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106 JOURNAL OF APPLIED METEOROLOGY VOLUME 43

An Experimental Study of Small-Scale Variability of Radar Re¯ectivity Using Disdrometer Observations

BENJAMIN J. MIRIOVSKY,A.ALLEN BRADLEY,WILLIAM E. EICHINGER,WITOLD F. K RAJEWSKI, ANTON KRUGER, AND BRIAN R. NELSON IIHRÐHydroscience and Engineering, The University of Iowa, Iowa City, Iowa

JEAN-DOMINIQUE CREUTIN AND JEAN-MARC LAPETITE Laboratoire d'Etude des Transferts en Hydrologie et Environnement, Grenoble, France

GYU WON LEE AND ISZTAR ZAWADZKI Department of Atmospheric and Oceanic Sciences, McGill University, Montreal, Quebec, Canada

FRED L. OGDEN Department of Civil and Environmental Engineering, University of Connecticut, Storrs, Connecticut

(Manuscript received 21 October 2002, in ®nal form 18 August 2003)

ABSTRACT Analysis of data collected by four disdrometers deployed in a 1-km2 area is presented with the intent of quantifying the spatial variability of radar re¯ectivity at small spatial scales. Spatial variability of radar re¯ectivity within the radar beam is a key source of error in radar-rainfall estimation because of the assumption that drops are uniformly distributed within the radar-sensing volume. Common experience tells one that, in fact, drops are not uniformly distributed, and, although some work has been done to examine the small-scale spatial variability of rain rates, little experimental work has been done to explore the variability of radar re¯ectivity. The four disdrometers used for this study include a two-dimensional video disdrometer, an X-band radar-based disdrometer, an impact-type disdrometer, and an optical spectropluviometer. Although instrumental differences were expected, the magnitude of these differences clouds the natural variability of interest. An algorithm is applied to mitigate these instrumental effects, and the variability remains high, even as the observations are integrated in time. Although one cannot explicitly quantify the spatial variability from this experiment, the results clearly show that the spatial variability of re¯ectivity is very large.

1. Introduction it is not the variability of rainfall itself, but the vari- The National Weather Service's Weather Surveillance ability of radar observables, like radar re¯ectivity factor Radar-1988 Doppler (WSR-88D) network currently es- Z, that is relevant. timates re¯ectivity with a spatial resolution of 1 km ϫ A key assumption of the radar equation is that rain- 1Њ azimuth and a temporal resolution of about 6 min, drops are uniformly distributed within the radar beam the time necessary to complete one volume scan. Al- (Battan 1973). It has been shown (Zawadzki 1982) that though spatial resolution of 1 km ϫ 1Њ azimuth seems re¯ectivity gradients strongly in¯uence the meteorolog- high, even casual weather observers intuitively recog- ical interpretation of radar measurements. Thus, inves- nize that rainfall can vary signi®cantly over much small- tigations into the spatial variability of radar re¯ectivity er areas. Numerous studies have explored small-scale within the radar beam can provide insight for uncer- spatial variability of rain rate and rainfall accumulation tainty quanti®cation that is useful in a number of ap- ®elds (e.g., Krajewski et al. 2003; Habib and Krajewski plications, including radar rainfall estimation and data 2002; Georgakakos et al. 1994; Kitchen and Blackall assimilation efforts for predictive models. In fact, the 1992). For radar rainfall estimation purposes, however, precision and accuracy of precipitation estimates by radar is affected by the variability of drop size distributions (DSDs) at all scales, including those larger than Corresponding author address: Witold F. Krajewski, IIHRÐHydro- science and Engineering, The University of Iowa, Iowa City, IA 52242. the radar beam sample volume. With polarimetric radars E-mail: [email protected] it is possible to infer parameters of drop size spectra,

