3.1 Statistical results of different evapotranspiration methods
Figure 2 compares the daily scale statistical results of the 12 evapotranspiration calculation methods and the P-M method under default conditions. As observed, applicability of these methods varied significantly. The Turc method exhibited best applicability, followed by the P-T method. Indeed, RMSE of data by the Turc method and the P-T method were significantly lower than those of other methods, while d-index of data of the two methods were significantly higher than those of other methods. The medians of NRMSE of data obtained by the Turc method and the P-T method were 0.16 and 0.20, respectively. According to MBE, deviations of under-estimation and over-estimation were relatively small for both methods, indicating better performance of the Turc method. Nevertheless, outliers in box plots demonstrated that accuracy of data collected by certain stations by the Turc method was lower than that by the P-T method. The average RMSE of data by other 10 methods exceeded 0.75 mm/d and these methods were not applicable in Jiangxi. Especially, the 48PM method and the B-S method had identical requirements on the integrity of climatological data as the P-M method, but their accuracies were poor (the accuracy of the 48PM method is slightly higher than that of the B-S method). For the 48PM method, median of RMSE was approximately 2 mm/d, d-index was below 0.9, NRMSE was significantly higher than 0.4, and MBE was over-estimated by 1.5 mm/d. For the B-S method, most RMSE were above 6 mm/d, d-index was below 0.6, NRMSE was significantly higher than 0.4, and MBE was over-estimated by 4 mm/d. In summary, the methods other than the Turc method and the P-T method could not be directly applied in Jiangxi before localized redetermination.
3.2 Redetermined statistical results of different evapotranspiration calculation methods
Figure 2 compares the redetermined daily scale statistical results of 12 evapotranspiration calculation methods and the P-M method. As observed, the accuracy of all evapotranspiration calculation methods was significantly improved after redetermination and the amplitudes varied significantly. Herein, the B-S method showed the greatest improvement (minimum RMSE, all medians < 0.1 mm/d, d-index ≈ 1, NRMSE < 10%) and optimized applicability after model redetermination (close to that of the P-M method). Additionally, RMSE of data by the 48PM method was low and the NRMSE was below 10% (except Nanchang, which may be attributed to over-estimation, according to MBE).
The RMSE, d-index, and NRMSE of data by the P-T method, the I-A method, and the PVB method were close to each other. The accuracy of the P-T method, the I-A method, and the PVB method was lower than that of the B-S method and the 48PM method while higher than that of other methods. Herein, under-estimation of the P-T method was 0 ~ 1.4 mm/d, that of the PVB method was below 0.3 mm/d, and that of the I-A method was negligible. In terms of the climatological data required, the PVB method has the highest requirement on the integrity of data set: the data of relative humidity and wind speed (not required in the P-T method and the I-A method) were required for calculation of actual vapor pressures and canopy resistances. However, the PVB method exhibited no significant advantages in accuracy over the P-T method and the I-A method. Therefore, the PVB method was not applicable in Jiangxi.
The Mak method shared a similar structure with the P-T method. RMSE and NRMSE of the Mak method was slightly higher than that of the P-T method, the I-A method, and the PVB method; d-index of the Mak method was slightly lower than that of the P-T method, the I-A method, and the PVB method, while MBE of the Mak method indicated neither over-estimation nor under-estimation. In summary, the Mak method was slightly inferior to the P-T method in terms of applicability. Considering the fact that the requirements on climatological data by the Mak method and the P-T method were consistent, the P-T method was preferred in this area.
The accuracy of the Turc method and the Mak method were consistent, while parameter redetermination brought negligible improvements on the accuracy of the Turc method. For the Turc method, the defaults of RMSE and NRMSE were 0.389 mm/d and 0.157 mm, respectively, while the redetermined RMSE and NRMSE were 0.375 mm/d and 0.151 mm, respectively. In other words, the Turc method could be directly applied in Jiangxi (except Nanchang, where RMSE = 0.972 mm/d) without parameter redetermination. The average RMSE of data by the H-S method before and after parameter reetermination were 0.75 mm/d and 0.70 mm/d, respectively, indicating negligible accuracy improvement by parameter redetermination.
