Application of Scaling in Radar Reflectivity for Correcting Range ...

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Application of Scaling in Radar Reflectivity for Correcting Range-Dependent Bias in Climatological Radar Rainfall Estimates SIRILUK CHUMCHEAN School of Civil and Environmental Engineering, The University of New South Wales, Sydney, New South Wales, Australia

ALAN SEED Bureau of Meteorology Research Centre, The Bureau of Meteorology, Melbourne, Victoria, Australia

ASHISH SHARMA School of Civil and Environmental Engineering, The University of New South Wales, Sydney, New South Wales, Australia (Manuscript received 28 October 2003, in final form 29 March 2004) ABSTRACT This paper presents a method to correct for the range-dependent bias in radar reflectivity that is a result of partial beam filling and of the increase in observation volume with range. The scaling behavior of reflectivity fields as a function of range from the radar was explored in this study. It was found that a simple scaling paradigm was applicable, and a scale transformation function was proposed to ensure uniformity in the probability distributions of reflectivity at both near and far range. The scaling exponents of the power-law transformation function were estimated using a 6-month sequence of reflectivity maps in a Cartesian grid and two sets of instantaneous reflectivity maps in polar coordinates from 18 C-band and 1.68 S-band radars at Sydney, Australia. The averaging process that transforms the instantaneous polar reflectivity into the hourly Cartesian grid data leads to a lower scaling exponent of the hourly Cartesian reflectivity data compared with the instantaneous plan precipitation indicator (PPI) polar data. The scaling exponents for the hourly Cartesian case, the instantaneous polar case using the 18 C-band radar, and the instantaneous polar case using the 1.68 S-band radar data were estimated as 0.024, 0.10, and 0.22, respectively.

1. Introduction The frequency, accuracy, and resolution of hydrologic records is a major limitation in the accurate modeling of hydrologic events. The weather radar provides realtime spatially continuous measurements covering a large area at short time intervals. Radar rainfall estimation traditionally involves the use of a parametric relation formulated based on measurements of radar reflectivity and ground (point) rain gauge rainfall. However, such relations are often uncertain, and their use in practical scenarios often leads to significant uncertainty when compared with rain gauge rainfall (Austin 1987; Joss and Waldvogel 1990; Seed et al. 1996; Krajewski and Smith 2002). Numerous factors are responsible for this uncertainty, including the variability of vertical profile of reflectivity, the error in measuring radar reflectivity, the variability of rainfall drop size distribution, and use Corresponding author address: Dr. Ashish Sharma, School of Civil and Environmental Engineering, The University of New South Wales, Sydney, NSW 2052, Australia. E-mail: [email protected]

q 2004 American Meteorological Society

of point measurements of ground rainfall as a surrogate of pixel averaged values (Battan 1973; Wilson and Brandes 1979; Zawadzki 1984; Austin 1987; Joss and Waldvogel 1990; Smith et al. 1996; Ciach and Krajewski 1999; Chumchean et al. 2003). To some extent, the random errors due to sampling can be overcome through averaging in time and space (Seo et al. 1999). Systematic biases will remain after averaging, and these biases cause significant errors when the data are used for hydrological modeling (Joss and Waldvogel 1990; Seo et al. 1999; Krajewski and Smith 2002). The difference between observed and true reflectivity causes high variability in the fitted Z–R relationship; therefore, Rosenfeld et al. (1993) presented a probability matching method (PMM) to investigate an appropriate Z–R relationship by matching probability distributions between reflectivity cumulative distribution functions (CDFs) to gauge CDFs directly. They also considered the change of reflectivity CDFs with range, naming the modified method window PMM (WPMM) (Rosenfeld et al. 1994), whereby pairs of spatial windows small enough to maintain some physical relevance are used in comparing the two CDFs. An advantage of WPMM

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is that the selected Z–R relationship is related to the variability associated with radar range and other parameters, but the method suffers from the fact that the number of rainfall–reflectivity pairs used in estimating the CDF for each window is significantly reduced. The systematic biases in measured reflectivity are often caused by the climatology of the vertical profile of reflectivity, and the effect of radar beam geometry. The conical shape of the radar beam causes the volume of a radar bin to increase as the square of the distance to the radar. Therefore, the small intense features that are present in a rain field will be averaged out by the measurement process, thereby leading to an underestimation of the frequency of high-intensity echoes at far range and, consequently, a bias in measured reflectivity. The true scaling behavior of the polar bins of radar reflectivity is complicated by the fact that the range resolution of the bin is invariant with range and the fact that the three-dimensional rain field shows anisotropic scaling behavior between the vertical and horizontal directions, but as a starting hypothesis it is assumed that the biases can be modeled using a simple scaling model. Based on this assumption, a method to correct the range-dependent bias in measured reflectivity due to increases in observation volume with range is proposed. While our alternative is similar in aim to the WPMM (Rosenfeld et al. 1994), it approaches the problem from a different rationale. Consequently, the limitation of estimating CDFs based on data in a small spatial window is avoided, the bias correction being approached through a transformation function derived using scaling theory. Radar data are measured in polar coordinates, where each polar bin is an average of a number of radar pulses. There are two main techniques for projecting the 3D polar data onto a 2D surface. The first technique is the precipitation plan indicator (PPI) system. A PPI reflectivity map is extracted from the raw reflectivity data from the beam at a particular elevation angle. The curvature of the earth and the refraction of the radar beam through the atmosphere cause the height of the beam to increase as a function of range, leading to height sampling error. To reduce the error caused by different observation altitudes, a constant altitude plan precipitation indicator (CAPPI) is used as an alternative. The CAPPI reflectivity data are extracted from the volume scan by averaging the reflectivities (in mm 6 mm 23 ) returned from all of the range bins that fall within a given pixel. The beam angle that has its center closest in elevation to the nominal CAPPI elevation is chosen for each pixel. While reflectivity is measured in polar coordinates, it is transformed to Cartesian coordinates for ease of use. Additionally, the reflectivity is often aggregated in time and presented at hourly time steps. We hypothesize that the scaling behavior of reflectivity will be more profound when no such aggregation or conversion to Cartesian coordinates is done. It is also of practical intent to investigate the scaling behavior of instantaneous polar data so as to develop real-time methods to correct

