## 1. Introduction

[2] This article is concerned with rainfall intermittence in space and time. Rainfall intermittence is defined as the alternation from zero to positive rain rates and vice versa. Several studies have investigated the modeling of rainfall intermittence. They include studies in space at fixed times [*Gupta and Waymire*, 1993; *Over and Gupta*, 1994; *Onof et al.*, 1998], and in time at fixed spatial locations [*Schmitt et al.*, 1998]. Recently, *Pavlopoulos and Gritsis* [1999] have investigated space-time intermittence of rainfall. Our objective here is to develop a new model-free methodology for testing statistical scale invariance of rainfall intermittence in space and time. We illustrate this methodology on data from TOGA-COARE over the tropical Pacific [*Short et al.*, 1997].

[3] Space-time rainfall covers multiple space and timescales. In order to investigate empirical features across multiple scales, the concepts of scale invariance and scale dependence have emerged as fundamental features of multiscale hydrologic processes [*Sposito*, 1998]. Scale invariance was introduced into studies of rain fields nearly two decades ago [*Lovejoy and Mandelbrot*, 1985; *Lovejoy and Schertzer*, 1985; *Schertzer and Lovejoy*, 1987; *Waymire*, 1985]. It can be used to exploit information obtained at any one scale for making inferences at another scale within the range of scales where the invariance is valid. For example, scale-invariant stochastic models can be applied at multiple scales by simply rescaling the model parameters appropriately. Some applications of scale-invariant stochastic models include subgrid-scale downscaling of spatial rainfall in climate models [*Foufoula-Georgiou*, 1998], and downscaling of spatial rainfall on river networks to study scale invariance of peak river flows [*Gupta et al.*, 1996; *Gupta and Waymire*, 1998a, 1998b; *Menabde and Sivapalan*, 2001; *Troutman and Over*, 2001].

[4] In this study, we define regional processes of wet and dry epoch durations at several spatial scales, derived from time series records of regional rainfall at these scales. Given a subregion *A* of a region of study *S* (i.e., *A* ⊆ *S*), let an intermittent temporal random process of spatially averaged rain rate over *A* be denoted by {*R*_{A}(*t*); *t* ≥ 0}. This process can be partitioned into its wet epochs, or spells of positive values, and dry epochs or spells of zeroes, whose lengths are assumed to be sample values from random variables *W*_{A} and *D*_{A} respectively. Thus, by varying the sampled subregion *A* ⊆ *S*, one naturally obtains a regional random process of wet epoch duration, {*W*_{A}; *A* ⊆ *S*}, and another regional random process of dry epoch duration, {*D*_{A}; *A* ⊆ *S*}. Wet and dry epoch durations can also be recovered from the temporal evolution of the overall fractional wet area of *A*.

[5] We test for spatial homogeneity at each scale to investigate the nature of probability distributions of regional processes of duration. Then, formulae for tail quantiles are derived in terms of scale and probability level, based entirely on empirical statistics. These formulae are used to examine the form of tail probabilities. They provide a mathematical basis for investigating what type of scale invariance, if any, holds in wet and dry epoch durations. Moreover, tail probabilities allow us to examine the issue of finiteness of moments at each spatial scale, and to estimate bounds of orders above which moments do not exist. This is a key issue, because finiteness of moments is very sensitive to extreme observations, which are represented by the tails of probability distributions.

[6] A formal definition of stochastic scale invariance or stochastic scaling is given in section 2. Its implications on the scaling behavior of quantiles and on the existence of moments is discussed there. A part of TOGA-COARE data used in this study is described in section 3. Nonparametric tests to diagnose homogeneity assumptions in space and time are implemented in section 4. The core of statistical analyses and results of our study are presented in section 5. A summary of the key results is given in section 6, and potential implications of this work for modeling, and for estimation of rainfall from multiple sensors, are briefly discussed there.