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Aug 10, 2019

Comparative Study on Spatiotemporal Characteristics of Fixed-Area and Storm-Centered ARFs

Authors: Boosik Kang https://orcid.org/0000-0002-5246-6588 [email protected], Eunji Kim [email protected], Jin-gyeom Kim [email protected], and Soojin Moon [email protected]Author Affiliations
Publication: Journal of Hydrologic Engineering
Volume 24, Issue 10
https://doi.org/10.1061/(ASCE)HE.1943-5584.0001839
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Abstract

In the design flood estimation procedure, the areal reduction factor (ARF) is used to convert ground-level point rainfall records into areal design rainfall for a reference area. Practically, the ARF is estimated through the fixed-area (ARFfa) scheme but has limitations as a statistical approach based on sparse ground-observation density. The ARFfa indicates potential biases because of the unsynchronized frequency analysis between point and areal rainfall. The storm-centered ARF (ARFsc) is obtained directly from individual storm captured by high resolution radar. In this study, the ARFsc values were estimated during the monsoon season (June to September) during 2007–2012 that covered the entire nation of South Korea, and then expressed as a function of reference area, duration, and return period. Both the ARFfa and ARFsc are proportional to the reference area. However, the most distinct difference is their responses to the return periods. For the fixed specific duration, the ARFfa indicates insensitive response to the return period over the reference area, whereas the ARFsc indicates quite varied declining rates according to the return periods. The ARFfa’s invariant characteristics arise from its statistical scheme assuming identical distributions with similar shape parameters for the point and areal annual maximum rainfalls. The results can be used to avoid excessively conservative designs and assist in more economic and reliable uses of practical areal reduction factors.

Introduction

The scientific purpose of identifying the spatiotemporal distribution of rainfall clusters is to understand the evolutionary processes of different types of rainfall phenomena in different geographic layouts. However, the practical engineering purpose is to estimate design areal rainfall from records of rainfall at gauging sites using the areal reduction factor (ARF) by rescaling point rainfall to areal rainfall. The ARF is defined as the ratio between the areal rainfall with a specific frequency and the point rainfall with the same return period within the same area—called the fixed area ARF (ARFfa). In the ARFfa method, the point and the areal rainfall have the same return period but do not occur naturally at the same time by definition. The ARFfa could depend on the type, strength, and duration of the rainfall. In reality, the rainfalls at the gauge sites within a clustered storm area do not have the same return period, and the ARFfa must be different from that estimated from a real storm cluster—the so-called storm-centered ARF (ARFsc). In this paper, the main objective is to compare two different types of ARFs and discuss the relative feasibility of the ARFfa in terms of its duration, area, and frequency through the comparison with the ARFsc.
A typical ARFfa presented in Technical Paper #29 (TP-29) (USWB 1957, 1958) is estimated by dividing the average areal rainfall of the annual maximum rainfall for a specific area and the duration by the annual maximum point rainfall statistics for the same area and duration. The Natural Environment Research Council (NERC) of the United Kingdom proposed a method to estimate the ARFfa by averaging the ratio of areal rainfall to the annual maximum point rainfall at each station (NERC 1975). Bell (1976) estimated the fa for a specific return period using a frequency analysis of point rainfall and areal rainfall and compared with the ARFfa of NERC. To estimate an ARFfa that reflects the characteristics of actual rainfall, a method using an intensity-duration-frequency curve based on the spatial correlation structure of rainfall (Sivapalan and Blöschl 1998). Allen and DeGaetano (2003) re-evaluated the ARFfa of TP-29 using relatively longer 47 years (1949–1995) of rainfall data. The ARFfa using the spatiotemporal scale characteristics of extreme rainfall events is also discussed in the literature (De Michele et al. 2001, 2011). However, the spatial distribution of gauged rainfall is not reflected properly when the densities of the gauge stations are relatively low. The ARFfa might differ from that of an actual rainfall event. ARFfa values based on radar data were presented in many studies (Stewart 1989; Allen and DeGaetano 2005a, b; Gill 2005; Lombardo et al. 2006; Lincoln et al. 2016) Pavlovic et al. (2016) and compared with ARFfa using radar data and four methods: (1) empirical method; (2) methods based on spatial correlation structures; (3) method based on spatial and temporal scale characteristics; and (4) methods based on extreme values.
Instead, the storm-centered approach interprets the center of the rainfall as the center of the target area for a specific rainfall event. The ARFsc is estimated using the areal rainfall along the rainfall contour lines that change over time. Because the ARFsc is used to estimate the areal rainfall that reflects the spatial distribution characteristics of synchronized rainfalls, various methods have been proposed to estimate the areal rainfall using radar data during the ARFsc estimation process. Bacchi and Ranzi (1996) attempted to use radar rainfall to estimate stochastic ARFsc values under the theoretical framework based on the crossings of Poisson space-time processes. Durrans et al. (2002) estimated the rainfall depth-area relationship through the annual maximum rainfall using radar rainfall data. Jolly et al. (2005) suggested that ARFsc values derived from radar data could improve the spatial resolution of the ARF itself. Olivera et al. (2008) selected the optimal areal rainfall by changing the major and minor axis ratio of the ellipse from 1∶1 to 15∶1 for the shape of the rainfall area. Martins et al. (2014) estimated the areal rainfall for a range of up to a 100-km radius from the cells having the maximum recorded rainfall. Wright et al. (2014) searched for the maximum value among the grids adjacent to the point grid and calculated the areal rainfall more accurately by repeating the same steps as those for the reference area. Hulstrand and Kappel (2016) calculated ARFsc values using rain gauge rainfall and gridded rainfall data, whereby an 85% reliability was determined that indicated the ARFsc representing the study area. Kok et al. (2018) estimated the ARFsc using the maximum average areal rainfall (MAAR)—an improved areal rainfall estimation method—to derive a new empirical ARF equation that reflects climate change characteristics. Mineo et al. (2018) analyzed the relationship of duration-area and proposed a new ARF equation.
ARFfa assumes that the probabilistic rainfall occurs simultaneously for a specific return period at all points within the reference area. However, this assumption has limitations in that it does not reflect the spatial distribution of rainfall when the densities of rainfall stations are relatively low (Jolly et al. 2005). Furthermore, even for high station densities, the annual maximum rainfall does not occur directly at the gauge (Durrans et al. 2002). In this regard, the ARFsc using the gauge rainfall has a potential limitation. In contrast, ARFsc values using radar rainfall events present realistic values because they can effectively reflect the spatial distribution characteristics of spatially synchronized rainfall events. To comprehensively compare the ARFfa with the ARFsc, the characteristics of the reference area, duration, and return period are presented with a focus on hydrological responses from the estimated rain-radar rainfall.

