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A Robust Error-Based Rain Estimation Method for Polarimetric Radar. Part I: Development of a Method ACACIA S. PEPLER Macquarie University, Sydney, New South Wales, Australia
PETER T. MAY Centre for Australian Weather and Climate Research, Bureau of Meteorology, Melbourne, Victoria, Australia
MERHALA THURAI Colorado State University, Fort Collins, Colorado (Manuscript received 23 November 2010, in final form 23 May 2011) ABSTRACT The algorithms used to estimate rainfall from polarimetric radar variables show significant variance in error characteristics over the range of naturally occurring rain rates. As a consequence, to improve rainfall estimation accuracy using polarimetric radar, it is necessary to optimally combine a number of different algorithms. In this study, a new composite method is proposed that weights the algorithms by the inverse of their theoretical error. A number of approaches are discussed and are investigated using simulated radar data calculated from disdrometer measurements. The resultant algorithms show modest improvement over composite methods based on decision-tree logic—in particular, at rain rates above 20 mm h21.
1. Introduction The potential improvements for quantitative precipitation estimation (QPE) through use of polarimetric radar have become widely recognized over the past two decades (e.g., Zrnic´ and Ryzhkov 1999; Bringi et al. 2004; Ryzhkov et al. 2005; Lee 2006). The differential reflectivity ZDR and specific phase difference KDP, in addition to the conventional reflectivity at horizontal polarization Z, substantially increase the information available about rain and drop size characteristics. In combination with variables such as the correlation coefficient between horizontal and vertical polarizations, these offer significant advantages in radar quality control through distinguishing of rain echoes from ground clutter and other nonmeteorological signals. Polarimetric variables are also employed for identification of precipitation types—in particular, hail (Zrnic´ and Ryzhkov 1999)—and improvements in radar calibration procedures
Corresponding author address: Acacia Pepler, Macquarie University, North Ryde, NSW 2109, Australia. E-mail:
[email protected] DOI: 10.1175/JAMC-D-10-05029.1 Ó 2011 American Meteorological Society
(Illingworth and Blackman 2002). The relationship between ZDR and drop size diameter (Seliga and Bringi 1976, 1978) and the immunity of KDP to hail, calibration, and attenuation errors (Zrnic´ and Ryzhkov 1996) also offer significant improvements in rainfall measurements by reducing the impact of variability in the drop size distribution (DSD). A number of polarimetric algorithms for rainfall estimation have been developed using these variables (e.g., Ryzhkov et al. 2005; Lee 2006; Gorgucci et al. 2001). By including polarimetric variables such as ZDR, these algorithms are more robust to variations in the DSD and can significantly improve the accuracy of rainfall estimation at moderate and high rain rates (Chandrasekar et al. 1990). The increased algorithm complexity can make such algorithms more sensitive to measurement errors, however. This is particularly evident at low rain rates, for which the change in reflectivity between polarizations is small and the values of the polarimetric variables are on the same order as the measurement accuracy (Hagen and Meischner 2000). As a consequence, no single estimator is optimal at all rainfall rates; estimation using reflectivity alone is often best at low rain rates. The majority of studies
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therefore seek to optimize estimates through combining estimators to take advantage of these varying error characteristics. The most common method of combining algorithms is through some kind of decision-tree framework, in which different algorithms are applied depending on some if– else criteria. These combination methods can be based purely on rain rate or reflectivity (Gorgucci et al. 2001; Ryzhkov et al. 2005), with reflectivity-based algorithms applied at low rain rates and KDP used at high rain rates (e.g., Bringi et al. 2009). As an alternative, these can be based on all radar variables, with the rain rate R being derived by using reflectivity alone (units for Zh: mm6 m23) when the values for ZDR and KDP are sufficiently small as to add no value (Bringi et al. 2004). Such methods have been demonstrated to be more accurate than individual polarimetric algorithms. One prominent study, Ryzhkov et al. (2005), found an average point fractional bias of just 20.2% using a combined algorithm derived for the polarization-capable Weather Surveillance Radar-1988 Doppler (WSR-88D) in Norman, Oklahoma, as compared with a minimum bias of 25% for any individual polarimetric algorithm. This also reduces the areal fractional root-mean-square (FRMS) error to just 17.5%, as compared with 24% for the best polarimetric algorithm and 65% for the Next-Generation Weather Radar (NEXRAD) Z–R relation. Such methods consistently offer significant improvements on individual estimators, although the base algorithm and combination thresholds used are determined for an individual location. As a consequence, algorithms are not necessarily applicable beyond the study of interest, and no consensus exists between studies on the optimal format. Furthermore, these simple thresholds fail to account for the varying accuracy of algorithms over the full range of rain rates. Some alternative methods of estimating rainfall through polarimetric radar have been proposed, including the derivation of the rainfall from DSD parameters determined from radar variables, which may be more robust to DSD variations. The complexity of these relationships varies among studies, some of which also apply a number of methods depending on the value of the polarimetric variables in a similar approach to the combination methods described above (e.g., Gorgucci et al. 2001; Le Bouar et al. 2001; Brandes et al. 2003; Bringi et al. 2004). Brandes et al. (2003) found no significant improvement using such methods over an R(Zh, Zdr) algorithm tuned to the location, however, and the additional complexity in such relations may result in higher sensitivity to measurement errors. Note that Zdr is the conventional notation for ZDR when expressed in linear units (e.g., Ryzhkov et al. 2005) and will be used to distinguish the units used hereinafter.
