Errors in the Estimate of the Fractal Correlation Dimension of Raindrop Spatial Distribution

Marco Gabella Politecnico di Torino, Dipartimento di Elettronica, Turin, Italy

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Sebastiano Pavone Politecnico di Torino, Dipartimento di Elettronica, Turin, Italy

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Giovanni Perona Politecnico di Torino, Dipartimento di Elettronica, Turin, Italy

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Abstract

Theories on drop formation and quantitative rain estimation would require knowledge not only of the statistical size distribution of drops, but also of their statistical spatial distribution, which, in turn, determines the statistical fluctuations of the echoes detected by meteorological radar. Of particular interest is the question of whether such a spatial distribution can be assumed to be either statistically homogeneous or fractal. To analyze the spatial patterns of raindrops, a reasonable and immediate way of proceeding in the estimation of the fractal dimension is through the computation of the correlation integral. In any experimental observation of raindrop distribution, only a finite number of drops in a given space–time volume can be observed. Consequently, in this situation, the estimated value D of the fractal dimension differs from the true value Dt because of systematic (s) and random (r) errors. This paper shows that these errors can be ascribed to the finite number of raindrops and to“edge effects.” The order of magnitude of the error components that affect the estimate of D is investigated using numerical simulations for the inference of r and s, given a “reference” spatial distribution with Dt = 2. It has been found that the correlation dimension, estimated on the basis of a previous experimental observation (452 raindrops collected on a square blotting paper of 1.28 m × 1.28 m during a 1-s exposure to rain), could be compatible with a uniform random spatial distribution. The paper also presents the characteristics that new experimental setups have to possess to permit better estimates of the raindrop correlation dimension.

Corresponding author address: Marco Gabella, Politecnico di Torino, Dipartimento di Elettronica, Corso Duca degli Abruzzi 24, 10129 Turin, Italy.

gabella@polito.it

Abstract

Theories on drop formation and quantitative rain estimation would require knowledge not only of the statistical size distribution of drops, but also of their statistical spatial distribution, which, in turn, determines the statistical fluctuations of the echoes detected by meteorological radar. Of particular interest is the question of whether such a spatial distribution can be assumed to be either statistically homogeneous or fractal. To analyze the spatial patterns of raindrops, a reasonable and immediate way of proceeding in the estimation of the fractal dimension is through the computation of the correlation integral. In any experimental observation of raindrop distribution, only a finite number of drops in a given space–time volume can be observed. Consequently, in this situation, the estimated value D of the fractal dimension differs from the true value Dt because of systematic (s) and random (r) errors. This paper shows that these errors can be ascribed to the finite number of raindrops and to“edge effects.” The order of magnitude of the error components that affect the estimate of D is investigated using numerical simulations for the inference of r and s, given a “reference” spatial distribution with Dt = 2. It has been found that the correlation dimension, estimated on the basis of a previous experimental observation (452 raindrops collected on a square blotting paper of 1.28 m × 1.28 m during a 1-s exposure to rain), could be compatible with a uniform random spatial distribution. The paper also presents the characteristics that new experimental setups have to possess to permit better estimates of the raindrop correlation dimension.

Corresponding author address: Marco Gabella, Politecnico di Torino, Dipartimento di Elettronica, Corso Duca degli Abruzzi 24, 10129 Turin, Italy.

gabella@polito.it

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