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the increasing bias with range before the data are projected onto a Cartesian grid. Hence, our analysis evaluated the scaling behavior of (a) reflectivity data in Cartesian coordinates averaged to an hourly resolution and (b) instantaneous reflectivity data in polar coordinates. The vertical profile of reflectivity is a major source of both bias and uncertainty in radar measurements of rainfall with range from the radar. The bias is often due to the radar beam intersecting the bright band, which leads to an overestimation of rainfall rates over the range of the intersection, and thereafter the rainfall is underestimated due to the rapid decreases in reflectivity with height above the bright band. Many studies, such as Kitchen et al. (1994), Andrieu and Creutin (1995), Andrieu et al. (1995), Fabry and Zawadzki (1995), Kitchen (1997), Vignal and Andrieu (1999), Sanchez-Diezma et al. (2000), and Vignal and Krajewski (2001), have investigated methods to determine a mean vertical profile of reflectivity and have developed methods to use the determined vertical profile of reflectivity to correct the bias of a radar measurement at some height above the ground relative to a measurement at ground level. The CAPPI can reduce the impact of the vertical profile of reflectivity up to the point where the base scan intersects the bright band. Nevertheless, a range-dependent bias in the CAPPI radar reflectivity remains even if the CAPPI data that are obtained from an altitude below the bright band have been used. This is because the conical shape of the radar beam causes the diameter of the radar bin to increase with distance from the radar. This affects the statistics of the observed reflectivity since the small high-intensity echoes are averaged out at far range, thereby leading to an underestimation of both the mean and variance of the measured reflectivity. This paper analyzes the change in the probability distribution function of measured reflectivity as a function of range from the radar in the context of a simple scaling hypothesis. This paper is organized as follows. Section 2 discusses the scaling hypothesis for radar reflectivity. Section 3 presents an analysis of the dependence of the CDF of an hourly CAPPI Cartesian reflectivity with range. Thereafter we evaluate the effectiveness of the scaling transformation by comparing the radar data with a rain gauge network. Sections 4 and 5 present the scaling behavior of instantaneous polar data using the 18 Cband and the 1.68 S-band radars, respectively. Finally, conclusions of this study are drawn in section 6. 2. Simple scaling hypothesis for radar reflectivity Some of the theories regarding the rescaling of rainfall statistic are derived from the scaling hypothesis popularized by Mandelbrot (1982) and Lovejoy and Schertzer (1985). It is assumed (and subsequently verified) that ‘‘simple scaling in the strict sense’’ [defined by Gupta and Waymire (1990)] holds for the change in the probability distribution of measured reflectivity Z D with range from the radar. This can be written as

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CHUMCHEAN ET AL. dist

Z d 5 (d/D) 2h Z D ,

(1)

dist

where 5 denotes equality in the probability distribution of reflectivity; d/D is a scale factor, where d(km) is the reference observation range interval; D(km) is the observation range of the measured reflectivity Z D (dBZ); h is the scaling exponent; and Z d (dBZ) is the transformed reflectivity of the measured reflectivity (Z D ) to be equivalent to reflectivity at the reference observation range interval d. The above equation implies that the quantiles and the moment of any order of the measured reflectivity are scale-invariant. The relationship between the qth moments is obtained as ^Z qd & 5 (d/D) 2hq^Z qD &.

(2)

The brackets represent the expectation of moments of the reflectivity profile. The transformation of the two fields to ensure equality across the important moments of the variable has been referred to as simple scaling, in the wider sense, in the scaling literature (see Gupta and Waymire 1990). The scaling property of measured reflectivity in the ‘‘strict sense’’ can be written explicitly in terms of the reflectivity CDF as (Menabde et al. 1999) F d (Z) 5 F D [(d/D) h Z].

(3)

Z d 5 (d/D) 2h Z D .

