The total radiant energy, and the average depth of all the photons, in the water column

Using radiative transfer theory I show that for a given waveband, the total radiant energy, Q(t), present within the water column (per square meter of surface) is proportional to the scalar irradiance just below the surface [E-0(0)] and inversely proportional to the irradiance-weighted vertical attenuation coefficient for scalar irradiance (K-w(0, av)), in accordance with Q(t) = E-0(0) /c(w)(w)K(0,av) where c(w) is the speed of light in the water. Further, Q(t) = (E) over right arrow (0)/c(w)a where (E) over right arrow (0) is the net downward irradiance just below the surface, and a is the absorption coefficient of the water for the waveband in question. The average depth, (z) over bar (p), of all the photons in the water column in a given waveband is equal to [K-w(E, av)](-1), the reciprocal of the irradiance-weighted vertical attenuation coefficient for net downward irradiance. This conclusion, at which we arrive on theoretical grounds, is confirmed by numerical modeling of the underwater light field.