The average RMSE of data by the B-R method and the M-B-R method before parameter redetermination were 1.49 mm/d and 0.91 mm/d, respectively, while the rate-determined average RMSE of data by the B-R method and the M-B-R method were 0.85 mm/d and 0.55 mm/d, respectively. This demonstrated significant accuracy improvement by parameter redetermination for the B-R method and the M-B-R method. Herein, the B-R method required temperature data but its accuracy was lower than that of the H-S method, indicating that the B-R method was not applicable in Jiangxi. In the presence of data about relative humidity, the M-B-R method showed better accuracy compared with the H-S method. However, the M-B-R method required five empirical parameters and this may be a major factor limiting the model application. The accuracy of the Sch method was still relatively low after parameter redetermination: the average RMSE was approximately 0.8 mm/d, the d-index ranged in 0.85 ~ 0.95, and the average NRMSE was above 0.3. Hence, the Sch method was not applicable in Jiangxi.
The J-H method requires sunshine data and this increases the model complexity. However, RMSE of the J-H method was increased by 0.3 mm/d compared with the I-A model, which required similar data with the J-H method, and accuracy improvement over the H-S model, which requires temperature data only, was limited. Therefore, the J-H method showed poor applicability and was not applicable in Jiangxi.
3.3 Redetermined parameters of different evapotranspiration calculation methods
The parameters were determined based on the results by the P-M method, and the linear regressions of different evapotranspiration calculation methods to the P-M method were established using the least square method. To simplify calculation methods, the secondary regression was applied for some methods. Specifically, if CV is relatively small after primary regression, average of values collected by different stations is defined as the value of this specific parameter and other parameters are using the least square method. Table 2 summarizes redetermined parameters of different evapotranspiration calculation methods.
Table 2
Calibration parameter value of different reference crop evapotranspiration methods (48PM, H-S, P-T, I-A, Mak, PVB and B-R)
Station name
|
48PM
|
H-S
|
P-T
|
I-A
|
Mak
|
PVB
|
B-R
|
a
|
a
|
b
|
c
|
α
|
a
|
a
|
a
|
b
|
c
|
a
|
b
|
c
|
d
|
Xiushui
|
0.91
|
0.0013
|
0.70
|
15.6
|
1.08
|
-0.63
|
-0.15
|
113
|
-19.1
|
-0.36
|
0.09
|
0.07
|
0.04
|
1.82
|
Yifeng
|
0.81
|
0.0011
|
0.80
|
12.7
|
1.06
|
-0.68
|
-0.12
|
94
|
-20.0
|
-0.49
|
0.09
|
0.09
|
0.04
|
2.06
|
Lianhua
|
0.90
|
0.0015
|
0.72
|
9.1
|
1.08
|
-0.63
|
-0.16
|
152
|
-14.3
|
-0.31
|
0.10
|
0.08
|
0.05
|
2.14
|
Yichun
|
1.01
|
0.0013
|
0.74
|
17.6
|
1.12
|
-0.55
|
-0.24
|
122
|
-28.8
|
-0.91
|
0.10
|
0.11
|
0.04
|
1.82
|
Ji’an
|
1.17
|
0.0018
|
0.71
|
8.4
|
1.16
|
-0.44
|
-0.33
|
667
|
55.7
|
2.35
|
0.12
|
0.08
|
0.03
|
1.83
|
Jinggangshan
|
0.83
|
0.0009
|
0.81
|
28.5
|
1.05
|
-0.69
|
-0.08
|
-81
|
-52.0
|
-1.60
|
0.06
|
0.12
|
0.05
|
1.78
|
Suichuan
|
1.18
|
0.0017
|
0.67
|
12.4
|
1.17
|
