Extended triple collocation: Estimating errors and correlation coefficients with respect to an unknown target

Abstract

Calibration and validation of geophysical measurement systems typically require knowledge of the "true" value of the target variable. However, the data considered to represent the "true" values often include their own measurement errors, biasing calibration, and validation results. Triple collocation (TC) can be used to estimate the root-mean-square-error (RMSE), using observations from three mutually independent, error-prone measurement systems. Here, we introduce Extended Triple Collocation (ETC): using exactly the same assumptions as TC, we derive an additional performance metric, the correlation coefficient of the measurement system with respect to the unknown target, ρ t;Xi . We demonstrate that ρ 2 t;Xi is the scaled, unbiased signal-to-noise ratio and provides a complementary perspective compared to the RMSE. We apply it to three collocated wind data sets. Since ETC is as easy to implement as TC, requires no additional assumptions, and provides an extra performance metric, it may be of interest in a wide range of geophysical disciplines.

Water, Energy, and Carbon with Artificial Neural Networks (WECANN): a statistically based estimate of global surface turbulent fluxes and gross primary productivity using solar-induced fluorescence

“…TC is a method to estimate the RMSE (and, if desired, correlation coefficients) of three spatially and temporally collocated measurements by assuming a linear error model between the measurements (McColl et al, 2014;Stoffelen, 1998). This methodology has been widely used in error estimation of land and ocean parameters, such as wind speed, sea surface temperature, soil moisture, evaporation, precipitation, f APAR, and in the rescaling of measurement systems to reference system for data assimilation purposes (Alemohammad et al, 2015;D'Odorico et al, 2014;Gruber et al, 2016;Hain et al, 2011;Lei et al, 2015;Miralles et al, 2010Miralles et al, , 2011bParinussa et al, 2011), as well as in validating categorical variables such as the soil freeze-thaw state (McColl et al, 2016).…”

“…ε i is the random error in measurement i and TC estimates the variance of this random variable in each measurement. By further assuming that the errors from the three measurements are uncorrelated Cov ε i , ε j = 0, for i = j and the errors are uncorrelated with the truth (Cov (ε i , t) = 0), the RMSE of each measurement error can be calculated as (McColl et al, 2014)…”

An assessment of the performance of global rainfall estimates without ground-based observations

“…multiplicative) error model which almost certainly provides a more realistic description of rainfall accumulation errors at fine space/timescales. In addition, they calculated the theoretical correlation of each product with the unknown truth by using the extended TC (ETC) (McColl et al, 2014) by analysing the covariance matrix of the three data sets.…”

JASMIN: A prototype high resolution soil moisture analysis system for Australia

“…The automated QC scheme rejects about 2% of the observations. For surface soil wetness, verification McColl et al (2014) have developed the Extended Triple Collocation (ETC) method to estimate the temporal correlation between a model or observation system and the unknown truth. The ETC method makes the same assumptions as the more widely known Triple Collocation (TC; Scipal et al 2008;Vogelzang and Stoffelen 2012;Zwieback et al 2012;Dorigo et al 2010;Yilmaz and Crow 2014;Gruber et al 2016;Draper et al 2013) method.…”

“…McColl et al (2014) show that R i,T C is closely related to the unbiased signal to noise ratio SN R ub,i = R 2 i,T C /(1 − R 2 i,T C ) and the f RM SE metric defined by Draper et al (2013)…”