Data

Fixed-Area ARF

The ARFfa values are provided in the government report, Study on improvement and supplement of Probabilistic Rainfall in South Korea (MLTM 2011). The purpose of this report is to provide the regional probabilistic rainfall by duration and frequency in South Korea’s four major river basins. Therefore, data on 233 rainfall stations located in South Korea were used. The period of the selected rainfall data varied from the shortest at 18 years (1993–2010) to the longest at 36 years (1975–2010). The ARFfa values were estimated by analyzing the selected rainfall data for durations of 1–72 h and corresponding return periods of 2–500 years. The Thiessen areas were constructed (for areas ranging from 300 to 20,000  km2) for the estimation of the ARFfa values given the low rainfall station density in South Korea.

Radar Data

Estimating an areal rainfall at the storm event scale that can reflect the spatial distribution of rainfall in the ARF using the storm-centered method is essential. Radar data are essential when deriving ARFsc because they can effectively convey the understanding of the spatial variability of rainfall (Bacchi and Ranzi 1996). A weather radar is remote sensing equipment that detects echoes reflected from droplets in the atmosphere that provide information about the characteristics of the echo signal and predicts atmospheric phenomena (Skolnik 1962; Doviak 1993; Bringi and Chandrasekar 2001). A quantitative estimation of rainfall through the use of a weather radar is complex and error-prone (Krajewski and Smith 2002). For this reason, various studies have been conducted to increase accuracy. Rainfall data measured with radar equipment were mainly calibrated using ground-level rain gauge data (Seo et al. 1999).
The measurement of backscattered radar by raindrops is related to the physical amount of reflectance (Z) and is expressed as dbz = 10 log Z, where Z = radar reflectivity based on a single 1-mm diameter drop per unit volume (m3). The received radar reflection intensity is related to the number and diameter of the raindrops within the radar unit volume. The rainfall intensity (R) is related to the falling rate of rainfall and the number and diameter of the raindrops existing within a unit volume. A Z-R equation is proposed to estimate the relationship between the reflectivity factor (Z) and the rainfall intensity (R). Eq. (1) is generally expressed in the form of a power law function
Z=αRβ
(1)
where α and β = empirical constants that depend on the type of rainfall. Considering the geographical and meteorological characteristics of South Korea, Eq. (2) is used for the more common stratiform rainfall found in mid-latitudes (Marshall and Palmer 1948)
Z=200R1.6
(2)
In this study, radar rainfall data of 10-min resolutions were analyzed by converting the reflectivity measured by weather radar into rainfall intensity or depth (mm). The Korea Meteorological Administration (KMA) operates weather radars at 11 stations (Baekryungdo, Mt. Gwanak, Mt. Gwangdeok, Gangneung, Yeongjongdo, Mt. Ohseong, Mt. Myeonbong, Mt. Guddeok, Jindo, Gosan, and Seongsan) [Fig. 1(a)]. Three sites (Baekryungdo, Mt. Myeonbong, and Yeongjongdo) use the C-band and the rest of them use the S-band. The data used in this study are 1.5-km CAPPI (Constant altitude plan position indicator), and the maximum range is 240 km. The KMA provides composite images of the observed weather data at each station every 10 min [Fig. 1(b)]. The output is composed of 1153-longitudinal and 1441-latitudinal grids with a 1-km2 pixel area. The ARFsc values were calculated for rainfall events for the rainy seasons (June–September) of 2007–2012 and covered the Han river basin, which forms a part of the study area.
Fig. 1. (a) Location of KMA single polarization radars in South Korea; and (b) composite radar image provided by KMA.

Methodology

Fixed-Area ARF