-0.45
|
-0.33
|
624
|
45.4
|
1.88
|
0.11
|
0.10
|
0.04
|
1.99
|
Ganzhou
|
1.14
|
0.0022
|
0.62
|
7.6
|
1.15
|
-0.44
|
-0.31
|
550
|
42.9
|
2.22
|
0.12
|
0.07
|
0.04
|
1.94
|
Lushan
|
1.06
|
0.0017
|
0.72
|
24.4
|
1.15
|
-0.45
|
-0.19
|
-379
|
-113.7
|
-3.43
|
0.12
|
0.14
|
0.00
|
0.60
|
Wu’an
|
1.06
|
0.0012
|
0.73
|
21.8
|
1.12
|
-0.53
|
-0.24
|
131
|
-20.7
|
-0.22
|
0.09
|
0.10
|
0.04
|
1.70
|
Boyang
|
1.18
|
0.0018
|
0.73
|
13.2
|
1.18
|
-0.35
|
-0.37
|
439
|
17.1
|
1.48
|
0.13
|
0.11
|
0.02
|
1.61
|
Jingdezhen
|
1.14
|
0.0014
|
0.72
|
14.5
|
1.15
|
-0.47
|
-0.28
|
271
|
-4.0
|
0.38
|
0.11
|
0.09
|
0.04
|
1.93
|
Jing’an
|
1.15
|
0.0009
|
0.92
|
21.4
|
1.15
|
-0.46
|
-0.29
|
-115
|
-81.7
|
-3.01
|
0.10
|
0.14
|
0.04
|
2.08
|
Nanchang
|
1.19
|
0.0019
|
0.67
|
17.3
|
1.17
|
-0.37
|
-0.37
|
141
|
-38.7
|
-0.95
|
0.13
|
0.10
|
0.01
|
1.15
|
Zhangshu
|
1.03
|
0.0018
|
0.68
|
9.9
|
1.12
|
-0.54
|
-0.24
|
395
|
27.6
|
1.58
|
0.11
|
0.08
|
0.03
|
1.78
|
Dexing
|
1.00
|
0.0012
|
0.75
|
12.2
|
1.09
|
-0.60
|
-0.20
|
77
|
-29.2
|
-0.85
|
0.10
|
0.08
|
0.04
|
1.95
|
Guixi
|
1.21
|
0.0017
|
0.69
|
13.2
|
1.18
|
-0.41
|
-0.33
|
527
|
21.1
|
0.74
|
0.12
|
0.09
|
0.02
|
1.71
|
Yushan
|
1.14
|
0.0015
|
0.70
|
17.8
|
1.15
|
-0.45
|
-0.30
|
257
|
-11.3
|
0.01
|
0.11
|
0.09
|
0.03
|
1.69
|
Shangrao
|
1.08
|
0.0015
|
0.72
|
14.3
|
1.13
|
-0.50
|
-0.23
|
261
|
-5.7
|
0.13
|
0.11
|
0.09
|
0.03
|
1.93
|
Yongfeng
|
1.04
|
0.0016
|
0.68
|
9.3
|
1.11
|
-0.56
|
-0.25
|
257
|
-1.0
|
0.21
|
0.11
|
0.07
|
0.04
|
1.88
|
Nancheng
|
1.14
|
0.0019
|
0.68
|
9.8
|
1.17
|
-0.41
|
-0.37
|
593
|
37.3
|
1.56
|
0.12
|
0.08
|
0.03
|
1.76
|
Nanfeng
|
1.16
|
0.0020
|
0.65
|
7.6
|
1.16
|
-0.45
|
-0.33
|
682
|
60.2
|
2.63
|
0.12
|
0.06
|
0.03
|
1.86
|
Ningdu
|
1.11
|
0.0016
|
0.70
|
13.6
|
1.15
|
-0.44
|
-0.32
|
84
|
-43.7
|
-1.20
|
0.12
|
0.08
|
0.03
|
1.90
|
Guangchang
|
1.00
|
0.0016
|
0.70
|
9.0
|
1.11
|
-0.57
|
-0.22
|
260
|
1.8
|
0.44
|
0.11
|
0.07
|
0.04
|
1.93
|
Longnan
|
1.11
|
0.0016
|
0.63
|
19.4
|
1.09
|
-0.55
|
-0.25
|
-162
|
-65.1
|
-1.40
|
0.10
|
0.07
|
0.04
|
1.58
|
Xunwu
|
1.07
|
0.0013
|
0.68
|
21.0
|
1.08
|
-0.59
|
-0.19
|
-28
|
-33.0
|
0.22
|
0.09
|
0.08
|
0.04
|
1.78
|
Maximum
|
1.21
|
0.0022
|
0.92
|
28.47
|
1.18
|
-0.35
|
-0.08
|
681.9
|
60.2
|
2.63
|
0.13
|
0.14
|
0.05
|
2.14
|
Minimum
|
0.81
|
0.0009
|
0.62
|
7.59
|
1.05
|
-0.69
|
-0.37
|
-378.6
|
-113.7
|
-3.43
|
0.06
|
0.06
|
0.00
|
0.60
|
Average
|
1.07
|
0.0015
|
0.71
|
14.67
|
1.13
|
-0.51
|
-0.26
|
228.1
|
-10.5
|
0.04
|
0.11
|
0.09
|
0.03
|
1.78
|
The redetermined parameter a of the 48PM method ranged in 0.81 ~ 1.21 in Jiangxi and was lower than the default value (6.43), demonstrating that the contribution to the reference crop evapotranspiration by the aerodynamic term would be over-estimated when this method was applied in Jiangxi. Indeed, the result was over-estimated by 1 ~ 2.2 mm/d. After redetermination, under-estimations of most stations were below 0.05 mm/d while few were around 0.18 mm/d. Future studies may divide Jiangxi into several districts according to the climate and align parameters within the specific district. In this way, applicability of a model may be improved without affecting its accuracy (Table 2).
