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

Abstract

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 methods may complement traditional approaches for the error characterization of soil moisture products.

Reviews and syntheses: Systematic Earth observations for use in terrestrial carbon cycle data assimilation systems

“…Recently, the TCA has been intensively elaborated, e.g. to solve for colinearities between errors (Gruber et al, 2016b) and non-linear dependencies between data sets (Zwieback et al, 2016). The most remarkable advancement has been to express TCA-based error estimates as a signal-to-noise ratio, which facilitates a direct intercomparison of the skill of data sets independent of their dynamic ranges (Gruber et al, 2016a); see Fig.…”

Estimating time-dependent vegetation biases in the SMAP soil moisture product

“…In the absence of perfect reference data, any useful error estimation procedure must cope with errors in all input data. Triple collocation and its various extensions can provide consistent error estimates in these circumstances (Gruber et al, 2016;Zwieback et al, 2016). However, similar to the standard RMSE metric, they cannot directly separate non-constant systematic errors from quasi-random measurement noise.…”

Impact of Rescaling Approaches in Simple Fusion of Soil Moisture Products

“…Among them, the CDFM (Reichle & Koster, 2004) is arguably the most common. In addition, linear regression- (Crow & Zhan, 2007), variance matching- (Draper et al, 2009), TCA- (Yilmaz & Crow, 2013), copula- (Leroux, et al, 2014), wavelet- (Su & Ryu, 2015), quadratic polynomial- (Zwieback et al, 2016), and GP-, MAR-, SVM-, ANN- (Afshar & Yilmaz, 2017) based methods have been also proposed. Intercomparison of these rescaling methods has demonstrated that MARS and SVM result in more precise rescaled products relative to other methods (Afshar & Yilmaz, 2017).…”

“…To this end, CDFM simply matches the CDF of unscaled product (e.g., Y ) to the CDF of the reference product ( X ). There are different ways available to use CDFM in order to remove the systematic differences between unscaled and reference soil moisture data sets (Zwieback et al, ). In this study, the CDFM method is applied through the calculation of the CDF for unscaled and reference soil moisture products based on ranks of their observations and conveyed into the space of reference products through the inverse relation between real observation and the CDF of the reference product.…”