Calibration of weather radar using region probability matching method ...

Theor Appl Climatol DOI 10.1007/s00704-017-2266-7

ORIGINAL PAPER

Calibration of weather radar using region probability matching method (RPMM) Hooman Ayat 1 & M. Reza Kavianpour 1 & Saber Moazami 2 & Yang Hong 3,4 & Esmail Ghaemi 1

Received: 26 August 2016 / Accepted: 25 August 2017 # Springer-Verlag GmbH Austria 2017

Abstract This research aims to develop a novel method named region probability matching method (RPMM) for calibrating the Amir-Abad weather radar located in the north of Iran. This approach also can overcome the limitations of probability matching method (PMM), window probability matching method (WPMM), and window correlation matching method (WCMM). The employing of these methods for calibrating the radars in light precipitation is associated with many errors. Additionally, in developing countries like Iran where ground stations have low temporal resolution, these methods cannot be benefited from. In these circumstances, RPMM by utilizing 18 synoptic stations with a temporal resolution of 6 h and radar data with a temporal resolution of 15 min has indicated an accurate estimation of cumulative precipitation over the entire study area in a specific period. Through a comparison of the two methods (RPMM and traditional matching method (TMM)) on March 22, 2014, the obtained correlation coefficients for TMM and RPMM were 0.13 and 0.95, respectively. It is noted that the cumulative precipitation of the whole rain gauges and the calibrated radar precipitation at the same pixels were 38.5 and 36.9 mm,

* Saber Moazami [email protected]; [email protected]

1

Department of Civil Engineering, K.N. Toosi University of Technology, Tehran, Iran

2

Department of Civil Engineering, Environmental Sciences Research Center, Islamshahr Branch, Islamic Azad University, Islamshahr, Tehran, Iran

3

Advanced Radar Research Center, University of Oklahoma, Norman, OK, USA

4

School of Civil Engineering and Environmental Sciences, University of Oklahoma, Norman, OK, USA

respectively. Therefore, the obtained results prove the inefficiency of TMM and the capability of RPMM in the calibration process of the Amir-Abad weather radar. Besides, in determining the uncertainty associated with the calculated values of A and B in the Ze–R relation, a sensitivity analysis method was employed during the estimation of cumulative light precipitation for the period from 2014 to 2015. The results expressed that in the worst conditions, 69% of radar data are converted to R values by a maximum error less than 30%.

1 Introduction The uncertainty in rainfall is the most important factor influencing stream flow response (Kuczera and Williams 1992), soil erosion (Yang and Yu 2015), and other hydrological processes. Over the developing countries, gauge networks are mostly too sparse to measure rainfall, especially in heavy convective storms (Morin et al. 1995; Sun et al. 2000; Tantanee and Prakarnrat 2008). Besides, the sparse rain gauge network fails to catch the intense center of the rainfall which leads to large errors in estimating floods (Morin et al. 1995; Sun et al. 2000; Xie and Arkin 1995). To overcome these limitations, satellites and weather radars, as a complement of ground stations, have been employed over different parts of the world (Wlison and Brandes 1979). However, for Iran, the studies of weather radar and satellite precipitation data are new and the most important ones have been carried out after 2010. Moazami et al. (2013a, b) applied the Copula method to evaluate uncertainty of bias from estimated satellite rainfall in the southwestern part of Iran. Moazami et al. (2013a, b), Moazami et al. (2016), and Javanmard et al. (2010) evaluated the estimated precipitation of TRMM 3B42 over Iran. Golian et al. (2015) carried out imperative researches by merging widely used satellite rainfall estimates (SREs)

