The advancement of technologies in the fields of satellite communications, multimedia applications, the Internet, and mobile communications has increased the demand for a high data transmission rate. Consequently, in the field of data transmission technology, the emphasis has shifted to high-frequency microwave bands. However, the problem of rain is significant for operation frequencies of more than 10 GHz. A reduction in the transmitted signal amplitude because of an atmospheric mechanism such as the absorption and scattering of radio waves results in rain attenuation [1]. Further, a communication system designer faces a problem in the prediction of the effects of rain on radar and the remote sensing of a location on or above the Earth’s surface. Although numerous rain attenuation models exist, their application to the prediction of experimental local data remains challenging [2]. Efforts to localize rain attenuation models however depend on the knowledge of the rainfall rate and the experimental data for the chosen location. The data further enhance the development of the local rain attenuation model. In this paper, the experimental result of rainfall rate statistics of certain selected locations of Korea is described. According to climatologists, recently, major cities in Korea have experienced unusually heavy rainfall, which might be attributed to urban heat island phenomena [3]. A comprehensive effort for the characterization of the 1-min rainfall rate has been made by International Telecommunication Union Radiocommunication Study Group 3 (ITU-R, formerly the CCIR). Moreover, to convert a relatively high integration time’s rainfall data to the equivalent 1-min distribution, various procedures have been enforced considering the relevant physical, analytical, and empirical models [4]. Because of the experimental dependence and for simplicity, in this study, empirical models are selected as an appropriate method for determining the distribution of the rainfall rate at the 1-min integration time. Further, in this paper, we emphasize on the difference between the measured rainfall rate statistics and the ITU-R P.837-6 [5] model along with the applicable empirical nature methods.

The rest of the paper is organized as follows: Section II presents a brief overview of the predicted point rainfall rate. The experimental setup is described in Section III. Section IV highlights the performed regression analyses for the applicable empirical methods. Similarly, Section V details the performance evaluation of the proposed technique. Finally, Section VI draws some useful conclusions.

Several studies have been conducted to measure the effect of this natural phenomenon on the radio propagation path above 10 GHz, using the various rain attenuation statistical models either globally or locally. Further, a considerable amount of research work has been carried out in this area over the years locally [3, 6-10]. Interestingly, the experimental duration of the rainfall rate data is relatively low, and more research needs to be carried out to ensure a reliable signal transmission. Furthermore, the rainfall rate research carried out in the neighboring country Japan [11] relies on the thunderstorm ratio and the regional climatic parameters of only one location of Korea as Daejeon. Because of the variations in the performed experiment, this model has not been analyzed, but the exponential nature of rain rate variation has been studied. Rainfall rate models are classified into global and localized rainfall rate models [12]. Global rainfall rate models depend on climatic parameters and geographical locations. In this study, we consider the proposed global coefficient values as listed in [13], which extend global coefficients value’s application to rain rate conversion methods in temperate, tropical, and cold climates, along with the ITU-R P.837-6 [5] method. ITU-R P.837-6 contains the software that implements the conversion of rain rate statistics with different integration times, adopted by Study Group 3 in Recommendation P.837-6, Annex 3. Of the two operational modes as recommended in the software, Mode A is chosen for inputting data for various times and percentage rain rate values. Similarly, the latitude and longitude information of the sites along with various source integration times, such as 5 min, 10 min, 20 min, 30 min, and 60 min, are provided as additional input parameters. The EXCELL Rainfall Statistics Conversion (EXCELL RSC) method is used for the conversion of the rainfall rate statistics from a long integration time to a 1-min integration time. This method is based on the simulated movement of rain cells over a virtual rain gauge, during the given integration time T, whose translation velocity depends on both the type of precipitation and the observation period. The rainfall conversion is obtained using a virtual rain gauge according to the local mean yearly wind velocity, which is extracted from the ERA-40 database. The prediction approach is based on the annual rainfall amount of the convective type and the stratiform type along with the probability of 6-hr rainy periods. Depending on the type of precipitation, rain structures appear to move mostly because of the wind speed (convective rain) or because of the time evolution (stratiform rain) [14]. The theoretical concept of this model is explained using a physical model-based methodology in [15] and in an update of Recommendation ITU-R P.837-5, Annex 3 [5]. EXCELL RSC is globally applicable and has strong physical soundness for measurements with an arbitrary integration time, T. The effectiveness of the EXCELL RSC model was tested using various error analyses against other global models in [16], where the EXCELL RSC model was found to be suitable for the estimation of the 1-min rainfall rate.

