Imperfect scaling of time and space–time rainfall Daniele Veneziano a, * , Pierluigi Furcolo b , Vito Iacobellis c a Department of Civil and Environmental Engineering, MIT, Cambridge, MA 02139, USA b Dipartimento di Ingegneria Civile, Universita ` degli Studi di Salerno, Fisciano (SA), Italy c Dipartimento di Ingegneria delle Acque e di Chimica, Politecnico di Bari, Bari, Italy Received 5 February 2004; revised 28 September 2004; accepted 8 February 2005 Abstract Scale invariance is the most fertile concept to be introduced in stochastic rainfall modeling in 15 years. In particular, a form of scale invariance called multifractality has been exploited to construct parsimonious representations of rainfall in time and space and address fundamental problems of hydrology such as rainfall extremes, downscaling, and forecasting. However, several authors have observed that rainfall is scale invariant only in approximation and within limited ranges. Here, we make a systematic analysis of the deviations of time and space–time rainfall from multifractality. We use a flexible multiplicative cascade model, which produces multifractality as a special case while allowing deviations from scale invariance to occur. By fitting the model to rainfall records from different climates and over land or ocean, we find significant and consistent departures from multifractality in both the alternation of wet and dry conditions and the fluctuations of precipitation intensity when it rains. The fractal dimension of the rain support increases with increasing rain rate and the (multiplicative) fluctuations are larger at smaller scales and for lighter rainfall. A plausible explanation of these departures from scaling is that the rate of water vapor condensation in the atmosphere is a multifractal process in three space dimensions plus time, but multifractality is destroyed when the condensation rate is integrated to produce rainfall intensity at fixed altitudes. q 2005 Elsevier B.V. All rights reserved. Keywords: Rainfall models; Scale invariance; Multifractal processes 1. Introduction During the past two decades, stochastic models of rainfall have increasingly exploited the property of multifractal scale invariance. This property states that rainfall fields are statistically invariant under a group of transformations that involve contraction of the support and multiplication of the field by a non- negative random factor (Gupta and Waymire, 1990; Veneziano, 1999). Rainfall models based on multi- fractal scale invariance have been proposed by Schertzer and Lovejoy (1987), Gupta and Waymire (1990, 1993), Tessier et al. (1993), Over and Gupta (1994, 1996), Svensson et al. (1996), Perica and Foufoula-Georgiou (1996), Menabde et al. (1997), Olsson and Berndtsson (1998), Harris et al. (1998), Schmitt et al. (1998), and Deidda et al. (1999), among others. Journal of Hydrology 322 (2006) 105–119 www.elsevier.com/locate/jhydrol 0022-1694/$ - see front matter q 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.jhydrol.2005.02.044 * Corresponding author. Tel.: C1 617 253 7199; fax: C1 617 253 6044. E-mail address: venezian@mit.edu (D. Veneziano).