H. Ayat et al.

namely PERSIANN, TMPA-3B42, TMPA-3B42RT, and CMORPH over the complex terrain of Iran. Ghaemi et al. (2017) determined the uncertainty of radar rainfall estimates over two different climates in Iran by employing the Copula method. Due to the low spatial resolution of the satellite, many problems may have appeared by applying them in small basins; however, a weather radar has enormous potential to measure rainfall in real time with high spatial and temporal resolutions (Atencia et al. 2002; Borga 2002; Hong and Gourley 2017; Linsley et al. 1988; Morin et al. 1995; Yang et al. 2004). In addition, because most of the satellite-based precipitation retrieval algorithms are still based on using IR imagery (Yilmaz et al. 2005), they are prone to errors greater than radar-based ones (Anagnostou et al. 2000; Tesfagiorgis et al. 2011). It should be mentioned that weather radars also create errors due to attenuation, electromagnetic interference, ducting, bright band, overshooting, anomalous propagation, etc. (Reinhart 1997). For this reason, Z–R relationship should be calibrated in each region and for each type of precipitation. Calheiros and Zawadzki (1987) and Rosenfeld et al. (1990) utilized the regression analysis technique known as the traditional matching method (TMM) to establish a relationship between datasets measured by the ground station and the recorded reflectivity factors at the corresponding time and pixels to rain gauges (Piman et al. 2007). Due to the difficulties of determining spatial and temporal synchronizations of data in this method, pairing the precipitation intensities occurring in the station and the recorded reflectivity values in the same pixels is often impossible in many cases. To overcome the limitations of TMM, Calheiros and Zawadzki (1987) introduced a method known as the probability matching method (PMM) in which Ze and R are matched with the same cumulative distribution function (CDF) values. Pairing Ze and R using this method removes a large portion of systematic errors and eliminates the non-linear behaviors in the log Ze–log R plots (Calheiros and Zawadzki 1987). This method does not optimize the measurement of individual rain rates; however, it aims to provide the estimates of rainfall accumulations (Calheiros and Zawadzki 1987). They also assumed that unconditional probabilities for any precipitation intensities are constant everywhere in the radar field (Rosenfeld et al. 1993). However, this assumption leads to unrealistic precipitation estimations in cases where ground stations are proportionately spread across the region during the conversion process of Ze. To solve this problem, Rosenfeld et al. (1993) proposed another method termed the window probability matching method (WPMM), the integral assumption of which was the elimination of the PDF similarity across the region. Matching the process of data takes place only within the small windows where the station is located in its center in terms of temporal and spatial resolutions (Rosenfeld et al. 1993). One of the limitations of this method is the necessity of ground data with sub-hourly temporal resolution (for instance, 5 min);

therefore, employing this method in regions like Iran where the best temporal resolution of ground stations is 6 h and above is almost impossible. In addition, the accuracy of this method is particularly for reflectivity factors higher than 30 dBZ, and hence, in light precipitation, estimated rain rates are associated with large errors. Piman et al. (2007) proposed the window correlation matching method (WCMM) for calibrating the weather radars over a mountainous area in the north of Thailand. The pairing of Ze and R values in the WCMM method is implemented by matching the Ze values within the space and time windows corresponding to reference gauge rainfall intensity and finding the values of Ze of the radar pixel that give the maximum correlation coefficient (Piman et al. 2007). In that research, a comparison was made between the proposed method and the previous ones as well as the results expressed to be more acceptable in comparison with both PMM and TMM; however, it was slightly better than the WPMM method. Taking into account the abovementioned studies, the aims of this research are to propose a new method entitled the region probability matching method (RPMM) to overcome the limitations of the aforementioned methods and to estimate the cumulative precipitation with a reasonable precision occurring across a region in Iran during a specific period.

2 Study area This study has been conducted on an area in the north of Iran. Due to the proximity of this region to the Caspian Sea and Alborz mountains, it has a moderate temperature over the most area. Hot and humid summers and mild and humid winters are the dominant characteristics of this climate which is similar to the Mediterranean climate. Modarres and Sarhadi (2010) determined Iran’s regional precipitation patterns in which Iran was divided into eight types of areas using cluster analysis with L moments. According to Fig. 1, Golestan and Mazandaran provinces located in the G6 group have an average annual precipitation about 739 mm (Modarres and Sarhadi 2010). These regions are under the coverage of the C-band radar of the Amir-Abad weather radar which is located on a 10-story tower with the height of 32 m at −23 m above sea level with geographic coordinates of 53° 23′ N, 36° 51′ E in the vicinity of the eastern Caspian coast. Mazandaran province is located in the south of the Caspian Sea from 35° 47′N to 36° 35′ N latitude and 50° 34′E to 54° 10′E longitude with an area of 23,842 km2, and Golestan province in the southeast of the Caspian Sea is spread from 36° 30′N to 38° 8′N and 53° 57′E to 56° 22′E with an area of 20,367 km2.

