Abstract

The estimation of reference evapotranspiration (ET₀) with the FAO-Penman-Monteith method faces challenges in some places due to its high data demand. To overcome this challenge some methodologies recommended by FAO. However, sharing the nearby station’s data is another way to estimate ET₀ more accurate in some cases than that of using the FAO’s recommendation. In this paper, the important matter is the determination of an effective distance (Xc) which is the upper limit of distance for data sharing between the stations. ∆ET_0(st) which is the average errors between the two stations given by the measured data is theoretically very small if the distance is zero. ∆ET_0(Alt) which is the error produced from the alternative data given by FAO’s recommendation is equal to ∆ET_0(st) at Xc. By using the data form 48 metrological stations in Japan, we examined this concept in the case of three kinds of data. The results confirmed, there was Xc exited along the investigated distance at which ∆ET_0(st) was smaller than ∆ET_0(Alt). This was the case corresponding to the solar radiation and actual vapor pressure. Xc was found smaller than the minimum distance in the case of wind data. It is, therefore, possible to use the FAO’s alternative wind data.

1. Introduction

When we discuss the climate condition for plants growth, not only soil characters but also climate conditions are essential, especially when we calculate crop water requirement. The FAO Penman-Monteith method, abbreviated as FAO-56PM in this study, is one of the well-known models for estimating requires minimum air temperature (), maximum air temperature (), wind velocity (), solar radiation () and relative humidity () ¹. However, the availability of the complete set of measured data is a big challenge for estimating in some locations worldwide ^{2, 3}. This is an extreme restriction to the application of the Penman-Monteith method ⁴.

To overcome the problem of the data lucking, especially in the case when , and are missing, there are some procedures proposed by FAO, allowing the alternative’s data to be estimated. The validity of some alternative data in the estimation was confirmed in variety of locations worldwide by many researchers ^{4, 5, 6, 7}. However, some of the alternative data were not valid in some locations, depends upon the climatic regime of a place. Ganji and Kajisa ⁸ reported that the estimation yielded with relatively higher errors when alternative and were used in the calculation compared to the alternative , in the case of humid climate of Japan. This may be the case for many locations over the globe.

To estimate more accurate than that of using the FAO’s alternative data there is a possibility to use the nearby station’s measured data when the data of a given station is missing. However, the important matter is the determination of a effective distance () which is the upper limit of distance for data sharing between the stations. This is the distance inside of that range sharing data leads smaller error than that of using the FAO’s alternative data as we are thinking. might be different of the range which can be determined by using different kind of models. One of the successful technique is using optimal approximation, which is applied in a geostatistical technique termed kriging ⁹. is the upper limit that longer that point data are no longer correlated. In this paper, from this approximation model equation and , we attempted to determine the spatially for sharing the data of , and when they are missing. The existence of was not clear before analyzing.

In this paper, is the average errors between the two places produced from the actual measured data. is theoretically very small in a case if the distance between two places is zero, and it may increase for the increasing of the distance. While is the error produced using the alternative data those given by FAO’s methodology in a given station. might be equal to at the based on our prediction. At the distance larger than , could become larger than .

The typical concept proposed in this study is illustrated in Figure 1. In Figure 1, the -axis shows the distance between the stations in (km), -axis shows , shows the model equation, shows the proper range in which data are no longer correlated, and shows the effective distance at which crosses the theoretical model equation’s graph which is given by . Considering

2. Methodology

The average meteorological data for a 30-year period used in this study were collected from the Japan metrological agency recorded in 48 places those are almost located in different prefectures over Japan, shown in Figure 2. The numbers in Figure 2 are in line with the numbers giving for each locations in Table 1. Details on elevation, coordinates and climate conditions of the locations are shown in Table 1.

Structural analysis of estimates was initially used in order to identify the spatial variability features of over Japan. As of the first step, we began with getting , computed with the values obtained from Eq. 1 for all pairs of locations separated by distance. The right side of Eq. 1 consists of two components, one is the variables’ differentiation () produced from the average difference between the measured data of two stations, given as Eq. 2 in which x is , or . The second component is the slope of the functions obtained from the average values of station 1 and 2 given as Eq. 3.

The value of the partial differential is the derivation of with respect to the variables. The second step was fitting of model equation. According to the Delhomme (1978), the well-known models are the monomial, spherical, exponential and Gaussian. In this paper, the spherical model was experimentally selected (see Eq. 4).

(1)

(2)

(3)

(4)

where, is the average error between the two places produced from the actual measured data (mm d^-1), and are the measured values in the first and second locations, respectively, 1 and 2 are the suffixes of each place first and second, is nugget effect which we considered very small in this study, is the distance between the two locations (km), means range in this paper, and means sill.