The redetermined parameter a of the H-S model ranged in 0.0009 ~ 0.0022 in Jiangxi and was lower than the default value (0.0023); parameter b of the rate determined H-S model ranged in 0.62 ~ 0.92 and was higher than the default value (0.5), indicating that the daily temperature range contributed significantly to the reference crop evapotranspiration in Jiangxi; parameter c of the rate determined H-S model ranged in 7.59 ~ 28.5 and the highest and second highest value appeared in the Lushan Mountain and the Jinggangshan Mountain, indicating significant effects of altitude on this parameter (Table 2).
Parameter a of the P-T method ranged in 1.05 ~ 1.18 and was lower than the default value (1.26), indicating that the contribution on the reference crop evapotranspira1tion by the aerodynamic term was relatively low when this method was applied in Jiangxi. Specifically, the contribution by the aerodynamic term was less than 20% of that by the radiation term without considering errors (Table 2).
In this study, after parameter redetermination of the I-A method, parameter b and c were set as 0.32 and 0.015, respectively, and parameter a ranged in -0.69~-0.35. Similar to the I-A method, parameter b of the Mak method ranged in -0.37~-0.08 with parameter a being set to be 0.66. Additionally, terms in the I-A method and the Mak method are constant and could only be used for correction of overall deviations of the ultimate results. As a result, MBE of the I-A method and the Mak method was 0 (see Fig. 3d).
Parameter a, b, and c of the PVB method varied significantly within − 378.6 ~ 681.9, -113.7 ~ 60.2, and − 3.43 ~ 2.63, respectively. Hence, the esp calculated based on average temperature was highly unreliable, resulting in poor model accuracy (Table 2).
The B-R method has four parameters. According to the calculation equation structure, coefficients of the high temperature term, the daily temperature range term, and the extraterrestrial radiation term were above 0.1. Despite their specific units, contributions of these three parameters to reference crop evapotranspiration were in similar scale. Also, redetermined coefficients of the extraterrestrial radiation term were basically less than half of the high temperature term, the daily temperature range term for all stations, indicating the dominant role of temperature in application of this model in Jiangxi. Contributions of the extraterrestrial radiation terms to reference crop evapotranspiration were small and the accuracy was relatively low (Table 2).
By introducing the difference term of the saturation vapor pressure and the actual vapor pressure into the B-R method, the M-B-R method has five parameters and average coefficient of the extraterrestrial radiation term was 20% higher than that of the daily temperature range term (Table 3). In other words, unlike the B-R method, the contribution of radiation on reference crop evapotranspiration was significant in the M-B-R method and its accuracy was significantly higher than that of the B-R method.
