Beyond triple collocation: Applications to soil moisture monitoring

Triple collocation (TC) is routinely used to resolve approximated linear relationships between different measurements (or representations) of a geophysical variable that are subject to errors. It has been utilized in the context of calibration, validation, bias correction, and error characterization to allow comparisons of diverse data records from various direct and indirect measurement techniques including in situ remote sensing and model-based approaches. However, successful applications of TC require sufficiently large numbers of coincident data points from three independent time series and, within the analysis period, homogeneity of their linear relationships and error structures. These conditions are difficult to realize in practice due to infrequent spatiotemporal sampling of satellite and ground-based sensors. TC can, however, be generalized within the framework of instrumental variable (IV) regression theory to address some of the conceptual constraints of TC. We review the theoretics of IV and consider one possible strategy to circumvent the three-data constraint by use of lagged variables (LV) as instruments. This particular implementation of IV is suitable for circumstances where multiple data records are limited and the geophysical variable of interest is sampled at time intervals shorter than its temporal correlation length. As a demonstration of utility, the LV method is applied to microwave satellite soil moisture data sets to recover their errors over Australia and to estimate temporal properties of their relationships with in situ and model data. These results are compared against standard two-data linear estimators and the TC estimator as benchmark.