How Ångström–Prescott Coefﬁcients Alter the Estimation of Agricultural Water Demand in South Korea

: The Food and Agriculture Organization (FAO) Penman–Monteith equation, recognized as the standard method for the estimation of reference crop evapotranspiration (ET 0 ), requires many meteorological inputs. The Ångström–Prescott (A-P) formula containing parameters (i.e., a and b) is recommended to determine global solar radiation, one of the essential meteorological inputs, but may result in a considerable difference in ET 0 estimation. This study explored the effects of A-P coefﬁcients not only on the estimation of ET 0 , but also on the irrigation water requirement (IWR) and design water requirement (DWR) for paddy rice cultivation, which is the largest consumer of agricultural water in South Korea. We compared and analyzed the estimates of ET 0 , IWR, and DWR using the recommended (a = 0.25 and b = 0.5) and locally calibrated A-P coefﬁcients in 16 locations of South Korea. The estimation of ET 0 using the recommended A-P coefﬁcients produced signiﬁcant overestimation. The overestimation ranged from 3.8% to 14.0% across the 16 locations as compared to the estimates using the locally calibrated A-P coefﬁcients, and the average overestimation was 10.0%. The overestimation of ET 0 corresponded to a variation of 1.7% to 7.2% in the overestimation of the mean annual IWR, and the average overestimation of the IWR was 5.1%. On average, the overestimation was slightly reduced to 4.8% in DWR estimation, since the effect of A-P coefﬁcients on the IWR estimation decreased as the IWR increased. This study demonstrates how the use of A-P coefﬁcients can alter the estimation of ET 0 , IWR, and DWR in South Korea, which underscores the importance of their proper consideration in agricultural water management.


Introduction
The estimation of agricultural water demand is very important in long-term water resources planning, because agricultural water use accounts for the largest portion of total freshwater use. Globally, about 70% of freshwater is consumed by agricultural production [1], and agriculture will use more water in the future [2]. In order to estimate agricultural water demand, the reference crop evapotranspiration (ET 0 ) needs to be calculated. The Food and Agriculture Organization (FAO) Penman-Monteith (P-M) equation, which combines both energy and mass balances based on physical principles, is recommended as the standard method for estimating ET 0 in a variety of climate types [3].

Study Sites
We selected 16 meteorological stations that had their calibrated A-P coefficients and provided complete and reliable weather data for estimating ET 0 as study sites in South Korea (Figure 1). The selected sites are located across South Korea from the coast to inland and they are classified as having humid subtropical and continental climates by the Köppen-Geiger climate classification [17] (Table 1). To estimate ET 0 , IWR, and DWR, we collected daily meteorological data including precipitation, maximum and minimum temperatures, wind speed, relative humidity, and sunshine hours from 1983 to 2007 from the Korea Meteorological Administration (https://data.kma.go.kr). This time period is the same one used by Choi et al. [12] to derive the calibrated Ångström-Prescott coefficients. Detailed geographical characteristics of the study sites and the calibrated Ångström-Prescott coefficients are given in Table 1.

FAO Penman-Monteith Equation
The FAO P-M equation used to estimate grass reference crop evapotranspiration (ET ) is given as follows [3]: where ET is in mm day , R is the net radiation at the crop surface (MJ m day ), G is the soil heat flux density (MJ m day ), T is the mean daily air temperature at 2-m height (℃), u is the wind speed at 2-m height (m s ), e is the saturation vapor pressure (kPa), e is the actual vapor pressure (kPa), ∆ is the slope vapor pressure curve (kPa ℃ ), and γ is the psychrometric constant (kPa ℃ ).

FAO Penman-Monteith Equation
The FAO P-M equation used to estimate grass reference crop evapotranspiration (ET 0 ) is given as follows [3]: where ET 0 is in mm day −1 , R n is the net radiation at the crop surface (MJ m −2 day −1 ), G is the soil heat flux density (MJ m −2 day −1 ), T is the mean daily air temperature at 2-m height (), u 2 is the wind speed at 2-m height (m s −1 ), e s is the saturation vapor pressure (kPa), e a is the actual vapor pressure (kPa), ∆ is the slope vapor pressure curve (kPa −1 ), and γ is the psychrometric constant (kPa −1 ). The detailed procedures for calculating all the parameters used in Equation (1) are described in Allen et al. [3].

