1. Introduction
The weather radar is a widely used basis for measuring rainfall at fine spatial and temporal resolutions (Collinge and Kirby 1987; Sun et al. 2000; Uijlenhoet 2001; Vieux 2003). Nevertheless, the weather radar does not measure rainfall directly; rather, it infers the rainfall based on the power of electromagnetic waves backscattered by raindrops in the atmosphere and intercepted by the radar. This backscattered power is represented as the radar reflectivity (Z) and is related to a rainfall rate (R) through a power-law relationship Z = ARb (referred to as the Z–R relationship). This relationship requires the specification of parameters A and b, which are functions of both radar and rainfall characteristics, as discussed in Collier (1996).
Various forms of Z–R relations have been suggested in the literature (Marshall and Palmer 1948; Joss and Waldvogel 1970; Battan 1973). However, these relationships cannot be directly used in any region (Mapiam and Sriwongsitanon 2008). This is because the A and b parameters of the Z–R relationship vary depending on many factors, including their dependence on the rainfall drop size distribution (DSD), which varies in both space and time. We can estimate Z and R directly from the DSD measured using a disdrometer. These Z and R values can then be used to construct a Z–R relationship (Atlas 1964; Battan 1973; Krajewski and Smith 2002; Russo et al. 2005). However, disdrometers are relatively expensive and complicated equipments to operate; hence, it is uncommon for more than one (or even any) of them to be operated in conjunction with a weather radar. The use of DSD to ascertain a Z–R relationship is not possible at all locations. In places where accurate measurements of the DSD are not possible, reflectivity data measured by the radar and the rainfall recorded in rain gauges within the radar coverage are generally used.
The approach followed to specify the Z–R relationship at a location involves collating ground rainfall data at the finest temporal resolution possible, accumulating the reflectivity to the same resolution and ascertaining the parameters using a suitable optimization rationale. This relationship is generally used at a resolution finer than the ground-measured rainfall, under the assumption that the relationship is independent of the temporal resolution that it is developed at.
Although past studies have calibrated the Z–R relationship at a range of temporal rainfall resolutions such as hourly, daily, weekly, monthly, seasonal, or even longer (Hitchfeld and Bordan 1954; Smith et al. 1975; Wilson and Brandes 1979; Klazura 1981; Steiner et al. 1995), little has been done to investigate the sensitivity of the relationship to the temporal resolution that is used. Although the assumption that a Z–R relationship developed using hourly rainfall data is not different from application at finer resolutions (such as 6 min) may be appropriate, can the same assumption be made if the relationship is developed using only coarse daily observations instead? This is the main question we investigate in this paper, proposing a rationale for scaling the Z–R relationship developed at one temporal resolution to another, thereby providing an option for specifying the Z–R relationship in locations where subdaily rainfall measurements on a dense rainfall network are not available.
There are two objectives in this study. The first objective is to study the effects of using rain gauge data of different temporal resolutions for calibrating climatological Z–R relationships. Different climatological Z–R relationships were estimated using rainfall aggregated over 1–24 h. Radar reflectivity data from the Kurnell, Mount Stapylton, and Pasicharoen radars located in Sydney and Brisbane, Australia, and Bangkok, Thailand, respectively, as well as corresponding rain gauge data in the three cities, were used in the analysis. The second objective in this study is to propose a generic scaling rule that can be used to estimate radar rainfall at fine temporal resolution for the cases in which only daily rain gauge rainfall data are available for use in the Z–R calibration. A simple scaling hypothesis was applied to construct a scaling transformation equation to ascertain the A parameters at finer temporal resolutions.
The rest of the paper is as follows: the next section outlines the logic adopted to ascertain the Z–R relationship across a range of rainfall temporal resolutions. Section 3 presents the five datasets used to test the proposed logic and presents the results obtained. Section 4 presents the rationale adopted to develop a simple scaling relationship for the A parameter of the Z–R relationship as a function of the temporal resolution of gauge rainfall. In this section, the scaling logic is applied to the rainfall datasets used and a validation of the proposed logic using alternate rainfall attributes is presented. Conclusions from the study are presented in section 5.
2. Calibration of the climatological Z–R relationship
To study the effects of using rain gauge data of different temporal resolutions on Z–R relationships, different Z–R relationships were estimated using rainfall aggregated over 1–24 h. The logic used was as follows:
Convert instantaneous radar reflectivity into an initial radar rainfall intensity using the relationship Z = 200R1.6 (Marshall and Palmer 1948). Note our earlier comment on the relative insensitivity of results to changes in the b exponent as our rationale for keeping it fixed equal to 1.6 in our study.
