Estimating the scaling of rain rate moments from radar and rain gauge



[1] Rain rate is a physical parameter that is only defined over some spatial or spatial-temporal integration volume. The moments of rain rate fields calculated from measurements derived from integration volumes of the same shape and a range of sizes are commonly used as a summarizing statistic of the rain rate process. Ranges of scales are known as stochastically scaling, either simple or multiscaling, when the moments are a power law of integration volume size. The identification of scaling ranges provides information about the dominant physical processes leading to rain rate variation and allow simulation models to be devised. The spatial moment scaling statistics of rain fields have been estimated from radar data many times. In this paper, three real and potential problems with the reported statistics are identified; that is, the existence of moments is not verified, the effects of interpolation have not been considered and the inhomogeneity introduced by variation in radar sample volume with range has been ignored. Radar data from the Chilbolton CAMRa radar in the UK are analyzed, and the existence of positive moments of all orders is demonstrated. An algorithm has been implemented for the calculation of moments from spatially averaged rain data on a regular polar grid. The resulting moments are consistent with cosited rain gauge data and show smooth variation across the scales considered, 300 m to 10 km and 10 s to 6 hours. The resulting moments are well approximated by two multiscaling ranges with a scale break around 3 km or 300 s.

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 μm) 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 MλR(q) be the qth moment of Rλ the rain rate measured over an integration volume of diameter λD0 where D0 is the diameter of the largest integration volume, i.e., MλR(q) = E(Rλq) where E(•) is the expected value. If MλR(q) ∝ λξ(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.

2. Gauge and Radar Data

[6] Three complete years of data from three Rutherford Appleton laboratory (RAL) Rapid Response Drop-Counting Gauges ( are used to estimate rain rates over 10 s intervals. Two gauges are located at Chilbolton Observatory (51.1445°N, 358.563°E), one is situated on the flat roof of a one-story building while another is sited on the ground a short distance away. The third gauge is situated on the flat roof of a two-story building, 9 km away at Sparsholt (51.0847°N, 358.607°E). Where averages over 9 gauge years are discussed the average is over data from three gauges over 3 a. Rainwater collected in the 150 cm2 gauge funnel passes through a sump to a device that produces equally sized drops. These drops are detected optically as they fall to a drain. The gauges record the number of drops in each 10 s interval, where each drop corresponds to a rain rate of 1.43 mm/hr. A rain gauge measurement can be thought of as rain rate averaged over the funnel collection area and an interval of time. Alternatively it can be treated as a spatial average over a cylinder of height V(D)T, where V(D) is the fall speed of drops of diameter D and T is the gauge integration time.

[7] The radar data used was generated by the Chilbolton Radar Interference Experiment (CRIE). This was a 2-a rain measurement campaign between 1987–1989, designed primarily for development and testing of rain scatter interference models as part of the COST 210 Project [1991]. It aimed to record an unbiased sample of the rain events occurring near Chilbolton. Rain fields were scanned using the Chilbolton Advanced Meteorological Radar, CAMRa, a 25 m steerable antenna equipped with a 3 GHz, Doppler-Polarization radar. The climate in the region is temperate maritime with an average annual rain rate exceeded 0.01% of the time of approximately 30 mm/hr.

[8] For the 2-a period the radar was operated on a 28 day duty cycle. A set of near-horizontal (PPI) and vertical (RHI) radar scans, measuring both horizontal and vertically polarized radar reflectivity, ZH and ZV, were recorded in a 10 min cycle for 9 out of the 28 days. If no rain was detected (defined by a measured radar reflectivity less than 25 dBZ) the data was discarded and this was recorded by the operator. The days of operation were chosen well in advance and with no reference to weather forecasts.

[9] The PPI scans were acquired with an elevation of 1.5° and covered an area approximately 50° in azimuth centered south west of the radar. As the scan rate of the radar is 1°/s it took less than 1 min to complete a PPI scan. Hence the scan duration is well within the 20–30 min duration for the lifetime of a rain event [Zawadzki, 1973] and each scan represents a good snapshot of the rain field before any significant structural change has taken place.

