The Radiation Field

The fundamental properties describing the radiation field are defined in precise physical terms, but their symbolic notations in the literature are far from universal. To minimize confusion and to adhere to the increasingly popular conventions emerging from the field of hydrologic optics, the symbols and definitions used here (Table 1) will conform to those of Kirk (1994) and Mobley (1994). The reader is encouraged to consult these excellent texts for a significantly deeper understanding of the fundamental concepts of hydrologic optics introduced here. Although values for the terms and functions defined below are spectrally dependent, the parenthetical notation (A) has been omitted from the equations for simplicity. Direction within the light field is generally expressed in terms of the zenith, nadir, and

A. WD. Larkum et al. (eds.), Seagrasses: Biology, Ecology and Conservation, pp. 295-301. © 2006 Springer. Printed in the Netherlands.

Fig. 1. A schematic diagram of the various processes that contribute to the signal as measured by a remote sensor in optically shallow water where the substrate has a significant effect on the water-leaving radiance (source: Dekker et al., 2001).

azimuth angles (9, 9n, and 0, respectively) as illustrated in Fig. 2.

The radiant flux represents the time rate of flow of radiant energy, and is denoted by the symbol O. Radiant flux can also be expressed in molar units, where 1 mol = 6.02 x 1023 photons (Avogadro's number). According to Kirk (1994), terms such as 'fluence rate' or 'photon fluence rate' should be avoided when describing properties of the radiation field. The field radiance (L) is the radiant flux projected onto a surface dA oriented at right angles to the incident beam (Fig. 2) and has units of W (or quanta s-1) m-2 steradian-1. The surface radiance is the radiant flux in a given direction per unit solid angle projected onto a horizontal surface (dS) oriented at some angle (9, 0) to the beam. It is denoted by the symbol L (9, 0) and also has units of W (or quanta s-1) m-2 steradian-1. The surface radiance is related to the field radiance according to the Cosine Law because the horizontal surface area (d£) is defined by cA.

Irradiance is the total radiant flux incident upon a surface of defined area. It is denoted by E, and has units of W (or quanta s-1 ) m-2. The scalar irradiance

Abbreviations: See Table 1 for a list of symbols, definitions, and units used in this chapter.

(Eo) integrates the radiance distribution equally over all directions of a sphere:

Scalar irradiance can be partitioned into downward and upward scalar irradiances, but it will be more useful to consider downward and upward plane irradiances in the discussion of seagrass-light interactions found in Chapter 13. The downward and upward plane irradiances (Ed and Eu) illuminate the upper and lower faces of a surface, respectively. Like scalar irradiances, they are obtained by integrating the radiance over all solid angles (m) of the upper and lower hemispheres, separately. Unlike scalar irradiances, however, plane irradiances are affected by the Cosine Law, which simply states that the irradi-ance incident on a plane surface is proportional to the angle between the photon direction and the surface normal (Fig. 2). Hence:

Table 1. Symbols, definitions, and units of some common terms used in hydrologie optics.

Symbol

Definition

Dimensions

Fundamental quantities

<P

Azimuth angle

deg or rad

e

Polar angle

deg or rad

On

Zenith polar angle

deg or rad

en

Nadir polar angle

deg or rad

f1

Cosine of polar angle

Dimensionless

Z

Depth

m

Radiometric quantities

L

Radiance

W (or quanta s-

) m-

-2 -1 2 sr 1 nm

L u

Upwelling radiance

W (or quanta s-

)m-

2 -1 2 sr 1 nm

L w

Water-leaving radiance

W (or quanta s-

) m-

2 -1 2 sr 1 nm

E

Irradiance

W (or quanta s-

)m-

2 -1 2 nm 1

Eo

Scalar irradiance

W (or quanta s-

) m-

2 -1 2 nm 1

Ed

Downwelling plane irradiance

W (or quanta s-

)m-

2 -1 2 nm 1

Eu

Upwelling plane irradiance

W (or quanta s-

) m-

2 -1 2 nm 1

Inherent optical properties

A

Beam absorptance

Dimensionless

B

Beam scatterance

Dimensionless

T

Beam transmittance

Dimensionless

D

Optical density

Dimensionless

a

Beam absorption coefficient

m-1

b

Beam scattering coefficient

m-1

c

Beam attenuation coefficient

m-1

P

Volume scattering function

m-1 sr-1

Apparent optical properties

R

Irradiance reflectance

Dimensionless

Rrs

Remote sensing reflectance

sr-1

Kd

Coefficient of downwelling attenuation

m-1

Ku

Coefficient of upwelling attenuation

m-1

Md

Average cosine of downwelling irradiance

Dimensionless

An

Average cosine of upwelling irradiance

Dimensionless

Similarly, the upward plane irradiance is obtained by integrating the radiance over all solid angles of the lower hemisphere, with respect to the Cosine Law:

Application of the Cosine Law means that Eo is always greater than the sum (Ed + Eu).

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