Ångström-Prescott Coefficients
The coefficients (i.e., a and b) of the Ångström-Prescott formula given below are required to estimate (global) solar radiation (R s ), which is used for calculating the net radiation (R n ) in Equation (1); the recommended coefficients from Allen et al. [3] and calibrated coefficients from Choi et al. [12] were used in this study.
where R s is in MJ m −2 day −1 , n is the actual duration of sunshine (h), N is the maximum possible duration of sunshine or daylight (h), n N is the relative sunshine duration, R a is the extraterrestrial radiation (MJ m −2 day −1 ), a is the regression constant, expressing the fraction of extraterrestrial radiation reaching the earth on overcast days (n = 0), and a + b is the fraction of extraterrestrial radiation reaching the earth on clear days (n = N).

Irrigation Water Requirement
IWR is the fraction of crop water requirements that is not satisfied by rainfall, soil water storage, and the groundwater condition [14]. The crop water requirements are defined as the depth of water required by a disease-free crop growing in large fields to compensate for water loss through evapotranspiration and to achieve the full production potential for a given growing environment [14]. The IWR for paddy rice in this study is the net irrigation water requirement, which does not take into account the losses that occur in the irrigation process. The IWR was calculated using a simplified water balance equation modified from Jensen et al. [18] as follows, considering that the irrigation water used for leaching and miscellaneous water requirements in ponding paddy fields in South Korea is negligible [13].
where ET c is the actual crop evapotranspiration (mm), DP is the deep percolation (mm), and EFR is the effective rainfall (mm). The ET c of paddy rice is calculated by multiplying the empirical 10-day crop coefficients given by Yoo et al. [19] and 10-day ET 0 values estimated by Equation (1). The deep percolation of paddy fields was assumed to be 5.0 mm day −1 , reflecting the monitoring results of previous studies in South Korea [20]. The effective rainfall for paddy fields is calculated using a freeboard model [21] to simulate the ponding water depth as follows: where PD is the ponding depth (mm), IR is the irrigated water (mm), RF is the rainfall (mm), SR is the surface runoff (mm), and t is time (day). We assumed that the outlet height of the paddy field is 80 mm and that irrigation water is supplied for a controlled ponding water depth of 40 mm, except for the period of midseason drainage (i.e., when the ponding water depth is 0 mm) from 25 June to 15 July. Rainfall of less than 5 mm day −1 is considered ineffective rainfall [22].
In the IWR estimation, the water requirement for transplanting was included and assumed to be 140 mm, as suggested by the Ministry of Agriculture and Forestry in South Korea [23]. The transplanting date and irrigation periods were defined as 26 May and from 27 May to 11 September, respectively.

Design Water Requirement
DWRs to determine a reference year (i.e., drought reference design year (DRDY)), which is used for designing agricultural water facilities (e.g., irrigation canals and agricultural reservoirs) in South Korea, were calculated by following the guidelines suggested by Yoo et al. [13]. DWRs were determined from the frequency analysis of the calculated time-series of the 25 IWRs at each study site, and the reference return period was 10 years. The generalized logistic (GLO) distribution, which was the probability distribution function recommended by Yoo et al. [13] and verified again through the same procedure used in their study (not presented here), was used as the optimal probability density function to determine the DWR of a 10-year return period through Chow's frequency factor method below [24]. The parameters of the GLO distribution function, which were tested and accepted by goodness-of-fit tests (i.e., Kolmogorov-Smirnov (K-S) and probability plot correlation coefficient (PPCC) methods), were estimated using the probability weighted moments method.
where x is a variate, µ is the mean, K is the frequency factor, C v is the coefficient of variation ( σ µ ), and T is the return period. For a given return period, the frequency factor can be determined by the K-T relationship for a given probability distribution (GLO distribution in this study).
Once the DWR is determined, the year in which the estimated IWR is closest to the DWR is determined as the DRDY. Detailed information on the IWR of the DRDY (e.g., daily maximum IWR and gross IWR considering irrigation losses) is used to design the agricultural water facilities.