(4)

Hence, The above variables are the same as mentioned before. Note that the relationship in (4) represents a distinct relationship between the variables Z d and Z D , and is a simplified representation of the complete distributional relationship in (1). 3. Scaling property of hourly CAPPI Cartesian reflectivity data Hourly accumulations of radar estimates of rainfall projected onto a Cartesian grid are standard input into operational hydrological models, and the archive of these data has accumulated into a substantial database. In principle, if the polar-to-Cartesian transformation is performed by averaging the polar data within a Cartesian pixel in units of reflectivity, one would expect that the resulting Cartesian field would not reflect smallerscale artifacts due to the increasing volume of the polar bins with range. However, the resolution of the Cartesian field is often set such that there is an oversampling of the polar data at far range. For example, a 1-km Cartesian field does not extend across a 18 beam for ranges that exceed about 60 km. The rest of this section first discusses the hourly Cartesian reflectivity and corresponding rain gauge data that were used to establish the simple-scaling transformation. This is followed by a discussion on the verification that was undertaken to establish the simple-scaling hypothesis. The results in this section pertain to the hourly case, with the second part

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of our investigation, a study of the presence of scaling in instantaneous reflectivity fields measured in polar coordinates, being presented in the sections that follow. a. Data The scaling analysis requires rainfall data that are independent of range from the radar. While one can expect that the rainfall field is stationary over a long record and short spatial scales, significant variations in both space and time are present in individual rainfall events. For the purpose of this study we have assumed that the rainfall record used is stationary over space and time. We have found this assumption to be valid in the limited analysis that was conducted to investigate stationarity in the rainfall. This study used the 1.5-km CAPPI data of 27 storms recorded by the Kurnell radar at Sydney, Australia, during November 2000–April 2001 and the corresponding hourly rain gauge data obtained from a 254 rain gauge network. The Kurnell radar is a C-band radar with a 3-dB width of 18. The reflectivity data are in a Cartesian grid with 256 km 3 256 km extent and a 1 km 2 , 10-min resolution. The hourly reflectivity values were obtained by converting the instantaneous 10-min reflectivity data into rainfall intensity and applying the accumulation method proposed by Fabry et al. (1994). This method assumes that the rain field moves at constant velocity 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. The hourly rainfall accumulations were converted back into dBZ using the inverse of the Z–R relation. Rain gauge data used in this study were obtained from a dense network of 254 hourly tipping-bucket gauge stations located within 100 km of the radar, as shown in Fig. 1. Eighty-five of these stations are owned and operated by the Australian Bureau of Meteorology. Most of these stations have a tipping-bucket size of 1.0 mm. The other 169 stations have a tipping-bucket size of 0.5 mm and are owned and operated by the Sydney Water Corporation. The tipping-bucket type of rain gauge records the time of bucket tips; hence, they are subject to significant quantization error at low rainfall intensity. Therefore, only the rainfall amounts that are greater than the volume of that gauge’s tipping bucket were used in this study. The effects of bright band and a different observation altitude at far range are the other sources of error that cause the range-dependent bias in radar reflectivity; therefore, only the reflectivity and rain gauge data that lie within 100 km from the radar were used in this section. The height of the center of the beam at 100 km from the radar is 1.8 km above the ground for the base scan, which can be considered to be not overly different from the 1.5-km height of the CAPPI. Sydney has a

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FIG. 1. Map of the Sydney area, showing rain gauges (small circles), the Kurnell radar (radar symbol), and range rings at 20, 40, 60, 80, and 100 km from the radar.

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climate that falls between the temperate and subtropical types. The lower freezing level during the study months is above 2.5 km. Therefore, we consider that the reflectivity data used in this section are free from the brightband effect. To avoid the effect of noise and high reflectivities caused by hail, the echoes less than 15 and greater than 53 dBZ (Fulton et al. 1998) were excluded from the analysis. It is necessary to verify that the rainfall mean and variance is indeed independent of range from the radar. The conditional mean of the rain gauge rainfall (rainfall . gauge’s resolution) within annular rings at 0–20-, 20– 40-, 40–60-, 60–80-, and 80–100-km range intervals (hereafter referred to as 20-, 40-, 60-, 80-, and 100-km range intervals, respectively) are 4.41, 4.19, 4.32, 4.02, and 4.27 mm h 21 , respectively. The slope of the straight line fitting these conditional means is considered to be insignificantly different from zero at the 90% confidence level. Rain gauge data of the 27 storms and the corresponding hourly 1.5-km CAPPI reflectivity in a Cartesian grid were used to investigate the dependence on range of CDFs of rain gauge rainfall and the corresponding reflectivity. We separated the hourly reflectivity and rain gauge data into 20-, 40-, 60-, 80-, and 100-km range intervals. The CDF of the gauge data and the radar data sampled at the gauge locations was calculated for each range interval. We have found that, in general, the probability of high reflectivity values decreases as a function of range in the measured radar reflectivity, but this dependence is not observed in the gauge data and must therefore be an artifact of the radar measurement process, as illustrated in Fig. 2. The same variation of the

FIG. 2. CDFs for 18 beamwidth hourly Cartesian reflectivity and rain gauge rainfall. The reflectivity’s CDFs are estimated using only the pixels that correspond to rain gauge locations. Note that dBR is 10 logR, where R is rainfall intensity (mm h 21 ).

