Radiative transfer theory

Having defined the properties of the light field and the optical properties of the medium we are now in a position to ask whether it is possible to arrive, on purely theoretical grounds, at any relations between them. Although, given a certain incident light field, the characteristics of the underwater light field are uniquely determined by the properties of the medium, it is nevertheless true that explicit, all-embracing analytical relations, expressing the characteristics of the field in terms of the inherent optical properties of the medium, have not yet been derived. Given the complexity of the shape of the volume scattering function in natural waters (see Chapter 4), it may be that this will never be achieved.

It is, however, possible to arrive at a useful expression relating the absorption coefficient to the average cosine and the vertical attenuation coefficient for net downward irradiance. In addition, relations have been derived between certain properties of the field and the diffuse optical properties. These various relations are all arrived at by making use of the equation of transfer for radiance. This describes the manner in which radiance varies with distance along any specified path at a specified point in the medium.

Assuming a horizontally stratified water body (i.e. with properties everywhere constant at a given depth), with a constant input of monochromatic unpolarized radiation at the surface, and ignoring fluorescent emission within the water, the equation may be written

The term on the left is the rate of change of radiance with distance, r, along the path specified by zenith and azimuthal angles 0 and f, at depth z. The net rate of change is the resultant of two opposing processes: loss by attenuation along the direction of travel (c(z) being the value of the beam attenuation coefficient at depth z), and gain by scattering (along the path dr) of light initially travelling in other directions (0', f') into the direction 0, f (Fig. 1.6). This latter term is determined by the volume scattering function of the medium at depth z (which we write b(z, 0, f; 0', f0) to indicate that the scattering angle is the angle between the two directions 0, f and 0', f') and by the distribution of radiance, L(z, 0', f'). Each element of irradiance, L(z, 0', f')dffl(0', f') (where drn(0', f') is an element of solid angle forming an infinitesimal cone containing the direction 00, f0), incident on the volume element along dr gives rise to some scattered radiance in the direction 0, f. The total radiance derived in this way is given by

If we are interested in the variation of radiance in the direction 0, f as a function of depth, then since dr = dz/cos 0, we may rewrite eqn 1.55 as dL(z, d, f )

b(z ; e ; f ; e' ; f') L(z ; ^ ; f') d^ ; f') (1.56)

dz = - c(z)L(z ; 0 ; f ) + L*(z ; 0 ; f) By integrating each term of this equation over all angles

Gain by scattering into path

Gain by scattering into path

Loss by scattering out of path

Fig. 1.6 The processes underlying the equation of transfer of radiance. A light beam passing through a distance, dr, of medium, in the direction 0, 0, loses some photons by scattering out of the path and some by absorption by the medium along the path, but also acquires new photons by scattering of light initially travelling in other directions (0', 0') into the direction 0, 0.

Loss by scattering out of path

Fig. 1.6 The processes underlying the equation of transfer of radiance. A light beam passing through a distance, dr, of medium, in the direction 0, 0, loses some photons by scattering out of the path and some by absorption by the medium along the path, but also acquires new photons by scattering of light initially travelling in other directions (0', 0') into the direction 0, 0.

.4p dz we arrive at the relation dz c(z)L(z, 6 , f )do +

originally derived by Gershun (1936). It follows that and

Thus we have arrived at a relation between an inherent optical property and two of the properties of the field. Equation 1.60, as we shall see later (§ 3.2), can be used as the basis for determining the absorption coefficient of a natural water from in situ irradiance and scalar irradi-ance measurements.

Exploration of the properties of irradiance-weighted vertical attenuation coefficients (defined in §1.3, above) has shown717 that the following relationships, analogous to the Gershun equation, also exist

where WKE(av) and WK0(av)are the irradiance-weighted vertical attenuation coefficients for net downward and scalar irradiances, respectively, ~pc is the integral average cosine for the whole water column, and ^(0) is the average cosine for the light field just below the water surface.

Preisendorfer (1961) has used the equation of transfer to arrive at a set of relations between certain properties of the field and the diffuse absorption and scattering coefficients. One of these, an expression for the vertical attenuation coefficient for downward irradiance,

we will later (§ 6.7) find of assistance in understanding the relative importance of the different processes underlying the diminution of irradiance with depth.

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