Reference Crop Evapotranspiration
The estimates of ET 0 using the recommended A-P coefficients were larger than the estimates using the calibrated A-P coefficients at all study sites (Table 2). Although there have been limited studies conducted to provide locally calibrated A-P coefficients in South Korea [12,25], the recommended coefficients have normally been used for the ET 0 estimation. This suggests that the ET 0 may have been overestimated so far. The results of this study indicate that the overestimation seems to have reached almost 10%. On a daily and annual basis, the ET 0 estimates using the recommended coefficients showed an average overestimation of 9.2%. In seven of the 16 study sites, the ET 0 estimates showed more than 10% overestimation. Except for one study site, Mokpo, all other sites presented more than 5% overestimation. The largest overestimation was found in Suwon (13.9%), followed by Seoul (13.0%), Chupungryeong (12.4%), and Jinju (12.4%). The smallest overestimation, 2.9%, was found in Mokpo ( Table 2). The t statistics suggested that all study sites showed a significant difference (p < 0.01) between ET 0 estimations using the different A-P coefficients on a daily basis, while 15 out of 16 sites showed a significant difference (p < 0.01) on an annual basis ( Table 2).
During the growing seasons (May to September) from 1983 to 2007, in nine of the 16 study sites, overestimation in the ET 0 estimates caused by the use of the recommended A-P coefficients was more than 10% and the average overestimation across all study sites was 10.0% ( Figure 2). The overestimation ranged from 3.8% to 14.0%. Chupungryeong showed the largest overestimation of 14.0% during the season, followed by Suwon (13.9%), Jinju (13.1%), and Seoul (12.9%), while the smallest overestimation was found in Mokpo (3.8%) ( Figure 2). During the growing seasons (May to September) from 1983 to 2007, in nine of the 16 study sites, overestimation in the ET estimates caused by the use of the recommended A-P coefficients was more than 10% and the average overestimation across all study sites was 10.0% ( Figure 3). The overestimation ranged from 3.8% to 14.0%. Chupungryeong showed the largest overestimation of 14.0% during the season, followed by Suwon (13.9%), Jinju (13.1%), and Seoul (12.9%), while the smallest overestimation was found in Mokpo (3.8%) ( Figure 3).

Irrigation Water Requirement
The effect of A-P coefficients on the estimation of IWR was smaller than that on the ET estimation. On average, the use of the recommended A-P coefficients overestimated IWR by about 5.1%, which is about half the percentage (10.0%) of the overestimation of ET during the growing season of paddy rice. This is because other important factors such as effective rainfall and deep  Table 2. Daily, monthly, and annual mean reference crop evapotranspiration (ET 0 , mm) at study sites for a 25-year period (1983-2007), and the difference in estimates between using the recommended and calibrated Ångström-Prescott (A-P) coefficients.    The monthly differences in the ET 0 estimates of the study sites located in the coastal or mountainous regions became larger during the paddy rice growing season (May to September), as shown in Table 2. Changes in the difference in the monthly mean estimates of ET 0 at the sites are indicative of a bell curve throughout the year (Figure 3). Particularly in Daegwallyeong and Chupungryeong, which are located at the relatively high altitudes (in mountainous regions) of 772.6 m and 243.7 m, respectively, the monthly variation of the difference between the ET 0 estimates was much larger than that of the other study sites. On the other hand, in the inland study sites, the monthly changes in the estimation difference showed a relatively flat shape without a large difference during the year, even though the monthly difference from October to January was relatively high as compared to the difference during spring. On average, the differences between the estimates by A-P coefficients of the inland sites were greater than the differences of the study sites located in the coastal regions ( Figure 3).

Irrigation Water Requirement
The effect of A-P coefficients on the estimation of IWR was smaller than that on the ET 0 estimation. On average, the use of the recommended A-P coefficients overestimated IWR by about 5.1%, which is about half the percentage (10.0%) of the overestimation of ET 0 during the growing season of paddy rice. This is because other important factors such as effective rainfall and deep percolation also influence the estimation of IWR besides the actual evapotranspiration calculated from the ET 0 , and they can reduce the impact of ET 0 on the estimation. The percentage of the overestimation of IWR ranged from 1.7% to 7.2%. The largest overestimation was observed in Chupungryeong and Seoul (7.2%), followed by Suwon (7.1%) and Jinju (6.6%), while the smallest overestimation was observed in Mokpo (1.7%) (Figure 4). However, only one site (Suwon) presented a statistically significant difference (p < 0.05) ( Table 3).