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Chandrasekar et al. (1993) suggested that weighted combinations of algorithms may optimize the estimation of rain rates, with weighting factors determined by the inverse of the total expected error for each estimator, including both measurement errors and physical variability. The study described a simple approach in which weightings were derived for only three categories of rain rate, and it showed no substantial improvement on simpler composite methods. In our study, we extend the Chandrasekar et al. (1993) method by combining estimators with weighting factors that vary according to the specific radar variables utilized, to better account for the variability in error characteristics. This paper uses simulations from 2D video disdrometer (2DVD) measurements and theoretical error characteristics to determine locally tuned rainfall estimators from radar measurements. These are used to determine a number of combination methods that perform well for a 10-cm-wavelength (S band) radar simulated in the region of the disdrometer. Using these disdrometer data, we then investigate the error characteristics of both local composite methods and weighted combinations of polarimetric algorithms, identifying any improvements associated with weighted combination methods.
2. Data and method a. Disdrometer data The 2DVD used for this study is located in southeastern Queensland, Australia, 16 km from the S-band radar (called the ‘‘CP2’’ radar) site. The disdrometer uses two orthogonal cameras to record the images of individual drops falling through its 10 cm 3 10 cm sensor area, the images being subsequently processed (Scho¨nhuber et al. 2008, chapter 1) to determine the 1-min DSDs for drop diameters ranging from nearly 0 to 10.25 mm, in increments of 0.25 mm. Such 1-min DSDs were obtained for 18 rain events between November 2008 and February 2009, which amounted to 2299 min of ‘‘nonzero’’ data. The DSDs were used to determine the corresponding 1-min rain rates R,1 the total drop number NT, and the mass-weighted median diameter DM. The majority of the database had 1-min rain rates below 10 mm h21 (Fig. 1), with a maximum rain rate of 131 mm h21. Radar variables (Z, ZDR, and KDP) were calculated using the same 1-min DSDs for a 10-cm-wavelength radar using T-matrix scattering techniques derived by Waterman (1971), and
1 Note that in determining R the measured fall velocity is used and not any formula that is given in the literature.
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FIG. 1. Distribution of rain rates in the disdrometer dataset. Bins are defined by the largest rain rate included.
later developed further by Mishchenko et al. (1996). The Beard and Chuang (1987) axis ratios are used [although the drop shapes are being investigated in an ongoing campaign, as in Thurai et al. (2009)], with a maximum diameter of 2.5DM. Following the work of Huang et al. (2008), the standard deviation of the canting-angle distribution was assumed to be 7.58. Other assumptions for the scattering calculations were 1) ambient/water temperature of 208C and 2) radar beam elevation of 08. Note also that all hydrometeors were assumed to be fully melted. In this study, the derived rain rates using these simulated radar variables will be compared with rain rates derived from the disdrometer DSDs, which are assumed to represent the corresponding ‘‘true’’ rain rate. As a consequence, the datasets used for algorithm derivation and validation are not independent. The impact of this will be investigated within the paper through a number of methods. The error between the disdrometer and derived rain rates is represented by a number of metrics, predominantly using the FRMS error and the fractional bias (FB) given by N #1/2 Ri 2 RTi 2 1 and FRMS 5 N i51 RTi
"
å
(1)
N
Ri 2 RTi 1 , FB 5 N i51 RTi
å
(2)
where N is the number of samples, Ri is the derived rain rate, and RTi is the true rain rate of comparison from disdrometer data.
