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It is important to note that the best fitted variogram model may not always give the best results in interpolation by kriging if it cannot satisfy the cross-validation criteria (Wackernagel 2003). Cross-validation test is often used in kriging to choose the ultimate form of the standard variogram models. Hence, the ultimate form of these models is chosen based on the cross-validation test and the adopted model is finally used to generate the rainfall map. In order to satisfy the cross-validation test criteria, the variogram parameters (i.e., nugget, sill, range) of the standard parametric variogram models are iteratively changed and interpolation results are re-evaluated. The cross-validation results are given in Table 3, which indicates that all adopted variogram models satisfy the cross-validation criteria and are thus appropriate for use in kriging interpolation and rainfall map production.

Table 3

Cross-validation results of estimating monthly and annual mean rainfall using different interpolation methods

Cross-validation statisticsInterpolation methodsJanuaryFebruaryMarchAprilMayJuneJulyAugustSeptemberOctoberNovemberDecemberAnnual
RMSE OKS 9.33 5.23 9.71 10.04 11.13 16.17 17.50 16.95 16.83 12.24 11.34 11.69 141.13 
OKE 10.65 6.20 11.84 12.38 12.41 19.62 21.68 21.28 20.57 15.20 13.43 13.89 171.75 
OKG 11.46 6.93 12.90 14.14 14.58 22.65 24.37 23.09 23.26 16.81 15.34 15.73 197.46 
KGP 11.10 5.48 11.52 13.16 11.49 16.42 23.14 21.30 22.98 13.17 12.23 14.70 177.26 
IDW 11.11 6.67 12.32 13.57 14.82 22.01 23.61 23.77 23.41 16.36 14.78 15.28 192.64 
ASE OKS −0.137 −0.118 −0.133 −0.140 −0.089 −0.111 −0.083 −0.077 −0.119 −0.117 −0.139 −0.145 −0.122 
OKE −0.102 −0.091 −0.096 −0.106 −0.087 −0.101 −0.087 −0.089 −0.103 −0.100 −0.097 −0.111 −0.101 
OKG −0.108 −0.102 −0.105 −0.113 −0.111 −0.114 −0.098 −0.100 −0.121 −0.105 −0.101 −0.111 −0.110 
KGP −0.369 −0.288 −0.136 −0.254 −0.201 −0.088 −0.027 0.004 −0.612 −0.247 −0.253 −0.121 −0.285 
IDW −0.155 −0.132 −0.143 −0.169 −0.172 −0.175 −0.165 −0.162 −0.174 −0.171 −0.142 −0.166 −0.169 
RMSS OKS 0.984 0.984 0.992 0.991 0.978 0.988 0.989 0.995 0.990 0.989 0.991 0.991 0.991 
OKE 1.000 1.012 1.001 1.029 1.019 1.036 1.056 1.039 1.033 1.054 1.030 1.045 1.042 
OKG 0.980 0.985 1.000 0.981 0.987 0.987 0.991 0.995 0.988 0.987 0.981 0.981 0.983 
KGP 1.144 1.127 1.108 1.080 1.047 0.896 1.107 1.131 1.427 1.023 1.069 1.053 1.127 
IDW 1.021 0.996 1.006 1.058 1.123 1.102 1.182 1.139 1.096 1.157 1.026 1.056 1.104 
Cross-validation statisticsInterpolation methodsJanuaryFebruaryMarchAprilMayJuneJulyAugustSeptemberOctoberNovemberDecemberAnnual
RMSE OKS 9.33 5.23 9.71 10.04 11.13 16.17 17.50 16.95 16.83 12.24 11.34 11.69 141.13 
OKE 10.65 6.20 11.84 12.38 12.41 19.62 21.68 21.28 20.57 15.20 13.43 13.89 171.75 
OKG 11.46 6.93 12.90 14.14 14.58 22.65 24.37 23.09 23.26 16.81 15.34 15.73 197.46 
KGP 11.10 5.48 11.52 13.16 11.49 16.42 23.14 21.30 22.98 13.17 12.23 14.70 177.26 
IDW 11.11 6.67 12.32 13.57 14.82 22.01 23.61 23.77 23.41 16.36 14.78 15.28 192.64 
ASE OKS −0.137 −0.118 −0.133 −0.140 −0.089 −0.111 −0.083 −0.077 −0.119 −0.117 −0.139 −0.145 −0.122 
OKE −0.102 −0.091 −0.096 −0.106 −0.087 −0.101 −0.087 −0.089 −0.103 −0.100 −0.097 −0.111 −0.101 
OKG −0.108 −0.102 −0.105 −0.113 −0.111 −0.114 −0.098 −0.100 −0.121 −0.105 −0.101 −0.111 −0.110 
KGP −0.369 −0.288 −0.136 −0.254 −0.201 −0.088 −0.027 0.004 −0.612 −0.247 −0.253 −0.121 −0.285 
IDW −0.155 −0.132 −0.143 −0.169 −0.172 −0.175 −0.165 −0.162 −0.174 −0.171 −0.142 −0.166 −0.169 
RMSS OKS 0.984 0.984 0.992 0.991 0.978 0.988 0.989 0.995 0.990 0.989 0.991 0.991 0.991 
OKE 1.000 1.012 1.001 1.029 1.019 1.036 1.056 1.039 1.033 1.054 1.030 1.045 1.042 
OKG 0.980 0.985 1.000 0.981 0.987 0.987 0.991 0.995 0.988 0.987 0.981 0.981 0.983 
KGP 1.144 1.127 1.108 1.080 1.047 0.896 1.107 1.131 1.427 1.023 1.069 1.053 1.127 
IDW 1.021 0.996 1.006 1.058 1.123 1.102 1.182 1.139 1.096 1.157 1.026 1.056 1.104 

RMSE, root mean square error; ASE, average standardized error; RMSS, root mean square standardized error.

OKS, ordinary kriging with spherical variogram model; OKE, ordinary kriging with exponential variogram model; OKG, ordinary kriging with Gaussian variogram model; KGP, kriging with genetic programming-based variogram model; IDW, inverse distance weighting.

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