Figure 4.
Estimates of the irrigation water requirement (IWR) for paddy rice using the recommended (black solid line) and calibrated A-P coefficients (red solid line) for the period 1983-2007. The average difference between the two estimates of IWR for the study period is presented as a percentage in sky blue (e.g., 3.5% for Daegwallyeong).
The difference between the minimum IWRs of the two estimations using the recommended and calibrated A-P coefficients was larger than that between the maximum IWRs ( Table 3). The percentage of the difference between the minimum IWRs ranged from 2.3% to 10.7%, while the percentage of the maximum IWRs ranged from 1.5% to 7.1%. This suggests that the effects of A-P coefficients on the estimation of IWR can be reduced during the drought period. Although agricultural drought can be defined in a different way, it is normally associated with either one or both of evapotranspiration and rainfall [26,27]. A year with a large IWR can be evaluated as an agricultural drought period, since IWR is the required irrigation water considering the amount of evapotranspiration and rainfall for the ideal growth of crops. The difference between the minimum IWRs of the two estimations using the recommended and calibrated A-P coefficients was larger than that between the maximum IWRs ( Table 3). The percentage of the difference between the minimum IWRs ranged from 2.3% to 10.7%, while the percentage of the maximum IWRs ranged from 1.5% to 7.1%. This suggests that the effects of A-P coefficients on the estimation of IWR can be reduced during the drought period. Although agricultural drought can be defined in a different way, it is normally associated with either one or both of evapotranspiration and rainfall [26,27]. A year with a large IWR can be evaluated as an agricultural drought period, since IWR is the required irrigation water considering the amount of evapotranspiration and rainfall for the ideal growth of crops. Table 3. Basic statistics of the irrigation water requirements (IWRs) for the study sites over a 25-year period (1983-2007).

Design Water Requirement
DWRs were determined by using a time-series of IWRs during the growing season of paddy rice from 1983 to 2007. The DWRs determined using the recommended A-P coefficients ranged from 693.5 mm to 927.3 mm, with an average of 846.3 mm. In the case of using the calibrated A-P coefficients, the DWRs ranged from 670.2 mm to 905.8 mm, with an average of 805.8 mm. In both cases, the minimum DWR was found in Daegwallyeong, but the maximum DWR was found in Jeju when the recommended A-P coefficient was used and in Mokpo when the corrected A-P coefficient was used (Table 4). As expected, the effect of A-P coefficients on the estimation of DWR decreased slightly compared to their effects on the estimation of IWR. The DWR, assumed to be for a ten-year return period, should be a value between the maximum and mean IWRs calculated using 25-year IWRs. As discussed above, the effect of the A-P coefficient on the estimation of IWR becomes smaller as the IWR increases. Therefore, the effect of A-P coefficients on the estimation of DWR should be less than that on the IWR estimation. On average, the use of the recommended A-P coefficients resulted in a 4.8% overestimation of DWR. The percentage of overestimation ranged from 1.5% to 6.9%. The largest overestimation was observed in Seoul and Suwon (6.9%), followed by Chupungryeong (6.7%) and Jinju (6.1%), while the smallest overestimation was found in Mokpo (1.5%) ( Table 4).
The DRDY, which enables engineers to determine a reference year for designing the size of agricultural facilities and is mainly used in the design of agricultural facilities in South Korea [13], is directly related to the estimate of DWR. The difference in DWR estimates can lead to different results in DRDY estimates, which in turn affect the economics of agricultural facilities. The DRDY was determined differently by the difference in DWR estimates at four (Seoul, Incheon, Jeonju, and Gwangju) out of 16 study sites (Table 4).