᭧ 2004 American Meteorological Society

Unauthenticated | Downloaded 09/27/21 05:38 PM UTC JANUARY 2004 MIRIOVSKY ET AL. 107 and, hence, their variability in space. From the evidence sure the small-scale spatial variability of DSDs. During currently available, it appears that polarimetry allows October and November 2001, we conducted the X-band detection of DSD variability at scales larger than a single Polarimetric Radar on Wheels (XPOL) ®eld experiment, radar-sampling volume, because of the degree of av- in Iowa City, Iowa, using both approaches. eraging required to have precise polarimetric measure- During the XPOL ®eld experiment we collected high- ments (e.g., Bringi and Chandrasekar 2001). Thus, we resolution polarimetric radar data using the National Ob- need small-scale in situ experiments to evaluate the DSD servatory of Athens (NOA), Greece, mobile dual-po- variability and its effects on radar-derived quantities. larization X-band radar, over well-instrumented sites. A Comprehensive characterization of DSD variability vertically pointing X-band Doppler radar (9.37 GHz), requires a multiscaling analysis (e.g., Venugopal et al. ®ve disdrometers, and several dual-gauge tipping-buck- 1999; Gupta and Waymire 1990). This is currently not et rain gauge platforms were deployed in an area about possible with in situ instruments. A limited, yet still 1.0 km ϫ 1.0 km (Fig. 1). The closest pair of disdro- useful description could focus on point-to-area variance meters is only separated by 240 m, and the most distant reduction, very much in the spirit of hydrologic rainfall pair is separated by 1.05 km. We used these instruments analysis (e.g., Bras and RodrõÂguez-Iturbe 1985). This to both augment and validate the data collected by the requires modeling spatial covariance functions at given polarimetric radar, which was located approximately 6 temporal scales. km to the west of this site. A high-density cluster of 10 Our knowledge of the DSD variability comes from dual-platform tipping-bucket rain gauges is located at ground-based point measurements with time intervals the Iowa City Municipal Airport, about 11 km east of on the order of 1 min. Like all atmospheric phenomena, the polarimetric radar, and three other platforms were the natural variability of the DSD is expected to be deployed with disdrometers, for a total of 13 dual-gauge strongly dependent on the scales at which this variability platforms. The ®ve disdrometers include the University is observed. The relationship between temporally in- of Iowa's two-dimensional video disdrometer (2DVD), tegrated disdrometric point samples and quasi-instan- a Precipitation Occurrence Sensing System (POSS) taneous radar volumetric samples is not clear, resulting from McGill University, a JWD provided by the Uni- in inadequate knowledge of the natural variability of versity of Connecticut, and two optical spectropluviom- DSDs. eters (OSP) from Laboratoire d'Etude des Transferts en There are also unresolved questions about the actual Hydrologie et Environnement in Grenoble, France. The shape of DSDs. A good deal of our past knowledge is area in which these instruments were deployed roughly derived from the Joss±Waldvogel disdrometer (JWD). corresponds to the size of one pixel from the Davenport, However, this instrument suffers from undersampling of Iowa, WSR-88D, located 85 km east of Iowa City, al- drop concentrations because of the acoustical noise pro- lowing exploration of the variability of re¯ectivity at duced by rain itself. This effect is ignored more often scales smaller than a typical radar pixel. While the main than not (Tokay and Short 1996). Acoustical noise af- goal of the XPOL ®eld experiment is aimed at exploring fects primarily, but not exclusively, the smaller drops, the advantages of dual-polarized X-band radar systems potentially creating a gamma-type distribution even if in radar-rainfall estimation, the experimental design the true one is exponential. The stronger the rain rate, readily facilitates characterization of the spatial vari- the more pronounced the effect. ability at sub-WSR-88D pixel scales, and comparison The physical process of drop sorting also contributes of different disdrometer types. to the arti®cial creation of gamma-type DSD shapes. As originally developed, the experimental setup Although this effect is real for ground-based disdro- yields 10 data points (one for each disdrometer pair) metric samples, it is not present in the volumetric radar over separation distances less than 1.05 km for con- sample, at least not to the same extent. However, gam- struction of the spatial covariance function necessary ma-type DSD models are often used in assessing our for point-to-area variance reduction. Unfortunately, the capability to take advantage of polarization diversity for original experimental setup was operational for only 7 improved rainfall estimation. These factors, and others, days; the easternmost OSP failed after 1 week of op- motivate us to take another look at DSD variability and eration. As such, this paper focuses on the intercom- instrumental effects. Campos and Zawadzki (2000) in- parison of the remaining four disdrometers. The removal vestigated several of these issues, but numerous ques- of the second OSP from the dataset reduces the number tions remain unresolved. of points available for estimation of the spatial covari- There are two approaches to quantifying the natural ance function to six, making point-to-area variance re- small-scale spatial variability of radar re¯ectivity. The duction analysis very dif®cult. With this knowledge, we ®rst involves another radar with a higher angular and focus on small-scale variability in terms of second-order range resolution, for instance, one operating at a higher moments and correlation/covariance between instru- frequency than the WSR-88D. The second approach in- ment pairs and among all instruments, without regard volves the aforementioned small-scale in situ experi- to spatial structure. ments, in which disdrometers and other instruments are The dataset collected during the XPOL ®eld experi- deployed in a small, heavily instrumented area to mea- ment is unique because of the spatial scales investigated

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FIG. 1. Aerial photograph of the densely instrumented area during the XPOL ®eld experiment. and the diverse instrumentation employed. In this paper, (2002). Below, we give only a brief description of its we focus on the results of an intercomparison of the principles of operation. The 2DVD consists of three disdrometer data and the resulting implications for the units: an outdoor sensor unit, an outdoor electronics question regarding small-scale re¯ectivity variability. unit, and an indoor user terminal. The 2DVD utilizes Speci®cally, we discuss the impact of instrument bias two orthogonal sheets of white light separated vertically on the quanti®cation of natural variability. by 6.2 mm. The light sheets are projected onto line- The paper is organized as follows: section 2 provides scan cameras. Falling hydrometeors passing through the a description of the disdrometers used. In section 3, we sensing area cast shadows on the photodetectors and are describe summary characteristics of the precipitation scanned twice, once by each light sheet, theoretically events during the ®eld experiment and present evidence providing three-dimensional information about the of the strong instrumental effects that permeate the data. drops. Comparing the scanned images from the two Section 4 addresses the issue of instrument bias and sheets and applying a matching algorithm yields drop discusses the correction algorithm applied to the data. diameter estimates. The vertical separation of the light The effects of natural variability are discussed in section sheets allows for estimation of vertical velocity. 5, both in terms of the variability itself and in terms of Each light sheet is 10 cm wide, resulting in a sensing gradients. In the last section, the ®ndings are summa- area of approximately 100 cm2, and the line-scan cam- rized and we offer concluding remarks. eras have 512 photodetectors; thus, each photodetector corresponds to approximately 0.2 mm. The ®nite num- 2. Description of disdrometers ber of photodetectors results in quantized drop sizes, and ®nite sampling by the cameras results in similar a. 2DVD quantization of fall speeds. When calibrated correctly, The 2DVD, manufactured by Joanneum Research at using small metal spheres of a known diameter, the the Institute for Applied Systems Technology in Graz, 2DVD provides reliable information for drops ranging Austria, is discussed in detail by Kruger and Krajewski from 0.2 to 8.0 mm. However, reliability decreases with

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Dmax S( f ) ϭ ͵ N(D)V(D)S ( f, D) dD, (1) Dmin where N(D) is the number concentration at drop diameter D per unit size interval per unit volume (mϪ3 mmϪ1), V(D) is the measurement volume (m3) for each diameter D and is proportional to D, and S( f, D)isthe volume-averaged Doppler power density at frequency f for a single drop of diameter D. The minimum de- tectable drop diameter, which for POSS is 0.34 mm, is