To determine the point, we computed using the error propagation approach. This approach was confirmed to approximate the root mean square error () of in Japan ⁸. was calculated using Eq. 5. This consist of, the variable’s differential yielded from the difference between measured data and alternative data at the same station (Eq. 6), and the partial differential of the function (Eq. 7). In Eq. 1 and 5, the FAO-56PM equation (Eq. 8) was transferred as Eq. 9. In Eq. 9 the components such as , and are independent, while those of to and are constant. The variables such as and were calculated with measured climatic data, given as Eqs. 10-11.

(5)

(6)

(7)

(8)

(9)

(10)

(11)

where, is the error produced from the application of the alternative data in a given station (mm d^-1), is the differentials between the measured data and alternative data in the same station, and are the measured and alternative variables in a given station, is the net radiation estimated with solar radiation data (MJ m^-2 d^-1), is the soil heat flux density (MJ m^-2 d^-1), is the psychrometric constant (kPa°C^-1), is the daily average air temperature (ºC), is the daily average wind speed (m s^-1), is the saturation vapor pressure (kPa), is the actual vapor pressure (kPa), is given by , is given by 0.340.408, is given by 0.140.408 , is given by , is equivalent to 0.35, is given by , is given by in which means the slope of the vapor pressure curve, is given by , is the albedo (0.23), is the Stefan-Boltzmann constant, is the clear-sky solar radiation (MJ m^-2 d^-1), is the mean relative humidity (%).

The FAO’s alternative methodologies are used in this paper to estimate the alternative data for the missing of , and are given as Eqs. 12-14.

(12)

(13)

(14)

where, is the solar radiation based on temperature (MJ m^-2 d^-1), is the maximum air temperature (°C), is the minimum air temperature (°C), is the extraterrestrial radiation (MJ m^-2 d^-1). is the adjustment coefficient proposed by Allen et al. (1998) as 0.16 and 0.19 for interior and coastal areas, respectively (°C^-0.5). In this study, was used for all locations since the air masses that dominates in the all locations have their origin from the surrounding sea water around, is the actual vapor pressure estimated using (kPa), and is the default world average value (ms^-1).

3. Results

Figure 3 from A to C shows the approximated curve, plots of versus the distance and as horizontal line. Table 2 listed the values for , , , , , and .

was confirmed within the investigated distance in the case of and only, shown in Figures 3A-B. While no exited within the investigated distance in the case of , shown in Figure 3C.

4. Discussion

As we expected before the analysis that , the results from the analysis met our expectation, however, was found out of the investigated distance. The results of the analysis found two different cases corresponding to the Figures 3A to C.

A and B) this is the case corresponding to the and shown in Figures 3A-B, respectively. In the case, any smaller than will mean the range inside of which sharing data will be effective, while any larger than will not mean so. Because, the approximated on the line, i.e. yielded below for , while it was yielded above for . This is implying that sharing the data among the stations within the rage of smaller than will be useful than that of using the FAO’s alternative data of and .

C) this case was found out of our expectation. was found very short and not effective. Therefore, applying the FAO’s recommended methodology for alternative was found useful. On the other hand, the average measured yielded 1.9 ms^-1 in the study area, given in Table 1 which is almost close to the FAO’s recommendation. In the case of missing we suggest to get the average in a given place if possible. Applying the average value should be very important which is free from the distance matter.

The fact that very smaller than means the alternative data recommended by FAO was much better than what we were thinking by seeing Figure 1.

5. Conclusion

Availability of the complete set of data is an extreme restriction to the application of the Penman-Monteith method in some places. Although, some producers have been recommended by FAO to estimate missing data using air temperature only, however, there is a possibility to use the nearby station’s measured data when the data of a given station is missing. The important matter is the determination of an effective distance () for data sharing. In this paper, by using the error propagation theory and experimental approximation equation we attempted to determine the spatially for sharing the data of , , and when they are missing. The existence of was not clear before the analyzing. In a examined cases of Japan, the analysis leads to the following conclusions:

1) The existence of was confirmed in the cases of and .

2) In our case, the was in the range of the measured data for and . Therefore, the shared data can be recommended at a distance smaller than , while the alternative data recommended by FAO can be selected at a distance larger than . The s were given as 2363 km and 341 km for and , respectively.

3) was smaller than any in the case of . Therefore, the alternative data recommended by FAO can be selected for the investigated distance. was given as 20.11 km which was smaller than which was 26.13 km.

References