[10] The resulting database contains 3199 scan sets, and 30590 records of no rain. It has been used for radio system engineering studies, see Goddard and Thurai [1996, 1997] and Tan and Pedersen [2000].

[11] Data were collected between the ranges of 4.8 km and 158 km from the radar and averaged over 300 m intervals. For this paper, only the data between 10 km and 70 km are used. This is to avoid sample volumes being within the freezing level and to limit differences in volume averaging due to beam spreading. This yields 181 reflectivity measurements along each ray. The beam width is 0.25° yielding 210 rays in each scan.

[12] Each set of PPI scans have been range corrected, calibrated, and correction made for absorption by atmospheric gasses. Reflectivities below the noise floor of 10 dBZ were assigned zero rain rates. Negative values of differential reflectivity are assumed to be due to nonliquid hydrometeors or anomalous propagation and are eliminated from the data set. The rain-hail algorithm of Leitao and Watson [1984] has been used to eliminate other data points where nonliquid hydrometeors may have influenced reflectivities. Finally, dual polarization reflectivity data was transformed into rain rate fields, see Usman [2005].

3. Quantile Scaling and Existence of Moments

[13] In sections 3 and 4 we develop the notion of rain rate averaged over temporal or spatial volumes with a range of diameters, and define simple and multiscaling. The development presented here is not rigorous; but summarizes parts of the much more detailed description by Pavlopoulos and Gupta [2003]. Consider the gauge-measured rain rate process Rλ(t) measured with an integration interval of length T = λT0 where λ ∈ Λ a subset of (0, 1] and T0 is the longest integration time considered. The cumulative probability distribution of rain rate is defined as:

equation image

[14] This distribution is continuous and strictly increasing over the range u∈ [0, +∞). As a consequence, the quantile functions:

equation image

are also continuous and strictly increasing for p∈ (0, 1], and so is the inverse function of the corresponding distribution. We will also use the inverse survival function, ZλR(p), which is related to the quantile function by ZλR(p) = QλR(1 − p). The moments of nth order are defined:

equation image

and exist only when this integral converges. The moments can be calculated from the quantile function:

equation image

Many authors have speculated on the shape of the extreme rainfall tail of the rain rate cumulative density function FλR(u) and the associated probability density function (pdf) fλR(u) [e.g., Cho et al., 2004; Kedem et al., 1994]. Where the pdf tail approaches a negative exponential, i.e., fλR(u) ∝ eequation image: u0 > 0, all positive moments exist. However, where the tail is rational, i.e., fλR(u) ∝ uα; α > 1, then only positive moments q < α − 1 are finite. In practice, arbitrarily high rain rates do not occur. In the fine-scale limit, rain becomes quantized into raindrops. The largest dynamically stable raindrops, with a diameter of approximately 1 cm, fall with a terminal velocity of approximately 10 m/s. Rain rate averaged over spatial-temporal volumes entirely within the largest drops, is approximately 10 m/s. Any spatial temporal volume that includes smaller drops, or regions between raindrops, will yield a lower averaged rain rate. In general, larger volumes will include more regions outside these extreme raindrops and so will yield a lower maximum rain rate. As rain rate is bounded above by the maximum possible value of 10 m/s, all moments exist.

3.1. Temporal Moments

[15] Figure 1 illustrates the quantile functions for 9 gauge years of rain rate measurements, for integration times from 10 s to 7 min. For a 10 s integration period, and probabilities approaching 1, i.e., 1 − p → 0, the quantile function approaches linearity in ln(1 − p). The quantiles for probability: 1 − p < 5 × 10−7, equivalent to ZλR(p), are defined by less than 20 samples are so are not reliable. For integration periods longer than 1 min the quantiles approach a maximum and are bounded above by linear functions of ln(1 − p). If the quantile function is bounded above by a linear function of ln(1 − p) for all probabilities above some pc > 0: QλR(pc) = Rc, then from (4):

equation image

By substitution, the second integral can be shown to be equal to equation image imageequation imagedu which exists and is bounded for all finite Aλ, Bλ < 0 and q. Therefore all positive moments of point rain rate exist for all integration periods down to 10 s. These results are consistent with rain rate pdfs that are exponential in extreme rain rate before becoming equal to zero above a maximum rain rate that is determined by the size of the spatial-temporal integration volume.

