New spatial upscaling methods for multi-point measurements: From normal to p-normal

Careful attention must be given to determining whether the geophysical variables of interest are normally distributed, since the assumption of a normal distribution may not accurately reflect the probability distribution of some variables. As a generalization of the normal distribution, the p-normal distribution and its corresponding maximum likelihood estimation (the least power estimation, LPE) were introduced in upscaling methods for multi-point measurements. Six methods, including three normal-based methods, i.e., arithmetic average, least square estimation, block kriging, and three p-normal-based methods, i.e., LPE, geostatistics LPE and inverse distance weighted LPE are compared in two types of experiments: a synthetic experiment to evaluate the performance of the upscaling methods in terms of accuracy, stability and robustness, and a real-world experiment to produce real-world upscaling estimates using soil moisture data obtained from multi-scale observations. The results show that the p-normal-based methods produced lower mean absolute errors and outperformed the other techniques due to their universality and robustness. We conclude that introducing appropriate statistical parameters into an upscaling strategy can substantially improve the estimation, especially if the raw measurements are disorganized; however, further investigation is required to determine which parameter is the most effective among variance, spatial correlation information and parameter p.