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  • View in gallery

    (a) The NEXRAD National Mosaic Reflectivity Image for 5 Feb 2010 at 0500 UTC. (b) The WSR-88D radar image of reflectivity around the Huntsville area (pink box indicated in panel a). (c) Reflectivity time series from the C-band radar “dwell scans” over the disdrometer site (azimuth 52°); the dotted line represents the location of the 2DVDs (at around 15-km range). (d) One-min DSDs from the low-profile 2DVD; N(D) in mm−1 m3.

  • View in gallery

    Differential reflectivity measurements: light brown line represents the extracted raw (but attenuation corrected) data over the 2DVD site, the green line shows the corresponding data after FIR filtering in range, and the blue points show the Zdr after time filtering the raw data (i.e., time filtering the brown line).

  • View in gallery

    (a) Reflectivity comparisons between the radar and the two disdrometer measurements and (b) the corresponding Zdr comparisons.

  • View in gallery

    Retrieved (a) log10(NW), (b) D0, and (c) rain rate R corresponding to Fig. 1c.

  • View in gallery

    Radar retrievals (after time interpolation) extracted at the 2DVD location, shown as solid lines, compared with disdrometer-based estimates (circles for low profile and crosses for compact 2DVD-based estimates) of (a) log10(NW), (b) D0, and (c) R.

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    Spatial correlations determined from the dwell scans, and their fitted lines.

  • View in gallery

    Correlation coefficient vs time for log10(NW) and D0 from radar data and the two disdrometers.

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Estimating the Accuracy of Polarimetric Radar–Based Retrievals of Drop-Size Distribution Parameters and Rain Rate: An Application of Error Variance Separation Using Radar-Derived Spatial Correlations

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  • 1 Department of Electrical and Computer Engineering, Colorado State University, Fort Collins, Colorado
  • 2 NSSTC/University of Alabama in Huntsville, Huntsville, Alabama
  • 3 NASA MSFC, Huntsville, Alabama
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Abstract

The accuracy of retrieving the two drop size distribution (DSD) parameters, median volume diameter (D0), and normalized intercept parameter (NW), as well as rain rate (R), from polarimetric C-band radar data obtained during a cool-season, long-duration precipitation event in Huntsville, Alabama, is examined. The radar was operated in a special “near-dwelling” mode over two video disdrometers (2DVD) located 15 km away. The polarimetric radar–based retrieval algorithms for the DSD parameters and rain rate were obtained from simulations using the 2DVD measurements of the DSD. A unique feature of this paper is the radar-based estimation of the spatial correlation functions of the two DSD parameters and rain rate that are used to estimate the “point-to-area” variance. A detailed error variance separation is performed, including the aforementioned point-to-area variance, along with variance components due to the retrieval algorithm error, radar measurement error, and disdrometer sampling error. The spatial decorrelation distance was found to be smallest for the R (4.5 km) and largest for D0 (8.24 km). For log10(NW), it was 7.22 km. The proportion of the variance of the difference between radar-based estimates and 2DVD measurements that could be explained by the aforementioned errors was 100%, 57%, and 73% for D0, log10(NW), and R, respectively. The overall accuracy of the radar-based retrievals for the particular precipitation event quantified in terms of the fractional standard deviation were estimated to be 6.8%, 6%, and 21% for D0, log10(NW), and R, respectively. The normalized bias was <1%. These correspond to time resolution of ~3 min and spatial resolution of ~1.5 km.

Corresponding author address: M. Thurai, Dept. of Electrical and Computer Engineering, Colorado State University, Fort Collins, CO 80523-1373. E-mail: merhala@engr.colostate.edu

Abstract

The accuracy of retrieving the two drop size distribution (DSD) parameters, median volume diameter (D0), and normalized intercept parameter (NW), as well as rain rate (R), from polarimetric C-band radar data obtained during a cool-season, long-duration precipitation event in Huntsville, Alabama, is examined. The radar was operated in a special “near-dwelling” mode over two video disdrometers (2DVD) located 15 km away. The polarimetric radar–based retrieval algorithms for the DSD parameters and rain rate were obtained from simulations using the 2DVD measurements of the DSD. A unique feature of this paper is the radar-based estimation of the spatial correlation functions of the two DSD parameters and rain rate that are used to estimate the “point-to-area” variance. A detailed error variance separation is performed, including the aforementioned point-to-area variance, along with variance components due to the retrieval algorithm error, radar measurement error, and disdrometer sampling error. The spatial decorrelation distance was found to be smallest for the R (4.5 km) and largest for D0 (8.24 km). For log10(NW), it was 7.22 km. The proportion of the variance of the difference between radar-based estimates and 2DVD measurements that could be explained by the aforementioned errors was 100%, 57%, and 73% for D0, log10(NW), and R, respectively. The overall accuracy of the radar-based retrievals for the particular precipitation event quantified in terms of the fractional standard deviation were estimated to be 6.8%, 6%, and 21% for D0, log10(NW), and R, respectively. The normalized bias was <1%. These correspond to time resolution of ~3 min and spatial resolution of ~1.5 km.