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FIG. 3. Scaling of the moments for 18 beamwidth hourly Cartesian reflectivity at different moment orders (q).

reflectivity’s CDF with range can be observed when using all echoes between 15 and 53 dBZ for ranges within 100 km of the radar. b. Analysis of the simple-scaling hypothesis for hourly data The scaling of the moments in measured reflectivity, as expressed in Eq. (2), was verified using all measured reflectivity pixels of the 27 storms that lie within 100 km of the radar. The moments of the hourly 1.5-km CAPPI Cartesian reflectivity (dBZ units) were estimated and plotted against the corresponding observation range interval, as illustrated in Fig. 3. The slope of the straight line fitting these points gives the value of the scaling

FIG. 4. Scaling exponent for 18 beamwidth hourly Cartesian reflectivity.

exponent h q for each qth-order moment. The dependence on q of scaling properties of moments is linear, as illustrated in Fig. 4, thereby confirming the wider sense simple-scaling hypothesis in Eq. (2). The slope of the straight line fitting the scaling properties of the moments h q with the corresponding moment orders is the scaling exponent, which is equal to 0.024, as illustrated in Fig. 4. The proposed climatological scale transformation function of an hourly 1.5-km CAPPI reflectivity in a Cartesian grid obtained from the 18 radar beamwidth can be written as Z transformed(dBZ ) 5

1 2 20 D

20.024

Z D(dBZ ) ,

(5)

where D(km) is the observation range of the measured reflectivity beyond 20 km and Z D (dBZ) is the hourly 1.5-km CAPPI Cartesian reflectivity at that range. The reflectivity’s CDF at the 20-km range interval is selected as the reference CDF because the scaling behavior starts at the 20-km range interval (see Fig. 2) and the accuracy of the reflectivity CDF that is closest to the radar is the highest due to a smaller observation volume. This proposed scale transformation function was used to transform the measured reflectivity at different range intervals to have the same CDF as measured reflectivity at the 20-km range interval. The CDFs of transformed reflectivity of different range intervals were estimated by using all reflectivity pixels that have values between 15 and 53 dBZ and lie within 100 km of the radar, as illustrated in Fig. 5. The CDFs of transformed reflectivity that were calculated using only the onshore reflectivity pixels that

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FIG. 5. The reflectivity’s CDFs of the scale-transformed hourly Cartesian reflectivity of the 18 beamwidth. All pixels having values between 15 and 53 dBZ and lying within 100 km of the radar were used.

correspond to the rain gauge locations were compared with the CDFs of rain gauge rainfall, as presented in Fig. 6. It is shown that the transformed reflectivity’s CDF of the 20-, 40-, 60-, and 80-km range intervals can be considered to be independent with range. However, some bias remains in the transformed reflectivity’s CDF of the 100-km range interval, which suggests that the simple-scaling hypothesis may need to be superseded by other mechanisms that affect reflectivity profiles at ranges greater than 100 km. One possible explanation for the poor representation at the 100-km range is the effect of the topography of the study area, as the complex terrain of the Sydney area has some systematic effect on thunderstorm distribution (Matthews and Greerts 1995). The Blue Mountains and the Southern Highland area are located approximately beyond 80 km west and southwest from the Kurnell radar, respectively. Matthews and Geerts (1995) investigated the distribution of the thunderstorm in the Sydney area using approximately 1000 thunderstorms that occurred during 1965–89. They found a west-toeast progression of storms of the Sydney area and reported that the thunderstorms are more common over the Southern Highland and the Blue Mountains around noon, cross the Sydney metropolitan area in the afternoon, and stall offshore at night. Potts et al. (2000) also found that many thunderstorms develop over the mountains, intensify and move east over the coastal plain, and decay as they move over the ocean. Because most of the thunderstorms that occur in Sydney develop over the mountains, the probability of high convective rainfall intensity is more likely to occur at the range beyond 80 km than at the other range intervals. For this reason, the nature of rainfall events in the onshore 100-km range interval is considered to be different from the others.

Therefore, the remaining bias in the transformed reflectivity’s CDF of the onshore 100-km range interval implicitly shows that the scaling exponent of the powerlaw-scale transformation function varies according to the rainfall type. The effectiveness of the scaling transformation function was further evaluated by calculating the mean bias between gauge and radar estimates of rainfall at five different range intervals (0–20, 20–40, 40–60, 60–80, and 80–100 km). The bias was defined as

OG B5 , OR n

i

i51 n

(6)

i

i51

FIG. 6. Scale-transformed 18 beamwidth hourly reflectivity and rain gauge rainfall CDFs. The reflectivity’s CDFs here represent only the onshore pixels that correspond to rain gauge locations. Note that dBR is 10 logR, where R is rainfall intensity (mm h 21 ).

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FIG. 7. Bias in climatological radar rainfall estimates of the measured and transformed reflectivity.

