## 1. Introduction

[2] The variation of rain rate over distances as short as one meter is of interest in a number of industrial and environmental applications. One application of significant economic importance is the design and regulation of microwave telecommunications systems using frequencies above 10 GHz, for both terrestrial and earth-space communications. The volume sampled by a terrestrial radio link at these frequencies is approximately the first Fresnel zone, has a diameter of a few meters and a length between 100 m to 30 km. The Quality of Service (QoS) of such a link is determined by the sequences of “severely errored seconds” the system experiences, principally due to attenuation caused by rain. To estimate the QoS a node in an arbitrary network will experience requires knowledge of rain variation down to these scales. Similarly, the understanding of erosion processes, such as rill formation and water infiltration on inclined surfaces, requires knowledge of rain variation over scales as small as landscape features, e.g., plough furrows [*Wainwright and Parsons*, 2002; *Parsons and Stone*, 2006]. Urban hydrology usually focuses on “city block” catchment areas of a few square kilometers. However, the drainage systems of small areas of hard landscape or complex roof systems, and the design of microhydroelectric schemes are of increasing interest and require knowledge of rain variation over these scales.

[3] The variation of rain rate within rain events, particularly at the finest scales, is generally linked to variations in the vertical component of the wind field. The major updraughts and downdrafts on the scale of a few km or so, feed the energy at large scales which then cascade down the scale-spectrum to smaller eddies, following the Kolomogorov spectrum [*Kolmogorov*, 1941, 1991], until the energy is viscously dissipated as heat. *Lovejoy and Schertzer* [1995] describe rain as a “passive tracer injected at a certain scale in a turbulent flow.” At the large and small scales rain is not a passive tracer, but interacts with the atmosphere though condensation, coalescence and heat transport, e.g., small drops (<10 *μ*m) are formed by condensation in the updrafts [*Atlas and Williams*, 2003]. Droplet inertia leads to large concentration variation and differential drop velocities greatly increasing collision rates [*Bec et al.*, 2005]. The variation of water vapor due to cloud turbulence also needs to be accounted for [*Celani et al.*, 2005], and these processes lead to multiscaling distributions of water and ice in clouds. Radar and gauge studies have also reported multiscaling ranges in widespread rain rate variation and this has been qualitatively linked to turbulent energy cascades [*Lovejoy and Schertzer*, 1995; *Veneziano et al.*, 1996]. Noninertial models also predict multiscaling behavior with a scale break at the energy injection scale [*Falkovich et al.*, 2005]. At the smallest scales the inertia of drops is important while at increasing scales the stratification of the atmosphere becomes increasingly important and turbulent motions change from being three-dimensional to two-dimensional [*Lovejoy and Schertzer*, 1995]. Recently, *Veneziano et al.* [2006] have suggested that rain rate fields are only approximately multiscaling, because of these mechanisms, and that rain models need to allow for this deviation.

[4] The identification of simple or multiscaling ranges provides a useful summarizing statistic for stochastic fields and suggests a number of modeling algorithms. Analysis methods identify ranges of scales where the statistical moments of rain rate are a power law function of the size of the spatial or/and temporal integration interval. The summarizing statistic used as a basis of modeling is the moment scaling function. Let *M*_{λ}^{R}(*q*) be the *q*th moment of *R*_{λ} the rain rate measured over an integration volume of diameter *λD*_{0} where *D*_{0} is the diameter of the largest integration volume, i.e., *M*_{λ}^{R}(*q*) = *E*(*R*_{λ}^{q}) where *E*(•) is the expected value. If *M*_{λ}^{R}(*q*) ∝ *λ*^{ξ(q)}, where, *ξ*(*q*) is concave and does not depend upon *λ*, over some range of scales, then the rain rate is said to exhibit scaling. For simple scaling *ξ*(*q*) is linear in *q*, otherwise it is known as anomalous scaling. The function *ξ*(*q*) yields the multifractal exponents. Various experiments have demonstrated multifractal scaling of rain fields in one or more space-time dimensions, e.g., *Tessier et al.* [1993] over scales 200 m to 2000 km, *Schertzer and Lovejoy* [1995] over scales 6 min to 30 days, and *Deidda* [2000] over scales 15 min to 16 hours and 4 km to 256 km. A recent paper by *Peters et al.* [2002] has used a vertical pointing Doppler radar data to demonstrate several rain scaling results for integration intervals as short as 1 min.

[5] In this paper, temporal data from rapid response rain gauges at Chilbolton Observatory with a 10 s integration time and spatial data from the Chilbolton Advanced Meteorological radar (CAMRa) with 300 m resolution will be analyzed. The existence of positive moments of the underlying rain rate distribution will be determined by examination of the quantile scaling statistics of both rain gauge and radar data. Then the moment scaling statistics for both spatial and temporal rain rate variation will be calculated for a range of positive moment orders. The spatial moment scaling statistics are calculated using a method that avoids the problems associated with the interpolation of polar data onto Cartesian grids and the variation of integration region with range.