Corresponding author address: M. Thurai, Dept. of Electrical and Computer Engineering, Colorado State University, Fort Collins, CO 80523-1373. E-mail: merhala@engr.colostate.edu

1. Introduction

One important practical application of dual-polarized radar is the retrieval of the drop size distribution (DSD) parameters and rain rate using the combination of horizontal reflectivity, differential reflectivity, and specific differential phase (Zh, Zdr, and Kdp, respectively). There is a large body of literature on this topic but most studies have dealt with the estimation of primarily rain rate and its accuracy by comparison with gauges at different spatiotemporal scales (e.g., Seliga et al. 1981; Le Bouar et al. 2001; Ryzhkov et al. 2005; Matrosov 2010). The accuracy of retrieving DSD parameters has been considered via simulations—for example, by Gorgucci et al. (2002)—considering both parameterization and measurement errors (see chapter 8 of Bringi and Chandrasekar 2001). Validation of the retrievals of median volume diameter (D0) using disdrometers generally without explicitly considering error variance separation has been obtained from S-band radar (e.g., Seliga et al. 1986; Bringi et al. 1998; Schuur et al. 2001; Brandes et al. 2004) and from X-band radar (e.g., Matrosov et al. 2005; Anagnostou et al. 2008; Kim et al. 2010). Discrepancies between radar retrievals and disdrometer measurements, apart from consideration of parameterization and radar measurement errors, have largely been ascribed to differences between the areal and point measurements or to vertical evolution of the DSD from the height of the radar sampling volume to the surface.

It is now well known that this “point-to-area” variance can be an important contributor to the error variance separation when comparing radar-based areal retrievals with point measurements from gauges or disdrometers (Ciach and Krajewski 1999; Habib and Krajewski 2002). The point-to-area variance can be computed under the assumption of second-order stationarity and isotropy if the spatial correlation function can be estimated (Bras and Rodriguez-Iturbe 1994). Using dual-polarimetric radar measurements, Moreau et al. (2009) employed the so-called ZPHI method (Testud et al. 2000) to estimate the spatial correlation function of rain. They applied the radar-based spatial correlation function to the error variance separation for hourly rainfall accumulations and found remarkable agreement with a dense rain gauge network. A similar error variance separation study using a C-band dual-polarimetric radar and rain gauge network was conducted by Bringi et al. (2011). Others (e.g., Gebremichael and Krajewski 2004) have used Weather Surveillance Radar-1988 Doppler (WSR-88D) radars over shorter integration times (5–15 min) to estimate the spatial correlation function of rain rate, but have had discouraging results when compared with rain gauge networks. They partly ascribe this disagreement to the 2 × 2 km2 resolution of the WSR-88D rainfall product used, and thereby no spatial correlation values could be estimated for distances less than 2 km. The behavior of the spatial correlation at distances less than 2 km governs the shape of the functional fit, and depending on the temporal resolution, the point-to-area variance is mostly governed by the spatial correlation values at shorter distances (Gebremichael and Krajewski 2004). It takes numerous rain gauge measurements to estimate the spatial correlation function, and depending on the spatiotemporal scales of interest, the gauge network must be sufficiently dense (Moore et al. 2000). On the other hand, if it can be shown that radar-based spatial correlation functions can be derived with reasonable accuracy, then a much smaller database is required by virtue of the vast coverage area with high density of range resolution volumes available from radar. Research radars can be used to obtain highly focused measurements (i.e., “dwell”) over a small instrumented site at short range, providing a unique opportunity to collect data at very high temporal resolutions (<10 s) with spatial resolutions on the order of 250–500 m. In the past, such radar data have been from single-polarized radars either pointing vertically or performing rapid RHI or plan position indicator (PPI) scans (Rico-Ramirez et al. 2007; Fabry et al. 1994; Austin et al. 2010).

It is well known that the rain rate shows large spatiotemporal variability, and one reason for this might be that the rain rate is a product of at least two random variables: (i) the total concentration behaving as a Poisson distribution (e.g., Gertzmann and Atlas 1977), and (ii) the diameter behaving as a gamma distribution (e.g., Wong and Chidambaram 1985). More recent work of Jameson and Kostinski (1999) demonstrates that the large variability can also be due to “clustered” non-Poisson rain (i.e., triply stochastic process) and that the variability in drop concentration might play a more important role than variations in drop sizes. Thus the ratios of DSD moments, such as the mass-weighted mean diameter (Dm = ratio of fourth to third moments), would be less variable than R since the multiplicative concentration term, at least on average, “cancels” out. Since the measure of Zdr is an estimator of Dm (see section 7.1.1 of Bringi and Chandrasekar 2001), it is possible to test this hypothesis experimentally by estimating the spatial correlation function of Dm (or, equivalently, the median drop diameter D0) from high-resolution polarimetric radar measurements. The other parameter of interest is the normalized “intercept” parameter (NW) of an assumed gamma DSD form (Illingworth and Blackman 2002). The parameter NW is proportional to W/Dm4, where W is the rainwater content. The NW can be estimated from the measurement of reflectivity and D0 (Bringi et al. 2002), where D0 is the median volume diameter. There does not appear any past work related to the spatial variability of NW except the airborne data from Testud et al. (2000), who demonstrate that NW is “locally” invariant after stratiform–convective rain type separation is performed.