b. Simulated error characteristics To investigate better the performance of algorithms under real-world conditions, we also developed a dataset of measurements with simulated error characteristics derived from the theoretical errors, as described in
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the appendix. The characteristics of the CP2 radar are applied to these equations to derive the appropriate measurement errors for this location, with s(Zh) 5 0.479Zh, s(Zdr) 5 0.136Zdr, and s(KDP) 5 0.7458 km21. For each measurement in the original disdrometer dataset, the theoretical standard deviation for each simulated radar variable was applied to a set of 100 random numbers normally distributed about 0, which was added to the initial value. This resulted in a set of 229 900 simulated data points that represent the expected variance in measurements under measurement errors. This dataset with assumed error enables preliminary validation of results so as to test alternative approaches without the need to use a large radar database, which can be computationally expensive. The use of normally distributed data results in the occurrence of some very large outliers, however, which can result in anomalously high and variable errors. For instance, the maximum values of ZDR and KDP in the disdrometer dataset are 3.3 dB and 3.68 km21, with minimum values of 0. In comparison, the simulated error dataset has values as high as 4.8 dB and 5.88 km21, respectively, and as low as 22.4 dB and 23.68 km21, with almost 60% of the dataset having negative values for one of these variables. Such errors will have a large impact on resulting rain rates, and therefore results using this dataset are analyzed in terms of the trimmed FRMS error, which truncates the top and bottom deciles of the distribution of squared error. This gives the mean of only the median 80% of data, minimizing the impact of high outliers without skewing the error distribution. A third dataset was also developed to reduce the disproportionate effect of the large number of measurements for which rain rates and polarimetric variables are close to zero. An evenly distributed dataset was developed by randomly selecting data points from discrete bins by 5 mm h21 increments to achieve a more representative selection of numbers. This subsampled dataset theoretically has more uniformly distributed rainfall characteristics and was used for development of QPE algorithms that are unbiased by the large number of low rain rates in the disdrometer database. The datasets are summarized in Table 1.
3. Polarimetric algorithms To develop combination methods, it is first necessary to determine the optimal polarimetric algorithms for the location of interest. A large number of both reflectivitybased and polarimetric algorithms have been developed over time in a large variety of locations (e.g., Battan 1973; Ryzhkov et al. 2005). The constants employed in these algorithms can vary substantially among studies
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0:88 22:51 RD (KDP , Zdr ) 5 88:9KDP Zdr .
TABLE 1. Datasets used in this study. Name
Description
Disdrometer (‘‘original’’) Subsampled
All 2299 disdrometer data points with simulated radar variables. Subsampled disdrometer dataset (120 points) with equal proportions of rain rates for 5 mm h21 bins between 0 and 60 mm h21. Original disdrometer dataset with simulated measurement errors added to radar variables—229 900 data points.
Simulated
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on the basis of different rain types, DSDs, and radar specification as well as the method used. This is particularly true for Z–R relations, which can change from Zh 5 216R1.44 to Zh 5 396R0.82 for the same radar when moving from coast to ocean (Prat and Barros 2009), where Zh is in linear units. Polarimetric algorithms using the same algorithm form and shape can also vary substantially. For instance, using similar methods, R(Zh, Zdr) algorithms 24:76 using varied between R(Zh , Zdr ) 5 0:007 46Zh0:945 Zdr DSDs measured in Florida (Brandes et al. 2002) and 21:51 using Oklahoma DSDs R(Zh , Zdr ) 5 0:0144Zh0:761 Zdr (Ryzhkov et al. 2005), with smaller exponents of Zdr being less sensitive to both variations in the DSD and errors in Zdr. As a consequence, the most accurate estimations will be through an algorithm tuned to the local area using disdrometer data and simulated radar variables. There are three main forms of polarimetric algorithms that are commonly used, although other combinations have been employed (e.g., Lee 2006). These derive rain rate as either a function of KDP alone [R(KDP)] or as a combination of Zh and one polarimetric variable [R(Zh, Zdr) and R(Zh, KDP)] and are the focus of this study. Some alternative forms suggested in Lee (2006) were investigated, but showed no improvement in this region (not shown). For each of these forms, locally valid constants were derived by performing a linear regression between the logarithm of rain rate and the radar variables in logarithmic form. This was repeated 1000 times using bootstrapped resamplings of the database. Standard deviations of derived constants were consistently less than 1% of the value, suggesting that the derived algorithms are relatively robust. The relationships derived for this disdrometer (RD) are
The small size of the available dataset unfortunately prevented the use of truly independent datasets for validation. To investigate the impact of such variations on the constants derived, the method was repeated using various subsets of the database to simulate ‘‘independent’’ datasets. The database was separated using alternate groups of N data points such that ‘‘odd’’ groups formed one dataset and ‘‘even’’ groups formed the second, with N of 10, 20, or 50 data points. This separation method was chosen to reduce the impact of correlation between consecutive data points without significantly changing the distribution of rainfall at high rain rates, which has a disproportionate impact on regressions. Regardless of how the database was separated, the derived constants varied by less than 5% between samples, decreasing to 1% using multiparameter methods, indicating these methods are fairly robust to variations in the DSD. This suggests that the results achieved with the simulation study are robust to variations in the DSD and represent well those that would exist if a truly independent database were used, in particular at low rain rates. This method, however, does not truly validate the algorithms on independent data, which shall be remedied in further studies. The polarimetric algorithms developed above have constants within the range of variability shown in Ryzhkov et al. (2005). The Z–R relation, in comparison, has a lower coefficient than those in the majority of studies described in Battan (1973). This may be biased toward performance at low rain rates because of the paucity of measurements above 15 mm h21, with large errors at high rain rates even in the absence of measurement error (Fig. 2a). As a consequence, we performed a bootstrap regression with subsampling of data to remove such bias (Table 1). The algorithms based on these subsampled data (RS) have fairly similar constants in most cases—notably the R(KDP) relationship. The Z–R relation derived using the subsampled database is substantially different than that derived from all disdrometer data in Eq. (3), however: Zh 5 155R1:52 S ,
(7)
0:745 , RS (KDP ) 5 46KDP
(8)
22:70 , and RS (Zh , Zdr ) 5 0:021Zh0:785 Zdr
200R1:36 D ,
(3)
0:8 44KDP ,
(4)
24:47 , and RD (Zh , Zdr ) 5 0:017Zh0:84 Zdr
(5)
Zh 5 RD (KDP ) 5
(6)
0:84 21:15 RS (KDP , Zdr ) 5 67:5KDP Zdr .