Accumulate the initial instantaneous radar rainfall into 1–24-h rainfall resolutions using the accumulation algorithm proposed by Fabry et al. (1994). In this method, the rainfall field is assumed to move at constant velocity and to vary linearly in intensity during the sampling interval. The storm velocity was first computed for each time interval and then used to simulate a 1-min sampling rate by advecting the field observed at the start of the interval toward the field observed at the end of the interval. Represent the accumulated rainfall now via a variable A parameter, denoted Aa, where the subscript denotes the time resolution the rainfall is aggregated over. The resulting radar rainfall (which is a function of Aa) is denoted as Ri,t,a where the subscripts denote the gauge, time-step, and temporal resolution, respectively.
Ascertain the rain gauge rainfall at station i for time-step t for temporal resolution a (denoted Gi,t,a) by accumulating it from 1 h to the other durations (1–24 h) considered.
- Estimate the optimal value of Aa for each time resolution a considered by minimizing the mean absolute error (MAE) between the gauge and radar rainfall estimates. The mean absolute error is expressed aswhere Ri,t,a is the radar rainfall accumulation at the pixel corresponding to the ith rain gauge for hour t for a temporal resolution a, Gi,t,a is the corresponding gauge rainfall for hour t, NG is the number of rain gauges, and Nt,a is the number of time periods for each time resolution a.
It should be pointed out that this study used the mean square error (MSE) and the mean absolute error as two separate error criteria. Although the results from both criteria were similar, the MSE exhibited greater instability for results for larger time resolutions than the MAE, possibly because of fewer gauge–radar pairs being available and the tendency of the measure to magnify the larger differences. As a result, the results presented in later sections are based on the MAE error criterion alone.
3. Application
a. Radar and rain gauge data
Radar reflectivity data from the Kurnell, Mount Stapylton, and Pasicharoen radars located in Sydney, Brisbane, and Bangkok, respectively, and corresponding rain gauges data representing large networks in the three cities were used for the analyses presented in this paper. Three datasets of the 1.5-km constant altitude plan position indicator (CAPPI) reflectivity data at the Kurnell radar and hourly rain gauge data obtained from a network of 227 rain gauges during November 2000–April 2001, August–December 2006, and January–May 2007 were used for the Z–R development for the Sydney area. The Mount Stapylton radar data used represented the November 2006–March 2007 period, with corresponding rain gauge data obtained from a network of 202 rain gauges. For the Z–R calibration in Bangkok area, one dataset of the 0.5° plan position indicator (PPI) reflectivity data at Pasicharoen radar and 15-min rain gauge data obtained from a network of 61 rain gauges during June 2005–October 2006 were used.
The Kurnell radar is a C-band Doppler radar that transmits the radiation with the wavelength of 5.3 cm and produces a beamwidth of 0.94°, the Mount Stapylton radar is an S-band Doppler radar that transmits the radiation with the wavelength of 10.7 cm and produces a beamwidth of 1°, and the Pasicharoen radar is a C-band minimax Doppler radar that transmits the radiation with the wavelength of 5.4 cm and produces a beamwidth of 0.90°. The radar reflectivity data achieved from the Sydney and Brisbane stations are in a Cartesian grid with 256 km × 256 km extent with 1 km2 spatial resolution and 10-min temporal resolution, whereas the reflectivity data achieved from Bangkok station are in a Cartesian grid with 240 km × 240 km extent with 1 km2 spatial resolution and 10-min temporal resolution.
Rain gauge rainfall data used in this study were obtained from the networks of 227, 202, and 61 continuous tipping-bucket gauge stations located within 100 km from the Kurnell, Mount Stapylton, and Pasicharoen radars, as illustrated in Fig. 1. The rain gauges used for Sydney and Brisbane are owned and operated by the Australian Bureau of Meteorology and the Sydney Water Corporation, whereas the rain gauges used for Bangkok are owned and operated by the Bangkok Metropolitan Administration (BMA). The tipping-bucket gauge can systematically underrecord the true rainfall accumulation during the hour by the volume of water required to initially wet the funnel plus the volume of water stored in the tipping bucket at the end of hour. However, the amount of rainfall required to wet the funnel of the gauge before it starts to drain into the tipping bucket is very small and considered to be insignificant in this study. The rain gauges used in this study have tipping-bucket sizes of 1.0 and 0.5 mm. Because tipping-bucket rain gauges record the time of the tips, they are subject to significant high quantization error at low rainfall intensity (Chumchean et al. 2003, 2004, 2006a,b). Therefore, only the rainfall amounts that are greater than the volume of that gauge’s tipping bucket were used in this analysis. It should be noted that quality control of these data has been performed by considering rainfall data from adjacent gauges and the plots of time series. If unusual rainfall data were found, these data were excluded from the analysis.
