Edge effect causes apparent fractal correlation dimension of uniform spatial raindrop distribution

Abstract

Abstract. Lovejoy and Schertzer (1990a) presented a statistical analysis of blotting paper observations of the (twodimensional) spatial distribution of raindrop stains. They found empirical evidence for the fractal scaling behavior of raindrops in space, with potentially far-reaching implications for rainfall microphysics and radar meteorology. In particular, the fractal correlation dimensions determined from their blotting paper observations led them to conclude that "drops are (hierarchically) clustered" and that "inhomogeneity in rain is likely to extend down to millimeter scales". Confirming previously reported Monte Carlo simulations, we demonstrate analytically that the claims based on this analysis need to be reconsidered, as fractal correlation dimensions similar to the ones reported (i.e. smaller than the value of two expected for uniformly distributed raindrops) can result from instrumental artifacts (edge effects) in otherwise homogeneous Poissonian rainfall. Hence, the results of the blotting paper experiment are not statistically significant enough to reject the Poisson homogeneity hypothesis in favor of a fractal description of the discrete nature of rainfall. Our analysis is based on an analytical expression for the expected overlap area between a circle and a square, when the circle center is randomly (uniformly) distributed inside the square. The derived expression (πr 2 −8r 3 /3+r 4 /2, where r denotes the ratio between the circle radius and the side of the square) can be used as a reference curve against which to test the statistical significance of fractal correlation dimensions determined from spatial point patterns, such as those of raindrops and rainfall cells.

Raindrop interaction in interrill erosion for steady rainfalls: A probabilistic approach

“…In the literature, different processes have been used to describe the distribution of natural raindrops. Among these processes, the most used are the Poisson process [ Gillespie , ; Hosking and Stow , ; Uijlenhoet et al ., ; Ma et al ., ; Uijlenhoet et al ., ], the Cox process [ Cox and Isham , ; Sasyo , ; Smith , ], and the fractal description [ Lovejoy and Schertzer , ; Lavergnat and Golé , ; Lilley et al ., ]. In the present study, we focus on steady rainfalls which has been demonstrated to be consistent with the Poisson process [ Uijlenhoet et al ., ; Jameson and Kostinski , ].…”

<i>Preface</i> "Nonlinear and scaling processes in Hydrology and Soil Science"

“…They highlight that the dependency of these findings on climatological, geographical, sample size among other factors, cannot be ignored. Uijlenhoet et al (2009) analysed the fractal correlation dimension of homogeneously distributed raindrops. Their analytical results show that the empirical results presented on fractal scaling behaviour of raindrops in space (Lovejoy and Schertzer, 1990) can be explained by edge effects in the instrument used.…”