(9) (10)
The subsampled algorithms exhibited error characteristics that are very similar to those of the original
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FIG. 2. FRMS error vs rain rate using simulated radar data in the absence of measurement error for all (a) Zh-based and (b) KDP-based algorithms derived from both the original disdrometer dataset (RD) and the subsampled dataset (RS).
algorithms when applied to the entire disdrometer dataset. Correlations between the rain rates derived by the two algorithms exceeded 0.98 in all cases, despite the difference in constants. Some variations in error characteristics were noted, with the subsampled polarimetric algorithms showing slightly larger fractional biases and FRMS errors when compared with the disdrometer rain rates. This reflects the slightly higher errors such algorithms exhibit at low rain rates, although subsampling does show some slight improvements in algorithm accuracy at high rain rates (Fig. 2). In the R–Z case, although subsampling produced a small improvement of errors at high rain rates, resulting in a lower overall FRMS error, it caused significantly higher errors at low rain rates (below 5 mm h21), where such algorithms are most used. The next step is to examine the error characteristics of these methods in the presence of simulated measurement error. As expected from theoretical error characteristics and studies such as Ryzhkov et al. (2005), Zh-based methods have consistent or increasing error with rain rate, whereas a sharp decrease in error is seen at high rain rates for KDP-based methods (Fig. 3). In all cases, the subsampled algorithms have lower total FRMS errors—in particular, for multiparameter methods R(Zh, Zdr) and R(KDP, Zdr). As the trimmed FRMS error is also lower for subsampled methods in almost all cases, this is unlikely to be due solely to high outliers, and it suggests that the use of subsampled disdrometer data for deriving local algorithms may offer a useful improvement in the accuracy of radar estimates. This improvement is substantial at all rain rates for Zh-based methods (Fig. 3a), in particular at high rain rates; relatively little change in error characteristics is observed using algorithms based
on subsampled data for KDP-based algorithms (Fig. 3b), where subsampling resulted in similar algorithms.
4. Combination methods a. Composite method In this section we will develop a composite method for more robust estimation of rainfall, accounting for the observed variability in error characteristics. Because DSDs and rainfall error characteristics vary considerably among studies and locations, to validate accurately a new method it must be compared with a composite method optimized to the same conditions. Therefore, we must first develop a locally tuned composite method against which to validate combination methods. Such methods apply a selection of estimators using some criteria based on the radar variables. These criteria may use reflectivity alone (Ryzhkov et al. 2005) or some combination of the three radar variables (Bringi et al. 2004), to accommodate the poor performance of polarimetric estimators where polarimetric variables are close to zero. Investigation of the error characteristics of each locally derived algorithm using the simulated error database identified no consistent variability due to ZDR or KDP. Combination methods based on the Ryzhkov et al. (2005) reflectivity-based thresholds were thus more accurate than multiparameter thresholds such as those employed in Bringi et al. (2004). As a consequence, we examined algorithm error as a function of Z to derive the threshold values: RC 5 R(Zh )
where Z , 25 dB,
(11)
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FIG. 3. As in Fig. 2, but for trimmed FRMS error vs rain rate using simulated radar data with theoretical measurement error added. Note: y axis is truncated in (b) for ease of visibility because of the very high errors at low rain rates.
RC 5 R(Zh , Zdr ) where
25 # Z , 40 dB, and (12)
RC 5 R(KDP , Zdr )
where Z $ 40 dB.