To avoid the effects of the bright band and different observation altitudes in radar reflectivity, the CAPPI reflectivity data of the Kurnell and Mount Stapylton radars at the altitude below the climatological freezing levels of Sydney and Brisbane, respectively, and the PPI reflectivity data of the Pasicharoen radar at the lowest elevation angle within the radar range that gives the height of radar beams below the freezing level of Bangkok were used in the analyses.
The climatological freezing levels for Sydney, Brisbane, and Bangkok are about 2.5, 3.0, and 4.7 km, respectively (Chumchean et al. 2003, 2004). In this study, the 1.5-km CAPPI reflectivity data of both radars in Australia, the 0.5° PPI reflectivity data of the Bangkok radar, and only the reflectivity and rain gauge data that lie within 100 km of the radars were used. Also note that the height of the base scan beam center at this range is about 1.8 km above the ground, which is also below the freezing levels for the three cities and can be considered to not be overly different from the 1.5-km CAPPI height. Therefore, we consider that the reflectivity data used in this study are free from the effects of the bright band and different observation altitudes.
To avoid the effects of noise and hail in the measured radar reflectivity, the reflectivity values less than 15 dBZ were assumed to represent a reflectivity of 0 mm6 m−3 and the reflectivity values greater than 53 dBZ were assumed to be 53 dBZ. Additionally, the errors resulting from the effects of ground clutter were also removed from the reflectivity data by finding the clutter locations from the map and discarding the radar measurements in these areas.
b. Climatological Z–R relations for variable temporal resolutions
The five datasets considered in this study were used to develop Z–R relationships by following the calibration procedure outlined in section 2. Numbers of radar–gauge pairs used for Z–R calibration for each temporal resolution in each dataset are also presented in Table 1. As mentioned earlier, the results presented here correspond to the use of the mean absolute error as the objective function to be minimized. The A parameters of the Z–R relationships derived by using different temporal rainfall resolutions for the five datasets are illustrated in the Fig. 2. Note that the datasets Sydney 1, Sydney 2, and Sydney 3 represent data for the periods November 2000–April 2001, August–December 2006, and January–May 2007, respectively. The results in Fig. 2 illustrate clearly that the highest value of parameter A is obtained when the temporal resolution is 1 h, with the parameter value decreasing as the aggregation period increases. The difference between the calibrated A parameters for Sydney, Brisbane, and Bangkok are large, whereas there are only small differences among the Sydney 1–3 datasets. Given the differences in the dominant storm types that could be expected between the three cities, this is to be expected.
From the above results, it can be seen that the multiplicative term A of the Z–R relationship varies as a function of the temporal resolution of the rainfall used in the calibration. Derivation of the A parameter using coarse (e.g., daily) rainfall and subsequently applying it to estimate fine-resolution (e.g., hourly or subhourly) rainfall can lead to significant biases in the resulting estimates. For situations where only daily rain gauge rainfall data are available for use in the Z–R calibration, a transformation function for converting the A parameters to other resolutions is needed.
4. Simple scaling hypothesis for the multiplicative term A
a. Climatological scaling transformation function
Moment orders q equal to 0.5, 1.0, 1.5, 2.0, 2.5, 3.0, 3.5, 4.0, 4.5, 5.0, 5.5, and 6.0 were used to ascertain the optimal value of η for the five rainfall datasets used. Results from this analysis are illustrated in Fig. 3a. An optimal value of η was ascertained as the slope of ∂ log(Atq)/∂ log(t) plotted as a function of moment order q in Fig. 3b. Note that Fig. 3 only represents results for the Sydney 1 data. Similar results were obtained for the other four datasets used.