Table 3
Calibration parameter value of different reference crop evapotranspiration methods (M-B-R, Sch, Turc, J-H and B-S)
Station name
|
M-B-R
|
Sch
|
Turc
|
J-H
|
B-S
|
a
|
b
|
c
|
d
|
e
|
a
|
a
|
a
|
b
|
c
|
a
|
b
|
Xiushui
|
0.021
|
0.050
|
2.51
|
-0.017
|
1.20
|
10.2
|
15.84
|
1.026
|
0.018
|
-0.030
|
-0.972
|
1.209
|
Yifeng
|
0.037
|
0.052
|
2.78
|
-0.012
|
1.27
|
10.2
|
15.65
|
0.220
|
0.084
|
-0.139
|
-1.079
|
1.009
|
Lianhua
|
0.035
|
0.055
|
2.36
|
-0.022
|
1.48
|
10.3
|
15.92
|
0.021
|
1.020
|
1.747
|
-1.012
|
1.138
|
Yichun
|
0.033
|
0.050
|
2.61
|
-0.017
|
1.20
|
10.5
|
16.48
|
0.041
|
0.557
|
1.456
|
-1.288
|
0.743
|
Ji’an
|
0.032
|
0.055
|
2.60
|
-0.012
|
1.26
|
10.6
|
17.21
|
0.006
|
3.524
|
0.241
|
-1.141
|
0.953
|
Jinggangshan
|
0.035
|
0.044
|
2.48
|
-0.032
|
1.09
|
10.7
|
15.68
|
0.048
|
0.509
|
0.952
|
-1.271
|
0.673
|
Suichuan
|
0.040
|
0.050
|
2.48
|
-0.014
|
1.17
|
10.5
|
17.16
|
0.307
|
0.066
|
-0.054
|
-1.063
|
1.081
|
Ganzhou
|
0.038
|
0.056
|
2.33
|
-0.014
|
1.39
|
10.2
|
16.78
|
0.127
|
0.168
|
0.215
|
-0.992
|
1.220
|
Lushan
|
0.021
|
0.045
|
3.25
|
-0.027
|
0.98
|
11.4
|
16.28
|
0.518
|
0.041
|
-0.111
|
-2.355
|
0.237
|
Wu’an
|
0.050
|
0.052
|
2.52
|
-0.015
|
1.36
|
10.4
|
16.49
|
-0.067
|
-0.306
|
0.163
|
-1.059
|
1.008
|
Boyang
|
0.050
|
0.051
|
2.61
|
-0.019
|
1.28
|
11.1
|
17.37
|
-0.160
|
-0.127
|
0.130
|
-1.421
|
0.670
|
Jingdezhen
|
0.048
|
0.055
|
2.23
|
-0.021
|
1.45
|
10.4
|
16.77
|
0.536
|
0.041
|
0.049
|
-1.031
|
1.101
|
Jing’an
|
0.060
|
0.053
|
2.64
|
-0.016
|
1.48
|
10.7
|
16.89
|
0.073
|
0.318
|
0.714
|
-1.079
|
1.006
|
Nanchang
|
0.042
|
0.050
|
2.62
|
-0.018
|
1.20
|
10.8
|
17.21
|
0.035
|
0.663
|
1.852
|
-1.424
|
0.665
|
Zhangshu
|
0.052
|
0.054
|
2.23
|
-0.024
|
1.43
|
10.5
|
16.51
|
0.279
|
0.068
|
-0.108
|
-1.167
|
0.886
|
Dexing
|
0.039
|
0.045
|
2.56
|
-0.020
|
1.20
|
10.3
|
16.15
|
0.286
|
0.071
|
0.005
|
-1.036
|
1.108
|
Guixi
|
0.042
|
0.051
|
2.30
|
-0.010
|
1.12
|
10.2
|
17.15
|
-0.340
|
-0.057
|
0.110
|
-1.091
|
1.045
|
Yushan
|
0.032
|
0.046
|
2.48
|
-0.017
|
1.01
|
10.7
|
16.93
|
0.086
|
0.269
|
0.657
|
-1.252
|
0.814
|
Shangrao
|
0.027
|
0.052
|
2.44
|
-0.019
|
1.21
|
10.6
|
16.59
|
0.043
|
0.556
|
2.124
|
-1.168
|
0.893
|
Yongfeng
|
0.031
|
0.049
|
2.75
