Multifractals as a tool to overcome scale problems in hydrology

Abstract

In hydrology, meteorology, oceanography, climatology and, more generally, in geophysics—all sciences which are involved in water resources management, researches and studies—technological and operational developments come up against a fundamental difficulty which is the extreme variability of geophysical fields. This extreme variability is often associated with natural hazards such as earthquakes, tornados, droughts, or floods and with a very large range of scales. This range extends from millimetres to thousands of kilometres in space, from milliseconds to geological eras in time. The traditional approaches to studying geophysical fields tend to ignore these basic empirical facts by making drastic truncations, that is to say studying one scale independently of the others, transforming partial differential equations into ordinary differential equations, arbitrarily hypothesizing the homogeneity of fields, and performing ad hoc and unjustified parameterizations. Alternative approaches to nonlinear variability are based on a fundamental property of nonlinear equations: scale invariance. In this paper, an attempt is made to give some examples of such scale invariant behaviour of hydrological processes and fields—mainly from empirical evidence. Furthermore, it is shown why and how the multifractal approach, which is still at the developmental stage, is a very promising domain. This approach may provide hydrology with useful concepts that could tomorrow turn into essential tools for water resource management.