represented by Dmin; Dmax corresponds to the diameter of the largest drop in the sensing volume. POSS esti- FIG. 2. (bottom) The 1-min instrument-averaged rain rates (mm mates the DSD by inverting a discrete approximation hϪ1) for the entire XPOL ®eld experiment and (top) the ASOS-averaged wind speed (m sϪ1) associated with each precipitation event. of (1); N(D) is estimated at 34 diameters, ranging from Note that wind speeds are only plotted in association with precipi- 0.34 to 5.34 mm. Because POSS does not measure the tation. fall velocities of individual drops, an assumed relationship between drop diameter and fall velocity is incor- porated into the processing software. Speci®cally, POSS increasing wind speeds, because the wind ¯ow around utilizes an empirical ®t from Atlas et al. (1973) to data the instrument can prevent drops from passing through from Gunn and Kinzer (1949): the sensing area and can distort the spatial distribution ␷(D) ϭ 9.65 Ϫ 10.30eϪ0.6D, (2) of drops in the sensing area. NesÏpor et al. (2000) describe the ¯ow ®eld in and around the 2DVD and char- where ␷(D) is the fall velocity (m sϪ1) and D is the acterize DSD errors arising from the ¯ow, but supply drop diameter (mm). For consistency, (2) is applied to no correction algorithm. For some precipitation events, data from all instruments, even those that estimate fall de®ned as periods with at least 20 consecutive nonzero, velocities for individual drops. 1-min observations from at least three instruments, the To facilitate more meaningful quantitative DSD in- event-averaged Automated Surface Observing System tercomparisons between the 34-bin POSS DSD and the (ASOS) wind speeds are greater than 8 m sϪ1 (Fig. 2). 20-bin JWD DSD, we transform the 34-bin distributions These speeds are still unlikely to affect medium- to into the coarser 20-bin distributions. Two approaches to large-sized drops (D Ͼ 1 mm), which are the primary this transformation were considered. The ®rst involves contributors to re¯ectivity. Thus, we believe the ob- linear interpolation of the 34-bin spectra to the 20 JWD served wind speeds do not result in signi®cant re¯ec- bin average diameters. In the second approach, drops tivity estimation errors. For this experiment, drops from are redistributed from the original 34 bins into 20 bins. the 2DVD were distributed into the same drop size in- The resulting spectra are similar to one another, and to tervals as the JWD (section 2c), with all drops larger the original 34-bin distribution, but integrated values than the upper JWD limit being placed in the largest (e.g., rain rate R and radar re¯ectivity factor Z) obtained sized bin. from the linearly interpolated distributions have smaller mean-squared differences with respect to the integrated values computed from the original spectra. As such, b. POSS POSS results presented in this paper are based on lin- POSS is a bistatic, continuous wave X-band radar early interpolated 20-bin drop size spectra. initially developed by the Atmospheric Environment Service of Canada for use as an automated present c. JWD weather sensor (Sheppard 1990). The instrument is ori- ented upward and detects any re¯ecting particles passing The JWD (Joss and Waldvogel 1967) is manufactured through the sensing volume, which is located several by Distromet, Ltd., of Basel, Switzerland. It is an im- centimeters above the instrument and is on the order of pact-type instrument, which essentially converts the mo- 2m3. In rain, each drop falling through the POSS-sens- mentum of falling hydrometeors to a drop diameter ing volume produces a voltage signal, the frequency of based on the amplitude of the electrical pulses generated which is proportional to its Doppler velocity. The am- by the impact. Physical limitations constrain the de- plitude of this voltage signal is a function of the location tectable drop diameter range to 0.31 mm at the lower in the bistatic antenna beam pattern and the drop di- bound of the ®rst sized bin to 5.6 mm at the upper bound ameter. of the largest sized bin, and drops are distributed into The measured Doppler power density spectrum S( f ), 20 size intervals, which are not of uniform size. The at frequency f, for a collection of drops with diameter intervals increase from about 0.1 mm for the smallest D is drops to 0.5 mm for the largest drops.

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TABLE 1. Rainfall accumulation for each disdrometer, with and without application of the 0.5 mm hϪ1 rain-rate threshold, as well as for the ASOS gauge at the Iowa City Airport and the dual-bucket rain gauge platform collocated with the 2DVD, for different time periods during the XPOL ®eld experiment. All accumulations are in millimeters. Entire experiment Days with four operational instruments Accumulation Accumulation Accumulation Accumulation Instrument (no threshold) (with threshold) (no threshold) (with threshold) 2DVD Ð Ð 70.3 67.7 POSS 108.7 102.9 48.3 45.2 JWD 135.3 120.8 65.6 56.7 OSP 68.1 64.2 30.7 28.2 Dual-bucket gauge 123.5 123.5 51.7 51.7 ASOS gauge 118.9 118.9 52.6 52.6

The mechanical nature of the JWD instrument creates tensity of the transmitted light. As raindrops pass what is known as disdrometer dead time. After drops through the beam of light they reduce the transmitted impact the cone, a short period of time is required for intensity, resulting in a decay of the voltage signal. The the cone to stabilize, during which subsequent impacts magnitude of this decay is proportional to the drop di- are not accurately recorded. An algorithm to correct for ameter, and the residence time of the drop is estimated dead time exists (Sheppard and Joe 1994), but the dead- based on the duration of the decay. When fewer than time correction is not universally utilized within the ®eld three drops are passing through the beam at the same (e.g., Tokay et al. 2001), and it is not applied to this time, the OSP software can obtain reasonable estimates work. No on-site calibration was performed on the JWD, of drop diameter and fall velocity (Salles et al. 1998). because the sensor head used during the experiment was The OSP, like the 2DVD, requires careful calibration brand-new and was calibrated by the manufacturer. using small metal spheres of known diameter. The fol- The data collected by the JWD during the XPOL lowing correction was applied to the measured drop experiment are contaminated with noise signals of an diameters: unknown origin, which are predominately present in the two smallest drop size bins. This noise is present on Dtmϭ 1.0422D Ϫ 0.0705, (3) days with and without precipitation (we could not dis- where Dt is the true diameter (mm) and Dm is the mea- cern any temporal pattern of the noise occurrence), ne- sured diameter (mm). Calibration for this experiment cessitating a threshold to discriminate between rain and revealed sensitivity issues for drops smaller than 0.7 noise signals. Many possible options for this threshold mm in diameter, from pollution of the signal by a per- exist, but, for consistency, we want to apply this thresh- manent noise signal of unidenti®ed source. The OSP old to all of the instruments. Tokay et al. (2001) use a processing software outputs diameter and vertical ve- combination of drop counts and rain rate to mask ques- locity for each individual drop, making it possible to tionable data. We cannot apply such a threshold equally construct a DSD for any drop size classi®cation scheme. to all instruments because the POSS drop size distri- For this experiment, DSD for drops from the OSP were bution is dissimilar from the JWD scheme. Instead, we created using the same drop classi®cation scheme em- focus solely on rain rate as a threshold. We select R ϭ ployed by the JWD. 0.5 mm hϪ1, which corresponds to the 95% nonex- ceedance level for known noise signals (i.e., signals occurring on days without precipitation) in the JWD 3. Precipitation characteristics data. Because our rain-rate threshold is more stringent than the one employed by Tokay et al. (2001), we are a. Rainfall con®dent that it adequately masks questionable data. Precipitation was recorded on 20 of the 54 days, for a total of nearly 30 h, of the XPOL ®eld experiment, d. OSP which began on 3 October 2001 (Fig. 2). Only three disdrometers (POSS, JWD, and OSP) were operational The OSP was designed and built by the Centre for all 20 days. Rainfall accumulations for the entire d'Etude des Environnements Terrestre et Planetaire, Par- experiment vary greatly between instruments (Table 1). is, France. The basis for measurement is occultation of However, it is clear that the impact of the noise threshold a parallel beam of infrared light by falling hydrometeors on the JWD accumulation is much larger than the impact (Hauser et al. 1984). The sensing area of this instrument on POSS or OSP.All four disdrometers were operational was reduced from 100 to 60 cm2 by shields designed on only 12 of the 20 days with precipitation, or for to mitigate the effects of drops collecting on the optical nearly 20 h of rain. As with rainfall accumulations for lenses. The infrared light beam is focused on a photo- the entire experiment, rainfall accumulations for these diode that produces a voltage signal related to the in- 12 days vary greatly between instruments (Table 1).