1987), and it is considered to be a severe problem for the measurement of high-intensity rainfall. The study of Hilderbrand (1978) indicated that C-band radar measurements of a storm with reflectivity maxima $50 dBZ will be seriously attenuated. Because the conditional mean rain gauge’s rainfall rate of the 6-month-long data is about 4.2 mm h 21 and the conditional mean of measured reflectivity (reflectivity . 15 dBZ) is only 25.6 dBZ, the effect of attenuation can be considered to be insignificant on average but can be a major problem in highly convective situations. To prove this assumption, we used the relationship between rainfall rate and attenuation (dB km 21 ) of a 5-cm wavelength proposed by Burrows and Attwood (1949) to calculate the attenuation as a function of range from the radar: K r 5 2(0.0018R1.05 ),

where G i is rain gauge rainfall (mm) at the ith gauge in the range ring, n is the total number of gauges in each range ring, and R i is radar rainfall (mm) at the pixel corresponding to the gauge location. The default Z–R relationship, Z 5 200R1.6 (Marshall and Palmer 1948), was used to convert the reflectivity into rainfall intensity. It was found that the bias in radar rainfall that was estimated from the hourly Cartesian reflectivity increases as a function of range, as illustrated in Fig. 7. The slope of the straight line fitting these radar rainfall biases with range is 8.9% and is significantly different from zero at the 90% confidence level. In contrast, the slope estimated based on the transformed reflectivity values is only 0.3%, which is not significantly different from zero at the 90% confidence level. It should be pointed out that even though the G/R results vary significantly with range (e.g., the G/R for the 60– 80-km range is smaller than the rest), the difference between the ‘‘scaling’’ and the ‘‘no-scaling’’ case is consistent, which adds strength to our hypothesis that range-dependent biases can be described through a simple-scaling formulation such as the one proposed here. It should be mentioned that this study focuses on the effect of radar beam spreading on the measured reflectivity. The Kurnell radar data used in this study have not been corrected for attenuation. Attenuation by rain or a wet radome is appreciable for C-band radars (Austin

(7)

where K r is the two-way attenuation (dB km 21 ) and R is the rainfall intensity (mm h 21 ). Based on the above equation, we have found that the increase in attenuation with range of the data used in this section is only 1.6% (on average), which can be considered to be insignificant when compared to the slope of increases in bias of radar rainfall estimates (8.9%), as shown in Fig. 7. Nevertheless, the estimated scaling exponent is overestimated, since attenuation was not accounted for in the analysis. It should also be noted that the 20-, 40-, 60-, 80-, and 100-km range rings have been adopted to ensure a reasonable number of rain gauges in each interval. While it is possible that a larger number of rings may give a more stable estimate of the scaling exponent, systematic variations are not likely. In summary, we have found that the proposed scale transformation function of the hourly CAPPI Cartesian reflectivity is able to remove range-dependent bias in the hourly Cartesian reflectivity at ranges within 100 km of the radar, as shown in Fig. 5. It helps to remove the range-dependent bias in the onshore hourly Cartesian reflectivity at the range within 80 km, and it can reduce the bias in the onshore hourly Cartesian reflectivity at the range beyond 80 km from the radar, as shown by the results presented in Table 1. We also have found that the scale transformation function helps re-

TABLE 1. Hourly rain gauge rainfall and hourly Cartesian reflectivity data before and after scaling correction (based on pixels that represent rain gauge locations). Reflectivity data

Rain gauge data No correction

Scaling correction

Range (km)

Conditional mean rainfall (mm h21)

No. of gauges

Mean (dBZ)

Variance (dBZ 2)

Mean (dBZ)

Variance (dBZ 2)

No. of Z–R (pairs)

