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Taking eqns 1.16 to 1.18 to constitute an alternative definition of pc, then an appropriate alternative name for the average cosine of all the photons in the water column would be the integral average cosine of the underwater light field.

The remaining parameter that provides information about the angular structure of the light field is the irradiance reflectance (sometimes called the irradiance ratio), R. It is the ratio of the upward to the downward irradiance at a given point in the field

In any absorbing and scattering medium, such as sea or inland water, all these properties of the light field change in value with depth (for which we use the symbol z): the change might typically be a decrease, as in the case of irradiance, or an increase, as in the case of reflectance. It is sometimes useful to have a measure of the rate of change of any given property with depth. All the properties with which we have dealt that have the dimensions of radiant flux per unit area, diminish in value, as we shall see later, in an approximately exponential manner with depth. It is convenient with these properties to specify the rate of change of the logarithm of the value with depth since this will be approximately the same at all depths. In this way we may define the vertical attenuation coefficient for downward irradiance

upward irradiance

Ku dln Ed dz dln Eu dz

Ed dz

1_àEu Eu dz net downward irradiance

radiance

Eo dz

In recognition of the fact that the values of these vertical attenuation coefficients are to some extent a function of depth they may sometimes be written in the form K(z). For practical oceanographic and limnological purposes it is often desirable to have an estimate of the average value of a vertical attenuation coefficient in that upper layer (the euphotic zone) where light intensity is sufficient for significant photosynthesis to take place. A commonly used procedure is to calculate the linear regression coefficient of ln E(z) with respect to depth over the depth interval of interest (§5.1). Choice of the most appropriate depth interval is inavoid-ably somewhat arbitrary. An alternative approach is to use the irradiance values themselves to weight the estimates of the irradiance attenuation coefficients.717 This yields K values applicable to that part of the water column where most of the energy is attenuated. If we indicate the irradi-ance-weighted vertical attenuation coefficient by wK(av) then wK(av) =

E(z)dz where E(z) can be Ed(z), Eu(z), E(z), or E0(z) and K(z) can be Kd(z), Ku(z), KE(z) or K0(z), respectively. The meaning of eqn 1.25 is that when we calculate an average value of K by integrating over depth, at every depth the localized value of K(z) is weighted by the appropriate value of the relevant type of irradiance at that depth. The integrated product of K(z) and E(z) over all depths is divided by the integrated irradiance over all depths.

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