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FIG. 3. Time series of rainfall accumulations for each disdrometer FIG. 4. The cumulative distribution functions for 1-min rain-rate and a dual-gauge platform collocated with the 2DVD during the 12 (dBR) data from each disdrometer, an instrument-averaged reference, days with four operational instruments. The dotted line is the cor- and the dual-gauge platform collocated with the 2DVD during the responding accumulation upon application of the 0.5 mm hϪ1 rain- 12 days with four operational instruments. rate threshold. Thus, each dotted line corresponds to the solid line immediately above it. The rain-rate threshold was not applied to the rain gauge accumulations because they are not based on rain rate, large random error (Ciach 2003); however, for our pur- and, therefore, there is no dotted line corresponding to the rain gauge accumulation trace. poses, these 1-min errors are not depicted because the mean error is zero and we are only analyzing the distribution of rain rates, rather than directly comparing 1- Again, the impact of the rain-rate threshold is most se- min rain rates from rain gauges and disdrometers. Al- vere on the JWD accumulations, while the other dis- though differences are not unexpected, the magnitudes drometers are affected less severely. In fact, the JWD of the observed differences are larger than we expect, accumulation curve is never truly horizontal because of based on intuition and previous work. Given that the the presence of this noise (Fig. 3). instruments are not collocated and that each has its own In comparing the rainfall accumulations from the dis- principle of operation, there are two possible sources drometers with the rainfall collected by a dual-gauge for the observed differences: natural variability and in- tipping-bucket platform collocated with the 2DVD, dif- strumental effects. The natural variability source in- ferences between each instrument and the rain gauge cludes storm motion (advection), microscale storm are not unexpected. Quality assurance checks indicate structure, and storm evolution. The instrumental effects that the rainfall record for this dual-gauge platform is include differences in operational principles, com- missing one event, so data from a nearby gauge are pounded by the fact that none of the instruments are substituted. The 2DVD accumulates about 30% more ``perfect;'' that is, none of these instruments measures rainfall than the 51.7 mm collected by the dual-gauge the true DSD for a given location. The effects of sam- platform, and the JWD accumulates about 10% more pling variability on a sample consisting of 20 h of pre- rainfall. POSS accumulates about 87% of this 51.7 mm cipitation are likely large and are included in the in- total and the OSP accumulates only 55% of what the strumental effects. dual-gauge platform records (Table 1). The discrepan- The magnitudes of the observed differences are much cies between instruments most noticeably arise during larger than we expect if natural variability were the sole periods of rapid accumulation, or heavy rain (Fig. 3). cause of the differences. Because these marginal dis- The 2DVD estimates are highest because it is not subject tributions are composed of a coincident set of obser- to saturation during heavy rainfall, like the JWD, and vations made by instruments in close proximity, the can physically detect large drops (D Ͼ 5.0 mm). The strong temporal correlation of precipitation processes, dual-platform gauge compares very favorably with the evident to even the most casual observer, suggests that tipping-bucket rain gauge at the Iowa City airport ASOS the likelihood of observing such different rain-rate dis- site, located nearly 4.5 km from the disdrometer sites. tributions is small, unless precipitation displays no spa- Just as the accumulations for the different instruments tial correlation. Again, casual observation and intuition are different, so too are the distributions of rainfall in- tell us that precipitation displays spatial correlation, and tensity (Fig. 4). We note here that the 1-min rain gauge that this correlation is high over short distances. In fact, rain rates are computed using the tip-interpolation a study of rainfall spatial correlation functions by Kra- scheme advocated by Ciach (2003). We also point out jewski et al. (2003), using the high-density cluster of that 1-min rain gauge rain rates are subject to relatively dual-platform rain gauges at the Iowa City Municipal

