Abstract
Abstract. Lovejoy and Schertzer (1990a) presented a statistical analysis of blotting paper observations of the (two-dimensional) 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 (πr2−8r3/3+r4/2, 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.
Highlights
Detailed knowledge of the microstructure of precipitation is important from both a fundamental and an applied point of view (e.g., Uijlenhoet and Sempere Torres, 2006)
This is consistent with earlier observations by Gabella et al (2001), who concluded on the basis of a limited Monte Carlo simulation study that “the correlation dimension estimated from the raindrop distribution observed by Lovejoy and Schertzer could be compatible with a uniform random spatial distribution”
Our analytical results confirm previously reported Monte Carlo simulations (Jameson and Kostinski, 1998; Gabella et al, 2001) showing that as a result of instrumental artifacts the empirical results presented by LS are not statistically significant enough to reject the Poisson homogeneity hypothesis in favor of a fractal description of the discrete nature of rainfall
Summary
Detailed knowledge of the microstructure of precipitation is important from both a fundamental and an applied point of view (e.g., Uijlenhoet and Sempere Torres, 2006). One consists of generalizing the restrictive homogeneous Poisson process to a Poisson process with a randomly varying mean, that is a so-called doubly stochastic Poisson process or Cox process (e.g., Cox and Isham, 1980) This type of approach was pioneered by Sasyo (1965) and has later been applied by Smith (1993). Kostinski and Jameson (1997) and Larsen et al (2005) have proposed alternative non-Poissonian yet statistically homogeneous descriptions of rain Another approach is the (multi-)fractal description of rain, based on models which have originally been used to describe turbulence. The first application of fractal geometry to describe the discrete nature of rainfall was the statistical (correlation dimension) analysis of the (two-dimensional) spatial distributions of raindrop stains on pieces of blotting paper reported by Lovejoy and Schertzer (1990a, LS hereafter). We concentrate on the blotting paper experiment of LS, which has lead to a lively debate in the scientific literature concerning the statistical significance of the supposed fractal nature of the microstructure of rain (Jameson and Kostinski, 1998; Gabella et al, 2001; Jameson and Kostinski, 2001b; Gabella and Perona, 2001)
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