The ARFfa values are estimated using the relationship between point and areal probability rainfall. First, the mean annual maximum series of the individual gauging sites in the reference areas are used to estimate the point probabilistic rainfall as a function of duration, return period, and reference area. Then, the annual maximum series of areal rainfalls in the same reference areas are estimated. The ARFfa values are calculated by dividing the areal rainfall for a given return period by the point rainfall for the same return period. The ARFfa formula can be derived through a regression equation with respect to various durations, return periods, and corresponding areas. The entire process is summarized in Fig. 2.
Fig. 2. Schematic diagram for estimating ARFfa values using gauge data.

Storm-Centered ARF

The ARFsc is estimated by dividing the areal rainfall by the point rainfall measured at the storm center. Unlike the ARFfa, in a storm-centered analysis of the same event, radar rainfall data enable the synchronized analysis between the point and the areal rainfall, and more accurate areal rainfall can be estimated as reflecting the spatial distribution of the rainfall observed through the radar image. The schematic process diagram for deriving these ARFsc values is indicated in Fig. 3.
Fig. 3. Schematic diagram of estimating ARFs using radar data.
The radar data measured in 10-min increments were accumulated for a specific duration. The radar images of the annual maximum rainfall were used to more efficiently find the associated maximum events and their occurrence times in the study area. The rainfall events and central coordinates are selected through images of annual maximum rainfall at a 1-km resolution. In the real rainfall cluster images, the central locations are commonly uncertain. Therefore, the mean values of the central and eight surrounding grids are determined as the central rainfall, which is defined as the representative maximum rainfall of ARFsc with a 9  km2 resolution.
In most previous ARFsc studies, the shape of the reference area was fixed as a circular or squared area when calculating the area rainfall (Bacchi and Ranzi 1996; Durrans et al. 2002; Martins et al. 2014). However, for ARFsc estimations, calculating the areal rainfall by fixing the areal of the rainfall in a circle or square in the event of frontal rainfalls, which have precipitation areas with narrow banded shape—typical in South Korea—is not appropriate. In this study, the rainfall area was set to be elliptical considering the shape and direction of rainfall events Fig. 4. To calculate the area of the ellipse, the direction angle of the major axis must be determined. The analysis was implemented for the direction angle ranging from 0° to 175° with 5° increments, and the angle at which the maximum negative skewness occurs was determined to be the optimum direction angle. When the direction angle is determined, the ratio of the major axis to the minor axis is changed from 1∶1 (circle) to 15∶1 (ellipse) to search for the optimal ratio of the ellipse. The optimal ratio is determined as the maximum areal rainfall found for the reference area. The areal rainfalls for the 30-, 150-, 380-, 700-, 900-, 1,250-, and 2,000-km2 areas were estimated from the center of the rainfall ellipses with the optimal ratio of the major axis to the minor axis for each rainfall event. Average areal rainfall was calculated by dividing the areal rainfall by each reference area. The ARFsc is calculated by dividing the areal average rainfall by the point rainfall, as indicated in using Eq. (3)
ARFsc(A)=∫ARPdARP0
(3)
where A = area of the circle or ellipse determined from the rainfall shape (km2); RP = radar rainfall (mm) located in the circle or ellipse; and RP0 = central rainfall (mm).
Fig. 4. (a) Conceptual diagram for estimating point rainfall; and (b) areal rainfall.