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FIG. 6. The cumulative distribution functions for 1-min re¯ectivity FIG. 5. The logarithm of the average DSD for each disdrometer, (dBZ) data from each disdrometer and an instrument-averaged ref- as well as an instrument-averaged reference, computed for obser- erence during the 12 days with four operational instruments. vations for which the rain rate of the reference is greater than 0.5 mm hϪ1. discrepancies in the poorly sampled ranges at both ex- Airport, shows that for Iowa City, correlation decays tremes of the distributions. Small drops (D Ͻ 1 mm) very slowly over spatial scales of 1 km. At the 15-min are subject to turbulent motions, which can mask the time scale, rainfall correlation at 1.2 km is greater than true terminal velocity of the drops and lead to mea- 0.9, which is certainly much higher than we could expect surement errors. Large drops (D Ͼ 4 mm) are so rare from these data. Further, the variability of cloud micro- that even instruments physically capable of measuring such drops cannot begin to accurately represent them physics over an area of 1 km2 over 20 h is unlikely to produce the observed discrepancies. All of these ar- in a sample size of 20 h. In contrast, the OSP does not guments point toward the presence of substantial in- behave like the other instruments. The OSP estimates strumental effects. of N(D) are signi®cantly lower for drop diameters less than 3 mm, which is likely a result of the ``pollution of the signal'' discussed in section 2d. Again, the observed b. Drop size distribution differences are larger than would be expected for 20 h of precipitation over an area of 1 km2. The drop size distribution N(D)(mϪ3 mmϪ1) for all instruments but POSS is calculated according to the following, which is given in the JWD documentation c. Re¯ectivity (Distromet 1975): Because the radar re¯ectivity factor Z is the sixth n(D ) moment of drop diameter, the observed DSD differences N(D ) i , (4) i ϭ should translate into larger differences in re¯ectivity Ft␷(Dii)⌬D values. Radar re¯ectivity is obtained by where n(Di) is the number of drops in a given size 2 Z ϭ N(D )D6 ⌬D , (5) interval, F is the sampling area (m ), t is the time sam- ͸ ii i Ϫ1 i pling interval (s), ␷(Di) is the fall velocity (m s ) for drops of size Di under the assumed relationship of (2), where N(Di) is de®ned in (4), Di is the diameter (mm) and ⌬Di is the size interval (mm) of drop class i. Recall, of an assumed spherical drop, and ⌬Di is the size in- the 20-bin POSS N(D) are linearly interpolated from terval (mm) of drop size class i. Because the distribution the original 34-bin spectra. Analysis of the average DSD of Z is highly skewed, we apply the logarithmic trans- collected by each instrument during the ®eld experiment formation, which reduces the skewness, and yields the reveals systematic differences between the sensors sim- familiar re¯ectivity units of dBZ. As expected from ilar to those seen in rain rate. analysis of the marginal rain-rate distributions, system- For medium-sized drops (1 mm Ͻ D Ͻ 3 mm), the atic differences exist in the re¯ectivity distributions for spectra from the 2DVD, JWD, and POSS show good each instrument (Fig. 6). As with rain rate, this type of agreement (Fig. 5). This is expected, because drops of marginal cumulative probability distribution analysis these sizes are well sampled; they are very common and largely reduces the impact of natural variability on the are measured with reasonable accuracy by each of the differences between the distributions. The rainfall cor- disdrometers. However, sampling variability leads to relation functions from the Iowa City Municipal Airport

Unauthenticated | Downloaded 09/27/21 05:38 PM UTC JANUARY 2004 MIRIOVSKY ET AL. 113 rain gauge cluster (Krajewski et al. 2003) are also relevant here, because the same DSD are responsible for both rain rate and re¯ectivity. But, again, we ®nd that the magnitude of the observed differences exceeds our expected differences over such a small area. The rainfall, DSD, and re¯ectivity data presented here clearly illustrate the signi®cance of instrumental bias on the observed differences. In order to analyze the natural variability component of the observed differences, we must ®rst attempt to remove these systematic instrumental effects.

4. Bias correction We begin removal of the instrumental effects by es- tablishing a reference set of measurements, to which individual instruments are compared in assessing the extent of the systematic errors present in each instru- FIG. 7. Bias correction factor Bm as a function of drop size class, ment. A number of choices exist for the reference set. represented at the mean diameter of each class, for each disdrometer. It could consist of observations from a single instrument, or from an average of a set of instruments. We performed the following analyses using several different an individual sensor. Thus, a bias correction factor Bmi reference sets, but our results are largely unaffected by exists for each drop size class. The bias correction is the choice of the reference. For the purposes of this applied to the DSD because such corrections translate work, we de®ne the reference set as the arithmetic mean to all integrated variables; a correction made to the DSD of the 2DVD, JWD, and POSS measurements during is realized in all higher-order moments considered, such all 20 h of precipitation. Our con®dence in the reliability as rain rate, re¯ectivity, or kinetic energy ¯ux, and at of the 2DVD and POSS data is high, especially in the any temporal integration scale, making it a robust bias well-sampled drop size ranges (1 mm Ͻ D Ͻ 3 mm); correction method. The multiplicative approach is cho- and the JWD has long been the community standard. sen for several reasons. First, it is a naturally weighted The OSP is omitted from the reference because of the method of bias correction; the impact of the correction sensitivity issues (section 2c) that lead to large devia- increases as N(Di) increases. The values of Bmi are non- tions in DSD from the other instruments in the well- dimensional, which eases interpretation of the results. sampled ranges (Fig. 5). We fully recognize that this In fact, Bmi provides an intuitive mechanism by which reference set is not equivalent to the true N(D) over the to compare accumulated DSDs (Fig. 7). Values greater area; rather, it serves as a mechanism by which we can than unity indicate an underestimation in comparison quantify the instrumental effects. with the reference, while values less than unity suggest In order to limit our study to coincident observations, overestimation. Comparing DSDs in this manner also the rain-rate noise threshold is applied such that only eases quantitative discrimination between instruments. Ϫ1 The multiplicative DSD bias correction is highly ef- observations with RRef Ͼ 0.5 mm h are considered. This effectively removes from consideration well over fective; it produces unconditionally unbiased observa- 98% of the questionable JWD observations. The re- tions, that is, the unconditional sample mean for each mainder of the paper focuses on analyses of data con- instrument is the same. The correction translates well strained in the manner described. to integral parameters, speci®cally, rain rate and re¯ec- Instrumental effects are an obvious source of differ- tivity, derived from DSDs (Fig. 8). The increased sim- ences and can cloud or overwhelm the natural vari- ilarity of the OSP is most signi®cant, because it dis- ability, necessitating removal of these effects. We pro- played extreme differences from the reference in the pose a multiplicative DSD bias correction, where the original data. However, the improvement does not com- correction factors have the following form: pletely remove the bias, because OSP continues to record low values of re¯ectivity and rain rate more fre- N(D ) ͸ i Ref quently than the other instruments. At larger values, B ϭ i , (6) especially above 30 dBZ, the bias-corrected OSP esti- mi N(D ) ͸ iK mates agree reasonably well with the reference, as well i as the other instruments. Improvements to the JWD re- where Bmi is the nondimensional bias correction factor ¯ectivity estimates are not as obvious as with the OSP; for drop size class i, ⌺ N(Di)Ref is the average drop however, the upper tail of the bias-corrected JWD re- concentration from the reference set of individual spec- ¯ectivity distribution agrees more strongly with the ref- tra in drop class i, and ⌺ N(Di)K is similar, except for erence distribution. The changes to the POSS re¯ectivity