(13)
Here, R(KDP) was not the most accurate algorithm for any reflectivity threshold, because R(KDP, Zdr) was more accurate at rain rates above 35 mm h21 where KDPbased algorithms are applied. Such algorithms are most accurate in hail conditions (Balakrishnan and Zrnic´ 1990) and may be applied at very high reflectivities, conditions not represented in the disdrometer database. Both the original and subsampled locally derived algorithms described above were applied for the selected criteria. We also attempted to improve this method by optimizing each algorithm to the range of reflectivity at which it is applied, which could theoretically increase accuracy. Such tuned methods had consistently lower FRMS errors and fractional biases than those using algorithms derived over the entire database when compared using the simulated radar dataset. These improvements were observed at all rain rates. When theoretical measurement error was added to the dataset (Fig. 4) these results changed, with the tuned methods showing significantly higher errors than untuned methods at rain rates below 20 mm h21 and no improvement at high rain rates. This is likely due to the small range of rain rates used to derive the tuned Z–R algorithms, as the threshold Z of 25 dB is associated with maximum rain rates of just 2.7 mm h21, making accurate regressions difficult. The high errors for tuned methods remain when a trimmed FRMS error is used, in particular at rain rates below 30 mm h21. Because the
lowest errors are consistently observed for the untuned method using the subsampled algorithms, this composite method will be used for comparative purposes with the weighted methods.
b. Weighted methods In this section, we describe the error characteristics of polarimetric algorithms, leading to the development of weighted combination methods for optimizing QPE. Two methods are proposed, with different strengths and weaknesses, and a number of options are investigated to improve accuracy.
FIG. 4. FRMS error for local composite methods vs disdrometer rain rates using simulated radar variables in the presence of theoretical measurement error.
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1) THEORETICALLY WEIGHTED METHOD A weighted estimate of rain rate can be obtained by adding the derived rain rate for each disdrometer algorithm weighted by the inverse of its theoretical error multiplied by a normalization constant a; that is, 4
R5
å sa Ri ,
i51
where
i
1 5 a
4
å s1 .
i51
(14)
i
This approach requires the derivation of errors as a function of radar variables. In the absence of physical biases the total error in the radar measurement of rainfall is the combination of measurement [s(«M)] and parameterization [s(«P)] errors, which can have very different characteristics. Of these, the measurement error can be easily derived from theoretical characteristics of the component radar variables as described in the appendix. The measurement error is related to the coefficients of the specific algorithm, with the theoretical measurement error corresponding to the algorithms in Eqs. (3)–(6) given by s[R(Zh )]/R 5 0:3525, s[R(KDP )]/R 5 0:596/KDP , s[R(Zh , Zdr )]/R 5 0:729, and 2 s2 [R(KDP , Zdr )]/R2 5 (0:43/KDP 1 0:1165)1/2 .
(15) (16) (17) (18)
It is apparent that reflectivity-based algorithms have constant measurement errors while measurement errors for KDP-based algorithms decrease substantially with rain rate (Fig. 5). In comparison, the parameterization error of an algorithm depends on its accuracy in representing the true rain rates, which is difficult to derive from theoretical calculations and varies with the DSD characteristics of the sample. If this error is excluded, the theoretical measurement errors alone indicate inferior performance by multiparameter algorithms at all rain rates, which does not agree with observed results (e.g., Ryzhkov et al. 2005). To determine a more appropriate weighting factor, it is therefore necessary to determine an approximate parameterization error. In this case, we attempt to determine this in the absence of measurement error using the simulated radar variables derived from disdrometer data. The parameterization error is best represented by the FRMS error, which can be determined for each algorithm as a function of the derived rain rate. The FRMS error was chosen over the standard deviation because it combines both errors relating to the bias and the standard deviation in a simplified form. The equations of
FIG. 5. Fractional error for literature algorithms as a function of rain rate, as derived from Bringi and Chandrasekar (2001).
error were determined relative to the derived rain rate to avoid multiple regressions and assumed relationships with radar variables while accommodating the absence of true rain-rate information for operational purposes. Errors are highly variable with rain rate, however, and cannot be represented perfectly by any single equation. We therefore attempted a number of approximations of the error to simplify equations, using a constant value, a linear relationship, or a power relationship. In each case, constants were determined using a bootstrap regression over 100 samples. Although traditionally the variance or square error would be used for such calculations, such methods were less accurate when applied to the disdrometer database and are not discussed here. In the absence of measurement error, all theoretically weighted methods have very low fractional biases of 13%. FRMS errors are constant with rain rate, with the inclusion of parameterization errors resulting in lower errors regardless of the approximation used. For these weighted combinations, methods based on the subsampled algorithms did not show any improvement over those using the original algorithms—in particular, at rain rates below 10 mm h21 at which rate the original algorithms had substantially lower errors (not shown). When theoretical measurement error was included, all such methods show significant increases in error at low rain rates, resulting in very high overall FRMS errors (Fig. 6), with little variation among the different approximations of parameterization error. This is also observed using trimmed FRMS errors and so cannot be attributed to outliers alone. Under such conditions, using subsampled algorithms again makes little difference, suggesting such weighting may reduce the impacts of individual algorithms.