0–20 20–40 40–60 60–80 80–100

4.41 4.19 4.32 4.02 4.27

38 103 61 31 21

24.0 23.8 23.5 23.4 23.2

33.7 33.3 29.2 29.9 29.3

24.0 24.3 24.3 24.4 24.4

33.7 34.7 31.2 32.5 32.2

2318 7494 3863 2198 1521

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duce range-dependent bias in the onshore hourly reflectivity’s variance by about 34%. From the above analysis, it is evident that a simplescaling paradigm is useful in removing range-dependent bias in measured reflectivity caused by increases in observation volume with range. A small but significant value of the scaling exponent of the hourly Cartesian reflectivity data (h 5 0.024) has been estimated for this case. It would be better to apply the scaling adjustment to the instantaneous polar data before converting to a Cartesian grid and accumulating to hourly accumulations. For this reason, a sample of instantaneous PPI polar reflectivity data was used to estimate the scaling exponent for instantaneous polar data. To reduce the effect of bright band and a difference observation altitude at far range, the PPI polar reflectivity data obtained from the radar beam at the lowest elevation angle (0.78) within 100-km range from the radar were used. The errors due to the effect of ground clutter were removed from this dataset by using a map of known clutter locations and discarding the radar measurements in these areas. Only cases with no sea clutter were selected for this analysis. 4. Scaling property of an instantaneous PPI polar reflectivity Due to the size of the dataset, only the PPI radar reflectivity of four significant rainfall events recorded from the Kurnell radar were used to investigate the scaling exponent of instantaneous PPI polar reflectivity. These data included three stratiform events: 30 January– 1 February 2001, 19–22 April 2001, and 5–10 May 2001, and one convective event, 3 November 2000. We assume that rainfall characteristics of the merged four rainfall events are similar to the climatological rainfall of the Sydney area. The polar data have 18 3 0.25 km resolution. The reflectivity of the radar bins that have values between 10 and 53 dBZ and lie within the 20-, 40-, 60-, 80-, and 100-km range interval were used to estimate the CDF for the radar echoes as a function of range. Figure 8 shows the CDFs for the 20–100-km range rings. The 20-km range CDF does not follow the trend shown by the other range CDFs. This is possibly due to the four rainfall events in this range being from a different population than the rest of the data, possibly due to the enhanced coastal effects noticed in regions adjacent to Sydney’s coastline. It is also possible that an ensemble of four events is insufficient to characterize the full variability of the rainfall, and a more extensive record is needed. Based on the above considerations, the 20-km range data have been excluded in developing the scaling relationship for this case. The scaling properties of moments of the instantaneous PPI polar reflectivity for different moment orders were examined using the reflectivities that lie within the 40-, 60-, 80-, and 100-km range intervals. Due to a high

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FIG. 8. Variation of CDFs of the 18 beamwidth 5-min PPI polar reflectivity with range. All radar bins having reflectivity values between 10 and 53 dBZ and lying within 100 km of the radar are used.

uncertainty in scaling property of moment of a high moment order, the scaling exponent was estimated based on the scaling properties of the moments of the 0.5– 5th-moment orders only. The straight line fitting these scaling properties of the moments with the corresponding moment orders has the correlation coefficient (r 2 ) of 0.99, and the scaling exponent was found to be equal to 0.1. As expected, the scaling exponent of the instantaneous PPI polar reflectivity is higher than the hourly CAPPI Cartesian grid reflectivity data. The proposed scale transformation of the instantaneous PPI polar reaflectivity of the Kurnell radar is therefore Z transformed(dBZ ) 5

1 2 20 D

20.10

Z D(dBZ ) ,

(8)

where Z D is an instantaneous PPI polar reflectivity datum and D(km) is the observation range of the measured reflectivity Z D . The proposed scale transformation function of the polar reflectivity was used to transform the measured instantaneous PPI polar reflectivity at the 40-, 60-, 80-, and 100-km range intervals. We found that the CDFs of transformed reflectivity for the above various range rings exhibit uniformity across the various ranges, as illustrated in Fig. 9. The exceedance probability of the measured and scale-transformed reflectivity was also plotted to investigate the tails in greater detail. We found that scaling transformation is equally effective in the tails of the distribution. We also test the effectiveness of the proposed scale transformation function of the instantaneous PPI polar reflectivity by comparing the hourly PPI Cartesian reflectivities that were obtained from the measured and scale-transformed instantaneous PPI polar reflectivity. Figures 10a and 10b illustrate the CDFs of the hourly PPI Cartesian reflectivity before and after scale correction, respectively. It is shown that the CDFs of scaletransformed reflectivity of the 40-, 60-, and 80-km ranges are similar; however, the CDF of the 100-km range

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FIG. 9. CDFs of scale-transformed 18 beamwidth 5-min PPI polar reflectivity at different range intervals. All measured radar bins having reflectivity values between 10 and 53 dBZ and lying within 100 km of the radar were used.

interval is slightly different from the others. We also found that the 10% probability of exceedance of the transformed reflectivities of the 40-, 60-, and 80-km range intervals are almost the same. The transformed reflectivity of the 100–120-km range ring (thereafter referred as 120-km range interval) was also used to verify the scaling transformation function. These data had not been used in estimating the scaling exponent. Note that the observation altitude at the 120km range from the radar is about 2.3 km above the ground, which is still below the climatological freezing level during the study months. Figure 10b shows that the proposed scale transformation function can remove the range-dependent bias in the hourly PPI Cartesian reflectivity at the 120-km range interval. A small difference between scale-transformed reflectivity of the

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100- and 120-km range rings and the other ranges is possibly a result of a higher hourly rainfall intensity at these two range intervals compared to the other ranges. It could be argued that the scaling behavior observed in the previous analysis is merely an artifact of the attenuation of the C-band signal. Also, it is interesting to estimate the sensibility of the scaling exponent to beamwidth and bin length, as the scaling exponent of the power-law-scale transformation function is expected to depend on radar beamwidth and the nature of the rainfall event. A wider beamwidth is expected to have a higher scaling exponent. To eliminate the attenuation problem and to prove the above assumption, the scaling exponent of the 1.68 S-band Sydney radar located close to the Kurnell radar was investigated and the results are presented in the following section. 5. Scaling property of the S-band radar reflectivity A 7-month dataset record from the Sydney radar during November 2002–May 2003 was used. The S-band Sydney radar is situated 40 km southwest of the Kurnell radar. This radar transmits radiation with a wavelength of 10 cm and produces a 3-dB beamwidth of 1.68. The 7-month-long instantaneous 1.5-km CAPPI reflectivity data used in this section are in polar coordinates with a 1-km bin length and 10-min resolution. CDFs of measured reflectivity of the different range intervals were estimated and plotted in Fig. 11, from which it is shown that the scaling behavior of the measured reflectivity could not be seen at the range 0–80 km, but it is evident in the range 80–256 km from the radar. This is considered to be due to the topography of the area under the radar umbrella. About 20–80 km west

