Background Theory
2.2 Atmospheric Correction Problem Formulation
2.2.1 Optical Oceanography
Lw(λ)
cos(θ0)t0(λ, θ0) (2.9) where Lw(λ) is the water-leaving radiance just above the sea surface, R is the Earth-Sun distance at the time of measurement, and R0 is the mean Earth-Sun distance. The (R/R0)2contribution corrects for the Earth-sun distance, which can vary up to8 %over a year due to the Earth’s elliptical orbit [13]. t0(λ, θ0)is the diffuse transmittance in the Sun’s direction accounting for attenuation effects. The corresponding normalized water reflectance, [ρw(λ)]N, is given as:
[ρw(λ)]N ≡ π F0
[Lw(λ)]N = R
R0
2 π Lw(λ)
F0cos(θ0)t0(λ, θ0) (2.10) where the denominator more commonly is written asEd(λ, θ0), and given by:
Ed(0+, λ, θ0) =F0 R0
R 2
cos(θ0)t0(λ, θ0) (2.11) and is known as the spectral downward plane irradiance (radiant flux received by a surface per unit area).F0is the extraterrestrial solar irradiance and is expressed inW/m2. It can be seen as the value of solar power reaching the earth without and disturbance from the Earth’s atmosphere.
The goal with AC is to derive sea-level properties like water-leaving radiance,Lw(λ), from the measured TOA radiance,Lt(λ). A common procedure is to simulate and estimate the various surface and atmospheric-radiances terms in Eq. 2.4 and subtract them from the measured TOA radiance. AC algorithms vary in how they calculate the different terms in Eq. 2.5.
All the equations described so far give a quantitatively way of describing light itself, but it should further be explained how to describe and measure the optical properties of the medium where the light is propagating through. This is commonly known as optical oceanography.
2.2.1 Optical Oceanography
When it comes to aquatic research, the state of the water can be retrieved by looking at its optical properties [13]. There is a connection between the biological, chemical and geological constituents of natural waters and the optical properties. Optical properties of water can be divided into two mutually exclusive classes: Inherent Optical Properties (IOP) and Apparent Optical Properties (AOP). IOPs are properties that only depend upon the medium, thus are independent of the surrounding light field within the medium. The two fundamental IOPs are the volume scattering function and the absorption coefficient [13]. The properties that share the same properties as IOP, as well as the geometric (di-rectional) structure of the surrounding light filed, are categorized as AOP. Also, AOP must
display enough stability and features to be useful descriptors of the water body [13]. A widely used AOP is the spectral irradiance reflectance, which is given as:
R(z, λ)≡Eu(z, λ)
Ed(z, λ) (2.12)
whereEd andEudenotes the downwelling and upwelling irradiance, respectively, illus-trated in Fig. 2.7. Radiometric variables such as radiance and irradiance depend on IOPs, but fail to be stable enough to be categorized as AOP. Measurements at the ground ofEd andEuwould both change drastically if there would appear a cloud that would block the light, thus they do not separately categorize as AOPs. On the other hand, R(z,λ) for the same case would not change much, thus it is also regarded as an AOP. Other common AOPs are average cosines, diffuse attenuation coefficient, and different reflectances. In later years, the AOP of choice for remote sensing of ocean properties [27, 28] has been the spectral remote sensing reflectance given as:
Rrs(θ, φ, λ)≡ Lw(0+, θ, φ, λ)
Ed(0+, λ) (2.13)
where0+means thatLwandEdare evaluated just above the sea surface. Notice, that Eq.
2.10 and Eq. 2.13 would give the following relationship between the normalized water-leaving reflectance and the remote sensing reflectance: [ρw(λ)]N =πRrs(λ). Rrs(λ)is the desired property because it is less sensitive to environmental conditions, such as sun angle or sky conditions when compared to R(z,λ). This property is the ratio of how much of the downwelling irradiance that would penetrate the sea surface and be backscattered by oceanic constituents and returned through the surface onto a small solid angle∆Ω cen-tered on a particular direction (θ, φ)[13]. This is further illustrated in Fig. 2.8.
Z
( , ) ( , )
Figure 2.7:Illustration of the downwelling and upwelling irradiance.
( , , ) ( , )
ΔΩ
Figure 2.8:Illustration of the light rays contributing toRrs(λ).
Two things can happen when a photon interacts with matter, which are absorption and scat-tering. Absorption happens when the energy of the photon is being converted completely to another form like heat or energy contained in an energy bond. Scattering happens when the photon changes its direction and/or energy [1]. These two properties only depend on the water itself and the substances in it, and are regarded as the two fundamental IOPs [13]. More specifically, the two fundamental IOPs are called the absorption coefficient
2.2 Atmospheric Correction Problem Formulation and the volume scattering function. If a small volume∆V of water would be illuminated by a collimated beam of monochromatic light at some wavelengthλ, then by conservation of energy, this could be written as:
Φi(λ) = Φa(λ) + Φs(λ) + Φt(λ) (2.14) whereΦi(λ),Φa(λ),Φs(λ), andΦt(λ)are the incoming, absorbed, scattered and trans-mitted spectral radiant power (inW nm−1), respectively [13]. These terms are illustrated in Fig. 2.9.