Calibration of weather radar using region probability matching method (RPMM)

Fig. 1 a Iran is divided into eight different climates in which the study area falls in the G6 category (Andrews 1992). b Mazandaran and Golestan provinces (red boundaries), radar position (red circle), and synoptic stations (blue circles)

3 Precipitation data 3.1 Ground data For calibrating the radar, 23 gauges were employed from the network of synoptic stations with a temporal resolution of 6 h. Datasets of these stations (depicted with blue color in Fig. 1) were obtained from the Islamic Republic of Iran Meteorological Organization (IRIMO). It should be noted that the selected stations located in the zone of the radar cover up to 250 km from the radar central site. The geographic coordinates and elevation of each station are displayed in Table 1. In Table 2, the 29 dates chosen for calibrating the radar are presented. In all these days, all of the 18 stations in italics in Table 1 had recorded precipitations simultaneously which are the most important available events for calibration. 3.2 Radar data The radar data used for calibration are raw data with (*.Vol) format. These files include the information of the radar volume scan. The spatial resolution of the information recorded by the radar at any elevation angle is 1.5° at 1 km, and the temporal resolution of a volume full scan is 15 min.

4 Data processing In order to use the 6-h ground data in the Ze–R relationship, the unit of R should be converted to hr/mm; therefore, the average

Table 1 Synoptic station details in Mazandaran and Golestan provinces within the zone of Amir-Abad radar coverage Station code Name

Y

X

Z

40759 99357 99299 40737

Sari Baladeh Galugah Gharakhil

52.98667 51.8 53.83694 52.77167

36.53667 36.2 36.73778 36.45417

11 2109 −5 42

40760 99361 99309 99348 99360 40735 40736 40734 40732 88116 99300

Kiyasar Alasht Amol Kojur Polsefid Siahbisheh Babolsar Nowshahr Ramsar Sari (Dasht-e-naz Airport) Aliabad-e-katul

53.54639 52.85 52.38333 51.73333 53.08333 51.30306 52.65306 51.46694 50.68333 53.19528 54.88333

36.24889 36.08333 36.46667 36.38333 36.13333 36.23111 36.72 36.66139 36.90444 36.63694 36.9

1265 2095 95 1477 1142 1868 − 29 − 21 − 20 9 247

99242 99241 99271 99237 99304 88113 40738 99240

Bandar-e-Torkaman Gorgan (Hashemabad) Inchehborun Minudasht Bandar-e-Gaz Kalaleh (airport) Gorgan Gonbad-e-kavus

54.06667 54.26667 54.72049 55.37248 54.06667 55.45944 54.41361 55.21278

36.9 36.85 37.45516 37.22857 36.9 37.385 36.905 37.26722

− 24 −6 10 154 − 24 127 − 10 43

H. Ayat et al. Table 2 Selected events for the calibration of the Amir-Abad radar during the two years 2014–2015 (In these events, all 18 stations in italics depicted in Table 1 have recorded precipitations simultaneously.) Dates of Events 3/22/2014

11/3/2014

2/26/2015

2/1/2014

1/31/2014

3/6/2015

2/2/2014 3/13/2014

11/5/2014 11/19/2014

3/9/2015 3/17/2015

3/31/2014 4/3/2014

11/23/2014 11/29/2014

4/12/2015 7/19/2015

6/4/2014

1/8/2015

7/20/2015

10/11/2014

1/23/2015

9/1/2015

10/20/2014

2/12/2015

9/2/2015

10/26/2014

2/21/2015

values of R for an hour are calculated by dividing the cumulative precipitation value of 6 h by 6. In addition, the values of Ze must be averaged in a 6-h period. For this reason, the Ze values were converted into precipitation rate (R) by employing the well-known coefficients of Marshall and Palmer (A = 200, B = 1.6) and then the resulting R values in 6-h intervals were averaged to obtain the mean values of R for the 6-h interval. Finally, the equivalent values of Ze for 6-h intervals were calculated through the inverse conversion of R to Ze using the aforementioned coefficients. In order to work with raw data format, the authors provided an application in the Python programming language by employing the wradlib library. This free library is programmed in such a way that it is useful for most stages of data processing in hydro-meteorological and hydrological applications (Heistermann et al. 2013). Fig. 2 a Frequency distribution of recorded reflectivity factors during 2014–2015 at the stations’ pixels (values smaller than 0 dBZ were filtered). b Frequency distribution of ground-recorded precipitation rates which occurred during 2014–2015 (values smaller than 0.1 mm/h were filtered)