b. Investigation of impact of attenuation on the temporal scaling relationship
Attenuation for C-band radar is considered to be a severe problem for measurement of high-intensity rainfall (reflectivity >50 dBZ; Hildebrand 1978; Austin 1987). However, the impact of attenuation can be ascertained by studying the relationship between gauge and radar rainfall as a function of distance from the radar (Burrows and Attwood 1949). In the results reported below, we investigate the effects of attenuation on the temporal scaling behavior of the Z–R relationship by considering distance from the radar site as a surrogate for possible attenuation. We assume that radar data within a given range interval have common attenuation effects. According to the spatial distribution of the three rain gauge networks of the three cities, the three datasets for Sydney and one dataset for Brisbane were separated into 0–50-km and 50–100-km range intervals and the dataset for Bangkok was separated into 0–15-km and 15–50-km range intervals because the farthermost gauge of the Bangkok network is located at around 46 km from the radar. To investigate the effects of attenuation on the temporal scaling of the Z–R relationship, the A parameters of each range interval for each radar were calculated; the results are presented in Fig. 4. Differences in the A parameters of each range interval for the same radar might be due to the differences in rainfall characteristic of each range, the differences being partly due to attenuation. Once the A parameters of each range interval were derived, the scaling transformation equations for each dataset were also estimated; the results are presented in Table 2. Note that the Sydney 1, Sydney 2, and Sydney 3 datasets represent data for the periods November 2000–April 2001, August–December 2006, and January–May 2007, respectively.
From the results presented in Table 2, it can be seen that the scaling exponents (η) of the scaling transformation Z–R equations obtained from the data lying within far range intervals of the C-band radar (50–100 km for Kurnell radar and 15–50 for Pasicharoen radar) are lower than using the data lying in the inner range intervals. This is possibly because of attenuation. For a situation where an intense rainfall cell passes between the radar and a rain gauge location, the reflectivity recordings from the pixel above the gauge will be underestimated. The calibration procedure would therefore return higher values of the coefficient A of the Z–R relationship than if there was no signal attenuation. Attenuation is a more severe problem during convective events and there may be a bias toward the prevalence of convective events at shorter durations over longer durations. Attenuation is more likely to produce lower A values at shorter durations than longer durations and consequently a lower value of the scaling exponent, η. This was indeed observed from both of the C-band radar, with much lower values of η for the outer range band, where attenuation (if not corrected for) is more pronounced. The lower scaling exponent indicates an overestimated scaling exponent because attenuation has not been corrected for the data at far ranges. From this result, it is evident that attenuation has affected the temporal scaling behavior of the Z–R relations of the C-band radars. However, for the S-band radar, the scaling exponents of the two range intervals are not appreciably different. This confirms the temporal scaling hypothesis of the Z–R relationship because the attenuation problem can be neglected for the S-band radar.
c. Verification of proposed scaling transformation function
Because the proposed scaling transformation is based on the assumption of distributional equality, it can be expected that the function leads to reasonable results when applied to ascertain specified quantiles of the data. Consequently, the scaling function was verified by using it to ascertain the probability distribution of the maximum intensity of rainfall burst as a function of rainfall duration and compared to the distribution of similar maximum intensity of rainfall burst observed in each rain gauge location. It is to be noted that the term “rainfall burst” has been used to represent rainfall bursts of fixed durations. The 24-h A-parameter value was used as a reference, based on which parameters for other temporal aggregation periods were estimated. Additionally, to show an effectiveness of the proposed scaled Z–R transformation equation, the frequency distributions of the maximum radar rainfall obtained from the scaling and 24-h Z–R relationships were also compared with rain gauge data, as presented in Figs. 5 –7. It can be noted that the distributional attributes of the estimated maximum radar rainfall are more similar to those of the rain gauge rainfall than the maximum radar rainfall obtained from the 24-h Z–R relationship. This lends further credibility to the assumptions that were used to formulate the proposed scaling transformation function.
The mean square errors for the maximum intensity of rainfall bursts for six time resolutions (1, 2, 4, 6, 12, and 24 h) for the combined Sydney data, the Brisbane data, and the Bangkok data are presented in Table 3. In addition to this, the mean square error for the full rainfall data, excluding zero radar rainfall, was also calculated. The “optimal” case in the table refers to the estimation of the optimal A coefficient based on mean absolute error as described in section 2. For contrast, values of the error that would be expected were the 24-h A parameter to be used are also given. Percentages of errors in radar rainfall estimates based on two different A parameters that were derived from the temporal scaling transformation equation and the 24-h A parameter have been calculated, as shown in Table 3. From this result, it is evident that for all rainfall durations of all three radars, errors in extreme radar rainfall of the scaling case are less than the 24-h case. The 24-h A parameter gives higher errors, especially at high rainfall intensities and low temporal resolutions. Using the scaling Z–R relationship can reduce error in extreme radar rainfall, especially at the finer temporal resolution. However, the improvements in radar rainfall estimates, when considering all radar data excluding zero values, are not significant if the scaling Z–R relationship has been used.