Sampling of Rainfall Events

The major difference between the ARFfa and ARFsc estimation methods is the frequency of rainfall during the estimation process. The ARFfa is calculated using the relationship between the point and the areal rainfalls estimated from the frequency analysis. Therefore, using a regression equation for the specific duration and the return period is possible. In contrast, the ARFsc is estimated as the ratio of the maximum areal rainfall to the maximum point rainfall for each rainfall event. Although the ARFsc is estimated based on the actual rainfall, the frequency of the rainfall needs to be identified to compare with ARFfa for rainfall with the same corresponding frequency. The point rainfall of the ARFsc is the maximum value at the storm center for different durations of a specific rainfall event. The corresponding return period is assigned to the entire storm. Because the ARFsc is estimated for the individual event, the resulting ARFs are scattered and finding any meaningful tendency or relationship is difficult. The identification of the return period of the ARFsc was implemented through matching the central maximum rainfall with the probabilistic rainfall in the national report (MLTM 2011) found using the intensity-duration-frequency formulas derived from the ground-gauge data. The return period is categorized into five classes (0–10, 10–20, 20–50, 50–80, and 80–100 years). Table 1 provides the number of rainfall events classified by return period. The number of rainfall events is negatively proportional to the return period. The 80–100-year return period with a 1-h duration was excluded because of the absence of the corresponding sample event.
Table 1. Classification of duration and return period for rainfall events during rainy seasons (June–September) from 2007 to 2012
Duration (h)Return period (year)Sub total
0–1010–2020–5050–8080–100
1251174120210566
315310557133331
6635836127176
122737242617131
242923292411116
Sub total52339726696381,320

Selection of Representative ARFsc

Because the ARFsc’s are calculated for each rainfall event, the ARFsc values indicate scattered representation on the ARF curve graph. To compare ARFfa’s with ARFsc’s, ARFsc’s corresponding return period and duration should be selected first. In several studies on ARFsc’s, the results were presented as averages of the values of duration and return period (Durrans et al. 2002; Olivera et al. 2008; Martins et al. 2014; Wright et al. 2014). However, from the engineering point of view, the conservative value enveloping the scattered points is more feasible than the simple average. In this study, the value with a 95% nonexceedance probability and the upper limit of 1 is used as the representative ARFsc under the assumed Welbull distribution. The Weibull distribution, which is also known as the GEV-III, is an effective distribution type for extremes, such as flood and drought events. The cumulative probability density function (CDF) and the probability density function (PDF) of the Weibull distribution are known as Eqs. (4) and (5) (Johnson and Kotz 1970)
F(x)=1−exp[−(x−x0α)β],x≥x0
(4)
f(x)=βα[x−x0α]β−1,x0≤x<∞
(5)
where α = scale parameter (α>0); β = shape parameter (β>0); and x0 = location parameter. In this study, a modified Weibull distribution is applied to scattered ARFsc values, which are categorized by duration and return period. Because the Weibull distribution has a lower limit of 0 and is distorted in a positive direction, it is applied by subtracting the ARFsc values from 1. In addition, a goodness-of-fit test was performed to statistically check the validity of the estimated probability distribution. The Kolmogorov–Smirnov (K–S) test is used to compare the maximum value of the difference between the cumulative histogram and the theoretical cumulative probability density function. The n-th function of the cumulative probability density function F(x) of the Weibull distribution is expressed as Fn (x), and the K–S statistic (Dn) is expressed as Eq. (6)
Dn=supx|Fn(x)−F(x)|
(6)
where supx is the upper bound of the difference between Fn (x) and F(x).
The goodness-of-fit test was conducted when more than 100 rainfall event samples were obtained for each class by return period. Table 2 provides the results of the test for significance levels equal to 2% of the ARFs with a reference area of 380  km2 and the duration of 1 and 3 h. For all events, the null hypothesis (Ho) of the “following Weibull distribution” fails to be rejected. In other words, the assumption of the Weibull distribution was revealed to be effective.
Table 2. Kolmogorov–Smirnov test result of ARFs
Duration (h)Return period (year)Area (km2)Sample size (N)Significance level (α)Critical level (Dn0.02)K-S static (Dn)Decision
10-103802510.020.095820.05411Fail to reject the Ho
110-203801740.020.115080.04946Fail to reject the Ho
120-503801200.020.138570.05508Fail to reject the Ho
30-103801530.020.122720.11220Fail to reject the Ho
310-203801050.020.148140.11800Fail to reject the Ho