In this research, the algorithm proposed by Gabella and NotarPietro (2002) and provided by the wradlib library was used for removing the clutters from the radar data. In addition, in this library, it is possible to correct the attenuation problem before the clutter removing stage, the process of which is carried out by two different criteria mentioned in Pfaff (2013). Figure 2 indicates the frequency distribution of the reflectivity factor (dBZ) and precipitation rate (mm/h) during 2014– 2015 for the dates mentioned in Table 2. According to Fig. 2a, b, the reflectivity values in the selected events are almost less than 30 dBZ; therefore, based on Table 3, which provides classification of digital video integrator and processor (DVIP) levels, it could be found that most of the selected events have almost been at the light precipitation level. It should be noted that the reason of this selection refers to the shortage of the Amir-Abad radar data in severe precipitations due to technical difficulties. Thus, continuous and useful data for calibration of the radar were available only for light precipitation events. For calibration, the ground-recorded precipitation values less than 0.1 mm/h and reflectivities less than 0 dBZ have been removed to prevent input errors in radar measurements and ground stations due to the non-important precipitation. To project stations on the radar image, characteristics like geographic coordinates of the radar building and ground stations and their altitudes, distances, and position angles are transferred to spherical coordinates to allow the highest possible use of the data resolution (Fig. 3). During the conversion of these characteristics, the curvature of the earth is also considered (Fig. 4). After projecting the stations in the spherical coordinate, according to Fig. 4, a point, which is located in an altitude of 1 km higher than the location of the related station, is considered as a point between eight nodes in the space of the radar’s scan. In order to determine the reflectivity at the

Calibration of weather radar using region probability matching method (RPMM)

Fig. 3 Location of stations depicted by red circles in the PPI image by employing wradlib at the elevation angle of 0

location of the stations, the recorded positive values in all the eight nodes are obtained during 6 h and are considered to calculate the equivalent Ze for this interval.

5 Methodology Calheiros and Zawadzki (1987) suggested that the probability density functions of both the Ze and R parameters are better paired with each other when the relation of t = 1.3A0.5v−1 is established. Here, t is the temporal resolution of the ground station (6 h in this research), A is the necessary spatial resolution of the radar coverage, and v is the horizontal component of the precipitation velocity that is assumed to be 25 km/h. Therefore, based on this equation, A = 13313 km2. Thus, according to the low temporal resolution of the ground stations, the window around each station could even be as large as the entire region. Therefore, to match Ze and R instead of plotting the CDF graphs only for a station and the surrounding pixels, these CDF graphs are plotted across the whole area for all stations and their related pixels in order to pair the averaged 6-h R values and the equivalent 6-h Ze values recorded during a precipitation event (Fig. 5). Hence, coefficients A and B calculated in this method can calibrate the Ze–R relationship for reflectivity factors, which are equivalent to 6 h. For a deeper understanding of the process of calibration using the proposed method, the following radar calibration for an event which occurred on March 22, 2014, is presented. Figure 7a indicates the process of calculation of A and B coefficients for 42 pairs through the TMM method on March 22, 2014. The intercept calculated from the graph indicates log (A), and the slope of the fitted line is equivalent to B.