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variable speci®c. The additive DSD correction method is unsatisfactory in that the DSD described by the collection of additive correction factors results in a nonzero baseline present in all corrected spectra. The nonzero baseline translates into a cutoff for all integrated variables, below which there are no occurrences. The natural weighting characteristic of the multiplicative DSD approach makes the nonzero baseline problem a nonissue, further justifying our decision to apply this method. For these reasons, we focused on the multiplicative DSD correction method.

5. Natural variability of radar re¯ectivity We now use the bias-corrected N(D) to consider the natural variability of radar re¯ectivity from two different approaches. We ®rst examine direct measures of variability and association within the 1-km2 area and then focus on statistics of re¯ectivity gradients, as in Tor- laschi and Humphries (1983). We also consider the effects of temporal integration on variability. In comparisons of radar and ground-based observations, the need to temporally integrate the ground-based observations is intuitive and well established (e.g., Habib and Kra- jewski 2002; Kitchen and Blackall 1992; Zawadzki 1975). While the task of comparing the disdrometer observations with either WSR-88D or XPOL radar observations is beyond the scope of this paper, the foundation for such work is discussed. Theoretically, integration of ground-based observations allows all of the drops illuminated by a radar beam to fall to the surface. Assuming that drops, on average, fall at 4 m sϪ1, and that the radar beam width is 0.95Њ, as with the WSR- 88D, integration times can theoretically be related to range. For example, 5 min of integration roughly cor- FIG. 8. Cumulative density functions of 1-min (a) rain rate (dBR) responds to a range of 72 km. and (b) re¯ectivity (dBZ) after application of the multiplicative bias We begin our analysis of the natural variability of correction. radar re¯ectivity by comparing the joint density functions of 1- and 15-min re¯ectivity data. The high variability of the measurements and the smoothing effects distribution are similar. Overall, the differences in the of temporal averaging are clearly illustrated (Fig. 9), distributions of both re¯ectivity and rain rate have been because the scatter of the 1-min bias-corrected data is greatly reduced by the bias correction. large, indicating high spatial variability at short time Although we choose to implement a multiplicative scales. In the mean, the agreement between instruments DSD bias correction, it is not the only approach. Other is strong, but individual observation pairs show large methods of bias correction include a regression-based sampling differences, with differences as large as 35 method speci®c to re¯ectivity (dBZ; or any integrated dBZ. As expected, the cloud of points constricts sig- variable desired), an additive bias model conditioned on ni®cantly in moving from 1- to 15-min data, with all the reference instrument, which is also speci®c to a sin- differences less than 20 dBZ, but considerable vari- gle integrated variable, and an additive DSD correction ability still exists. method. We examined these other approaches, but none Given that Z is the sixth moment of the drop size are as robust as the multiplicative DSD approach. The distribution and R is only the 3.67th moment, it follows regression-based technique is highly effective, but its that re¯ectivity should exhibit more variability than rain speci®city to a single integrated variable requires a new rate. We use the coef®cient of variation (CV), de®ned relationship be ®tted for each integrated measure con- as the ratio of the sample standard deviation to the sam- sidered. Further, the form of the functional relationship ple mean, to compare the variability of radar re¯ectivity can vary between instruments and with time integration. and rain rate, relative to their central locations. The high Similarly, the additive conditional bias model is also skewness of re¯ectivity should not affect this compar-

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FIG. 9. Joint density functions, with intervals of 2 dBZ, of bias-corrected re¯ectivity (dBZ) for each instrument paired with the 2DVD: (a)±(c) 1-min data, and (d)±(f) 15-min data. Individual points in intervals too sparsely populated for contouring are shown as light gray circles.

