The relationship between α and the cross-correlation of cloud fraction

The cloud overlap weighting parameter, a, relates the averaged combined cloud fraction between two altitude levels to the averages derived under the random and maximum overlap assumptions. In several recent studies, it is found that alpha decreases approximately exponentially with separation distance between the two altitudes. This note provides a mathematical basis for these observed behaviours. An analytic expression for alpha is derived. It is found that a is solely a function of the cross-correlation function between cloud fractions at the two altitudes, and the individual averages and variances in cloud fraction at the two altitudes. Only the first of these is a joint function of the two altitudes of interest and therefore determines the form of the dependence of a on the height separation between the two altitudes. Where the variances in cloud fraction are small compared to their respective averages, it is found that a is approximated by the cross-correlation function. As the cross-correlation function is exponential wherever and whenever the physical thickness of clouds follows an exponential distribution, it is suggested that, where observations show a strong exponential dependence of a on separation distance between two altitudes, this indicates times and regions where cloud thickness follows an exponential distribution.