Chapter 2 Marine Optics and Radiometry
Radiance,L, in units of W m−2 sr−1, is the area and solid angle density of radiant flux L= d 2Φ dωds = d2Φ dωds0cosθ (2.1.3) whereds=ds0cosθ is a quantity called projected area in the direction of propagation onto the plane perpendicular to this direction, dω is an element of solid angle in the specified direction and θ is the angle between this direction and the normal to the surface at the specified point (see Fig. 2.1). Thus radiance is a function of both position and direction. The integral of all radiance elements over a solid angle Ω, gives the irradiance.
Figure 2.1: Concept of radiance.
The solid angle, which is a basic quantity in radiomery, is formed by the straight lines from a single point (the vertex) and it is defined as the area intercepted on the surface of a unit hemisphere centered at the vertex (Wyatt, 1978). Considering the element of solid angle dω in units of sr in spherical coordinates on a unitary sphere,
Chapter 2 Marine Optics and Radiometry
with φ and θ indicating the azimuth and zenith angles,
dω= ds
r2 =sinθdθdφ, (2.1.4) the solid angle Ω is determined by integrating dω according to
Ω = Z 2π 0 Z θ 0 sinθdθdφ= 2π(1−cosΘ) (2.1.5) where Θ is half-angle of the circular cone defining Ω.
Irradiance and radiance are commonly expressed as spectral quantities. Spectral irradiance,E(λ), in units of W m−2 nm−1, is the spectral concentration of irradiance
E(λ) = dE
dλ = d2Φ
ds0dλ
. (2.1.6)
Spectral radiance, L(λ), in units of W m−2 sr−1 nm −1, is the spectral concen- tration of radiance L(λ) = dL dλ = d3Φ dωds0cosθdλ . (2.1.7)
The fundamental relationship linking E(λ) and L(λ) for a point at the surface on which they are defined is
E(λ) =
Z
Ω
L(θ, φ, λ)cosθdω. (2.1.8) When considering the whole hemispherical solid angle, i.e., Ω = 2π,
E(λ) = Z 2π 0 Z π/2 0 L(θ, φ, λ)cosθsinθdθdφ. (2.1.9) If L(θ, φ, λ) is constant (i.e., isotropic) over the range of integration, then
E(λ) =πL(λ). (2.1.10) Relevant to marine optics is the law of radiance invariance at an interface. This describes the change in radiance distribution across two media of refractive indices
Chapter 2 Marine Optics and Radiometry
Figure 2.2: Geometry of light refraction.
n1 and n2, assuming radiance is not absorbed or scattered at the interface between the two media. Explicitly, if ρ is the reflectance for the given angle of incidence at the interface for the radiance L1 in the medium with refractive index n1, then the fraction of radiance that enters the medium with refractive index n2 is L2 = (1−ρ)L1(n2
2/n21). Following McCluney (1994) and making reference to Fig. 2.2, this can be demonstrated by introducing the elements of fluxdΦ1 anddΦ2for the radiance terms L1 and L2
dΦ1 =L1 cosθ1 ds dω1 =L1 ds cosθ1 sinθ1 dθ1 dφ (2.1.11) and
Chapter 2 Marine Optics and Radiometry
The element of transmitted flux is then
(1−ρ)dΦ1 = (1−ρ)L1dscosθ1sinθ1dθ1dφ. (2.1.13) Being (1−ρ)dΦ1 =dΦ2 for conservation of energy,
(1−ρ) L1 cosθ1 sinθ1 dθ1
L2 cosθ2 sinθ2 dθ2
= 1. (2.1.14) Now differentiating Snell’s law with respect to angles (Snell’s law provides the rela- tionship linking the angle of incidence and refraction for rays crossing media with different refractive indices),
sinθ1 sinθ2 = cosθ1 dθ1 cosθ2 dθ2 = n2 n1 . (2.1.15) Combining (2.1.15) with (2.1.14) (1−ρ)L1 n2 1 = L2 n2 2 . (2.1.16)
This relationship states that when ignoring reflection losses (i.e., ρ = 0), the ratio
L/n2 remains invariant for a light beam crossing the interface between two media with different refractive indices.
Specific spectral radiometric quantities of relevance for marine optics are the up- welling radiance Lu(z, λ) at depth z and wavelength λ, the downward irradiance
Ed(z, λ), and the upward irradiance Eu(z, λ). Additional radiometric quantities are
the scalar upwardanddownward irradianceE0u(z, λ) andE0d(z, λ) which result from
the integral of radiance contributions over a hemispheric surface (different from the plane surface considered in Eq. 2.1.8 for irradiance). These latter quantities, in units of W m−2 nm−1, indicate the volume density of radiant flux. They are related to
Chapter 2 Marine Optics and Radiometry
(Aas and Højerslev, 1999; Voss, 1989). They however are not considered in the fol- lowing chapters focussed on the marine radiometric quantities more strictly related to remote sensing applications.