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FIG. 10. Comparison of the mean re¯ectivity of the four disdrometers (dBZ) and the std dev (dBZ) of those corresponding re¯ectivity measurements for all 20 h of 1-min data. The std dev and mean 6 are presented in decibels of Z but were computed in mm6 mϪ3. FIG. 11. Spatial correlograms of radar re¯ectivity factor Z (mm mϪ3) at three different time scales. Each point corresponds to one of the six possible disdrometer pairs. The (ⅷ) are 1-min data, the (Ⅺ) are 15-min data, and the (᭡) are 60-min data. ison because rain rate is also highly skewed. The disdrometer observations from this experiment indicate that re¯ectivity is more variable than rain rate. For 1- measurements, speci®cally, of the OSP observations. In min radar re¯ectivity factor data, CVZ,1 ϭ 0.636, while fact, the three lowest correlation coef®cients at the 1- CVR,1 ϭ 0.475. As discussed, temporal integration re- min time scale correspond to instrument pairs involving duces the variability: CVZ,15 ϭ 0.393 and CVR,15 ϭ the OSP (Fig. 11). As such, we turn our focus to de- 0.319. scriptions of the variability without regard to separation The small-scale cellular nature of intense precipita- distance. tion results in a direct relationship between variability Re¯ectivity gradients have been used for a variety of and intensity. Comparing instrument-averaged re¯ectiv- purposes, including general descriptions of precipitation ity with absolute interinstrument variability highlights patterns (Riley and Austin 1976) and removal of non- the strength of this relationship (Fig. 10). However, for precipitating echoes (Steiner and Smith 2002). Torlaschi any area-averaged re¯ectivity, the variability within the and Humphries (1983; hereinafter TH83) present dis- area can vary by up to one order of magnitude. Although tributions of re¯ectivity gradients computed from radar the sample size from which the variability is computed data and show the range dependency of these gradients. is small (four instruments), the large number of obser- We qualitatively compare distributions of XPOL dis- vations (1200 1-min observations) increases the reli- drometer re¯ectivity gradients with the TH83 data. The ability of these results. XPOL gradient distributions are computed for each ob- We now consider the natural variability of radar re- servation and all instrument pairs, at several time scales. ¯ectivity in terms of spatial correlation. The high skew- So, the gradient distribution for any given time scale is ness of the radar re¯ectivity factor Z leads to overes- composed of approximately 7200 observations (20 h per timation of the correlation by the traditional Pearson's instrument pair times six instrument pairs). While the formula. To mitigate this overestimation problem, we distributions of gradients computed from the XPOL dis- use a transformation-based procedure outlined by Habib drometers never exactly match the distributions from et al. (2001). No clear trend in correlation with sepa- TH83, they share many similarities (Fig. 12). In fact, ration distance is apparent at any time scale, due in part the linear shape (when plotted on a semilog scale) of to the small sample size (only six points are available the gradient distributions is consistent at all spatiotem- to estimate the spatial correlation function) and in part poral scales considered. A similar scaling also exists in to the high spatial variability, evidenced by the extreme the two datasets. As temporal integration (which is in scatter of these correlation coef®cients (Fig. 11). Spatial some sense equivalent to range) increases, the distri- correlation for rain rate (not shown) exhibits similar butions tend toward smaller gradients, while shorter behavior. The primary reason for the low correlation at temporal scales (and shorter ranges) exhibit large gra- short time scales is the high variance of the re¯ectivity dients (Ͼ10 dBZ kmϪ1) more frequently.

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FIG. 12. Density functions of radar re¯ectivity gradients (dBZ kmϪ1) from TH83 and the XPOL disdrometers at three different time FIG. 13. Re¯ectivity rmsd measurements from the XPOL disdro- scales. The dot±dash curves were taken from TH83. meter pairs at integration times of 1 and 60 min, which can roughly be equated with ranges from 14 to 800 km. We use a moving-average window to temporally integrate the bias-corrected data to identify an appropriate length of time for averaging. Analysis of the root-mean- tegration of 15 min, indicating that additional factors square difference (rmsd) at different time scales indi- are acting to cause the observed differences. We have cates that at time scales less than 15 min, the re¯ectivity chosen not to quantitatively address the issue of sam- gradient is highly dependent on the temporal integration pling variability because regardless of the source of the (Fig. 13). Averaging for longer than 15 min results in instrumental effects, they need to be accounted for in considerable smoothing. In fact, decreases in rmsd be- some way before proceeding with additional analysis. yond 15 min of averaging are negligible, suggesting that To this end, a multiplicative DSD bias correction 15-min temporal integration minimizes the differences scheme, based on the premise that N(D) averaged over due to natural variability as well as longer periods of several storms should not vary considerably at scales integration. below 1 km2, is applied to mitigate these instrumentation effects. The approach is valid across temporal scales and is naturally weighted, affecting more strongly 6. Concluding remarks those observations with large N(D). The XPOL ®eld experiment yielded a unique dataset Using the bias-corrected observations, we examined and presented numerous avenues by which to examine the variability of re¯ectivity in space. We found no dis- small-scale spatial variability of re¯ectivity. This ®eld cernible spatial structure, suggesting that whatever spa- experiment employed an X-band polarimetric radar, col- tial structure exists is clouded by the small sample size, lecting data over a densely instrumented site with four and our inability to fully remove the instrumental effects disdrometers, a vertically pointing Doppler radar, and through the bias correction scheme. Still, the variability several tipping-bucket dual-gauge platforms. This paper of re¯ectivity is considerable at spatial scales smaller focuses on analyses of the disdrometer data collected than a WSR-88D pixel. Even at temporal integration from October to November 2001 near Iowa City, but scales of 15 min, which should largely eliminate the comparisons of the XPOL data with these disdrometer variability due to advection, the spatial variability is data and WSR-88D data are forthcoming. signi®cant, and reduces little upon further integration. Strong instrumental bias permeates the data collected As expected, the absolute spatial variability tends to during the experiment. The existence of such bias is increase with increasing rainfall intensity, and re¯ectiv- readily apparent through analysis of rainfall accumu- ity displays more variability than rain rate. Examination lation, accumulated DSD, and marginal cumulative dis- of re¯ectivity gradients showed qualitative consistency tributions of rain rate and re¯ectivity. Of course, sam- with the work of TH83 in both the temporal/range de- pling variability contributes to these instrumental effects pendency of gradients, and in the behavior of the dis- because of the limited nature of our precipitation sample tributions of re¯ectivity gradients. (20 h); however, temporal integration theoretically mit- Although we are unable to make any quantitative igates the effects of sampling variability, as the number statements about small-scale spatial variability of re- of drops sampled increases, thereby providing more ro- ¯ectivity because of the overwhelming instrumental ef- bust estimates of the true DSD. We observed the per- fects, it is clear that spatial variability of re¯ectivity is sistence of instrumental effects even upon temporal in- high, suggesting that this is a likely source of error in