According to the calculated correlation coefficient (R2 = 0.13), it could be found that by using this line, no meaningful relationship could be established between Ze and R, and this fitted line is not the representative of 42 pairs in this event. This indicates the weakness of the TMM approach in calibrating the Amir-Abad radar. Perhaps one of the most important reasons of this weakness is due to the inefficiency of this method for Ze ranges less than 30 dBZ. Figure 6 indicates the CDF for the 6-h averaged R values and the equivalent 6-h reflectivity factors for the aforementioned event. Both red and blue lines were plotted in a logarithmic format using 42 pairs. Some of them, especially on the red line, have overlapped because of their similarities. Therefore, in order to match Ze and R, the CDF of log (Ze) was the basis and the equivalent values of log (Rg) were interpolated from the blue graph. Figure 7b illustrates the fitted regression line on the paired data obtained from Fig. 6 with a correlation coefficient of 0.95. According to the results, the RPMM method improved the correlation coefficient of 0.13 obtained from TMM to 0.95, which indicates the efficiency of this method in establishing a meaningful relationship between the radar data and the precipitation intensities recorded by the synoptic stations. However, in order to employ this method, it should be considered that the calibrated coefficients are related to converting the reflectivity factors which have the same CDF values of ground-recorded precipitation intensities. Therefore, using these coefficients in the Ze–R relationship, one can convert the Ze recorded in a specified time and place into R in another time and place. However, a proper estimation of precipitation cannot be provided at a specified time and place. By converting all the existing reflectivity factors (Ze) in a set of paired data, all computed R values within the set are estimated more accurately but at the wrong place and time. Therefore, accumulation of the estimated R values in the current set of paired data is close to the accumulation of ground-recorded R values in the set. The calibrated coefficients are only useful for pixels in which the ground stations are located and for the times that ground station data are available. If these calculated coefficients can be used for the rest of the radar image pixels on which there are no ground stations or at times that there are no equivalent ground data, a suitable estimation of cumulative Table 3 VIP levels provided by the National Weather Service (http:// w1.weather.gov/glossary/index.php?word=VIP) Reflectivity (dBZ)

Comments

18–30 30–38 38–44 44–50 50–57 > 57

Light precipitation Light to moderate rain Moderate to heavy rain Heavy rain Very heavy rain; hail possible Very heavy rain and hail; large hail possible

H. Ayat et al. Fig. 4 Schematic illustration of projecting stations by considering the curvature of the earth (the sample station is depicted by a black circle, and the selected point for extracting Z values is shown by a blue circle located 1 km higher than the station’s height and between the green grids of the slices n and n + 1)

precipitation over the entire region during the event will be achievable. Several studies have declared that distribution of precipitation intensity is fitted to the log-normal type (Kassim and Kottegoda 1991; Kedem et al. 1990; Krajewski and Smith 1991). Likewise, distribution of the reflectivity factor of Ze is similar to that of the precipitation intensity. According to this assumption (Xin et al. 1997; Xudong et al. 1999), ! logðRÞ−μlogðRÞ pffiffiffi F ðlogðRÞÞ ¼ F ðlogðZ ÞÞ→erf σlogðRÞ 2 ! logðZ Þ−μlogðZ Þ pffiffiffi ¼ erf σlogðZ Þ 2 2 x 2 ∂erf ðxÞ erf ðxÞ ¼ pffiffiffi ∫0 e−t dt→ π ∂x 2 2 ðAndrews 1992Þ ¼ pffiffiffi  e−x π  2  2 logðRÞ−μlogðRÞ logðZ Þ−μlogðZ Þ pffi pffi logðZ Þ−μlogðZ Þ − − σlogðRÞ 2 σlogðZ Þ 2 ¼e → e σlogðZ Þ ¼

logðRÞ−μlogðRÞ σlogðRÞ

Fig. 5 Matching process of the Ze and R values (Piman et al. 2007)

logðZ Þ ¼ B¼

  σlogðZ Þ σlogðZ Þ  logðRÞ þ μlogðZ Þ −  μlogðRÞ σlogðRÞ σlogðRÞ

σlogðZ Þ σlogðRÞ

A ¼ μlogðZ Þ −

ð1Þ σlogðZ Þ  μlogðRÞ σlogðRÞ

ð2Þ

Referring to Eqs. 1 and 2, it could be found that the calibration coefficients calculated using the RPMM method are a function of the standard deviation and the mean value of the logarithm of Ze–R pairs. Thus, A and B calculated for a set of data pairs of Ze–R can be used by another set only if the standard deviation and mean of log (Ze) and log(Rg) of both sets are similar. In this study, a radar image consists of a large number of pixels where a limited number of them contain synoptic stations. If both the mean and standard deviation of log (Ze) and log (R) for these stations and for all pixels of the radar image in an event are the same, A and B calculated from the pixels of the included stations for that event can be used for other pixels without any stations. In other words, these stations can be a suitable representative of all the pixels of a radar image. Therefore, the cumulative precipitation which occurred during an event may be estimated accurately for all pixels in the radar image and, consequently, over the entire region.

