 1988):
ρ λ−1( 1)×ρ λ( 3)∝chl-a (18�8) where λ1 is in the red region and λ3 in the NIR region beyond 730 nm�
Another two-band model, which is different in its formulation from the previously mentioned two-band model (Equation 18�8), is (Gitelson 1992; Gitelson, Szilagyi, and Mittenzwey 1993)
ρ λ−1( 1)×ρ λ( 2)∝chl-a (18�9) where λ1 is in the red region and λ2 is in the region of the reflectance peak, around 700–710 nm� Recently, a four-band model was suggested (Le et al� 2009) for the estimation of chl-a concentration in productive waters with very high concentrations of inorganic suspended matter:
[ρ−1(662) –ρ−1(693)] [× ρ−1(740) –ρ−1(7 50 )]−1
The three-band NIR–red model was modified by including one more spectral band, ρ705, in order to reduce the effect of variations in scattering by suspended matter� The Medium Resolution Imaging Spectrometer (MERIS), the Moderate Resolution Imaging Spectro- radiometer (MODIS), and the Sea-Viewing Wide Field-of-View Sensor (SeaWiFS) are three commonly used spaceborne optical sensors, whose data may be used for the estimation of chl-a concentration using NIR–red models� The spectral bands in the red and NIR regions for these sensors are as follows:
MERIS—Spectral bands centered at 665 nm (band 7), 681 nm (band 8), 708 nm
•
(band 9), and 753 nm (band 10)
MODIS—Spectral bands centered at 667 nm (band 13), 678 nm (band 14), and
•
748 nm (band 15)
SeaWiFS—Spectral bands centered at 670 nm (band 6) and 765 nm (band 7)
•
The proximity of the 681-nm MERIS band and the 678-nm MODIS band to the chl-a flu- orescence wavelength at 685 nm means that the variable quantum yield of fluorescence (Dall’Olmo and Gitelson 2006) might affect the accuracy of chl-a concentration estimated using these bands� Therefore, these bands were eliminated as candidates for inclusion in NIR–red models for estimating chl-a concentration� Thus, the NIR–red models for estimating chl-a concentration using satellite data are as follows:
Three-band MERIS NIR–red model based on Equation 18�7:
chl-a∝[(ρband7)−1−(ρband9) ] (−1 × ρband10) (18�10) Two-band MERIS NIR–red model based on Equation 18�9:
chl-a∝(ρband7)−1×(ρband9) (18�11)
Two-band MODIS NIR–red model based on Equation 18�8:
chl-a∝(ρband13)−1×(ρband15) (18�12)
An appropriate equivalent for Equation 18�12 for SeaWiFS should be based on bands 6 and 7 of that sensor� Note that the two-band MERIS NIR–red model (Equation 18�11) is
fundamentally different from the two-band MODIS NIR–red model (Equation 18�12), yield- ing significantly different results