Analysis of underwater radiance observations: Apparent optical properties and analytic functions describing the angular radiance distribution

A primary data set consisting of 70 series of angular radiance distributions observed in clear blue western Mediterranean water and a secondary set of 12 series from the more green and turbid Lake Pend Oreille, Idaho, have been analyzed. The results demonstrate that the main variation of the shape of the downward radiance distribution occurs within the Snell cone. Outside the cone the variation of the shape decreases with increasing zenith angle. The most important shape changes of the upward radiance appear within the zenith angle range 90 degrees-130 degrees. The variation in shape reaches its minimum around nadir, where an almost constant upward radiance distribution implies that a flat sea surface acts like a Lambert emitter within +/-8% in the zenith angle interval 140 degrees-180 degrees in air. The ratio Q of upward irradiance and nadir radiance, as well as the average cosines mu(d) and mu(u) for downward and upward radiance, respectively; have rather small standard deviations, 110%, within the local water type. In contrast, the irradiance reflectance R has been observed to change up to 400% with depth in the western Mediterranean, while the maximum observed change of Q with depth is only 40%. The dependence of Q on the solar elevation for blue light at 5 m depth in the Mediterranean coincides with observations from the central Atlantic as well as with model computations. The corresponding dependence of mu(d) shows that diffuse light may have a significant influence on its value. Two simple functions describing the observed angular radiance distributions are proposed, and both functions can be determined by two field observations as input parameters. The epsilon function approximates the azimuthal means of downward radiance with an average error less than or equal to 7% and of upward radiance with an error of similar to 1%. The alpha function describes the zenith angle dependence of the azimuthal means of upward radiance with an average error less than or equal to 7% in clear ocean water, increasing to less than or equal to 20% in turbid lake water. The alpha function suggests that the range of variation for mu(u), falls between 0 and 1/2, and for Q it is between pi and 2 pi. The limits of both ranges are confirmed by observations. By combining the epsilon and alpha functions, a complete angular description of the upward radiance field is achieved.