Calibration of weather radar using region probability matching method (RPMM)

Fig. 7 a Calculating coefficients A and B using TMM. The value of R2 demonstrates the inefficiency of the TMM method in the calibration of weather radar. b The fitted regression line with the correlation coefficient

of 0.95 on the new paired data. The red line can be a suitable representative for paired data

A simple test is employed in this study to assess the sufficiency number of the data pairs (Ze–R) collected during an event to be as the representative of all the pixels in the study area. In this test, at first, five Ze–R pairs out of all pairs are randomly selected, and the coefficients of A and B are calculated using RPMM. Then, another pair is randomly added to the five previous selected members and again A and B are calculated with the six new pairs. The process will be continued until all pairs are taken into account. The fluctuations of the computed A and B values are plotted in terms of the increasing number of members. If the graph remains almost constant after selection of n pairs, it shows that by increasing the population of the selected members, the values of A and B are almost constant. Thus, A and B calculated with n randomly selected members are approximately equivalent to A and B calculated with all pairs and, finally, are equivalent to A and B computed by all pixels of the radar image during that event. As represented, on March 22, 2014, 42 pairs were selected for calibration and A and B were plotted in terms of increasing the number of pairs. According to Fig. 8a, b, after choosing 30 pairs, the fluctuation of A is restricted in the range from 32 to 37 and the variation of B is limited in the range from 1.95 to 2.1. It means that A and B will have a value between these ranges over the entire region.

6 Results and discussion

Fig. 6 Cumulative distribution functions for recorded data by the synoptic stations and weather radar

In this study, the Amir-Abad radar was calibrated for 29 events shown in Table 2, using the TMM and RPMM methods. The results are depicted in Table 4. Two blue columns in this table show the correlation coefficient of the fitted regression line in the two methods. It is clear that the TMM method provides inappropriate correlation coefficients due to the inability of the method in calibrating the Amir-Abad radar; however, the correlation coefficients presented by RPMM are close to 1 in all events. As it was mentioned in the previous sections, an appropriate estimate of precipitation at a specified location and in a given time cannot be achieved by the RPMM method. On the other hand, the cumulative precipitations of each event are accurately predicted by the calculated coefficients over the entire region. The last two red columns are the accumulated rainfalls which occurred at the location of all stations during each event. It is noted that the calculated cumulative precipitations belong to the period of a day and only for the pixels in which the ground stations are located. The test presented in the previous section is used to determine the uncertainty of calculated coefficients A and B for all pixels of the radar image in each event. The selected event for this purpose took place on March 22, 2014, where 42 pairs of available data are selected for radar calibration. Gradual random selection from 5 to 42 members out of 42-member sets was performed in 10 different ways for finding the uncertainty of A and B coefficients by increasing the number of samples. Figure 9a, b demonstrates these variations. To extract information from this figure, it should be considered that reliable results cannot be derived from the last parts of the graphs, since the members of the 10 randomly selected categories were very similar and it is clear to have close answers in this part of the graph. Therefore, in order to define a reliable uncertainty range for the A and B coefficients, it should be more focused on the initial parts of both charts. In Fig. 9a, fluctuations are dramatically reduced after selection of 16 pairs, and the variation range of B values is restricted

H. Ayat et al.

Fig. 8 a Fluctuations of calculated A by increasing the number of paired data. b Fluctuations of calculated B by increasing the number of paired data

between 1.8 and 3. With respect to Fig. 9b, fluctuations of A values after selection of 16 pairs are limited between 25 and 55. By considering the uncertainty range for the A and B values, the uncertainty range for R values could be Table 4

determined. Table 5 illustrates the upper (R1) and lower (R0) ranges of estimated R at each value of Ze. The closer maximum and minimum values of R means that by selecting 16 pairs, the uncertainty ranges of R values are insignificant over