Some of the early work on spatial variability of the radar reflectivity was done by Miriovsky et al. (2004) using four different types of disdrometers deployed within an area of 1 × 1 km2. They concluded that instrument differences could not be ignored and tended to “…cloud the natural variability of interest.” They also concluded that spatial variability of reflectivity can be “very large.” In a later study, Lee et al. (2009) examined the spatial variability of DSD moments, the coefficient/exponent of ZR relations, as well as the characteristic number density and characteristic diameter in stratiform rain with four Precipitation Occurrence Sensor System (POSS) instruments, but the separation distances were not optimal (1.3 km, 15.5 km, and four distances close to 30 km). They found that the ZR relations generally varied substantially within a single stratiform rain event. They also found that the characteristic number density (similar to NW) had a greater decorrelation distance (1.3 km) than the characteristic diameter (similar to Dm). Tokay and Bashor (2010) estimated the spatial correlation of NW and Dm (including other quantities such as Z, W, and R) using data from three impact-type disdrometers (spaced 0.65, 1, and 1.7 km) and at time resolutions of 1–15 min. Not surprisingly, they found that the correlation coefficient decreased (increased) with distance (time averaging), and in contrast with expectations, they found a larger decrease in the correlation for Dm compared with R. The previous studies of Miriovsky et al. (2004), Lee et al. (2009), and Tokay and Bashor (2010) can be considered as fairly limited in scope in so far as the spatial correlation of DSD or R are considered. There is now an increasing trend toward deploying dense disdrometer networks to study the small-scale variability of the DSD (Berne et al. 2010; Tapiador et al. 2010) with or without accompanying polarimetric radar coverage. Tapiador et al. (2010), who were able to measure the spatial correlation function of reflectivity for different storm events at separation distances ranging from 200 to 3200 m, found that the coefficient–exponent of ZR relations could vary more within a single storm event (i.e., intrastorm), as opposed to interstorm variability, which is in agreement with Lee et al. (2009).

In this paper we derive the spatial correlation function of D0, log10(NW), and R using polarimetric radar data [Advanced Radar for Meteorological and Operational Research (ARMOR); see Petersen et al. 2007] acquired at high temporal resolution from a long-duration (4 h) cool-season precipitation event near Huntsville, Alabama. The rain event was widespread, with frequent occurrences of embedded convection. The retrieval algorithms for D0, NW, and R were derived from scattering simulations using DSDs measured by two collocated 2D video disdrometers (Schönhuber et al. 2008) for this same event. A comprehensive error variance separation is conducted, including (i) the point-to-area variance, (ii) parameterization or algorithm errors, (iii) radar measurement errors, and (iv) sampling errors from the two collocated disdrometers. We describe the relative importance of these various error sources for estimating D0, NW, and R. We also discuss our findings with other related work.

2. Radar and disdrometer measurements

The precipitation event considered herein (5 February 2010) was very widespread with somewhat high larger-scale spatial uniformity (in relative terms) in all directions. The Next Generation Weather Radar (NEXRAD) National Mosaic Reflectivity Image for 5 February 2010, at 0500 UTC, is shown in Fig. 1a, and the corresponding WSR-88D radar (KHTX) image of reflectivity around the Huntsville area is shown in Fig. 1b. The “red dot” in Fig. 1b indicates the location of the two video disdrometers (2DVDs) (which are separated by no more than 5 m) and the “blue star” indicates the radar site. The distance between the radar and the instrumented site is 15 km. The ground temperature at Huntsville was around 43°F—the event being a low bright-band case.

Fig. 1.
Fig. 1.

(a) The NEXRAD National Mosaic Reflectivity Image for 5 Feb 2010 at 0500 UTC. (b) The WSR-88D radar image of reflectivity around the Huntsville area (pink box indicated in panel a). (c) Reflectivity time series from the C-band radar “dwell scans” over the disdrometer site (azimuth 52°); the dotted line represents the location of the 2DVDs (at around 15-km range). (d) One-min DSDs from the low-profile 2DVD; N(D) in mm−1 m3.

Citation: Journal of Hydrometeorology 13, 3; 10.1175/JHM-D-11-070.1

The radar azimuth was almost fixed to dwell over the disdrometer site, with the elevation set to 1°. Given that the half-power antenna beamwidth is close to 1°, the cross-beam resolution will be around 250 m at the range of 15 km. The height of the radar pixel above ground will also be around 250 m.

The radar task configuration was set in such a way that the range profiles of radar reflectivity were recorded at a time sampling between 4 and 8 s. The scanning strategy and the pertinent radar specifications are given in Table 1.

Table 1.

Scan strategy and the relevant radar specifications.

Table 1.

The range-time variation of the radar reflectivity is shown in Fig. 1c for a 4-h period. The location of the 2DVD site is indicated with a dotted line. There is evidence of “streaks” of reflectivity advecting over the 2DVD site as well as cell evolution. As will be explained later, the spatial correlations of the DSD parameters for this event were determined using the 2DVD location as the reference point. Figure 1d shows the time series of 1-min DSDs recorded by one of the 2DVD units (the low profile). As seen, the event is dominated by small- to moderate-sized drops.

a. Radar data processing

The initial processing of radar data comprised several standard procedures, as follows:

  • For each recorded range profile, the meteorological/nonmeteorological echo separation is made, primarily using the standard deviation of the differential propagation phase Φdp; this is followed by the use of the so-called hail detection ratio (often abbreviated as HDR and defined in Aydin et al. 1986, but adapted here for C band) to qualitatively ensure no hail or melting graupel is present. (Since this event did not contain drops larger than 3 mm—as seen from Fig. 1d—it became possible to utilize the C-band version of HDR without any complications arising because of resonance scattering.)
  • The Φdp range profile is finite impulse response (FIR), filtered using the technique described in Hubbert and Bringi (1995) to determine the specific differential propagation phase (Kdp); this method has the advantage of quantifying and removing any backscatter differential phase contribution.
  • An attenuation-correction scheme is employed in order to correct for copolar attenuation and differential attenuation. Although there are several methods available in the literature, we have chosen (i) the iterative ZPHI method (Bringi et al. 2001; Testud et al. 2000) to determine the attenuation-corrected reflectivity (Zh) range profile (in brief, the ZPHI method uses a power law between the specific attenuation and reflectivity, a linear relation between specific attenuation and specific differential phase, and uses the path differential phase as a constraint) and (ii) the gate-to-gate correction scheme (Tan et al. 1995) to determine the attenuation-corrected differential reflectivity (Zdr) range profile. For (i), the coefficients of the (linear) attenuation versus Kdp relation are determined using the iterative procedure (Bringi et al. 2001), whereas for (ii) the coefficient–exponent of the power law relating differential attenuation to Kdp are determined based on scattering calculations–simulations using the measured 1-min DSDs from the 2DVD for the precipitation event of 5 February 2010.
  • Consistency checks are used to determine Zh and Zdr calibrations from the “two-dimensional histograms” representing the contoured frequency of occurrence plots for Kdp versus Zh and Zdr versus Zh, following the procedure described in Bringi et al. (2006). The calculations based on the aforementioned 1-min DSDs are superimposed in order to ensure the accuracy of the calibrations.
For our present study, one additional step needed to be implemented. Because the measured Zdr is generally “noisy,” the FIR filtering (i.e., the same technique used for Φdp filtering) is used for smoothing in order to filter out the high-frequency gate-to-gate fluctuations. An illustrative example is given in Fig. 2. The thin brown line indicates the attenuation-corrected Zdr time series extracted over the 2DVD site and the green line indicates the corresponding Zdr after the FIR filter. Also included is the time-filtered Zdr variation (solid blue circles) by applying the so-called “Lee-filter algorithm” (a standard procedure used for speckle noise filtering; see, e.g., Lee 1980) to the noisy Zdr data. As can be seen, the range-filtered Zdr agrees very well with the time-filtered Zdr. The agreement not only implies that both methods are equally applicable, but also indicates that there is some form of a small-scale equivalence in the space–time domain. The FIR range filtering is essentially a weighted moving average filter of order 20 and here the window is 10 gates on either side of the window center, whereas the Lee filtering is applied over 23 time samples (window with 11 time samples on either side of the window center).
Fig. 2.
Fig. 2.

Differential reflectivity measurements: light brown line represents the extracted raw (but attenuation corrected) data over the 2DVD site, the green line shows the corresponding data after FIR filtering in range, and the blue points show the Zdr after time filtering the raw data (i.e., time filtering the brown line).

Citation: Journal of Hydrometeorology 13, 3; 10.1175/JHM-D-11-070.1

b. Comparison with DSD-based calculations

The attenuation-corrected Zh and Zdr (from the range-filtered profiles) extracted over the 2DVD location are shown as time series in Figs. 3a and 3b, respectively, and compared with the 2DVD data-based calculations. For the latter, scattering (T matrix) calculations based on 1-min DSDs were made assuming our reference drop shapes (Thurai et al. 2007) as well as the standard Gaussian distribution for drop canting angle distributions (Huang et al. 2008). The events considered here are not severe, nor intense to warrant more detailed calculations using the scattering matrix for each individual drop shape and orientation (as was done, e.g., in Thurai et al. 2009 for an intense event). Moreover, the agreement between the radar data and the 2DVD-based calculations further corroborate the sufficient validity of the “bulk” assumptions.

Fig. 3.
Fig. 3.

(a) Reflectivity comparisons between the radar and the two disdrometer measurements and (b) the corresponding Zdr comparisons.

Citation: Journal of Hydrometeorology 13, 3; 10.1175/JHM-D-11-070.1

3. DSD retrievals

The retrieval of DSD parameters from the radar data (after correction, filtering, and smoothing) was made based on the assumption that the DSDs can be represented by the gamma distribution with the three standard parameters, namely NW, D0, and μ—the definitions of which can be found, for example, in Illingworth and Blackman (2002). Simulations using the 1-min fitted gamma DSDs using the disdrometer measurements for the first event gave rise to the following equations for retrieving the DSDs from the radar data:
e1a
e1b
e1c
e1d
e1e
where Zdr in (1a) is in dB, and Zdr in (1b) and (1e) are expressed as the ratio term. Also note Zh_linear represents the radar reflectivity in linear units. The assumptions for the scattering simulations were (i) water temperature = 7°C (consistent with ground temperature for this event) and (ii) 0° elevation angle.

The above set of equations have the same form as those used in previous C-band studies (see, e.g., Bringi et al. 2011), with the exception of (1b), which has been included as an additional estimation method for D0 for DSDs with relatively high concentration of small drops. The main reason for using a combination of Zh and Zdr for estimating D0 in the “small-drop” region is the insensitivity of Zdr to drops having axis ratios close to 1, as is the case for smaller drops. Comparisons made for other events (not considered here) do show the improvement in D0 estimates when using (1b). At the other extreme, an upper limit of 1.8 mm for D0 is quoted since the range of D0 values used in the fitting procedure did not go beyond this limit.

Figures 4a and 4b show, respectively, the retrieved log10(NW) and D0 from the radar data for the event corresponding to Fig. 1 (on 5 February 2010). Again, the black lines represent the range of the two 2DVD locations. Higher D0 values prior to 0530 UTC can be seen that correspond to the higher Zdr from Fig. 3b. Also seen from Fig. 4a is the advection of the high D0 region as it traverses the 2DVD location and beyond (observed as a streak). The retrieved rain rates are shown in Fig. 4c [as log10(R)], which also shows the advection in the space–time domain, with rain rates reaching just above 10 mm h−1 in some areas.

Fig. 4.
Fig. 4.

Retrieved (a) log10(NW), (b) D0, and (c) rain rate R corresponding to Fig. 1c.

