Block Kriging With Measurement Errors: A Case Study of the Spatial Prediction of Soil Moisture in the Middle Reaches of Heihe River Basin

Block kriging (BK) is a common method of predicting the true value at the pixel scale when validating remote sensing retrieval products. However, measurement errors (MEs) increase the prediction uncertainty. In this letter, an extended interpolation technique - BK with MEs (BKMEs) - is developed. The properties of BKME are proven through derivation and demonstrated in a case study of soil moisture (SM) upscaling. Three prediction scenarios - one without MEs (BK), BK with homogeneous MEs (BKHOME), and BK with heterogeneous MEs (BKHEME) - are considered for the upscaling of SM data observed by a distributed wireless sensor network, and the results are compared. Both BK and BKHOME yield the same upscaling results, which differ from those of BKHEME, and the prediction results of BKHEME show less bias than those of the other scenarios. Because both BKHOME and BKHEME consider MEs, their prediction results show smaller kriging variances than do the BK results. Three primary conclusions are drawn. The first is that the optimal kriging coefficients assigned to the observations are affected not only by spatial distance but also by the MEs when the MEs of the samples are unequal. The second is that when the MEs are equal, it may not be necessary to consider the MEs to predict the value for an unobserved location. The third is that although the prediction uncertainty can be reduced by considering MEs, it is more meaningful to consider unequal MEs than equal MEs in the prediction process. BKME is an advanced upscaling method that achieves improved prediction accuracy by considering MEs.