Results of RPMM calibration method at the Amir-Abad weather radar

Events

No. of pairs

A calc

B calc

R2 RPMM

R2 TMM

Σ Rg

Σ Rr

3/22/2014

42

35.95

1.95

0.95

0.13

38.44

36.89

2/1/2014 2/2/2014

17 12

20.64 28.18

1.25 1.04

0.97 0.89

0.00 0.33

9.73 5.32

8.83 4.65

3/13/2014 3/31/2014

25 43

13.94 58.49

1.35 1.48

0.90 0.98

0.00 0.08

14.10 45.27

14.67 44.54

4/3/2014 6/4/2014

10 33

63.94 26.10

1.28 0.86

0.97 0.92

0.55 0.06

2.23 99.10

1.94 100.52

10/11/2014 10/20/2014 10/26/2014

17 25 10

14.35 6.59 9.06

1.06 0.86 1.15

0.95 0.94 0.90

0.02 0.00 0.11

10.11 80.83 8.61

9.54 102.59 8.86

11/3/2014 1/31/2014 11/5/2014 11/19/2014 11/23/2014 11/29/2014 1/8/2015 1/23/2015 2/12/2015 2/21/2015 2/26/2015

18 29 17 15 17 12 16 33 18 18 16

67.30 41.40 63.13 27.59 19.65 23.86 44.81 50.11 33.87 34.67 43.73

1.42 1.12 1.88 1.44 1.48 1.10 2.55 1.78 1.35 2.03 1.84

0.95 0.84 0.85 0.96 0.81 0.94 0.93 0.93 0.97 0.93 0.95

0.07 0.02 0.10 0.06 0.03 0.19 0.29 0.03 0.14 0.03 0.02

14.48 24.64 9.67 12.02 12.14 9.99 21.83 20.36 6.17 8.44 7.29

12.92 28.00 9.70 11.79 14.48 9.24 21.11 20.00 6.05 8.02 6.90

3/6/2015 3/9/2015 3/17/2015 4/12/2015 7/19/2015 7/20/2015 9/1/2015 9/2/2015

20 20 11 17 32 12 14 16

1.84 18.66 36.09 11.74 50.95 5.38 28.37 37.32

2.86 1.07 1.50 0.61 1.02 1.58 1.33 1.56

0.94 0.79 0.92 0.97 0.88 0.94 0.71 0.91

0.09 0.03 0.47 0.13 0.24 0.00 0.09 0.06

48.11 8.08 5.27 12.89 30.28 55.83 35.17 17.53

47.42 8.67 5.34 12.89 26.99 55.49 31.17 17.92

Calibration of weather radar using region probability matching method (RPMM)

Fig. 9 a Fluctuations of coefficient B by increasing the number of paired data in 10 different random selections. b Fluctuations of coefficient A by increasing the number of paired data in 10 different random selections

conditions, about 54% of the estimated R values by selecting 16 pairs have a maximum error of 30%. In addition, an attempt was made to estimate the cumulative light precipitation which occurred over the entire region from 2014 to 2015. According to Fig. 10, it is clear that the number of paired data is 667, and it can be concluded that by selecting 230 or more pairs, the A and B variations will be reduced greatly, and A fluctuations are restricted in the range from 22 to 32, and B is limited in the range of 1.25 to 1.55. Based on Fig. 11, in the worst conditions, about 69% of the estimates have a maximum error of 30%. Although the uncertainty ranges of the A and B coefficients in this period are much more limited than the daily estimation; however, the uncertainty of the estimated R values is not as limited as expected because the estimated R

the region. It should be noted that in order to determine the uncertainty ranges of A and B by selecting 42 pairs, it is necessary to have more paired data. To determine the maximum errors of estimated R values, a parameter is introduced in Table 5 with the following equation: . Bias ¼ jR0 −R1 j minðR0 ; R1 Þ ð3Þ

This parameter represents the maximum errors of estimated R for each value of Ze. In Table 5, Ze values located within the bold zone have the maximum bias less than 30%. The cumulative distribution of Ze on March 22, 2014, is shown in Fig. 11. According to Table 5, in the worst

Table 5 Maximum bias calculated by boundary values of A and B (R0 and R1 are the boundary calculated precipitation rates from the mentioned coefficients, and the maximum bias is (|R0 − R1|)/ min(R0, R1) ) 3/22/2014 (25 < A < 55 and 1.8 < B < 3)

2014 and 2015 (22 < A < 32 and 1.25 < B < 1.55)

dBZ

R0 (mm/h)

R1 (mm/h)

Bias (%)

dBZ

R0 (mm/h)

R1 (mm/h)

Bias (%)