Figure 1.

Rain rate quantile function Q(1-p) for rain gauge integration periods, from top to bottom, of 10 s and 1, 2, 3, 4, 5, 6, 7 and 8 min. Thick lines indicate linear functions of ln(1-p) as upper bounds for quantile variation.

[16] The question remains as to whether the same is true in the short integration time limit, T → 0. The slope of Bλ, the linear part of quantile QλR(1 − p) = ZλR(p) as a function of ln(1 − p), may be calculated as a function of integration period. If Bλ tends to a finite limit as the integration period reduces to zero, this implies that a model for rain rate distributions should always be bounded. The slope Bλ is approximately linear for integration periods between 50 s and 400 s. At shorter integration times the coefficient diverges rapidly from linear. As the current data only support four integration periods below this period, it is impossible to extrapolate to the zero integration time limit. There is also considerable uncertainty in the measurement of the extreme rains at these short integration times.

3.2. Calculation of Spatial Moments

[17] The moments of spatially averaged rain rate have been calculated from radar images on many occasions. Terrestrial radars measure rain reflectivity averaged over voxels with corners on a regular polar grid, centered on the radar. Typically, the radar scans near horizontally at a fixed rate of rotation. A measurement voxel is defined by the angles and ranges over which measurements are averaged. The height of a voxel is determined by the width of the primary lobe of the antenna pattern and for CAMRa this is 0.25°. Therefore rain radar data is averaged over voxels with constant radial scale but tangential and vertical scales that grow linearly with range.

[18] The usual process used to estimate the moment scaling statistics begins by interpolating the rain field derived from polar radar data onto a Cartesian grid [e.g., Lovejoy, 1982; Rhys and Waldvogel, 1986; Deidda, 1999; Féral and Sauvageot, 2002]. The rain rate samples are then accumulated to yield integration volumes, with square footprints, with a range of sizes. This process is significantly easier on a Cartesian grid than a polar grid. However, there are several problems with this procedure. The statistics of the radar derived rain field are not homogeneous, because of the changes in the shape and size of the integration volume, but are a function of range. The design of an interpolation scheme that preserves the original statistics, before they are measured, is a significant problem. Even if this problem were addressed, the inhomogeneity in the polar data will be present in the Cartesian data and so it is not correct to accumulate samples with equal weights, for widely separated ranges. Finally, the increasing height of the radar voxels with range has not been addressed. Although radar voxels can be accumulated into voxels with the same footprint, those at greater range will have greater volume. It is likely that the vertical variation of rain rate will be much smaller than the horizontal variation, but this needs to be tested.

[19] This paper uses an accumulation process without interpolation. Analysis voxels are defined with square footprints and sides of lengths that are multiples of 300 m. Radar voxels are accumulated to yield volumes as close to the size and shape of analysis volumes as possible. Figure 2 illustrates radar data on a polar grid centered on a radar. Also indicated are two square, target volumes and the best approximation formed by accumulating voxels on the polar grid. The rain rate sample is calculated as a weighted sum of radar rain rates with weights proportional to the radar voxel footprint area. Each rain rate sample calculated by accumulation is stored along with information on the mean range and a measure of the difference between the analysis and actual sample voxel shape and size.

Figure 2.

Accumulation of radar voxels on a regular polar grid to yield approximately square voxels. Ranges are in kilometers.

[20] Estimates of quantile functions can be calculated from the accumulated rain rate database. The CRIE database contains over 120 million rain samples from the regular polar grid, with each sample being an average over a voxel 300 m × 0.25° × 0.25° range × azimuth × elevation; in the ranges used for analysis. After accumulation this yields approximately 2000 million overlapping samples with footprints 300 m square, decreasing to 100 million samples 10 km square. Moments and quantiles can be calculated using subsets of the accumulated rain rate database to check for sensitivity to range or accumulation voxel size and shape.