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10 mm h21, however, and high FRMS errors are observed at rain rates between 5 and 10 mm h21 where KDP-based algorithms are first incorporated. The advantage of the theoretically weighted combination method is the robust representation of error, calculated individually for each data point, and the ability to accommodate different estimators with little change in accuracy. The assumptions of some static relationship between error and rain rate may not best represent the true variability in the database, however, because no clear pattern in the relationship between error and rain rate was observed (Fig. 2). Furthermore, the calculated weightings cause high errors at low rain rates, making such combinations invalid for much of the rain in the dataset.
FIG. 6. Trimmed FRMS error for weighted methods vs disdrometer rain rate using simulated radar variables with theoretical measurement error.
The increase in errors at low rain rates (below 5 mm h21) likely is due to the very high errors of KDP-based algorithms at such rain rates (Fig. 3b). As a consequence, even small weighting factors for such algorithms have a significant impact on the derived values. An attempt was made to combine such weighting methods with an element of hybridization, employing the Z–R relation alone at low rain rates to reduce this effect. Because the relationship between FRMS error and reflectivity is logarithmic, the choice of a threshold reflectivity at which to apply the Z–R relationships is somewhat arbitrary. A rain rate of 20 mm h21, where the error decreases to nearly 0.5 in Fig. 5, corresponds to a reflectivity of 40 dBZ using either algorithm, approximately where the error as a function of Z reaches 1. Because 92% of the disdrometer dataset gave rise to reflectivities below 40 dBZ, including rain rates as high as 25 mm h21, such a high threshold significantly limits the use of the weighted method. Therefore, a reflectivity threshold of 25 dBZ was chosen for consistency with the criteria for the disdrometer-derived decision-tree combination. This corresponds to a trimmed FRMS error of approximately 10. The use of hybridized combinations has little impact on the fractional bias for any method, with a slight increase in FRMS error but no substantial change in the error characteristics (not shown). In the presence of theoretical measurement errors, the hybridization approximately halves the total trimmed FRMS error for rain rates above 1 mm h21 in all cases, predominantly because of a significant decrease in errors at rain rates below 5 mm h21. No change in error characteristics is observed when restricting analysis to rain rates above
2) DISCRETELY WEIGHTED METHOD Because of the poor performance of the theoretically weighted method at low rain rates, we developed a second approach for weighting algorithms that reduces some of these sources of unreliability. Rather than assuming relationships between parameterization error and rain, this method instead consolidates both measurement and parameterization errors through use of the simulated error database. These were used to determine the FRMS errors for each estimator as a function of all three variables. In this method, instead of assuming a form for the error equation, the errors were determined for discrete boxes as a function of one or more radar variables to create a set of matrices of weighting coefficients. The correct weighting factor for each data point can then be determined from the values of the radar variables at that point. The divisions used (Table 2) were determined through trial and error to optimize the range of data represented while remaining robust to variations in error characteristics. These result in an average of 170 nonzero weighting factors for the two-parameter combinations and 380 nonzero weighting factors for the three-parameter combination. The advantages of this method are greater robustness to variations in error rather than assuming a simple relationship with rain rate, in addition to reduced weighting factors for KDP-based algorithms at low rain rates where such algorithms have very high errors. The discrete nature of the boxes results in a potential problem at values infrequently sampled in the disdrometer dataset, however—in particular, at very high rain rates. In the absence of measurement error, the discrete weighting method has similar error characteristics to the theoretical weighting method, although a greater decrease in error is observed at high rain rates. All discretely weighted methods have fractional biases smaller
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TABLE 2. Divisions used for weightings by discrete boxes. Version
Zh
Zdr
Zh Zh, Zdr Zh, KDP KDP, Zdr All
1 dB 5 dB 5 dB
0.2 dB
5 dB
0.3 dB 0.5 dB
KDP
0.28 km21 0.38 km21 0.58 km21
than 63%, increasing to 67% when the subsampled algorithms are used. FRMS errors are also lower for the original algorithms at all rain rates (not shown), with a mean FRMS error over all methods of 0.23 as compared with 0.26 for subsampled algorithms. The majority of weighting divisions had similar error characteristics, lowest for that weighted for all variables. An anomalous increase in error is noted at high rain rates for that weighted by Z and KDP, however. In the presence of measurement error, this method successfully removed the large increase in error at low rain rates seen for the theoretically weighted method. Substantial variability in errors is seen between methods (Fig. 7), with constant error with rain rate when weighted by just ZDR and KDP, but significant decreases in error at high rain rates for other choices. The increased error at low rain rates is more significant using the subsampled algorithms, resulting in mean FRMS errors over all methods of 0.52, as compared with 0.48 for the methods using the original algorithms. In both cases, the lowest errors were again for the method weighted by all three variables for all rain rates above 10 mm h21, whereas the method using only ZDR and KDP is superior below this. This is a surprising result, because polarimetric variables are noisy at low rain rates. It is expected that discretely weighted methods incorporating Zh-based thresholds will perform better than that using ZDR and KDP alone if applied to radar data, in particular at low rain rates.