Discussion and Conclusions
The FAO P-M equation is recognized as the standard method for the estimation of ET 0 and has been widely used in a variety of climates. The A-P formula is recommended to determine (global) solar radiation, one of the essential meteorological inputs of the P-M equation. However, A-P coefficients used in the equation could have a considerable impact on the estimation of ET 0 , and thus influence the estimation of IWR and DWR in agricultural water resources planning. In this study, we explored the impact of A-P coefficients on the estimation of ET 0 , IWR, and DWR by analyzing and comparing their estimates using the recommended and locally calibrated A-P coefficients in 16 locations of South Korea.
Based on our results, considerable overestimation from using the recommend A-P coefficients (i.e., a = 0.25 and b = 0.50) was verified in the estimation of ET 0 across all 16 study sites in South Korea. The overestimation ranged from 3.8% to 14.0% as compared to the estimates using the locally calibrated A-P coefficients and the average was 10.0%. All study sites were significantly different (p < 0.01) on a daily basis, and 15 out of 16 sites showed a significant difference (p < 0.01) on an annual basis. In contrast to other studies [7,10], which presented possibilities of both the over-and underestimation of ET 0 , the use of the recommended A-P coefficients in this study only resulted in ET 0 overestimation. This is presumably because of the climate region. The study sites included in the current study are located in the humid subtropical and continental climate regions, according to the Köppen-Geiger climate classification [17], while the other studies covered more various climate regions such as tundra and tropical and subtropical desert. Nevertheless, the study sites of Liu et al. [7] and Sabziparvar et al. [10], which are located in the same climate region as this study, also presented overestimated results for ET 0 estimation. The estimation of ET 0 is the basis for predicting the demand for agricultural water use in water resources planning and many other applications, which require water partitioning (e.g., hydrological modeling) or the estimation of crop water consumption (e.g., crop modeling). Given the important role and frequent use of the P-M equation in the ET 0 estimation, the accurate estimation of evapotranspiration using the locally calibrated A-P coefficients should be recommended when available.
The 10% overestimation in the ET 0 estimation during the growing season of paddy rice resulted in a 5.1% overestimation of the IWR, on average. A variation of 3.8% to 14.0% in the overestimation of the mean annual ET 0 estimation during the growing season corresponded to a variation of 1.7% to 7.2% in the overestimation of the mean annual IWR estimation. This suggests that the use of the recommend A-P coefficients can have a level of uncertainty similar to the impact of future climate change in predicting the agricultural water demand for paddy rice production in South Korea. Yoo et al. [28] predicted that climate change could lead to a change of −2.7% to 2.7% in agricultural water demand for paddy rice production in South Korea for this century. Although only one out of 16 study sites presented a significant difference (p < 0.05) in the IWR estimation, the possible overestimation for paddy irrigation would reach about 625 million tons (5% of 12.5 billion tons), considering the amount of agricultural water used for paddy rice cultivation [14]. It reaches about 12.5 billion tons a year, which accounts for almost 30% of the industrial water use (2.1 billion tons a year) in South Korea [16]. The effect of the A-P coefficients on the IWR estimation showed a tendency to decrease as the IWR increased. Therefore, the effect of A-P coefficients on the DWR estimation was slightly reduced, and it was about 4.8% on average. The DRDY, which is defined by the DWR, was determined differently by the use of A-P coefficients at four sites out of the 16 study sites. Therefore, it is necessary to examine the impact of the use of A-P coefficients more closely in terms of engineering and economics perspectives, in that the DWR estimates and the resulting DRDY directly affect the design of irrigation facilities.
There can be many uncertainties associated with the use of the P-M equation other than those concerning the A-P coefficients [29][30][31]. However, as shown in the results of this study, the use of A-P coefficients can cause considerable uncertainty in estimating ET 0 using the P-M equation and its applications. As such, this study underscores the need for the accurate consideration of the A-P coefficients in agricultural water management. As the FAO recommends [3], if the A-P coefficients can be locally calibrated, then the use of the calibrated coefficients should be considered when using the P-M equation. Estimates of other variables (e.g., the crop coefficient) being used in the process of calculating the actual evapotranspiration from ET 0 should be reestablished in order to properly use the calibrated A-P coefficients.
We have also identified some directions for future studies. The effects of climate change and calibration method on the estimation of A-P coefficients need to be explored. Climate change is expected to make a significant difference in the various meteorological factors associated with the estimation of ET 0 . Therefore, it is necessary to see how A-P coefficients will affect changes in future ET 0 driven by climate change. In addition, since the A-P coefficients can differ depending on the data and method used in their estimates [7], it is necessary to further examine the effect of the method on ET 0 estimation. Last but not least, as argued by Liu et al. [7], in the estimation of ET 0 , a similar argument can be raised. The effects of A-P coefficients on the estimation of agricultural water demand and the design of agricultural water resources system should be explored in a variety of climates.

Conflicts of Interest:
The authors declare no conflict of interest.