Results

Shape and Direction Angle of Reference Area

The ARFs were calculated for individual rainfall events with corresponding durations and return periods. The ARF varies not only with the rainfall type, for example, frontal, local convective, and typhoon, but also the shape of the reference area. For example, the frontal rainfall with a long, narrow rainfall area indicates higher ARF for the elliptical area than for the circular area. Even the local convective or typhoon rainfall rarely indicate higher ARF for the circular reference area. In this study, the direction angle and the ratio of the long and short axes of the ellipse leading to maximum ARF were found for the individual storm event.
Fig. 5 indicates the ARFs for several radar images of representative rainfall events with various durations from 1 to 24 h and intensities. The concentric ellipses in the radar images show the 150, 700, and 1,250  km2 reference areas. In principle, the ARFs are relatively small when intensive rainfall clusters are concentrated at the center of the rainfall and increase when rainfall clusters are scattered in space. However, in nature, the ARF relationship with the rainfall intensity and the area is very complex and irregular, and the ARF varies with high variability. Events #2 and #3 of different types and intensities indicate comparative ARF patterns. The ARFs of Event #2, which has more intense rainfall, decreases more rapidly with the increasing reference area than Event #3. In contrast, Events #3 and #4, with similar intensity but different types, indicate similar ARF variation patterns. Although the overall mean behavior of ARF directly estimated from the real storm events follows the expected pattern, the high variability of ARF is the nature of storm-centered ARF. The ARFfa presents relatively low variability because it is the result of the ratio of the frequency between point and areal rainfall. However, the ARFfa can show irregular and abnormal patterns when the sample size is insufficient for a properly fitting probability distribution function.
Fig. 5. Estimating storm-centered ARF of rainfall event by each duration.
Fig. 6 compares the ARFs and the corresponding radar images of two contrasting distributed and intense rainfall events with 3 h of duration, a similar size of 57 mm, and a 10-year return period. Their values were standardized to between 0 and 1 for a comparison of different rainfall types. Both rainfall clusters are elliptical, but the direction angle and the ratio of the major and minor axes are different. The ARFs of two events show a maximum difference of 0.23 for the 380-km2 reference area. Their difference decreases with a larger reference area.
Fig. 6. Comparison of ARFs for two rainfall events for 10-year return period (and for 3-h duration).