Δ
( )
Φ Φ ( )
Δ
Δ Δ ΔΩ
( , ) Φ
( ) Φ
Figure 2.9:Illustration of absorption, scattering and transmission of an incoming beam with geometry used to define volume scattering and inherent optical properties.
Then the three properties absorptance, A(λ), scatterance, B(λ), and transmittance, T(λ), can be expressed as:
A(λ)≡ Φa(λ)
Φi(λ), B(λ)≡ Φs(λ)
Φi(λ), T(λ)≡Φt(λ)
Φi(λ) (2.15) where the A(λ)is the fraction of incident power that is absorbed within the volume, B(λ) is the fraction of incident power that is scattered out into all direction, T(λ)is fraction of non-interacting incident power and the sum of them should be 1 due to conservation of energy. The optical depth of a material is another used property in remote sensing and is defined as:
τ(λ) =ln
Φi(λ) Φt(λ)
=−ln(T(λ)). (2.16) whereτ(λ)is the atmospheric optical path along a vertical path with nadir viewing direc-tion for the sensor and includes all effects of atmospheric absorpdirec-tion and attenuadirec-tion by all atmospheric constituents [13]. For a general path (off-nadir viewing directionθ), the direct transmittance (from Eq. 2.5) would be:
T(θ, λ) =exp
− τ(λ) cos(θ)
. (2.17)
Aerosol optical depth (AOD) is the optical depth coming from the aerosols in the atmo-sphere and is a well used property in remote sensing. More precisely, it is a dimensionless number indicating how much aerosol there is along a vertical column of the atmosphere and yields the amount of the direct sunlight that is prevented from going through the at-mosphere.
In optical oceanography, the IOPs usually employed are the absorption and scattering co-efficients and single-scattering albedo, a(λ), b(λ) andω0(λ)respectively. The absorption and scattering coefficient is defined as the absorbance and scattering per unit distance in the medium, respectively, given as:
a(λ)≡ lim
∆r→0
∆A(λ)
∆r =dA(λ)
dr , b(λ)≡ lim
∆r→0
∆B(λ)
∆r =dB(λ)
dr [m−1]
(2.18) and the single scattering-albedo is given as:
ω0(λ)≡ b(λ)
c(λ) (2.19)
wherec(λ)is the beam attenuation coefficient defined as the sum of the absorption and scattering coefficient. ω0can be interpreted as the probability that a photon will be scat-tered rather than absorbed in any given interaction [13].
The second fundamental IOP is the volume scattering function (VSF),β(ψ, λ). To de-fine this property, two assumptions must hold. First, the medium must be isotropic, which would be reasonable for water where turbulence leads to randomly oriented particles. Sec-ond, the incoming light must be unpolarized. If these two assumptions hold, the scattering would only depend on the scattering angleψ, andB(ψ,λ) would be the fraction of the incident power scattered out of the beam through an angleψinto a solid angle∆Ωcentred onψ. Lastly, the volume scattering function (VSF) is given as:
β(ψ, λ)≡ lim
∆r→0 lim
∆Ω→0
B(ψ, λ)
∆r∆Ω = lim
∆r→0 lim
∆Ω→0
Φs(ψ, λ)
Φi(λ)∆r∆Ω (2.20) The VSF describes angular distribution of light that is scattered towards a direction, ψ, illustrated in 2.9, at a wavelengthλ. The previously defined scattering coefficient, b(λ), is a measure of the overall magnitude of the scattered light, without regard to its angular distribution [13]. It therefore follow that the scattering coefficient would be the integral of β(ψ, λ)over all solid angles (4π) [13]:
2.2 Atmospheric Correction Problem Formulation
b(λ) = Z
β(ψ, λ)dΩ = 2π Z π
0
β(ψ, λ)sinψ dψ (2.21)
whereψis the azimuth angle. In ocean optics, the scattering is often described by another property, known as the phase function [13]. The phase function,β, provides information˜ about the shape of the VSF regardless of the scattered light intensity and is defined as the VSF normalised to the total scattering, b, given as:
β˜≡ β
b. (2.22)
This property is often desired of oceanic waters, and instruments have been build to mea-sure them [13]. It can be interpreted as that if a scattering event has occurred,β˜dψ/4πis the probability that a light beam traveling in the directionΩˆis scattered into a cone of solid angle arounddψthe directionΩˆ0. HereΩˆandΩˆ0are unit vectors that are following the incoming direction ofΦi(λ)andΦs(λ)(shown in Fig. 2.9), respectively. It follows from the unity vectors that their dot products yields, cos(ψ) =Ωˆ ·Ωˆ0. It is also usually desired to have an analytic formula that approximates the shape of an actual phase function. The Fournier-Forand phase function is commonly used in oceanography for this case to serve as such an approximation, as it is more realistic than other phase functions that have been used in the past [13].
IOPs together with initial conditions of the environment can be used to achieve radio-metric variables and AOPs with radiative transfer theory.