Citation: Journal of Hydrometeorology 13, 3; 10.1175/JHM-D-11-070.1

The extracted D0 and log10(NW) are compared in Figs. 5a and 5b with those based on the 1-min, gamma-fitted DSDs from the disdrometer (low profile as well as the compact unit) measurements. The two disdrometer-based estimates are very close to each other. Figure 5c shows the rainfall rate comparisons. In all three panels of Fig. 5, the radar-based estimates have been “time interpolated” to correspond to the time stamps of the disdrometer data. Moreover, a “running average” over 3 consecutive minutes was employed for both sets of 2DVD estimates. Another point to note is that the rainfall rate from the 2DVD is calculated as the volume of drops crossing the 10 × 10 cm2 sensor area per unit time. This method overcomes the need to assume any fall velocity versus drop diameter relation. Earlier comparisons (Thurai and Bringi 2011) made in terms of event total rain accumulations with a collocated Geonor rain gauge (within a few meters) had shown excellent agreement when the rain rates from 2DVD had been calculated using the volume of drops crossing the sensor area per unit time.

Fig. 5.
Fig. 5.

Radar retrievals (after time interpolation) extracted at the 2DVD location, shown as solid lines, compared with disdrometer-based estimates (circles for low profile and crosses for compact 2DVD-based estimates) of (a) log10(NW), (b) D0, and (c) R.

Citation: Journal of Hydrometeorology 13, 3; 10.1175/JHM-D-11-070.1

The hourly rain accumulations are compared in Table 2a for 3 individual hours for this event, as well as the 3-h total. The agreement is within a few percent between the low profile and the compact 2DVD. The differences are expressed in percentage terms in Table 2b. The total difference between the two disdrometers is only 0.2%, and between the Geonor and the disdrometers is around 2%. A previous study (Thurai and Bringi 2011) had compared the daily totals for 37 events between the two 2DVD units and the Geonor over a period of 9 months. In every single case (not shown here), there was excellent agreement between the three sets of accumulations.

Table 2a.

Hourly rainfall accumulations (mm).

Table 2a.
Table 2b.

Percentage differences in hourly rainfall accumulations.

Table 2b.

Returning to Figs. 5a–c, we see that, in general, there is close agreement between the radar retrievals and the 2DVD measurements over the entire analysis period. However, for short periods we note a systematic underestimation of the radar-derived R relative to the 2DVDs for 6 min between 0448 and 0454, which is related to an underestimation of radar-measured reflectivity during the same time period (see Fig. 3a). Near the end of the analysis period (after 0642) there is a systematic overestimation in radar-derived log10(NW), but this can be correlated with a few dB overestimation of the radar-measured reflectivity (see Fig. 3a); moreover, this occurs during a period of rapidly decreasing reflectivity from around 32 dBZ (at 0642) to around 20 dBZ at the end of the analysis period. For the purposes of quantifying the error, we only consider the entire analysis period. Comparisons of temporal correlations between the 2DVD-based DSD parameters and the corresponding radar-based estimates also showed close agreement (see appendix: Fig. A1).

To quantify the comparisons in Fig. 5, we show in Table 3a the mean values over 200 consecutive minutes—that is, from 0400 to 0720 UTC—determined from the two 2DVD datasets and the corresponding radar-based estimates. For the latter, the “interpolated points” were used so that the effects due to any high-frequency fluctuations are kept to a minimum. The agreement is well within a few percent for all three parameters: D0, log10(NW), and R. The three rows are denoted by D1, D2, and RE, representing compact 2DVD measurements, low profile 2DVD measurements, and radar estimates, respectively.

Table 3a.

Mean values for 200 consecutive minutes.

Table 3a.

Values of variances for the differences between the estimates are given in Table 3b. The first two rows show the variances between the disdrometer-derived values and the radar estimates, while the third row shows the variances between the two disdrometer-based estimates. As will be seen later, the values in the third row will be used to quantify the sampling errors associated with disdrometer measurements. In the next section, we will discuss the various error components contributing to the variances Var(D1 – RE) and Var(D2 – RE) in Table 3b.

Table 3b.

Variances calculated using the 3-min average (running) DSDs for D1 and D2 corresponding to Table 3a.

Table 3b.

4. Error variance separation

There are several quantifiable factors that contribute to Var(D1 – RE) and Var(D2 – RE) given in Table 3b. In the following we describe these contributors, namely: (i) point-to-area variance, (ii) parameterization or algorithm error variance, (iii) variance due to radar measurement errors, and (iv) disdrometer sampling errors. The following describes the estimation methods used for each of these components.

a. Point-to-area variance

This component is attributed to the disdrometer representativeness error, which arises from the fact that disdrometers provide “point” measurements whereas the radar estimates represent a larger volume in space, the dimensions of the volume being defined by (i) the range gate length, which is 250 m; and (ii) the beamwidth (in azimuth and in elevation), which, at the range of the disdrometer site (15 km), also amounts to 250 × 250 m2. The point-to-area variance can be estimated from the spatial correlation using an approach analogous to radar-gauge comparisons given in Ciach and Krajewski (1999):
e2a
where var(Rd) is the variance of the point disdrometer measurement, A is the areal domain of the radar pixel with xy coordinates, ρ is the spatial correlation, and xd is the location of the disdrometer(s) within the radar pixel. From (2a), one can determine the variance reduction factor (VRF) using the equation (Habib and Krajewski 2002; Moore et al. 2000)
e2b
To calculate the spatial correlations of the three parameters, we make use of the radar data corresponding to Fig. 1c in the following way. From the calibrated and corrected radar measurements of Zh and Zdr, we construct the time series of D0, log10(NW), and R for different ranges with respect to the disdrometer site. The Pearson correlation coefficient is then calculated using the time series corresponding to each of the range separation, and from the range variation of the correlation coefficient, a three-parameter exponential model is fitted given by
e3
where d represents the separation distance, R0 is the correlation radius, F is the shape parameter, and ρ0 is the nugget parameter.