Generic ARF Formula

The ARFs are based on Eq. (7), an exponential fitting equation (Lee 1987). The parameters of these curves for each duration and the return period are indicated in Table 3
ARF(A)=1−Mexp[−(aAb)−1]
(7)
where M, a, and b are regression constants, and A is the reference area (km2).
Table 3. Regression constants of ARFsc for duration and return period
Return period (year)Regression constantDuration
1361224
0–10M2.4964.9974.8094.9664.949
a0.0830.0800.0830.0770.066
b0.2720.1860.1510.1290.146
10–20M1.6713.9254.1873.8904.956
a0.0710.0780.0710.0730.089
b0.3510.2110.1730.1440.104
20–50M1.4372.5143.8634.3614.872
a0.0700.1000.0720.0600.070
b0.3920.2210.1860.1670.124
50–80M2.6961.2008.5243.2174.572
a0.1180.1020.0760.0660.064
b0.2430.3170.1640.1800.157
80–100M—0.8770.9824.5043.767
a—0.0620.0680.0860.076
b—0.5060.3500.1750.173
The ARFs calculated for each rainfall event, with the associated return period and duration in the specific area, were fitted to the Weibull distribution. The 95% nonexceedance probabilistic values are determined as the upper maximum values of the ARFsc for all reference areas, which compares with the ARFfa. In its nature, the ARFfa is the possible maximum value of ARF from real events because the ARFsc is estimated from real events and the central maximum rainfall and areal rainfall occurs synchronized at the same time, but the ARFfa is the ratio of unnecessarily synchronized probabilistic rainfalls with the same frequency between point and areal rainfall.
The ARFsc curves for each return period corresponding to the 1-h duration and four return periods are indicated in Fig. 7. As observed in Fig. 7, the statistical maximum (95% nonexceedance probabilistic) ARFsc curves are mostly lower than the ARFfa curves. The ARFfa was originally developed for practical use in engineering design, and the comparative overestimating characteristics can result in a conservative design considering the high variability of the ARF in real storm events. However, the excessively conservative design can bring the consequences of economic disadvantages.
Fig. 7. ARFsc curves by duration and return period: (a) D=1  h, T=0–10  years; (b) D=1  h, T=10–20  years; (c) D=1  h, T=20–50  years; and (d) D=1  h, T=50–80  years.
The declining trend of the ARF with respect to the reference area varies for each duration and return period. The relationship between the ARF and the return period is provided in Fig. 8 using reduction factor-area-return period (RAT) curves for 3-h durations. The ARFfa versus area curve indicates the same for all return periods, but the ARFsc versus area curves are scattered with respect to the return period. As the reference area increases, the ARF difference for the lowest and highest return periods is indicated as 0.38 at the 2,000  km2 reference area.
Fig. 8. Reduction factor–area–return (RAT) period curve (D=3  h): (a) fixed-area ARF; and (b) storm-centered ARF.
Under the assumption of an identical distribution between the point and areal rainfall except for their scale differences given the absolute values, their ARFfa had an unexpected invariable pattern along the frequencies, as supported by the empirical evidence (Fig. 8). This result might be unexpected against the conventional hydrological principle and was actually the methodology of the ARFfa. The ARFfa’s invariant characteristics arise from its statistical scheme assuming the identical distributions with similar shape parameters for the point and areal annual maximum rainfalls. The ARFsc from a real storm cluster from a hydrologic radar reflects the realistic spatial distribution of actual storm clusters. The ARFfa conforms mostly to the ARFsc with a 20–50-year return period. The ARFsc is higher than the ARFfa for events with lower frequency and lower in the opposite case. In this study, the real storm-based ARFsc was estimated as a function of area, duration, and return period using radar images covering the entire national area and was compared with the conventional ARFfa to guide more reliable use of practical areal reduction factors.
In addition, the relationship between the ARF and the duration can be confirmed using a reduction factor-area-duration (RAD) curve for the 20-year return period for the ARFfas and 10–20-year return period for the ARFscs. It was noted that both ARFfa and ARFsc values are proportional to the duration with different rates of change (Fig. 9). As the reference area increased, the range of ARF from 1 h to 24 h widened, and the rate of ARFsc change indicated a different pattern with respect to the duration. The ARFsc values indicated a relatively slower decline over the long (≥2  h) duration and a fast decline over the short (≤3  h) duration than the ARFfa values.
Fig. 9. Reduction factor-area-duration (RAD) curve [T=20  years (ARFfa), T=10–20  years (ARFsc)]: (a) fixed-area ARF; and (b) storm-centered ARF.
The additional noticeable property is the scaling pattern between the ARF curves with respect to the third variables; that is, the slopes of the ARF curves are the function of the third variables. Figs. 8 and 9 indicate that a scaling pattern exists between the ARF versus area curves and the return periods and durations. As previously stated, the ARFfa presents an overlapped pattern with respect to the return periods. In other words, the scaling pattern is not found in Fig. 8(a) for the ARFfa, which is not natural and exceptional. The overlapped pattern appears because of the statistical methodology of the ARFfa. The ARFfa is the ratio of probabilistic rainfalls with the same frequency between point and areal rainfall but is not necessarily synchronized. They present similar values for the same reference area regardless of nonexceedance probabilities.