1 2 3.7 5 8

0.28 0.31 0.35 0.39 0.49

0.19 0.22 0.27 0.32 0.47

0.49 0.42 0.30 0.22 0.04

1 3 5 7 9

0.12 0.16 0.22 0.30 0.40

0.10 0.14 0.21 0.30 0.44

0.23 0.14 0.06 0.01 0.09

11 14 16 18 20 21 23 25

0.61 0.77 0.90 1.05 1.22 1.32 1.54 1.79

0.68 1.00 1.29 1.67 2.16 2.45 3.17 4.10

0.12 0.30 0.44 0.60 0.77 0.86 1.06 1.29

11 12 14 17 18 21 23 25

0.55 0.63 0.85 1.34 1.55 2.43 3.28 4.43

0.64 0.77 1.11 1.94 2.34 4.08 5.92 8.58

0.17 0.21 0.30 0.45 0.51 0.68 0.80 0.94

H. Ayat et al.

Fig. 10 a Fluctuations of coefficient A by increasing the number of paired data in 10 different random selections during 2014–2015. b Fluctuations of coefficient B by increasing the number of paired data in 10 different random selections during 2014–2015

values strongly depend not only on the calculated A and B values but also on the magnitude of the Ze values which should be converted to R values.

7 Summary and conclusion In this research, a new method entitled RPMM was developed for calibration of weather radar. A comparison between the results of RPMM and TMM was carried out in Mazandaran and Golestan provinces located in the north of Iran with a temperate and wet climatic condition. To achieve this, 18 synoptic stations with a temporal resolution of 6 h are used for 29 events. Based on Table 4, the correlation coefficient of constructing a meaningful relationship between Ze and R values resulted from the RPMM method is much better than TMM. For instance, the correlation coefficients of TMM and RPMM

Fig. 11 Percentage of estimated precipitation rates with the maximum bias smaller than 30%

method were obtained at 0.13 and 0.95, respectively, on March 22, 2014. However, the A and B parameters calculated by the RPMM method cannot be used to estimate precipitation at a determined location in a given time and can only be used to estimate the cumulative precipitation over the entire region for the whole period. This period may vary from hourly to yearly based on the temporal resolution of ground station data. To assess the applicability of calculated A and B over the entire region, variation of these parameters in terms of increasing the number of pairs was plotted for the event which occurred on March 22, 2014. After choosing 30 pairs, the fluctuation of A was restricted in the range from 32 to 37 and the variation of B was limited in the range from 1.95 to 2.1. It means that the 42 paired data for this event are suitable representatives of all the pixels involved in this event. In order to determine the uncertainty of A and B using the method proposed here, an analysis was carried out by random

Calibration of weather radar using region probability matching method (RPMM)

selection of paired data in 10 different ways. The results showed that for the event on March 22, 2014, by selecting 16 pairs of R and Z, the uncertainty of A and B is limited in the range from 25 to 55 and from 1.7 to 3, respectively. Using the maximum and minimum values of A and B uncertainty ranges, the uncertainty range of R was estimated. The smaller range of estimated R indicates the better applicability of A and B coefficients. According to Fig. 11, in estimating the daily cumulative precipitation on March 22, 2014, by considering the uncertainty ranges of A and B, in the worst condition, 54% of the estimated R values expressed a maximum error of less than 30%. Consequently, to estimate the cumulative light precipitation which occurred over the entire region from 2014 to 2015, a sensitivity analysis was implemented for 667 pairs and the results showed that by selecting 230 pairs, in the worst condition, 69% of the calculated R values had a maximum error of 30%. The aim of developing the method proposed in this study was eliminating the limitations in the previous methods such as PMM, WPMM, and WCMM. Those methods were inefficient in light precipitation (less than 30 dBZ) and in using ground data with low temporal resolution (6 h or daily), especially in developing countries like Iran where it suffers from a dense ground network with high temporal resolution. Obviously, the precise estimated cumulative precipitation occurring over the entire of a region is a valuable input for hydrological forecasts. In this study, because of the lack of recorded radar data from severe precipitation, it was not possible to use the proposed method in hydrological forecasts. However, by accessing the heavy rainfall data with high temporal resolution, it may be able to calibrate weather radars in sub-hourly intervals with a high precision, as well as estimate extreme events more accurately.

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