Atmospheric correction

The total reflectance measured by the sensor at the top of atmosphere (TOA) was obtained from the calibrated Digital Numbers (DNs) (Chander et al., 2009), following the methodology explained in Appendix F (See Appendix F in the Index).

The TOA at a given wavelength λ , ,can be written as the sum of several components (Gordon, 1997): (3.3) Where Tg λ and Tv λ

are, respectively, the gaseous transmittance and the viewing diffuse transmittance from the water to the sensor; ρsfc(λ) is the sea-surface reflectance ; ρa(λ) is the aerosol reflectance resulting from multiple scattering by aerosol in the absence of air; ρr(λ) is Rayleigh reflectance resulting from multiple scattering by air molecules in the absence of aerosol; ρra(λ) is the reflectance from the interaction between air molecules and aerosol; ρw(λ) is the water-leaving reflectance resulting from the interaction between the light and the water column. To derive the water-leaving reflectance, ρw(λ), all other terms of Eq.3.3 must be quantified.

Gaseous transmittance Tgλ is calculated from ancillary data on ozone and water vapor concentrations by using the transmittance models of Goody (1964) and Malkmus (1967). For viewing angles lower than 60º, the viewing diffuse transmittance Tvλ is weakly dependent on aerosol and can be approximated following Gordon et al. (1983).

The diffuse transmittance Tvλ is approximated following the model of Wang (1999). The coupled term ρra(λ) can be neglected at the NIR part of the spectrum (Gordon and Castano, 1987). A detailed description of every step can be found in Salama and Chen (2010) and Salama et al. (2012). Rayleigh reflectance ρr(λ)is calculated from geometry (sun and sensor zenith and azimuth angles) and atmospheric pressure (Gordon et al., 1988a) as described below:

Where pr is the Rayleigh scattering phase function and is obtained from its relationship with the scattering angle:

Where Ψ is the angle between observed and reflected light and can be written as Gordon(1997):

Where θo , θ are sun zenith and satellite viewing angles and Фo , Ф the sun and satellite azimuth angles in radians, respectively.

The Rayleigh optical thickness τr at altitude z can be calculated from Handsen and Travis (1974):

Where λ is wavelength in μm and P(0)=1013.25mb is standard atmospheric pressure at sea level. P(z) is atmospheric pressure at altitude z that is obtained from a solution of pressure-elevation relationship.

toz is two way ozone atmospheric transmittance given by Viollier et al. (1980):

Where k(λ) is the ozone absortion coefficient from Neckel et al (1981) and U is the ozone total column content in cm-atm (http:/jwocky.gsfc.nasa.gov).

Sea-surface reflectance, ρsfc (λ)

, is estimated knowing than ρsfc (λ) = ρdr (λ) + ρdf (λ) ; and ρdf (λ) is assumed negligible. ρdr(λ) can be derive using de model of Cox and Munk (1954) based on Fresnel reflectance equation

Where η, η0 and ηn are the cosines of the observation θ, solar zenith θ0, and the normal of the facet θn angles respectively.

θr and θt (Eq. 3.11) are the zenith angle of the reflected and the transmitted light, respectively.

Basically, Eq.3.3 has two unknowns: the water-leaving reflectance, ρw(λ), and the aerosol reflectance, ρa(λ). The Rayleigh corrected reflectance is computed from

.

Aerosol scattering reflectance, ρa(λ) is directly estimated from the aerosol ratio at two NIR bands, s and l(s short, l long wavelengths), , which can be related to the aerosol optical thickness and type, and is usually considered to be constant in the whole image (Salama and Copin, 2004).

This ratio can be estimated in different ways (Salama and Chen, 2010). Ruddick et al. (2000), used visual inspection of the reflectance‟s scatter plot for its estimation. Hu et al. (2000), determined the values of aerosol ratio over clear water pixels and extrapolated them to adjacent turbid water pixels. Salama et al. (2004) suggested an automated approach to determine the aerosol ratio based on eigenvector decomposition of the NIR bands (Eq.3.12):

where Csl is the correlation between Rayleigh corrected reflectance of short band (s) and long band (l) of Landsat-ETM.; Cll is the correlation between Rayleigh corrected reflectance of long band(l) and long band (l) and Css is the correlation between Rayleigh corrected reflectance of short band(s) and short band (s). In this work, this approach was applied to data from Landsat-ETM bands 4(837 nm) and band 5 (1625 nm) (Danbara, 2014).

Once the value of ε(4,5) is estimated from Eq.3.12, and adopting the standard approach by Gordon and Wang (1994) which assumes that the water-leaving reflectance is negligible in the NIR (ρw(5)=0) for clearest waters, ρa(4)and ρw(4)can be derived from Eq.3.3:

(3.14)

(3.15)

ρw(5)=0 (3.16)

It should be noted that this approach is not directly applicable to high turbidity conditions (Kuchinke et al., 2009); in such cases, a second parameter must be included as the ratio of water-leaving reflectances (α) at band 4 and 5 (Carder et al., 1999a.b), that can be approximated as Gould (1999):

Where and represent the pure water absortion coefficients at 4 and 5 band, respectively. These values can be obtained from the work by Palmer et al. (1974). This ratio, , is assumed to be a constant value in space and time (Newkermans et al., 2009). ρw(4) and ρw(5) can then be obtained from the TOA reflectance, provided that good estimates of the ratios ε(4,5) and α(4,5) are available.

However, the great spatial variability of SPM in the study area, which is accentuated on certain dates coinciding with extreme turbidity events, makes it difficult to employ a spatially homogeneous ε(4,5) for the whole image and justifies the use of different ε(4,5) values for given conditions (Ruddick et al, 2000). In this work, different regions were identified in the corrected reflectance at band 4 versus corrected reflectance at band 5 plots to derive specific values for each group. The calculation of the ρw(4) for these areas

is made using a specific ε(4,5) for each of them and the most suitable algorithm depending upon water is clear or turbid.

The whole process followed to derive water leaving reflectance values from Landsat ETM images is resumed in Fig. 3.2.

Landsat ETM+ images Conversion of DN to Spectral Radiance Lλ Conversion of Spectral radiance to TOA Reflectance geometry & atmospheric pressure Rayleigh reflectance Obtention of Rayleigh corected reflectance Turbid water α (s,l) spatially homegeneous Yes Aerosol reflectance ρw (λ5)=0 No

Water leaving reflectance

Figure 3.2. Flowchart of the obtaining of water leaving reflectance values from Landsat ETM images at the study site.