Unauthenticated | Downloaded 09/27/21 05:38 PM UTC 118 JOURNAL OF APPLIED METEOROLOGY VOLUME 43 radar rainfall estimation. This is especially relevant Georgakakos, K. P., A. A. Carsteanu, P. L. Sturdevant, and J. A. when comparing radar estimates with ground-based Cramer, 1994: Observation and analysis of Midwestern rain rates. J. Appl. Meteor., 33, 1433±1444. point estimates. Further, several key lessons were Gunn, R., and G. D. Kinzer, 1949: The terminal velocity of fall for learned from the XPOL ®eld experiment that would aid water droplets in stagnant air. J. Meteor., 6, 243±248. in future experiments. While not necessary, future com- Gupta, V. K., and E. Waymire, 1990: Multiscaling properties of spatial parisons will bene®t from using multiple disdrometers rainfall and river ¯ow distributions. J. Geophys. Res., 95, 1999± 2009. of the same type. Obviously, if carefully calibrated, this Habib, E., and W. F. Krajewski, 2002: Uncertainty analysis of the should eliminate all instrumental biases. Second, the TRMM ground-validation radar-rainfall products: Application to experiment should be conducted in two phases: ®rst with the TEFLUN-B ®eld campaign. J. Appl. Meteor., 41, 558±572. the instruments collocated, and then separated. Collo- ÐÐ, ÐÐ, and G. J. Ciach 2001: Estimation of rainfall interstation cating the instruments initially is especially important correlation. J. Hydrometeor., 2, 621±629. Hauser, D., P. Amayenc, B. Nutten, and P. Waldteufel, 1984: A new in the event that the instruments are not all of the same optical instrument for simultaneous measurements of raindrop type. By beginning with the instruments collocated, the diameter and fall distributions. J. Atmos. Oceanic Technol., 1, natural variability is reduced or eliminated, isolating the 256±269. instrumental differences. This eliminates the need to try Joss, J., and A. Waldvogel, 1967: A raindrop spectrograph with au- tomatic analysis. Pure Appl. Geophys., 68, 240±246. to separate the instrumental effects from natural vari- Kitchen, M., and R. M. Blackall, 1992: Representativeness errors in ability and advection when the instruments are sepa- comparisons between radar and gauge measurements of rainfall. rated. J. Hydrol., 134, 13±33. In fact, during the summer of 2002 we conducted Krajewski, W. F., G. J. Ciach, and E. Habib, 2003: An analysis of another ®eld experiment with six collocated instruments small-scale rainfall variability in different climatological re- gimes. Hydrol. Sci. J., 48, 151±162. that included two vertically pointing radars, three optical Kruger, A., and W. F. Krajewski, 2002: Two-dimensional video dis- disdrometers, and a Joss±Waldvogel disdrometer. We drometer: A description. J. Atmos. Oceanic Technol., 19, 602± will report on the results of this study, in addition to 617. results of comparisons between the XPOL and WSR- NesÏpor, V., W. F. Krajewski, and A. Kruger, 2000: Wind-induced error of raindrop size distribution measurement using a two-dimen- 88D data, in the near future. sional video disdrometer. J. Atmos. Oceanic Technol., 17, 1483± 1492. Acknowledgments. The National Science Foundation Riley, G. F., Jr., and P. M. Austin, 1976: Some statistics of gradients Grant EAR00-03046, IIHRÐHydroscience and Engi- of precipitation intensity. Preprints, 17th Conf. on Radar Me- neering, and Rose and Joseph Summers Professorship teorology, Seattle, WA, Amer. Meteor. Soc., 419±420. Salles, C., J.-D. Creutin, and D. Sempere-Torres, 1998: The optical held by W. F. Krajewski at The University of Iowa sup- spectropluviometer revisited. J. Atmos. Oceanic Technol., 15, ported this project. An AMS Industry/Government 1215±1222. Graduate Fellowship funded by the NSF Division of Sheppard, B. E., 1990: Measurement of raindrop size distributions Atmospheric Sciences supported B. Miriovsky. We are using a small Doppler radar. J. Atmos. Oceanic Technol., 7, 225± 268. also indebted to Mary Ebert, Diana Thrift, Leanne Ei- ÐÐ, and P. I. Joe, 1994: Comparison of raindrop size distribution chinger, and Ching Long Lin for allowing us to use their measurements by a Joss±Waldvogel disdrometer, a PMS 2DG land for instrument deployment. spectrometer, and a POSS Doppler radar. J. Atmos. Oceanic Technol., 11, 874±887. Steiner, M., and J. A. Smith, 2002: Use of three-dimensional re¯ec- REFERENCES tivity structure for automated detection and removal of nonpre- cipitating echoes in radar data. J. Atmos. Oceanic Technol., 19, Atlas, D., R. C. Srivastavam, and R. S. Sekhon, 1973: Doppler radar 673±686. characteristics of precipitation at vertical incidence. Rev. Geo- Tokay, A., and D. A. Short, 1996: Evidence from tropical raindrop phys. Space Phys., 11, 1±35. spectra of the origin of rain from stratiform versus convective Battan, L. J., 1973: Radar Observation of the Atmosphere. University clouds. J. Appl. Meteor., 35, 355±371. of Chicago Press, 324 pp. ÐÐ, A. Kruger, and W. F. Krajewski, 2001: Comparison of drop Bras, R. L., and I. RodrõÂguez-Iturbe, 1985: Random Functions and size distribution measurements by impact and optical disdro- Hydrology. Addison-Wesley, 704 pp. meters. J. Appl. Meteor., 40, 2083±2097. Bringi, V. N., and V. Chandrasekar, 2001: Polarimetric Doppler Torlaschi, E., and R. Humphries, 1983: Statistics of re¯ectivity gra- Weather Radar: Principles and Applications. Cambridge Uni- dients. Preprints, 21st Conf. on Radar Meteorology, Edmonton, versity Press, 656 pp. AB, Canada, Amer. Meteor. Soc., 173±175. Campos, E., and I. Zawadzki, 2000: Instrumental uncertainties in Z± Venugopal, V., E. Foufoula-Georgiou, and V. Sapozhnikov, 1999: R relations. J. Appl. Meteor., 39, 1088±1102. Evidence of dynamic scaling in space±time rainfall. J. Geophys. Ciach, G. J., 2003: Local random errors in tipping-bucket rain gauge Res., 104, 31 599±31 610. measurements. J. Atmos. Oceanic Technol., 20, 752±759. Zawadzki, I., 1975: On radar±raingage comparison. J. Appl. Meteor., Distromet, 1975: Distrometer RD-69: Instruction manual. Distromet, 14, 1430±1436. Ltd., 18 pp. [Available from Distromet, Ltd., P.O. Box 33, CH- ÐÐ, 1982: The quantitative interpretation of weather radar mea- 4020 Basel, Switzerland.] surements. Atmos.±Ocean, 20, 158±180.

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