What is the Uncertainty of the Uncertainty and (Why) Does it Matter? Improving the Uncertainty Estimates of Merged Multi-satellite Soil Moisture Data Sets

Abstract This study aims to improve the uncertainty estimates of soil moisture datasets produced by merging various satellite products via inverse-variance weighting. In this scheme, the weight of each sensor depends on its specific uncertainty derived from Triple Collocation Analysis (TCA). However, the TCA-derived uncertainties are themselves uncertain due to finite sample sizes, introducing a second-order uncertainty we denote the ‘uncertainty of the uncertainty’. Here, we estimate it empirically by bootstrapping and find that it follows a power-law relationship as a function of the number of collocated observations, whose exponent is comparable to the analytical solution for simple error models. Furthermore, we propose an extended scheme that includes the resulting uncertainty of the weights in the uncertainty estimate of the merged dataset. The proposed scheme is tested on soil moisture retrievals from three different satellite sensors, the active Advanced Scatterometer (ASCAT), the passive Soil Moisture Active Passive (SMAP), and the passive Soil Moisture And Ocean Salinity (SMOS) sensors. Comparing the improved uncertainty estimates to skill metrics calculated against the global reanalysis product ERA5-Land confirms that they indeed better describe (spatial) uncertainty variations of the merged soil moisture product against the reference dataset. The findings of this study underscore the necessity of advancing uncertainty quantification methods in satellite-retrieved climate data sets.