3.3. Existence of Spatial Moments

[21] Figure 3 shows the quantile functions for the whole of the CRIE database, for a range of spatial integration volumes of linear size 300 m to 10 km. One again the quantile functions approach, or are bounded above by, linearity in ln(1 − p). Therefore moments of all positive orders exist by the same argument used to analyze the temporal quantiles.

Figure 3.

Rain rate quantile function Q(1-p) for a range of spatial integration volumes.

[22] The sensitivity of these results to the increasing height of voxels was checked by dividing the data set in two subsets of analysis voxels with ranges less and greater than 40 km. The resulting quantile functions exhibit the same QλR(1 − p) ≅ Aλ + Bλ ln(1 − p) interval but plateaux at different extreme rain rate values due to the different extreme events experienced by the two regions. The accumulated rain rate data set was also divided into two subsets depending on how well the square, target voxel was approximated by the accumulation of radar voxels. The quantile functions for the two sets had the same features but leveled off at different rain rates for the same reason. This suggests that the voxel accumulation method is sufficiently good for all the data to be treated as accumulations over equivalent areas. It does not imply that the accumulation shape is not important, as all accumulation areas were close to square.

4. Temporal and Spatial Moment Scaling

[23] In section 3 the existence was verified of moments of all orders of spatial and temporal averages of rain rate. In this section a selection of positive moments are calculated for a range of spatial and temporal integration volumes and these data are used to identify scaling ranges.

4.1. Simple and Multiscaling

[24] The rain rate process Rλ(t) is stochastically scaling if and only if there is a scalar process {Cλ}λ∈Λ, such that P(C1 = 1) = 1 and P(Cλ > 0) = 1 for all λ∈ Λ, and Rλequation imageCλR1. Here equation image denotes equality of probability distribution functions so that P(Rλr) = P(CλR1r). Special cases of scaling exist, depending on the form of the process Cλ. For simple scaling, also known as stochastic self-similarity, P(Cλ = λθ) = 1. In this case the finite moments scale as power laws, and the scaling exponents qθ are linear in moment order:

equation image

Although the rain rate process cannot be simple-scaling due to intermittence [Kedem and Chiu, 1987], it has been proposed that log rain rate, where raining, is well approximated as simple-scaling [Paulson, 2002]. The rain rate process is more commonly modeled as multiscaling [Gupta and Waymire, 1993], where:

equation image

The process {Zρ; ρ ≥ 0}λ∈Λ is such that P(Z0 = 0) = 1 and has stationary increments: imageequation imageimage + image Assuming independence of RT and Cλ, the finite moments still scale as power laws, but the scaling exponents are now an arbitrary concave function of moment order, i.e.,

equation image

Having identified a process as having the properties of a multiscaling process, a range of cascade processes can be used to simulate or downscale a given realization [Schertzer and Lovejoy, 1995; Ossiander and Waymire, 2000].

4.2. Temporal Moment Scaling

[25] Figure 4a illustrates the moment scaling function for the 9 gauge years of rain rate measurements, for integration times from 10 s to 1 day and for moment orders between q = 1/2 and q = 3. As all positive moments exist, the moments considered are those of most interest in applications. Moments of high order are increasingly determined by the extreme rain rates in the data set, with longer return times, and so are estimated to lower accuracy. The observed moments do not follow power laws across the range of scales investigated. However, they are well approximated by three scaling intervals: approximately 10 s to 200 s, 200 s to 10000 s and above 10000 s. These three regions correspond to scaling intervals where different physical processes dominate. Splitting the 9 gauge years of data into summer and winter periods yields moment scaling functions with the same scale breaks. The moments and scaling exponents vary with the season because of the higher incidence of heavy rain in summer convective storms, leading to higher moment values particularly at higher orders.

Figure 4.

(a) Moment scaling function for rain rate time series averaged over 9 gauge years of data and (b) moment scaling exponent as a function of moment order for integration periods less than 200 s (dashed) and greater than 200 s (solid).

[26] Figure 4b shows the scaling exponents over the two scaling ranges 10 s to 200 s and 200 s to 10000 s. For the moments of order q ≥ 1 the scaling exponents are concave and well approximated by quadratics.