5. Comparison between methods The simulated data were used to perform a preliminary validation of the weighted combination methods. The original composite method using the subsampled algorithms forms a basis of comparison, in combination with a method based on DSD characteristics derived by Bringi et al. (2004). Such DSD-based methods are expected to be robust to variations in DSD between locations and thus not require local optimization, and this is used to ascertain whether such methods offer improvements over local combinations. These are compared with the optimal version of each weighted method, in both cases employing the original algorithms. Results above suggest
FIG. 7. As in Fig. 6, but for discretely weighted methods.
that subsampling may not be a necessary approach for algorithm optimization, with weighted methods minimizing the impact of differences in algorithm coefficients. Algorithms are first compared in the absence of measurement errors using the original disdrometer dataset. In this case, both weighted combination methods have very low fractional biases—substantially lower than the composite method (Table 3). In comparison, the composite method has lower FRMS errors than theoretically weighted combinations, with the lowest error of any method for rain rates above 5 mm h21 (not shown). The discretely weighted method outperforms the theoretically weighted method by approximately 10% for all rain rates. All locally tuned methods have lower biases and errors than that derived by Bringi et al. (2004), which tends to overestimate rain rate, suggesting these perform substantially better in the absence of measurement errors. Because real rain estimation is always performed using radars that are subject to a number of physical and measurement errors, it is important to identify the sensitivity of each method to such errors. When using the disdrometer dataset with simulated measurement errors (see Table 1), theoretically weighted methods have very high errors at low rain rates, resulting in large overall errors. The composite method has the lowest total errors of any algorithm in terms of both fractional bias and trimmed FRMS error; all other methods overestimate rain rate (Table 4). This is a consequence of the low errors for the composite method at rain rates below 10 mm h21 (Fig. 8), with a large impact on total error due to the skewed rain-rate distribution (Fig. 1). In comparison, the discretely weighted method has slightly higher overall errors, and errors exceed those for
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TABLE 3. Errors for various combination methods using the simulated radar variables in the absence of measurement error where rain rate $1 mm h21. Method
Composite—untuned, subsampled
Theory—linear error
Discrete—all
Bringi et al. (2004)
FB FRMS error
0.06 0.30
0.02 0.36
20.01 0.28
0.12 0.43
the composite method at rain rates below 10 mm h21. Nonetheless, the discretely weighted method is substantially superior to any alternative for moderate rain rates between 20 and 40 mm h21, with almost constant errors for rain rates above 10 mm h21. Overall, the consistently low errors observed using the discretely weighted method make this the most useful composite method for radar estimation of rainfall, although theoretically weighted methods have slightly lower errors at rain rates above 40 mm h21. In comparison, the Bringi et al. (2004) DSD-based method continues to have higher errors than do locally optimized methods over the full range of rain rates, indicating that such methods fail to account for all DSD variability between locations. The simulated radar database under measurement errors therefore identifies a substantial improvement in QPE using weighted combination methods over both composite and DSD-based techniques. This is particularly true at moderate to high rain rates, whereas composite methods have the lowest errors at rain rates below 10 mm h21. The discretely weighted method is also shown to give significantly more accurate rain estimation than do theoretically weighted methods, particularly at low rain rates, with neither method being significantly affected by the choice of estimators. Therefore, this method may offer a useful improvement on traditional optimization techniques for QPE using polarimetric radars.
6. Conclusions In this paper we discussed the use of known theoretical measurement errors for radar variables and a simulated radar database based on disdrometer data to develop locally optimized rainfall estimators. The simulated radar database was first used to derive locally tuned radarrainfall algorithms, with a subsampled bootstrap found to offer significant improvement over standard linear
regressions in determination of constants. These algorithms were then combined in both traditional composite methods and two alternative methods that applied a weighted combination proportional to the inverse of the error for each algorithm. Although a similar method using three subdivisions was attempted by Chandrasekar et al. (1993), the more robust methods developed in this study are deemed a significant change in method. Initial results using simulated radar variables with theoretical measurement errors applied indicate that these weighting methods reduce the variation between algorithms, with no improvement using subsampled radarrainfall algorithms. These results suggest that such methods offer some improvement over both locally tuned composite methods and alternative methods based on derived DSD characteristics. This is particularly true when errors are experimentally derived for discrete factors as a function of one or more radar variables rather than through derived relationships. Improvements are particularly significant at moderate to high rain rates (exceeding 20 mm h21), where trimmed FRMS errors for weighted combinations are 20%– 40% lower than for a composite method derived from the same local disdrometer data. Weighted methods, in particular using discrete weighting factors, may increase the accuracy of radar-rainfall estimates and are potentially applicable to any radar with available disdrometer data for local optimization. The simulated error dataset may not accurately represent the true variability of rain measured by radar, however; in this paper, validation was performed using the same disdrometer data from which the algorithms were derived. As a consequence, it is essential to independently validate such methods using a radar–rain gauge network to confirm accuracy. This will be discussed in Part II of this paper (Pepler and May 2011, manuscript submitted to J. Appl. Meteor. Climatol.), in which the derived polarimetric algorithms and weighted combinations will be validated using an independent radar–rain gauge dataset in Queensland, Australia.
