The Impact of Quadratic Nonlinear Relations between Soil Moisture Products on Uncertainty Estimates from Triple Collocation Analysis and Two Quadratic Extensions

The error characterization of soil moisture products, for example, obtained from microwave remote sensing data, is a key requirement for using these products in applications like numerical weather prediction. The error variance and root-mean-square error are among the most popular metrics: they can be estimated consistently for three datasets using triple collocation (TC) without assuming any dataset to be free of errors. This technique can account for additive and multiplicative biases; that is, it assumes that the three products are linearly related. However, its susceptibility to nonlinear relations (e.g., due to sensor saturation and scale mismatch) has not been addressed. Here, a simulation study investigates the impact of quadratic relations on the TC error estimates [also when the products are first rescaled using the nonlinear cumulative distribution function (CDF) matching technique] and on those by two novel methods. These methods-based on error-invariables regression and probabilistic factor analysis-extend standard TC by also accounting for nonlinear relations using quadratic polynomials. The relative differences between the error estimates of the ASCAT remotely sensed product by the quadratic and the linear methods are predominantly smaller than 10% in a case study based on remotely sensed, reanalysis, and in situ measured soil moisture over the contiguous United States. Exceptions with larger discrepancies indicate that nonlinear relations can pose a challenge to traditional TC analyses, as the simulations show they can introduce biases of either sign. In such cases, the use of nonlinear