|
-0.016
|
1.19
|
10.6
|
16.54
|
0.111
|
0.189
|
0.198
|
-1.102
|
0.987
|
Nancheng
|
0.027
|
0.042
|
3.13
|
-0.014
|
0.91
|
11.3
|
17.48
|
0.005
|
4.529
|
0.220
|
-1.334
|
0.709
|
Nanfeng
|
0.021
|
0.050
|
2.79
|
-0.013
|
1.09
|
10.8
|
17.22
|
0.729
|
0.027
|
-0.036
|
-1.150
|
0.931
|
Ningdu
|
0.032
|
0.050
|
2.61
|
-0.021
|
1.21
|
10.9
|
16.92
|
0.653
|
0.031
|
-0.024
|
-1.393
|
0.674
|
Guangchang
|
0.025
|
0.049
|
2.73
|
-0.014
|
1.12
|
10.5
|
16.33
|
-0.242
|
-0.080
|
0.087
|
-1.068
|
1.051
|
Longnan
|
0.030
|
0.051
|
2.82
|
-0.018
|
1.34
|
10.2
|
16.05
|
0.167
|
0.126
|
0.107
|
-1.179
|
0.881
|
Xunwu
|
0.026
|
0.052
|
2.84
|
-0.017
|
1.40
|
10.1
|
15.73
|
0.062
|
0.383
|
1.326
|
-1.114
|
0.970
|
Maximum
|
0.06
|
0.06
|
3.25
|
-0.01
|
1.48
|
11.4
|
17.47
|
1.03
|
4.53
|
2.12
|
-0.97
|
1.22
|
Minimum
|
0.02
|
0.04
|
2.23
|
-0.03
|
0.91
|
10.0
|
15.65
|
-0.34
|
-0.31
|
-0.14
|
-2.36
|
0.24
|
Average
|
0.04
|
0.05
|
2.61
|
-0.02
|
1.23
|
10.6
|
16.59
|
0.18
|
0.49
|
0.46
|
-1.20
|
0.91
|
The parameters of the Sch method ranged in 10.0 ~ 11.4 and were lower than the default value (16). (Table 3) However, the accuracy of this method was low, suggesting that this method neglects some key factors although the effects of temperature and relative humidity have been considered. As a result, applicability of the Sch method in Jiangxi was poor. For the Turc method, parameter b ranged in 15.65 ~ 17.47 if parameter a was set to be 0.018. Considering the fact that redetermination led to limited improvement on the model accuracy, it was concluded that the Turc method could be directly applied in Jiangxi. Parameter a, b, and c of the J-H method ranged in -0.34 ~ 1.03, -0.31 ~ 4.53, and − 0.14 ~ 2.12, respectively, and the parameter values varied significantly. This demonstrated that the J-H method was not reliable, thus not applicable in Jiangxi (Table 3).
With parameter a of the B-S method set to be 0.69, parameter b and c ranged in -2.36~-0.97 and 0.24 ~ 1.22, respectively, and the contribution of the wind speed on the aerodynamic term was adjusted by these two parameters. Moreover, as the aerodynamic term in this method was negative, parameter b should be negative to bring positive contribution to the aerodynamic term (Table 3).