FIG. 10. CDFs of (a) measured and (b) scale-transformed hourly Cartesian PPI reflectivity with range. Note that the hourly reflectivity data were obtained by averaging of the measured and scale-transformed 5-min PPI reflectivity of the 18 beamwidth. All radar bins having reflectivity values between 10 and 53 dBZ and lying within 100 km of the radar were used.

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FIG. 11. CDFs of 10-min 1.68 beamwidth CAPPI polar reflectivity. All pixels having values between 10 and 53 dBZ and lying within 256 km of the radar were used.

from the radar is the mountain area, which could lead to a spatial nonuniform rainfall over the entire radar umbrella. Recall that a similar deviation from the scaling relationship was apparent in the case of the Kurnell radar reflectivity. In the Kurnell radar case, the mountain range was located at a distance of about 80–100 km west and southwest from the radar, which led us to exclude that range from the estimation of the scaling exponent. However, Fig. 11 shows a systematic bias in measured radar reflectivity at the range beyond 80 km from the radar. This is thought to be due to the effect of radar beam geometry if the observation altitude of reflectivity is below the brightband level and to the effect of both radar beam geometry and vertical profile of reflectivity if the observation altitude of reflectivity is above the brightband level. Interestingly, the CDF shapes shown in Fig. 11 can be divided into two groups. The first group is in the 0– 140-km range, and the second is in the 140–256-km range. Difference in the CDF shape between these two groups is possibly due to the fact that the observations between 140 and 256 km are above the climatological freezing level. At this range, the radar measures ice particles instead of raindrops, and the vertical profile of reflectivity decreases rapidly with height above the bright band. Since the main aim of this study is to investigate the scaling behavior of measured reflectivity from rainfall, only the reflectivities lying between 80 and 140 km were used in the analysis. The observation altitude at the 140km range is about 2.4 km above the ground, which is still below the brightband level (2.5 km). The scaling properties of moments of different moment orders were examined. The straight line fitting the scaling properties of the moments with the corresponding moment orders has the r 2 of 0.99. The scaling exponent was found to be equal to 0.22, which is higher than the scaling exponent of the Kurnell radar, as expected. These results

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FIG. 12. CDFs of 10-min transformed CAPPI polar reflectivity of the 1.68 beamwidth.

confirm the assumption that the scaling exponent of a wider beamwidth is higher than the narrow beam. The proposed climatological scale transformation function of the instantaneous 1.5-km CAPPI polar of the S-band Sydney radar is Z transformed(dBZ ) 5

1 2 80 D

20.22

Z D(dBZ ) ,

(9)

where Z D is an instantaneous 1.5-km CAPPI polar reflectivity and D(km) is the observation range of the measured reflectivity Z D . It should be noted that the constant ‘‘80’’ in the numerator of Eq. (9) is specified due to lack of an obvious scaling relationship for ranges less than 80 km. Equation (9) was used to transform the measured reflectivity at the range between 80 and 140 km. The transformed reflectivity CDFs at that range are plotted in Fig. 12. It is shown that CDFs of the transformed reflectivity of the 80–100-, 100–120-, and 120–140-km range intervals are almost the same. This confirms the effectiveness of the proposed scale transformation function of the Sydney radar for the selected range. We also investigated the performance of Eq. (9) for correcting range-dependent bias for echoes beyond 140 km. Figure 12 illustrates the CDFs of transformed reflectivity at the range beyond 80 km from the radar. It can be seen that the transformed CDFs can be divided into two groups: the 80–140- and 140–256-km ranges, as expected. The transformed reflectivity CDFs of the range between 140 and 256 km are closer to each other compared with the nonscaling correction. A range-dependent bias remains in the transformed reflectivity at these ranges. However, the proposed scale transformation function helps to reduce range-dependent bias in measured reflectivity at the range beyond 140 km, as the difference between the 10% probability of exceedance of the transformed reflectivity of the 80–100- and the 240–256-km range intervals was only 2.75 dBZ, while it was equal to 10 dBZ for the uncorrected data.

