Inherent Optical Properties

Consider a beam of light incident on a small volume AV of medium (e.g. water or a plant leaf) with a thickness Ar (Fig. 3). Some of the beam's radiant Fig. 2. Definitions of radiance and angular orientation at a point in space (after Kirk, 1994). L(9, 0) is the radiance incident on area dA at zenith angle 9 and azimuth angle 0, given the radiant flux \$ into the small solid angle d&>. The Cosine Law is illustrated by the relationship between the area of illumination normal to d&>, defined by dA, and the surface-normalized area of illumination, cA.

Fig. 2. Definitions of radiance and angular orientation at a point in space (after Kirk, 1994). L(9, 0) is the radiance incident on area dA at zenith angle 9 and azimuth angle 0, given the radiant flux \$ into the small solid angle d&>. The Cosine Law is illustrated by the relationship between the area of illumination normal to d&>, defined by dA, and the surface-normalized area of illumination, cA.

flux is absorbed within the medium (Oa), some is scattered out of the beam (Os), and some is transmitted unaltered through the medium (Ot). The beam absorptance (A) represents the fraction of the incident radiant flux absorbed by the medium: Fig. 3. Interaction of a beam of light with a thin optically active layer. The flux incident (\$i) on the medium of thickness Ar is dissipated by scattering out of the path (\$s) and by absorption (\$a) within the medium. The remaining flux (\$t) is transmitted out of the medium.

The beam scatterance (B) is the fraction of incident radiant flux scattered out of the beam:

Finally, the beam transmittance (T) is the fraction emerging from the medium:

It thus follows that A + B + T = 1 because these terms represent dimensionless ratios normalized to the incident flux. In hydrologie optics, the absorption and scattering coefficients have dimensions of inverse meters (i.e. m-1 ). Thus, the beam absorption and scattering coefficients are defined as the depth derivatives of the absorptance and scatterance over an infinitesimally small distance (Ar):

ArAr

Ar^0 Ar

Beam attenuation is defined by summing the absorption and scattering coefficients:

If we assume B = 0, as in the case of a transparent chemical solution subjected to spectrophotometric analysis, the transmittance represents the fraction of incident light that was not absorbed. In that case, the absorbance, or optical density, is defined as:

D s logio Tt

It follows that:

Unlike the transparent solution mentioned above, however, scattering has a major impact on the fate and distribution of light in natural waters. Scattering impedes vertical light penetration by prolonging the path length, which increases the probability of absorption. Scattering also propagates photons back into the direction of the down-welling light. Earth observation sensors measure the backscattered photons that survive the water column, the air-water interface and the atmosphere. In most cases, the aim of applying earth observation is to reconstruct the constituents causing that backscattering.

The way in which scattering affects the penetration of light into the medium depends on the angular distribution of the scattered flux. This angular distribution has a characteristic shape for any given medium and is specified in terms of the normalised volume scattering function (P). The scattering coefficient b can be obtained by integration of P over all directions (solid angles). The measurement of P, however, is not trivial and b is routinely calculated (not measured) from easier-to-obtain measurements of beam absorption and attenuation (b = c — a).

It is important to distinguish between forward vs. downward scattering and upward vs. backward scattering. These pairs are identical only for vertically incident light (sun at zenith of 0°) and a flat water surface. As soon as we deviate from these circumstance forward and backward scattering describe the scattering processes with respect to the angle of the incident light (which may be e.g. sunlight at 10, 20,... 80° from zenith) whereas upward and downward scattering describe the fractions of incident light (at any incident angle) that are scattered relative to the surface normal (Fig. 4). Although the mathematics are significantly more complicated for scattering than for absorption, their bulk effects on Fig. 4. Definitions of scattering direction. Forward and backward scattering are defined relative to the direction of the incident beam. Downward and upward scattering are defined relative to a horizontal surface. Thus, the upwelling light detected by remote sensing is derived from light scattered in both forward and backward directions.

Fig. 4. Definitions of scattering direction. Forward and backward scattering are defined relative to the direction of the incident beam. Downward and upward scattering are defined relative to a horizontal surface. Thus, the upwelling light detected by remote sensing is derived from light scattered in both forward and backward directions.

light attenuation in natural waters are often summarized by the average cosine (¡1), an apparent optical property discussed below.

The volume scattering function describes an elastic process in which the scattered photon has the same wavelength and polarization as the incident photon. Inelastic scattering, by contrast, implies a change in the wavelength and/or polarization of the scattered photon. Raman scattering and fluorescence represent two types of inelastic scattering that may be relevant in natural waters. Both processes cause a shift to longer wavelengths. Raman scattering, however, is virtually instantaneous. Although it involves some transfer of energy from the photon to the target molecule (hence the wavelength shift), the scattered photon retains its characteristic incident polarization. Raman scattering in natural waters results primarily from photon interactions with water molecules, which makes it a relatively constant factor that influences the underwater light slightly between 550 and 650 nm. Fluorescence is an absorption-emission process that requires at least 10-9 s (a long time in the particle physics world) and the emitted photon is unpolarized with respect to the incident photon. Fluorescent substances in natural waters include dissolved organic matter and photosynthetic pigments contained in phytoplank-ton, seagrasses, macrophytes, and symbiotic algae. Typically 1-5% of the photosynthetically absorbed photons are emitted by chlorophyll as fluorescence in a 25 nm band centered at 685 nm (Falkowski and Raven, 1997).