Several researchers around the globe have proposed regional rainfall rate models. These rainfall rate models have been developed from empirical equations using the results of field measurements collected over a long period of time [12]. Similarly, the mathematical theory based on the principles for de-integrating a T-min experimental probability distribution (pd) into the corresponding 1-min pd is presented in [17]. However, there is a need for more efficient propagation planning, based on the use of the number and the duration of rain events along with the fraction of the rain time. The experimental system used by Korea Meteorological Administration (KMA) provides the experimental record only for the 1-min rainfall amount as discussed later in this paper. Despite the numerous models proposed thus far, empirical techniques have been selected due to its simplicity as they are all based on conversion coefficients determined by means of measured data and their simple formulation, which allows one to extend their applicability to more than one climatic regions. Empirical conversion techniques provide analytical laws expressing the relationship between equiprobable rain rate values with the available 1-min and T-min integrated distributions [4]. In this study, we test the applicability of the following empirical conversion methods: the Segal method [18], which provides a systematic approach for obtaining a specified number of rain zones in countries such as Canada that have sufficiently large databases of short integration time data; the method proposed by Burgueno et al. [19] with data from Barcelona, which emphasizes the power law relationship that exists between equiprobable rain rates of two integration times; the Chebil and Rahman method [20] with data of Malaysia, which uses an empirical formula that combines the Segal method and the power law; the method proposed by Lee et al. [10] using the Electronics and Telecommunications Research Institute (ETRI) rainfall rate data measured through an optical rain gauge (ORG), which is a conversion model with zero interception for the logarithmic scale that shows an efficient measure for the estimated 1-min rainfall rate data; and the conversion method that employed an exponential function for equal rain rate in [11, 21, 22], which determined the probability distribution function of the M distribution. In this research, an exponential function is studied for equal time percentage ranges. Further, polynomial fit is considered the most preferred method to characterize the 1-min rainfall rate [23], which emphasizes that the conversion technique is dependable and consistent to be used for the Malaysian tropical climate. Furthermore, model application against the measured statistical data at 0.01% time exceedance results in good agreement. In addition, polynomial fit gives a better statistical result for the conversion of the rain rate from a 5-min to a 1-min integration time equivalent for South Africa and the surrounding islands. The suitable coefficients are proposed for this method in [24, 25]. The empirical methods selected in this study have been proposed at different moments in time as recommended by ITU-R P.311-15 [26] for rainfall statistics conversion.

In order to measure the 1-min rainfall amount, several experimental systems have been developed by KMA since 2004. The system for collecting and storing rainfall data at 1-min intervals was installed in 93 different locations, out of which nine major sites that are characterized as big metropolis are considered. The latitude and longitude information of these sites is listed in Table 1.

These major cities lie in a temperate climatic region whose experimental 1-min rainfall rate data for a decade are analyzed. Korea has a temperate climate with four distinct seasons. Winters are usually long, cold, and dry. Summers are very short, hot, and humid. Spring and autumn are pleasant but also short in duration. The country has sufficient rainfall; rarely does it have less than 75 cm of rainfall in any given year or more than 100 cm [27]. The maximum rainfall is noted from May to September of each year. The measurement setup includes a tipping bucket rain gauge, as shown in Fig. 1.

KMA uses a conducting vessel size of 0.5 mm to improve the shortcomings of the gutter. Once the collected water is more than 0.5 mm, it eventually fills the bucket. The bucket is mounted on a particular axis of rotation to shift the center like a seesaw. This bucket is in contact with the Reed switch on the rotation axis, which is operated by an electrical pulse generated by the tipping phenomenon. Finally, the signal generated through the Reed switch is recorded on a recording device that records the post-processing of the time stamp of each tip, which provides the measurement of the 1-min rainfall amount. The heater is installed inside the sewer for measurement under snowfall. Further, Table 2 presents the specifications of the rain gauge.

The tipping bucket has an unstably balanced twin bucket with a sensitivity of 0.1 mm per tip, which triggers an electronic impulse that is stored in the data logger, which scans the data at an interval of 1 min. The availability of the gauge is about 99.2%. The 0.8% unavailability is due to system maintenance. Fig. 2 shows the overall operation of the experimental system used for the rainfall data logging parts where the accumulated rainfall amount is first collected in a data logger and then stored in the data storage devices.

The calculated 1-min rain rate from the experimental 1-min rainfall amount obtained from the KMA database at 0.01% of the time lies between 60 to 90 mm/hr for all the considered regions. The Ulsan region shows greater rain rate variability because this region lies near the coastal area and the average rainfall accumulation is higher in this belt. At a time percentage higher than 0.1%, the 1-min rainfall amount values are extremely low and tend to be negligible. Analyses of these cumulative statistics indicate that the rainfall process is mostly of the convective type and shows the saturation limit in the rain mechanism. The rain rate at a higher time integration of 5 min, 10 min, 20 min, 30 min, and 60 min are used as the input parameter for the software recommended by ITU-R P.837-6 [5]. The value of the 1-min rain rate distribution at 0.01% of the time lies between 80 and 90 mm/hr for most of the prime locations in Korea. The rainfall rate at 0.01% of the time is preferred in communication system design and to precisely predict rain attenuation. ITU-R P.618-12 [28] emphasized the use of the 1-min integration time of the rainfall rate at this prominent integration time for the prediction of rain attenuation. The importance of the 1-min rain rate has been further studied for satellite and terrestrial rain attenuation predictions [29, 30, 33].