4.3. Spatial Moment Scaling

[27] Using the voxel accumulation process described in section 3.2, a selection of positive moments of spatially averaged rain rate have been calculated for voxels with a square base with diameter ranging from 300 m to 10 km, and for moment orders between q = 1/2 and q = 4.

[28] Figure 5a shows the moment scaling functions. As with the temporal moments, the observed moments do not follow power laws across the range of scales investigated. However, they are well approximated by two scaling intervals: approximately 300 m to 1 km and above 3 km. The scale break apparent in the temporal data at 200 s is less clear in the spatial data. This may be due to anisotropy in the statistics along lines parallel and perpendicular to advection, e.g., squall lines or fronts. Figure 5b shows the variation of the scaling exponents for these two intervals. The scaling exponent variation is concave, approximately quadratic, and so multiscaling.

Figure 5.

(a) Spatial moment scaling function derived from the CRIE database and (b) spatial moment scaling exponent as a function of moment order for polar data and for Cartesian data produced by interpolation from a polar grid.

[29] For comparison, the scaling moments were calculated on rain data bilinearly interpolated onto a regular Cartesian grid with samples separated by 300 m, before accumulation into larger sample volumes. The interpolation process reduced the extremes of measured rain rates and so reduced the values of higher-order moments. The log-log moment scaling curves became much closer to linear across the scale range, obscuring the possible scale break around 2 km. The moment scaling exponents were greatly reduced over the larger scale range identified in Figure 5a. Using interpolated measurements could have lead to the conclusion that the data were approximately multiscaling across the scale range considered, with scaling exponents much closer to zero.

5. Conclusions

[30] Nine gauge years of 10 s integration time rain rate measurements have been analyzed to examine the moments and the moment scaling functions. Physical arguments indicate the existence of a maximum rain rate as a function of integration period. Measurements indicate rain rate distributions with an exponential region limited above by a maximum rain rate. It was concluded that all temporal rain rate moments exist for finite integration periods. The moment scaling function was calculated over the temporal range 10 s to 80000 s. At least three multiscaling ranges were necessary to approximate the moment scaling function well, with scale breaks near 200 s and 10000 s.

[31] Similar analysis has been performed for rain rates integrated over spatial squares. These statistics are difficult to calculate from polar sampled data produced by ground-based radar. Previous authors have interpolated polar data onto Cartesian grids. However, there are concerns about anomalies introduced into the statistics by the interpolation process. A method has been developed for accumulating radar pixels into approximately square regions. Moment scaling statistics calculated with the polar accumulation method indicate two approximately multiscaling regions with a scale break around 2 km.

[32] The spatial and temporal scaling moments show a scale break around 200 s and 2 km. Assuming Taylor's frozen storm hypothesis [Taylor, 1938], the point temporal variation of rain rate over these relatively short periods of time, is principally due to the advection of rain events over the measurement point. This is consistent with an advection speed of approximately 1 km per minute. In Figure 4a the scale brake becomes better defined at higher moment orders implying that it is present in intense rain fall. Intense rain occurs in small convective cells, typically with diameters in the range 4 to 8 km [Harden et al., 1974] and spacing 3 to 7 km [Veneziano et al., 1996]. The break at 2 km implies a finer structure within intense rain cells. Sinclair [1974] reports a scale break in the vertical wind velocity at 0.5 km, derived from penetrating flights through thunder storms. This should be observable in rain variation at integration times of 30 s and at the fine-scale limit of spatial scaling. However, this effect was not observed.


stochastic scaling coefficient.


the expected value.


cumulative probability of rain rate r.


dimensionless scale factor.


probability of.


qth moment of Rλ.


quantile function of Rλ.


rain rate process over integration volume with scale factor λ.


notional point rain rate at time t.


rain gauge integration time.


inverse survival function.


scaling exponents.


[33] We would like to thank the Radio Communications Research Unit of Rutherford Appleton Laboratory for acquiring and providing access to the rain gauge and radar data. We are also very grateful to Harry Pavlopoulos for his very helpful interactions during the preparation of this paper.