5. Conclusions
The main conclusions of this paper are summarized below:
The A parameters of the Z–R relationship of Sydney, Brisbane, and Bangkok radar stations tend to decrease with a decrease in the rainfall temporal resolution.
The decrease in the A parameters can be described through a simple scaling law and a derived scaling transformation function.
The scaling exponents for five datasets representing three locations (Sydney, Brisbane, and Bangkok) lie in the vicinity of 0.055. Hence, a climatological scaling law with an exponent equal to 0.055 is proposed.
Attenuation might have some effects on the climatological scaling component for C-band radar at far range from the radar site if no corrections are made for attenuation in the data; however, the scaling hypothesis appears to be valid for the S-band radar.
The proposed scaling transformation equation with scaling exponent 0.055 exhibits significant improvements in estimating extreme rainfall, especially at fine temporal resolutions, in contrast to the accuracy obtained when using the A parameter, which is based on 24-h rainfall. For the extreme rainfall, this accuracy decreases as one proceeds to higher resolutions.
The proposed scaling relationship is consistent across multiple locations but exhibits variations with range in radars where attenuation is a significant issue. The use of this relationship is promising, especially in locations with limited short-duration rain gauge measurements and attenuation-corrected radar measurements.
Acknowledgments
The first and second authors gratefully acknowledge the Thailand Research Fund through the Royal Golden Jubilee Ph.D. program (Grant PHD/0118/2547) and Kasetsart University Research and Development Institute for financially supporting this research. We wish to thank Dr. Alan Seed (Australian Bureau of Meteorology, Melbourne, Australia) for many helpful suggestions. We also appreciate the Australian Bureau of Meteorology, the Sydney Water Corporation, the Bangkok Metropolitan Administration, and the Thai Meteorological Department (TMD) for providing the radar, rain gauge, and freezing-level data used in this study. Comments from three anonymous reviewers greatly helped to improve the quality of this paper.
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Locations of tipping-bucket rain gauges within the radii of the Kurnell, Mount Stapylton, and Pasicharoen radars’ range.
Citation: Journal of Atmospheric and Oceanic Technology 26, 7; 10.1175/2009JTECHA1161.1
Coefficient A of the Z–R relationship derived using a fixed exponent b = 1.6, for the Kurnell, Mount Stapylton, and Pasicharoen radars as a function of varying temporal rainfall resolutions.
Citation: Journal of Atmospheric and Oceanic Technology 26, 7; 10.1175/2009JTECHA1161.1
Verification of scaling hypothesis for Sydney 1 rainfall data representing the period November 2000–April 2001. (a) Scaling of the moments for A parameters. (b) Scaling exponent for A parameters at different moment orders (q).
Citation: Journal of Atmospheric and Oceanic Technology 26, 7; 10.1175/2009JTECHA1161.1
As in Fig. 2, but at different range intervals.
Citation: Journal of Atmospheric and Oceanic Technology 26, 7; 10.1175/2009JTECHA1161.1
Frequency of maximum gauge rainfall and the scaling transformation–based estimated radar rainfall for 1-, 6-, and 24-h durations for the combined Sydney datasets.
Citation: Journal of Atmospheric and Oceanic Technology 26, 7; 10.1175/2009JTECHA1161.1
As in Fig. 5, but for the Brisbane dataset.
Citation: Journal of Atmospheric and Oceanic Technology 26, 7; 10.1175/2009JTECHA1161.1
As in Fig. 5, but for the Bangkok dataset.
Citation: Journal of Atmospheric and Oceanic Technology 26, 7; 10.1175/2009JTECHA1161.1
Number of radar–gauge pairs used for Z–R calibration at each temporal resolution for (top) Sydney and (bottom) Brisbane and Bangkok datasets.
Scaling exponents at different range intervals for five datasets.
Effectiveness of the scaling function for estimation of radar rainfall at varying temporal resolutions for the (a) Sydney, (b) Brisbane, and (c) Bangkok data.