Furthermore, the mean rainfall rate distribution generated for the nine regions is shown in Fig. 3. The 1-min rainfall rate distribution, which is obtained after averaging over these sites, is used for further analyses with the application of six empirical conversion methods along with the second-order and third-order polynomial fits. The conversion from the 10-min and the 20-min curves to the 1-min curve shows a slightly higher rain rate at a lower time percentage when P ≤ 0.05%, which might be due to the higher rainfall distribution obtained in the Ulsan region. Similarly, Fig. 4 indicates the mean 1-min rain rate distribution along with the average 1-min rain rate generated after averaging over several integration times, namely 5 min, 10 min, 20 min, 30 min, and 60 min, to the 1-min conversion time from ITU-R P.837-6 [5]. This indicates that ITU-R P.837-6 [5] shows a fair estimation of the 1-min rainfall rate at a lower time conversion, particularly 5 min to 1 min.

However, as integration times increase, the probability of overestimation also increases. Further, Fig. 5 presents the overall distribution of the 1-min rain rate over nine regions. These graphs clearly depict that the 1-min rain rate at 0.01% of the time lies between 60 mm/hr and 90 mm/hr for most of the major cities in Korea.

A regression analysis was performed to match the data to the known mean distribution using six applicable empirical conversion methods along with the second-order and third-order polynomial fits. The regression models summarize a large amount of data with a minimum modeling error [2]. Regression analysis is a statistical method to estimate the values of dependent variables that correspond to certain values of new independent variables once the magnitude of the influence of independent variables on the dependent variables is measured, thereby determining the regression plane or line with respect to the independent variables. The expressions used in the equal probability method representing R₁(P) and R_τ(P) for the 1-min and τ-min integration times with an equal probability of time percentage, P, respectively, are given as follows:

(i) Segal method [18]:

where conversion factor ρ_τ(P) = aP^b, and the parameters a and b denote the regression coefficients that are derived from a statistical analysis of rainfall data.

(ii) Burgueno et al.’s method [19]:

where a and b represent the conversion variables obtained from a statistical analysis of rainfall data.

(iii) Chebil and Rahman’s method [20]:

Chebil and Rahman [20] introduced an experimental technique for estimating the precipitation rate conversion element by using the conversion process from 60-min to 1-min integration time as follows:

where R₆₀(P) denotes the precipitation rate for the 60-min integration time. ρ₆₀(P) is expressed as a mixed power-exponential law; p₆₀(P) = aP^b + ce^(dP) with regression variables represented by a, b, c, and d obtained from a statistical analysis of rainfall data. The suitability of this method has been further tested for other lower integration time intervals.

(iv) Logarithmic model [10]:

where a denotes the regression variable derived from a statistical analysis of the rainfall rate.

(v) Exponential model:

where a and b represent the regression coefficients obtained from a statistical analysis of the rainfall rate.

(vi) Proposed approach:

In this study, we statistically evaluate the different degree of polynomial fits as follows:

Second-order polynomial fit:

Third-order polynomial fit:

The suitability of the proposed techniques has been further analyzed in [31, 32]. The obtained regression coefficient values applicable for different conversion techniques against the mean 1-min rain rate distribution are tabulated in Table 3.

These coefficients are used for obtaining the mean 1-min rain rate distribution from the 5-min, 10-min, 20-min, 30-min, and 60-min integration times. Furthermore, the coefficient of determination is calculated for the applicable empirical nature of methods. The coefficient of determination, R², describes the proportion of the variance in the measured data explained by the model. The proportion of variability in a dataset is accounted for by this statistical model [35]. R² provide a measure of the accuracy of future predictions which ranges from 0 to 1, with higher values indicating lower error variance.

The values of R² are obtained through a statistical analysis whose results are calculated after averaging over the nine mentioned sites, as listed in Table 4.

As summarized in Table 4, the polynomial models of the second-order and third order has a relatively high correlation with the rainfall rate over the various integration times. Within these models, the proposed polynomial technique of the third order exhibits greater correlation values, which show its effectiveness in the estimation of the 1-min rainfall rate. Comparatively, the exponential model and the logarithmic model show a lower value of R². This confirms that the error variance increases when we apply the exponential and Burgueno et al.’s models for the prediction of the 1-min rainfall rate. Because of the dependability on the statistical analyses, the coefficient of determination values for the Segal, Chebil and Rahman, logarithmic, and global coefficients are not listed in Table 4.