3.4 Redetermined monthly values by different evapotranspiration calculation methods
Figure 4 shows deviations of monthly values by different evapotranspiration methods after redetermination. As observed, the monthly variation was less than 3 mm for the 48PM method, indicating good applicability. For the H-S method, deviations in April to August ranged in 4 ~ 10 mm and errors would be exacerbated in calculations of water demands of crops as crop coefficients were usually larger than 1 in this period. The absolute deviation of the P-T method was relatively high in October to December. However, this period was not the major water consumption period for crops, the P-T method was applicable in Jiangxi. The monthly absolute deviations of the I-A method were highly consistent (< 4 mm even in April to August), indicating the good applicability of the I-A method in Jiangxi. The absolute deviation of the Mak method was relatively high in July and August (under-estimation was 8.9 mm for July and 5.9 mm for August), demonstrating that the Mak method was inferior to the P-T method in terms of applicability in Jiangxi. The monthly absolute deviations of the PVB method were highly consistent (< 3 mm in May to September). The B-R method exhibited over-estimations of 6 mm in May to July and severe under-estimations in July and August. Compared with the B-R method, the M-B-R method exhibited significant reductions of monthly absolute deviations, although under-estimations of 5 mm were observed in July and August. The monthly absolute deviations of the Sch method were above 5 mm and maximized in July to November. Redetermination by month may enhance the applicability of this model. The monthly absolute deviations of the Turc method were highly consistent (< 5 mm) and maximized deviations appeared in January and July. The J-H method exhibited under-estimations of 10 mm in October to April and over-estimations of 5 mm in July and August, demonstrating that the J-H method was not applicable in Jiangxi. The monthly absolute deviations of the B-S method were small, demonstrating good applicability of this method in Jiangxi.
3.5 Annual values by improved evapotranspiration methods
In the background of global climate change, model applicability may vary significant with time. Figure 5 shows the model applicability in Jiangxi with interannual deviation, which facilitated the study. The annual ET0 calculated by various methods exhibited significant variation around 2003. Herein, the 48PM method exhibited over-estimations of 20 ~ 30 mm p.a. in 1961 to 2002 and 10 mm p.a. below after 2003. The H-S method exhibited under-estimation before 1990s, over-estimation of 10 ~ 40 mm p.a. in 1995 to 2002, and under-estimation of 0 ~ 25 mm p.a. after 2003. The P-T method exhibited negligible deviations before 2005 and under-estimation of 40 mm p.a. after 2005.
The Mak method, the PVB method, the B-R method, the M-B-R method, and the Sch method shared similar trends in long term: the results shifted gradually from under-estimation to over-estimation before 2003, and the trends changed significantly after 2004. More specifically, annual under-estimation of the PVB method and the Mak method ranged in 20 ~ 50 mm and the absolute deviation of the B-R method ranged in -30 ~ 30 mm. The M-B-R method exhibits significant over-estimation (70 ~ 110 mm) after 2003. The Sch method exhibits significant over-estimation (above 50 mm) after 1995. Limited by severe deviations in certain periods, neither of the five methods mentioned above could precisely reflect trends of reference crop evapotranspiration. The Turc method exhibited small annual deviations before 2003, but the significant under-estimation (25 ~ 50 mm) were observed after 2005. The J-H method exhibited annual under-estimations above 100 mm and it was further exacerbated after 2005. Therefore, the J-H method was not applicable in Jiangxi. The B-S method exhibited annual under-estimation in 0 ~ 10 mm, indicating that the overall deviation of the B-S method was minimum among that of all 12 methods. The long term ET0 curve of the B-S method was highly consistent with that of the P-M method.
3.6 Spatial distribution of errors in different models
Figure 6 shows spatial distribution of NRMSE of the PM method and other ET0 calculation methods in Jiangxi. NRMSE was maximized in the middle and north part of Jiangxi in all models, while minimized NRMSE appeared in different areas. For instance, minimized NRMSE was located in the east and west mountains in the H-S method, the P-T method, the I-A method, the PVB method, and the Turc method, while minimized NRMSE was located in southern Jiangxi in the 48PM method, the Mak method, the B-R method, the M-B-R method, the Sch method, and the B-S method. For the J-H method, NRMSE was maximized in northern and southern Jiangxi (north > south) and minimized in the middle part.
Overall, the B-S method and the 48P-M method shows optimized accuracy (maximized NRMSE < 5%), followed by the P-T method, the I-A method, the Mak method, the PVB method, and the Turc method (NRMSE = 0.1 ~ 0.2). In all these methods, overall surface radiation or net radiation has been considered and led to significant improvement for the accuracy. Among temperature-based methods, the J-H method was slightly better than the H-S method, while NRMSE of data by other methods ranged in 0.3 ~ 0.4, indicating that applicability of these methods was limited.