TABLE 4. Errors for various combination methods using the simulated radar variables with theoretical measurement error where rain rate $1 mm h21.
FB Trimmed FRMS error
Composite—untuned, subsampled
Theory—linear error
Discrete—all
Bringi et al. (2004)
0.13 0.45
4.20 3.88
0.27 0.48
0.56 0.88
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(Bringi and Chandrasekar 2001), and the autocorrelation r(k) is given by ! 28p2 s2D k2 T 2 , r(k) 5 exp l2
FIG. 8. As in Fig. 6, but for various combination methods.
Acknowledgments. This research was supported by a Macquarie University Research Excellence Scholarship and the Macquarie University Postgraduate Research Fund. Author MT acknowledges support from the National Science Foundation via AGS-0924622. The authors thank Justin Peter and Susan Rennie as well as three anonymous reviewers for helpful comments and feedback.
APPENDIX
where l is the wavelength of the signal, T is the radar sampling rate, and sD is the spread of the Doppler spectrum of the horizontal reflectivity approximated by a Gaussian shape. In the case of the CP2 radar with l of 10 cm, T of 1 ms, N of 40, and sD assumed to be 1 m s22, this is given by r(k) 5 exp(28p2k2 3 1024). From this, the theoretical measurement errors of the radar variables in linear units can be expressed as s(Zh) 5 0.479Zh and s(Zdr) 5 0.136Zdr. The error for the specific differential phase KDP is more complicated and is a function of both the error in the differential phase fDP and the smoothing length used to determine KDP. Error in uDP is related to both the variance in phase for each polarization, assumed to be the same, and also the covariance between the two polarizations, given by var(uDP ) 5 1=4[var(u1 ) 1 var(u2 ) 2 2 cov(u12 )],
where
1 2 r(1)2 r20 2N 2 r(1)2 r20
N21
å
(N 2 jkj)r(2k)2 and
k52(N21)
Theoretical Measurement Error Calculations
var(Zh ) 5
N21 Zh2 jkj jr(2k)j and 12 N N k52(N21)
å
(A1)
N21 2 2Zdr jkj var(Zdr ) 5 [ p(2k) 2 r(2k 1 1)r20 ]. 12 N N k52(N21)
å
(A4)
var(u1 ) 5 var(u2 ) 5
The equations below are derived primarily from Bringi and Chandrasekar (2001), in addition to Williams and May (2008), in which detailed information on derivations is available. If N is the number of consecutive radar pulses from which the variables are derived, and r(k) is the autocorrelation between radar signals at lag k, the proportion of error in radar measurements due to measurement errors can be described by the equations
(A3)
(A5)
cov(u12 ) 5
r20 2 r(1)2
N21
å
2N 2 r(1)2 r20 k52(N21)
(N 2 jkj)r(2k 1 1)2 . (A6)
Using the values for the CP2 radar given above, these then become var(u1,2) 5 4.8 3 1023 rad and cov(u12) 5 22.1 3 1023 rad, with var(uDP) 5 3.45 3 1023 rad. The standard deviation of uDP is therefore 5.87 3 1022 rad, or 3.378. The error in KDP can then be derived from errors in uDP following Bringi and Chandrasekar (2001) as 1/2 s(uDP ) 3 s(KDP ) 5 , NK Dr NK 21=NK
(A7)
(A2) Here r0 is the correlation between vertical and horizontal reflectivities at lag 0, assumed to be 0.98 for rain
where NK is the number of range gates over which KDP is smoothed and Dr is the length of each range gate. If we assume a smoothing length of 2 km and the 150-m range
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PEPLER ET AL.
TABLE A1. Four commonly used radar–rainfall estimators and their theoretical measurement errors. Algorithm R1 5 cZha b R2 5 cKDP
b R3 5 cZha Zdr
a b R4 5 cKDP Zdr
Error s1 («m ) s(Zh ) 5a 5 0:479a R Z h s2 («m ) s(KDP ) 0:745 5b 5b KDP R KDP " # " # 2 2 2 2 s3 («m ) 2 s(Zh ) 2 s(Zdr ) 5a 1b 2 R2 Zh2 Zdr 5 0:4792 a2 1 0:1362 b2 " # " # 2 2 s24 («m ) 2 s(KDP ) 2 s(Zdr ) 5 a 1 b 2 2 R2 KDP Zdr 2 0:745 1 0:1362 b2 5 a2 KDP
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