The mean 1-min rainfall rates estimated using the applicable models from the relatively high integration times such as 5 min, 10 min, 20 min, 30 min, and 60 min to the 1-min time interval are shown in Figs. 6–10, respectively. These plots indicate the suitability of the empirical method to characterize the 1-min rainfall rate. Although the ITU-R P.837-6 method shows a fair estimation from lower integration times of 5 min and 10 min to 1 min, as shown in Figs. 6 and 7, this method tends to overestimate for higher time integration of 20 min, 30 min, and 60 min to 1 min, as shown in Figs. 8–10. The relative error percentage deviations along with other statistical analyses are calculated for the proposed conversion technique.

In order to determine the adequacy of the proposed technique, the mean, standard deviation (STD), and root mean square (RMS) values of the absolute percentage relative error ε(P) are calculated, where they are compared to the performance of the ITU-R [5] and other conversion models. The RMS error values of 0 indicate a perfect fit.

The relative error percentages, ε(P), error values have been weighted over the probability levels of 0.001%, 0.002%, 0.003%, 0.005%, 0.01%, 0.02%, 0.03%, 0.05%, and 0.1% of the time, as recommended in ITU-R P.311-15 [26]. The 1-min rainfall amount values beyond 0.1% are very few and tend to be smaller as the percentage of time reaches 1%. Hence, following the recommendation from ITU-R P.311-15 [26], in this paper, we present the approaches to determine the goodness of fit for 10 years of observations at the Gwangju, Daegu, Daejeon, Busan, Seogwipo, Seoul, Ulsan, Incheon, and Chuncheon sites with an evaluation of the average rain rate values weighted over several probability levels. The error matrices are calculated by following the approaches presented in [31, 32].

The error variables thus obtained between the experimental result and the estimated 1-min cumulative distributions are presented in Tables 5 and 6.

As noted from Table 5, the ITU-R P. 837-6 and global coefficient methods generate higher error probabilities because of the increased RMS values. The error probabilities will be higher for higher integration times, which are justified by the increasing RMS values of 4.36%, 10.22%, 36.43%, 63.60%, and 102.76% for the conversion of 5-min, 10-min, 20-min, 30-min, and 60-min data to the 1-min data, respectively.

In contrast, the polynomial fits of the second-order and third-order show low error probabilities. In particular, the third-order polynomial exhibits low error probabilities because of the relatively low RMS errors of 1.15%, 0.81%, 3.10%, 2.98%, and 3.49% for the 5-min, 10-min, 20-min, 30-min, and 60-min times to the 1-min integration time, respectively. Similar trends of statistical values are obtained with the application of the second-order polynomial fit. Although the application of the Segal, Burgueno et al., Chebil and Rahman, logarithmic, and exponential formulas show a fair goodness of fit, the error probabilities remain high because of the relatively high RMS error values.

To further elaborate the accuracy of the proposed technique at 0.01% of the time, several tests are carried out, whose results are tabulated in Table 6. As shown in Table 6, although the ITU-R P.837-6 and global coefficient approaches exhibit a fairly good accuracy with a relatively small error of about 0.04% and 0.01% at a low time integration of 5-min to 1-min conversion time, respectively, the error probabilities increase from a higher time conversion to 1-min as justified from the increase in the RMS error. In contrast, polynomial fits of the second-order and third-order show a lower marginal change in the relative error percentages and RMS values than the other empirical formulas. The error probabilities still remain higher for other models such as the Segal, Burgueno et al., Chebil and Rahman, logarithmic, and exponential methods, which are justified by the increasing error values, as listed in Table 6. Hence, we can confirm that the proposed technique exhibits stable and relatively low RMS values of prediction and can be effectively used for the 1-min rain rate prediction for Korea.

The scope of work in this study is limited to the CDF analysis of the rainfall rate using the measured rainfall distribution as received from KMA for a 10-year period (2004–2013) over nine regions of Korea. The analysis of the 1-decade rainfall data has given an appropriate indication for the study of rainfall behavior over these regions. As observed, empirical formulations are simple yet powerful tools for the conversion of rainfall statistics among the available models. The performance of the proposed polynomial fits is considered to be the best for the conversion of the rain rate from various integration times to their 1-min equivalent because of the relatively low average error evaluation, which confirms the overall best-performing regression fit. In particular, the ITU-R P.837-6 and global coefficient approaches are unable to provide a reliable estimation against the experimental result of the 1-min rain rate. Through the evaluation of different error matrices, it hasbeen shown that the proposed technique adequately trails the nature of the 1-min rainfall rate. Overall, in this paper, we emphasize a suitable technique that can best describe the 1-min rainfall characteristics of Korea.