Figure 6 shows the spatial correlations calculated for the three parameters—D0, log10(NW), and R—together with their fitted curves. The calculations were performed from the time series constructed for each adjacent range gates—that is, every 250 m corresponding to the radar range resolution. This figure is central to the calculations presented in this analysis. One interesting aspect to note is that in terms of spatial correlation, the mean diameter fall is the slowest, and the rain rate is the fastest, at least beyond half a kilometer. Also to be noted is that D0 has a slower fall at short distances (<0.5 km) compared to the longer distances, indicating that its uniformity at the very small scale is relatively high. The fitted values of R0 and F for all three cases are included in Fig. 6. The nugget parameter ρ0 is assumed to be 1.

Fig. 6.
Fig. 6.

Spatial correlations determined from the dwell scans, and their fitted lines.

Citation: Journal of Hydrometeorology 13, 3; 10.1175/JHM-D-11-070.1

Using the fitted functions and, further, assuming that all three correlations are isotropic (i.e., in x and y), Eq. (2b) was evaluated for the three parameters to determine the variance reduction factor. A 500 × 500 m2 was chosen to represent radar “pixel” area to allow for two adjacent beams and range gates with respect to the 2DVD location. The “elemental area” dxdy in 2b was selected to be 6 × 6 m2. Furthermore, an extreme scenario was considered, with the disdrometer positioned at a corner of the area. The point-to-area variance values so obtained were 0.0008, 0.0017, and 0.476 for D0, log10(NW), and R, respectively. The values are given in the first row of Table 4.

Table 4.

Factors contributing to the variances. (Percentage of contribution to the total variance is provided in parentheses.)

Table 4.

b. Variance due to radar measurement error

The second component in the error variance analysis is the radar measurement error. Its effect on the retrieved DSD parameters can be evaluated in the same manner as in chapter 8 of Bringi and Chandrasekar (2001). In brief, the measurement errors are assumed to be additive with zero mean. These errors are then “propagated” via the retrieval algorithms to estimate the corresponding errors in the DSD parameters and R. To evaluate the errors in D0 arising from Zdr uncertainties, we assume for convenience a power-law relationship of the form D0 = αZdrβ, rather than Eq. (1a). This assumption is sufficient for calculating the variance of the retrieval error [as opposed to using (1a) and (1b); see also Bringi et al. 2009]. In such a case, it can be shown that the variance of D0—denoted as var[ɛm(D0)]—due to radar measurement error in Zdr, is given by
e4a
With α equal to 1.52 and β equal to 0.351 (as in Williams and May 2008), and assuming a Zdr uncertainty of 0.2 dB, the var[ɛm(D0)] is calculated to be 0.002 85 mm2.
The corresponding variance in NW due to radar measurement error in Zh and Zdr can be calculated from
e4b
where and represent the mean values of NW and D0. Using the appropriate values in Table 3a, and assuming 0.5-dB uncertainty in Zh, the var[ɛm(NW)] is calculated to be 0.025.

Finally, the corresponding variance in R due to radar measurement error {denoted as var[ɛm(R)]} was estimated to be 0.256 (mm h−1)2. All variances due to radar measurement errors are given in the second row of Table 4.

c. Algorithm error

The third contributor to the error variance values quoted in Table 3b is the error arising from the application of the various algorithms to determine the DSD parameters. Such “algorithm error” (or parameterization error) is a measure of the variability of the individual DSD-based calculations from the mean fits given in the set of Eqs. (1a)(1e). The appropriate variances {denoted as var[ɛp(D0)], var[ɛp(NW)], and var[ɛp(R)]} can be quantified from the retrievals using the scattering calculations compared with the direct estimates from the disdrometers. Section 2 of Bringi et al. (2006) describes this procedure in detail. The same procedure was applied to the 5 February 2010 Huntsville event, and the resulting variances are given in the third row of Table 4. Note the retrieval algorithm error is not sensitive to the precise form of the fitting algorithm (see, e.g., Bringi et al. 2006, 2009, 2011).

d. Disdrometer sampling errors

The final quantifiable contributor to the overall error variances is the sampling error for the disdrometers, which arises because of their finite sensor areas (10 × 10 cm2). Since we have the two sets of side-by-side disdrometer measurements for our event, it was possible to directly quantify the variances for the three DSD-related parameters. These were given earlier in Table 3b as Var(D1 D2). The variance var(ɛs) because of the sampling error will be one-half of Var(D1 − D2), and these are included in Table 4, in the fourth row. The correlations of the DSD moments M0M6 were also derived for this case and found to range from 0.94 to 0.99. The high values indicate the relatively low sampling errors, as was observed earlier in Figs. 5a–c, where the two sets of 2DVD-based estimates were seen to be highly correlated.

e. Total variance

To calculate the total variance due to the various contributors mentioned above, we assume they are all uncorrelated, and moreover, they all have mean equal to zero. Then, their total contribution, var(ɛtotal), will be a simple addition of the individual terms—that is,
e5
The fifth row in Table 4 shows the total var(ɛtotal) of the variances. These, in turn, are compared with the observed variances (i.e., disdrometer minus radar) in the sixth row of Table 4. (Note the sixth row contains the averages of the first two rows of Table 3b.) In the last row of Table 4, we present the percentage of the total quantifiable errors—that is, the ratio of var(ɛtotal) to the variance from the observations given by the average of Var(D1 − RE) and Var(D2 − RE).