ARF versus Return Period (T)

The RAT curve (Fig. 8) indicated the ARF variation with respect to the reference area, the return period, and the different pattern between ARFfa and ARFsc for the fixed specific 3-h duration. In contrast, the ARFfa and the ARFsc were compared with respect to the return period and duration for the fixed specific area of 1,250  km2 (Fig. 10). The 1,250  km2 area has the largest concentric ellipses. The ARFfas are almost constant with respect to the return periods. A similar pattern is found in Fig. 11, which indicates the relationship of ARFs with respect to the return period and the reference area. Noteworthy is that the sensitivity to the return period seen in the ARFsc is hardly visible in the ARFfa. As stated in the previous section, a parallel pattern is not natural, and the statistical distributions of the point and areal rainfall are understood as having the same shapes but just a shifted location. Additionally, a lower the sensitivity to the return period is obtained from a longer duration of the ARFsc. In other words, the scale characteristic of the derived ARF is viewed as not clear but is reflected to a certain degree. A comparison of the ARF reference area and the return period for a 3-h duration is provided in Fig. 11. As indicated in Fig. 10, the ARFfa has low sensitivity to the return period in all reference areas. However, the slope of the ARFsc versus the return period curves is proportional to the reference area. The scaling characteristics between the ARF and the reference area are much clearer than those between the ARF and the duration (Figs. 10 and 11). The scaling characteristics are known to be evidence of scale-invariant parameters associated with the process or function that estimates ARF in a parsimonious fashion.
Fig. 10. ARF-return period relationship by duration (A=1,250  km2): (a) fixed-area ARF; and (b) storm-centered ARF.
Fig. 11. ARF-return period relationship with reference area (D=3  h): (a) fixed-area ARF; and (b) storm-centered ARF.

ARF versus Duration (D)

For a fixed reference area of 1,250  km2, the ARF relationship with respect to the duration and the return periods are compared between ARFfa and ARFsc (Fig. 12). The ARFfa versus the duration curves indicate almost a single linear proportionality with respect to the return periods, but the ARFsc versus the duration curves indicate a scaling property. However, Fig. 13 indicates that both the ARFfa (and ARFsc) and the duration curves represent the scaling behavior. Figs. 8–13 indicate the relationship among ARF, duration, reference area, and return period (scale of the storm). Noteworthy to mention is that the derived ARFsc reflected the radar image of real-scale rainfall of similar events. This reflection confirms the variations in the return period, which is important when estimating ARF values for actual hydrological designs.
Fig. 12. ARF-duration relationship by return period (A=1,250  km2): (a) fixed-area ARF; and (b) storm-centered ARF.
Fig. 13. ARF-duration relationship with reference area (T=20  years, 10–20 years): (a) fixed-area ARF; and (b) storm-centered ARF.
As mentioned in Fig. 5, the ARF relationship between the rainfall intensity and the area is very complex and irregular, and the ARF varies with high variability. However, scaling properties are embedded in their high variability. Although the scaling patterns are not always clear because of the nonlinear features, the results present that the spatial and temporal distribution of rainfall clusters should follow the rule that the scaling property governs the relationship among ARF, area, duration, and return period. The spatial rainfall observation with high-resolution images of the hydrologic radar enables the capture of scaling properties, which was hardly difficult to deal with from the ground observation network.

