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Multifractal Analysis and Simulation of the Global Meteorological Network

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  • 1 Collège Militaire Royal de St-Jean, Département de Physique, Richelain, Quebec, Canada
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Abstract

Taking the example of the meteorological measuring network, it is shown how the density of stations can be characterized by multifractal measures. A series of multifractal analysis techniques are applied (including new ones designed to take into account the spherical geometry) to systematically test the limits and types of network multiscaling. These techniques start with a network density defined by grids or circles and proceed to systematically degrade their resolution (no a priori scaling assumptions are necessary). The multiscaling is found to hold over roughly the range 20 000 to 200 km (limited by the finite number of stations—here about 8000). Special attention is paid to qualitative changes in the scaling behavior occurring at very low and high density regions that the authors argue are associated with multifractal phase transitions. It is argued that the density was produced by a universal multifractal process, and the three corresponding universal multifractal parameters are estimated. The minimum and maximum orders of singularities present in the network are estimated, as well as the minimum- and maximum-order statistical moments that can be reliably estimated. The results are then used to simulate the effects of the finite number of stations on a network with the same statistical properties, and hence to quantitatively show that the observed breaks in the multiscaling can be accounted for by the finiteness. A growing number of geophysical fields have been shown to exhibit multiscaling properties over various ranges, and in this paper it is discussed how the bias introduced by the network clustering can be removed by new “multifractal objective analysis” procedures.

Abstract

Taking the example of the meteorological measuring network, it is shown how the density of stations can be characterized by multifractal measures. A series of multifractal analysis techniques are applied (including new ones designed to take into account the spherical geometry) to systematically test the limits and types of network multiscaling. These techniques start with a network density defined by grids or circles and proceed to systematically degrade their resolution (no a priori scaling assumptions are necessary). The multiscaling is found to hold over roughly the range 20 000 to 200 km (limited by the finite number of stations—here about 8000). Special attention is paid to qualitative changes in the scaling behavior occurring at very low and high density regions that the authors argue are associated with multifractal phase transitions. It is argued that the density was produced by a universal multifractal process, and the three corresponding universal multifractal parameters are estimated. The minimum and maximum orders of singularities present in the network are estimated, as well as the minimum- and maximum-order statistical moments that can be reliably estimated. The results are then used to simulate the effects of the finite number of stations on a network with the same statistical properties, and hence to quantitatively show that the observed breaks in the multiscaling can be accounted for by the finiteness. A growing number of geophysical fields have been shown to exhibit multiscaling properties over various ranges, and in this paper it is discussed how the bias introduced by the network clustering can be removed by new “multifractal objective analysis” procedures.

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