From Table 4, we note the following:

  • the error variance separation results show that 100% of the observed variance in D0 (i.e., variance of difference between radar and disdrometer estimates) can be explained by the four contributing factors, but only 57% of log10(NW) and 73% of R can be explained;
  • the retrieval algorithm error is the largest component of var(ɛtotal) for D0, whereas it is the radar measurement error for log10(NW) and for R;
  • the disdrometer sampling errors are the lowest for all three parameters; and
  • the point-to-area variance is significantly lower than the radar measurement errors or the algorithm errors for D0 and log10(NW), but for R, it is somewhat comparable.
The remainder of the percentages in the last row of Table 4—namely 43% for log10(NW) and 27% for R—can perhaps be attributed to other sources of unexplained random errors and by errors induced during the initial radar data processing (such as some inaccuracies in correcting for copolar and differential attenuation). Variations in the shape parameter μ could also have contributed to the remaining differences (from the 2DVD-based gamma-fitted DSDs, the values of μ showed a Gaussian distribution, with a peak at around 4–5, from both sets of data), as would some vertical discontinuities between the radar observations at 250 m AGL and the disdrometer measurements at ground level. In any event, and as mentioned earlier, the 5 February 2010 case considered here is a very widespread event with somewhat uniform reflectivity in all directions, and thus the variances given in Table 4 can be considered to represent the “best” possible scenario (minimal gradients in reflectivity or minimal nonuniform beam-filling effects).

f. Radar estimation error

Following Habib and Krajewski (2002), the radar estimation error—or simply the radar error—is defined as follows:
e6
where var(radar − disdrometer) is given in row six of Table 4, var(point-to-area) is given in row one, and var(ɛs) in row four. The fractional standard error (FSE) is then defined as [var(radar error)]1/2 divided by the mean of RE where the latter is given in the third row of Table 3a. The FSE of D0, log10(NW), and R are calculated to be, respectively, 6.8%, 6%, and 21%. The normalized bias was calculated to be less than 1%.

5. Discussion of results

One of the key results of this paper is the radar-based estimation of the spatial correlation function of D0, log10(NW), and R. Apart from its application to the calculation of the point-to-area variance, these results show that the spatial decorrelation distances (R0) and shape parameters (F) differ for D0, log10(NW), and R as shown in Fig. 6. For example, the decorrelation distance R0 is smallest for rain rate (4.74 km) and largest for D0 (8.24 km), with log10(NW) being in between (7.22 km). We can compare the R0 and shape parameter values for rain rate with Moreau et al. (2009), who obtained for 6-min rain accumulation over a 1 × 1 km2 area the values of R0 = 4.54 km and F = 1.3 (they used the ZPHI method using X-band radar to estimate rain rate and used 1 yr of radar data). It is quite remarkable that we obtain very similar R0 (4.74 km) and F (1.28) values from the C-band radar using a very different rain rate estimator and for just one event (see, also, Bringi et al. 2011 for a comparison of spatial correlation fits with Moreau et al. 2009 based on hourly rain accumulations). The other comparison we can make is with S-band radar using a fixed ZR relation from Gebremichael and Krajewski (2004), who obtained R0 = 4.7 km and F = 0.79 for 2 × 2 km2 areal pixel and 5-min accumulation. The main difference is in the shape parameter, which is <1, thus implying that their radar-based estimate of spatial correlation falls off significantly faster with distance as compared with either our fitted values or that of Moreau et al. (2009). In contrast, Gebremichael and Krajewski (2004) obtained R0 = 4.56 km and F = 1.51 from a dense gauge network, which is more comparable with what we and Moreau et al. (2009) obtain. Note that the shape parameter strongly controls the behavior of the spatial correlation at short distances, which in turn strongly affects the estimation of the point-to-area variance calculation. From these comparisons it appears the decorrelation distance of 4–5 km and shape parameter of 1.3–1.5 for rain rate may, in fact, be quite typical for widespread events.

There appear to be no radar-based estimates of the spatial correlation function of D0 and log10(NW) that we could find in the literature to compare with. However, there are some general trends that qualitatively agree with our fitted values. The fact that R0 is about a factor of 2 smaller for rain rate as compared to D0 is in agreement with Jameson and Kostinski (1999) that the variability of rain rate would be higher than for D0 (see, also, section 1). Lee et al. (2009) have made a detailed analysis of spatial correlations in stratiform rain. In this reference, the DSD moments were used to determine the spatial correlation versus separation distance (using four disdrometers) as well as for what they define as characteristic number density (similar to NW) and characteristic diameter (similar to D0). They found that the higher-order moments decorrelated faster with distance (had smaller R0 values) than the lower-order ones, which is in qualitative agreement with our lower R0 values for rain rate (3.67th moment of the DSD) as compared with D0 (for gamma DSDs, related to the ratio of fourth to third moments). They also found that the characteristic number density decorrelated more (at 1.3-km distance) than the characteristic diameter, which is in qualitative agreement with our result—that is, R0 for log10(NW) is somewhat smaller than for D0 (7.22 versus 8.24 km).

