A theoretical derivation of the dependence of the remotely sensed reflectance of the ocean on the inherent optical properties

Abstract

An expression for the ratio of the upwelling nadir radiance L (π, z ) and the downwelling scalar irradiance E od (Z) is derived from the following equation of radiative transfer. This expression is given by RSR( z ) = [ L (π, z )]/ E od ( z ) = [ ƒ b ( z ) b b ( z )]/2π[ k (π, z ) + c ( z ) − ƒ L ( z ) b f ( z )], where b b ( z ) is the backscattering coefficient, k (π, z ) is the vertical attenuation coefficient of the nadir radiance, c ( z ) is the beam attenuation coefficient, and ƒ b ( z ) and ƒ L ( z ) are shape parameters that depend on the shape of the volume scattering function and the radiance distribution. Successive approximations are subsequently applied to the above exact equation. These are ƒ b ( z ) = [2πβ(π − θ m , z )]/[ b b ( z )], where β(π − θ m , z ) is the volume scattering function at 180° minus the zenith angle of the maximum radiance, and k (π, z ) = am = c [1 − 0.52 b/c − 0.44 ( b/c ) 2 ], where m is a parameter that is numerically equal to the inverse of the average cosine of the asymptotic light field for a medium with the same inherent optical properties, α is the absorption coefficient, and b/c is the single scattering albedo. Together with ƒ L ( z ) = 1.05 and application of Gershun's equation, it is shown that for nearly all oceanic cases RSR( z ) ≡ L (π, z )/ E od ( z ) = [β(π − θ m , z )]/{ a ( z )[1 + m ( z )]}.