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6. Conclusions The conclusion of this work can be summarized as follows. 1) The conical shape of the radar beam introduces a bias in the CDF of radar reflectivity measurements with range. We have found that the hourly 1-km resolution CAPPI reflectivity (in dBZ units) from a 18 beamwidth has a simple-scaling property with range, even though the averaging process to transform instantaneous polar reflectivity into hourly Cartesian grid reflectivity has been performed. The scaling exponent was found to be equal to 0.024. Although the scaling exponent is small, it helps to correct the reflectivity by about 0.5–2.5 dB, depending on the observation range and the reflectivity value, which is significant. The value of the exponent is expected to be a function of the spatial resolution of the Cartesian grid. 2) The instantaneous polar reflectivity has a higher scaling exponent (h 5 0.1) than the hourly Cartesian data. 3) In the case where the volume scan reflectivity data are available, it would be better to correct for the range-dependent bias due to increases in observation volume with range while the data are still in the form of instantaneous measurements in PPI polar coordinates. For the case where only the reflectivity data in a Cartesian grid are available, the scaling correction can also be performed by using the scaling exponent that is derived from the Cartesian grid data. 4) We found that it was difficult to estimate the sensitivity of scaling exponent to the type of rainfall (stratiform, convective) due to the difficulties of using a relatively small volume of data. 5) The scaling exponent for the 1.68, 1-km beam (h 5 0.22, D 0 5 80 km) was much higher than that for the 18, 0.25-km beam (h 5 0.10, D 0 5 20 km), leading to significant adjustments at ranges beyond 100 km. 6) Even if climatological scaling corrections constitute only a small proportion of the instantaneous and short-term errors in radar rainfall estimates, it is useful to remove the range-dependent bias in measured reflectivity because it constitutes a systematic bias that will have a negative impact on many hydrological applications that use radar data. Once the reflectivity is free from range-dependent bias, bias correction strategies such as the removal of the mean field bias (G/R) can be used to estimate the rainfall in real time. Finally, it must be emphasized that we do not claim that the proposed scale transformations are valid for different radar beamwidths, bin lengths, and rainfall types other than those from which they were derived. We do find however, that the biases observed in a significant volume of data can be modeled using the sim-

ple-scaling hypothesis, and therefore this correction technique warrants further attention, at least in the warm climates where the bright band is not such a problem. The scaling exponent has been shown to be very sensitive to beamwidth and is expected to depend on the nature of the rainfall event as well. The scaling exponent of convective rain is expected to be higher than that of stratiform rain due to the smaller raining area and the higher rainfall intensity of the convective rain. Therefore, future work should be concentrated on investigating the effect of the nature of the rainfall event on the scaling exponent of the power-law-scale transformation function. Acknowledgments. The authors gratefully acknowledge Mahanakorn University of Technology (Thailand) for funding the first author’s Ph.D. studies at the University of New South Wales. We also thank the Australian Bureau of Meteorology and the Sydney Water Corporation for providing the radar and rain gauge data used in this study. We would like to thank the anonymous reviewers for their helpful comments. REFERENCES Andrieu, H., and J. D. Creutin, 1995: Identification of vertical profiles of radar reflectivity for hydrological applications using an inverse method. Part I: Formulation. J. Appl. Meteor., 34, 225– 239. ——, G. Delrieu, and J. D. Creutin, 1995: Identification of vertical profiles of radar reflectivity for hydrological applications using an inverse method. Part II: Sensitivity analysis and case study. J. Appl. Meteor., 34, 240–259. Austin, P. M., 1987: Relation between measured radar reflectivity and surface rainfall. Mon. Wea. Rev., 115, 1053–1070. Battan, L. J., 1973: Radar Observation of the Atmosphere. The University of Chicago Press, 326 pp. Burrows, D. R., and S. S. Attwood, 1949: Radio Wave Propagation. Academic Press, 219 pp. Chumchean, S., A. Sharma, and A. Seed, 2003: Radar rainfall error variance and its impact on radar rainfall calibration. J. Phys. Chem. Earth, 28, 27–39. Ciach, G. J., and W. F. Krajewski, 1999: On the estimation of radar rainfall error variance. Adv. Water Resour., 22, 585–595. Fabry, F., and I. Zawadzki, 1995: Long-term radar observations of the melting layer of precipitation and their interpretation. J. Atmos. Sci., 52, 838–861. ——, A. Bellon, M. R. Duncan, and G. L. Austin, 1994: High resolution rainfall measurements by radar for very small basins: The sampling problem re-examined. J. Hydrol., 161, 415–428. Fulton, R. A., J. P. Breidenbach, D.-J. Seo, D. A. Miller, and T. O’Brannon, 1998: The WSD-88D rainfall algorithm. Wea. Forecasting, 13, 377–395. Gupta, V. K., and E. C. Waymire, 1990: Multiscaling properties of spatial rainfall and river flow distribution. J. Geophys. Res., 95, 1999–2009. Hildebrand, P. H., 1978: Iterative correction for attenuation of 5 cm radar in rain. J. Appl. Meteor., 17, 508–514. Joss, J., and A. Waldvogel, 1990: Precipitation measurement and hydrology. Radar in Meteorology: Battan Memorial and 40th Anniversary Radar Meteorology Conference, D. Atlas, Ed., Amer. Meteor. Soc., 577–606. Kitchen, M., 1997: Toward improved radar estimates of surface precipitation rate at long range. Quart. J. Roy. Meteor. Soc., 123, 145–163.

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