2.1.2
Optical Properties of Seawater
Following Preisendorfer (1960, 1976), the optical properties of seawater are divided intoinherentandapparent. Inherent optical properties (IOP’s) are those depending on the medium only and not on the illumination conditions. Apparent optical properties (AOP’s) are those depending on both IOP’s and illumination conditions.
a. Inherent Optical Properties
The most common IOP’s are the spectral absorption coefficient, scattering coef- ficient, beam attenuation coefficient, single scattering albedo, volume scattering func- tion, back-scattering coefficient, scattering phase function.
Theabsorption coefficient,a(λ) in units of m−1, is the absorbance (i.e., the ratio of the absorbed to incident spectral radiant flux of a narrow collimated monochromatic beam in a elementary volume of a given medium) per unit distance.
The scattering coefficient,b(λ) in units of m−1, is the scatterance (i.e., the ratio of the scattered to incident spectral radiant flux of a narrow collimated monochromatic beam in a elementary volume of a given medium) per unit distance.
The beam attenuation coefficient,c(λ) in units of m−1, is given by
c(λ) =a(λ) +b(λ). (2.1.17) The single scattering albedo,ω0(λ), is dimensionless and given by
Chapter 2 Marine Optics and Radiometry
The volume scattering function, β(ψ, λ) in units of m−1 sr−1, is the scattered intensity per unit incident irradiance per unit volume with ψ indicating the angle of scattering with respect to the direction of the incident light. The scattering coefficient,
b(λ), is obtained by integrating β(ψ, λ) over all directions
b(λ) = Z Ω β(ψ, λ)dω = 2π Z π 0 β(ψ, λ)sinψdψ. (2.1.19) When restricting the integration ofβ(ψ, λ) to the intervalπ/2≤ψ ≤π, the result provides the back-scattering coefficient, bb(λ) in units of m−1.
Additional relevant IOP is the scattering phase function, βe(ψ, λ) in units of sr−1, which provides the angular distribution of the scattered light and is given by
e
β(ψ, λ) =β(ψ, λ)/b(λ). (2.1.20)
b. Apparent Optical Properties
Frequently measured and applied AOP’s are the spectral irradiance reflectance, remote sensing reflectance,normalized water-leaving radiance,diffuse attenuation co- efficient and the so calledQ-factor.
The dimensionless irradiance reflectance at depth z, R(z, λ), defined as the ratio of upward to downward irradiance, is given by
R(z, λ) =Eu(z, λ)/Ed(z, λ) (2.1.21)
The value of R(z, λ) at the so called depth z=0− just below the water surface, i.e.,
R(0−, λ), has particular relevance in marine optics and is determined usingE
u(0−, λ)
andEd(0−, λ) obtained from the extrapolation toz=0−of the log-transformedEu(z, λ)
and Ed(z, λ) measured at multiple depths z, respectively.
The remote sensing reflectance, Rrs(λ) in units of sr−1, is defined as
Chapter 2 Marine Optics and Radiometry
whereLW(λ) is the so calledwater-leaving radiancein units of W m−2 nm−1 sr−1,
i.e., the radiance leaving the sea and quantified just above the surface, and Ed(0+, λ)
the above water downward irradiance. In practice LW(λ) is determined as
LW(λ) = 0.543·Lu(0−, λ) (2.1.23)
where the factor 0.543 (Jerlov and Nielsen, 1974) accounts for the reduction in ra- diance from below to above the water surface (mostly due to the change in the re- fractive index at the air–water interface), andLu(0−, λ) is the upwelling radiance just
below the water surface extrapolated from log-transformedLu(z, λ) values at multiple
depths z.
The normalized water leaving radiance, Lwn(λ) in units of W m−2 nm−1 sr−1, is
then defined as
Lwn(λ) = Rrs(λ)ES(λ) (2.1.24)
with ES(λ) average extra-atmospheric sun irradiance.
This normalizationprocess removes fromLW(λ) the effects of illumination condi-
tion dependent from the sun zenith angle and the atmospheric transmittance (Mueller and Austin, 1995). Specifically, for both remote sensing reflectance and normalized water-leaving radiance, the minimization of the effects of illumination condition is ob- tained through normalization with respect to the above-water downward irradiance.
The diffuse attenuation coefficient,Kd(λ) in units of m−1, indicates the extinction
of irradiance in the water column and is determined as the slope term from the regression of the log-transformed Ed(z, λ) as a function of depthz.
The Q-factor, Q(θ, φ, z, λ) in units of sr, quantifies the light distribution in the water and is the ratio of the upward irradiance to the upwelling radiance in the generic
Chapter 2 Marine Optics and Radiometry
direction (φ, θ). Commonly used quantity in marine optics isQn(0−, λ), the Q-factor
at nadir view and depth 0−, defined as
Qn(0−, λ) =Eu(0−, λ)/Lu(0−, λ). (2.1.25)