Conclusion

In a conventional storm-runoff modeling procedure, the use of probabilistic areal rainfall by duration and return period is still a key process for securing reliable flood estimations. Through the remotely sensed radar observation with high resolution becomes readily available, the ARFs are still estimated from the ground-gauging observation. In estimating ARF, the use of ground-gauge rainfall has a number of disadvantages: first, the gauge has an irregular spatial distribution and, second, it has a very low density that is not adequate for capturing the spatial distribution effectively. In addition, the ARFfa method was developed independently using the statistical frequency analysis between point and areal rainfall, which embeds the potential biases relative to real storm events of similar size (frequency) under consideration between ARFfa and ARFsc.
By comparing the ARFfa values with those of ARFsc, more realistic hydrologic responses were achieved using hydrologic radar observations in South Korea. The following are the step-by-step findings from this study and recommendations for a design flood estimation procedure.
•
The simple definition of ARF is the ratio between the point maximum rainfall within a reference area and the areal rainfall. The ARFfa has been used conventionally in engineering design; however, disregarding the concurrence between the maximum point rainfall and the areal rainfall as a statistical model is naturally unrealistic. Yet, the ARFfa has been accepted for engineering purposes because it tends to provide significantly conservative values. From an economic point of view, it may be unfeasible.
•
With the aid of the hydrologic radar observations, the areal reduction properties of real rainfall clusters are readily identified and can be compared among different types of rainfall with different distributions in space. Actually, the ARF depends not only on the reference areas but also the shape (circle, ellipse) and the directional angle. The conventional ARFfa assumed only the area with a circular shape and disregarded the effects of an elliptical shape conforming the narrow band type of the frontal rain clusters. In this study, the ARFsc was estimated for the area of the elliptical shape with the optimal major and minor axes ratio.
•
The ARFfa versus area curves are almost similar for all return periods; that is, the ARFfas with respect to the return periods are invariant. The ARFfa’s invariant characteristics arise from its statistical scheme assuming identical distributions with the similar shape parameters for the point and areal annual maximum rainfalls. The location and scale parameters do not influence the ratio between the point and areal probabilistic rainfalls. Actually, the ARFsc indicates the difference of 0.3 between the 20–50 and 80–100 years of return periods for the 1,200  km2 of the reference area. However, the ARFfa does not show the invariant pattern with respect to the duration.
•
The ARFsc analysis indicates that there exists the scaling property; in other words, the slopes of the ARF curves are a function of the other variable’s scale. The ARF versus area curves indicate the semi-logarithmic scaling pattern with respect to the return periods or durations. The ARF versus return period curves indicate the scaling pattern with respect to the durations or the areas. The ARF versus duration curves indicate the scaling pattern with respect to the return periods and areas. Certain differences in degree exist; some are semi-logarithmic and others are natural, and scaling properties exist in ARF curves, which means that more parsimonious ARF expression could be possible.
The radar data used in this study span the six years from 2007 to 2012, were limited total sample size, that is, the sample sizes are uneven between the high and low return periods. Despite all of the challenges in the availability of observational data, more reliable results may be achieved by adding rainfall events of various frequencies through the accumulation of radar rainfall data in the framework of flood estimations.
The main objectives of this study are to understand the spatiotemporal distribution of the rainfall clusters and to improve the engineering design criteria by understanding the limitations of the existing ARF and finding a new way to enable the economic design. The results can be used to avoid excessively conservative designs and to help ensure a more economic and reliable use of practical areal reduction factors.

Acknowledgments

This work was supported by Korea Environment Industry & Technology Institute (KEITI) through Advanced Water Management Research Program, funded by Korea Ministry of Environment (Grant No. 83085).

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Go to Journal of Hydrologic Engineering
Journal of Hydrologic Engineering
Volume 24 • Issue 10 • October 2019

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This work is made available under the terms of the Creative Commons Attribution 4.0 International license, http://creativecommons.org/licenses/by/4.0/.

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Received: Jul 13, 2018
Accepted: May 17, 2019
Published online: Aug 10, 2019
Published in print: Oct 1, 2019
Discussion open until: Jan 10, 2020

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Boosik Kang https://orcid.org/0000-0002-5246-6588 [email protected]
Professor, Dept. of Civil and Environmental Engineering, Dankook Univ., Yongin 16890, Republic of Korea. ORCID: https://orcid.org/0000-0002-5246-6588. Email: [email protected]
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Eunji Kim [email protected]
Ph.D. Candidate, Dept. of Civil and Environmental Engineering, Dankook Univ., Yongin 16890, Republic of Korea (corresponding author). Email: [email protected]
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Jin-gyeom Kim [email protected]
Research Professor, Dept. of Civil and Environmental Engineering, Dankook Univ., Yongin 16890, Republic of Korea. Email: [email protected]
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Soojin Moon [email protected]
Manager, R&D Center, Korea KGT Consultant, Incheon 22144, Republic of Korea. Email: [email protected]
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