The error variance separation quoted in Table 4 indicates the significance of each contributing error component. For the error in the radar-based estimate of rain rate, 26% of the observed variance can be attributed to point-to-area variance (in fractional terms, ). It is difficult to compare our result with other values quoted in the literature for the contribution of the point-to-area variance to the total error variance since the time integrations (for rain accumulation) vary from 5–15 min to hourly. Perhaps the highest number quoted is by Le Bouar et al. (2001), who estimated that point-to-area variance could account for 70% of the total variance, but this figure refers to hourly accumulations and for 4 × 4 km2 pixels. More recently, Moreau et al. (2009) compared hourly rain accumulations from an X-band polarimetric radar with a dense gauge network comprising 25 gauges, all within a 25-km range from the radar. They estimated the point-to-area variance to account for 30%–40% of the total observed error variance. Habib and Krajewski (2002) found corresponding percentages of 30%–45% for light rain and 40%–75% for heavier rain (15-min accumulations and 2 × 2 km2 pixel using ZR relation).

In terms of the error variance separation for the DSD parameters, only the components of algorithm (or parameterization) error and radar measurement error are generally available in the literature. For example, Gorgucci et al. (2002) quotes—from simulations at S band—that on average, FSE of the radar estimation of D0 is around 10% while that of log10(NW) is around 6% (these FSE values include both parameterization and measurement errors). In comparison, from Tables 4 and 3a, we obtain corresponding FSE values of 7% and 4.4%, respectively, which we consider as reasonable agreement. Further, Williams and May (2008) estimate via simulations the FSE of D0 to vary between 6% and 12% depending on the mean value of D0.

For completeness, we consider the temporal representativeness error relating to the finite radar sampling interval. Austin et al. (2010) considered the sampling errors in a spatial–temporal domain (as was suggested earlier by Fabry et al. 1994). They found a substantial increase in the sampling error with decreasing spatial resolution and that “…at lower spatial resolutions, the temporal resolution is of lesser importance….” Further, they quote, “…for common operational resolutions, such as 2 km–5 minutes or 3 km–10 minutes, RMSE percentage errors are around 40% and 60%, respectively….” Considering the very small sampling interval used in our study, and the 250-m range resolution—which is considerably less than 2–3 km—we can assume that the temporal representativeness error would be, in essence, negligible (i.e., much less than those quoted in Table 4).

6. Summary

For a long duration and widespread cool-season precipitation event, we have compared the DSD parameters retrieved from C-band radar data with measurements from two collocated 2D video disdrometers. The radar data were obtained from “near-dwell” scans taken at a 4–8-s interval over the disdrometer site. After several steps of data processing, the retrieved DSD parameters from the radar measurements were found to be in close agreement with the disdrometer-based estimates, allowing for a radar-based estimation of the spatial correlation function of D0, log10(NW), and R. Error variances between the radar-based and disdrometer measurements were determined after time interpolation of radar-based estimates to match the disdrometer sampling times. Also calculated were the various contributing error variances such as (i) “point-to-area” variance (determined from the spatial correlation function), (ii) radar measurement error, (iii) retrieval algorithm error, and (iv) disdrometer sampling error. Assuming all four factors are uncorrelated, their combined errors explained 100% of the observed variance between radar and disdrometer estimates of D0 and, correspondingly, 57% of log10(NW) and 73% of R. The retrieval algorithm error was found to be the highest contributor for D0 (57% of total estimated error) followed by the radar measurement error (32%). In the case of log10(NW), the highest contributor was the radar measurement error (80% of total). In the case of rain rate, the highest contributor was radar measurement error (43% of total), followed by point-to-area variance (35%). The overall accuracy of the radar-based retrievals in terms of the fractional standard deviation (FSE)—see (6)—were estimated as 6.8%, 6%, and 21% for D0, log10(NW), and R, respectively. The normalized bias was found to be less than 1%. These results may be considered as a measure of the lower bound for expected accuracies when retrieving DSD parameters and rain rate from C-band radar observations (at temporal resolution of ~3 min and spatial resolution of ~1.5 km).

Acknowledgments

MT/VNB acknowledge support from NASA Grant Award NNX10AJ12G. MT also acknowledges support from the National Science Foundation via AGS-0924622. WAP/LDC/ES/PG acknowledge NASA Precipitation Measurements Missions Science Team research support provided by Dr. Ramesh Kakar, and also NASA Global Precipitation Measurement Mission Project Office support provided by Dr. Mathew Schwaller.

APPENDIX

Temporal Correlations from Radar and Disdrometers

While the radar-based estimates of the spatial correlation function of the DSD parameters and rain rate are used here to determine the “point-to-area” variance, for completeness we also show the temporal correlations of the radar-based estimates over the disdrometer location and have compared these with corresponding temporal correlations determined from disdrometer data. The time series variations from Fig. 5 are used to calculate the temporal correlations, which are shown in Fig. A1 for D0 and log10(NW). The agreement is very close and implies that the radar retrievals (which are inherently spatial averages) are comparable with point measurements with integration times of ~ few minutes as used here. The temporal correlation of log10(NW) falls off faster with time as compared with D0, which was also observed in the spatial correlation functions in Fig. 6. Decorrelation times for D0 are approximately 230 s and for log10(NW) are 154 s. One implication is that the radar scan strategy should enable a revisit time well below the decorrelation times as was the case for our study (i.e., “near-dwell” scans). The close agreement of the temporal correlations also provided confidence in using the radar-based spatial correlation functions for estimating the point-to-area variance.

Fig. A1.
Fig. A1.

Correlation coefficient vs time for log10(NW) and D0 from radar data and the two disdrometers.

Citation: Journal of Hydrometeorology 13, 3; 10.1175/JHM-D-11-070.1

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