The vertical attenuation coefficients for radiance, irradiance and scalar irradiance are, strictly speaking, properties of the radiation field since, by definition, each of them is the logarithmic derivative with respect to depth of the radiometric quantity in question. Nevertheless experience has shown that their values are largely determined by the inherent optical properties of the aquatic medium and are not very much altered by changes in the incident radiation field such as a change in solar elevation.59 For example, if a particular water body is found to have a high value of Kd then we expect it to have approximately the same high Kd tomorrow, or next week, or at any time of the day, so long as the composition of the water remains about the same.

Vertical attenuation coefficients, such as Kd, are thus commonly used, and thought of, by oceanographers and limnologists as though they are optical properties belonging to the water, properties that are a direct measure of the ability of that water to bring about a diminution in the appropriate radiometric quantity with depth. Furthermore they have the same units (m-1) as the inherent optical properties a, b and c. In recognition of these useful aspects of the various K functions, Preisendorfer (1961) suggested that they be classified as apparent optical properties (AOP) and we shall so treat them in this book. The reflectance, R, is also often treated as an apparent optical property of water bodies.

The two fundamental inherent optical properties - the coefficients for absorption and scattering - are, as we saw earlier, defined in terms of the behaviour of a parallel beam of light incident upon a thin layer of medium. Analogous coefficients can be defined for incident light streams having any specified angular distribution. In particular, such coefficients can be defined for incident light streams corresponding to the upwelling and downwelling streams that exist at particular depths in real water bodies. We shall refer to these as the diffuse absorption and scattering coefficients for the upwelling or downwelling light streams at a given depth. Although related to the normal coefficients, the values of the diffuse coefficients are a function of the local radiance distribution, and therefore of depth.

The diffuse absorption coefficient for the downwelling light stream at depth z, ad(z), is the proportion of the incident radiant flux that would be absorbed from the downwelling stream by an infinitesimally thin horizontal plane parallel layer at that depth, divided by the thickness of the layer. The diffuse absorption coefficient for the upwelling stream, au(z), is defined in a similar way. Absorption of a diffuse light stream within the thin layer will be greater than absorption of a normally incident parallel beam because the pathlengths of the photons will be in proportion to 1 /Md and 1 /Mu, respectively. The diffuse absorption coefficients are therefore related to the normal absorption coefficients by where Md (z) and Mu(z) are the values of Md and Mu that exist at depth z.

So far as scattering of the upwelling and downwelling light streams is concerned, it is mainly the backward scattering component that is of importance. The diffuse backscattering coefficient for the downwelling stream at depth z, bbd(z), is the proportion of the incident radiant flux from the downwelling stream that would be scattered backwards (i.e. upwards) by an infinitesimally thin, horizontal plane parallel layer at that depth, divided by the thickness of the layer: bbu(z), the ad(z) =

corresponding coefficient for the upwelling stream is defined in the same way in terms of the light scattered downwards again from that stream. Diffuse total (bd(z), bu(z)) and forward (bfd(z), bfu(z)) scattering coefficients for the downwelling and upwelling streams can be defined in a similar manner. The following relations hold bd (z) = b / ~Pd (z); bu (z) = b / Mu (z)

bd (z) = bfd (z) + bbd (z), bu (z) = bfu(z) + bbu(z)

The relation between a diffuse backscattering coefficient and the normal backscattering coefficient, bb, is not simple but may be calculated from the volume scattering function and the radiance distribution existing at depth z. The calculation procedure is discussed later (§ 4.2).

Preisendorfer (1961) has classified the diffuse absorption and scattering coefficients as hybrid optical properties on the grounds that they are derived both from the inherent optical properties and certain properties of the radiation field. I prefer the term quasi-inherent optical properties, on the grounds that it more clearly indicates the close relation between these properties and the inherent optical properties. Both sets of properties have precisely the same kind of definition: they differ only in the characteristics of the light flux that is imagined to be incident upon the thin layer of medium.

The important quasi-inherent optical property, bbd(z), can be linked with the two apparent optical properties, Kd and R, with the help of one more optical property, k(z), which is the average vertical attenuation coefficient in upward travel from their first point of upward scattering, of all the upwelling photons received at depth z.710 k(z) must not be confused with, and is in fact much greater than, Ku(z), the vertical attenuation coefficient (with respect to depth increasing downward) of the upwelling light stream. Using k(z) we link the apparent and the quasi-inherent optical properties in the relation

At depths where the asymptotic radiance distribution is established (see § 6.6) this relationship holds exactly. Monte Carlo modelling of the underwater light field for a range of optical water types710 has shown that k is approximately linearly related to Kd, the relationship at zm (a depth at which irradiance is 10% of the subsurface value) being k(zm)«2.5 Kd (zm) (1.52)

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