Publications

Tsamalis, C (2022). Clarifications on the equations and the sample number in triple collocation analysis using SST observations. REMOTE SENSING OF ENVIRONMENT, 272, 112936.

Abstract
The triple collocation (or three-way error) analysis is a widely used approach to infer the random uncertainty of three datasets measuring the same geophysical variable. It has been applied primarily for studying the quality of satellite observations for several variables, including sea surface temperature (SST) among others. Different algorithms have been developed in the literature to calculate the random uncertainties with sometimes distinct assumptions of the underlying error model or different equations. Four of the available triple collocation (TC) algorithms have been examined here in order to clarify the assumptions behind them and investigate their mathematical equivalence. In addition, triple collocations of SST observations from drifting buoys, and satellite observations from the ATSRs and microwave (MW) imagers have been used to examine the divergence among the TC algorithms with real data. It has been found that the maximum absolute difference is less than 0.009 K for the data considered here, indicating that the choice of TC algorithm is not significant to first order. This result permitted the investigation of the impact of the number of collocations by performing statistical experiments with real observations selecting them randomly without replacement. It is established that the ideal number is at least 500. Numbers below 200 on average would underestimate the true value of random uncertainties, while for numbers of about 100 or below numerical instabilities appear preventing the convergence to a solution with this issue becoming more acute as the number becomes smaller. For the SST instruments considered, the dependence of the random uncertainty on the sample number is described adequately by a theoretical equation given by Zwieback et al. (2012). This theoretical equation has been used to study the robustness of the differences in AMSR-E random uncertainties between day and night. The differences are mainly due to wind speed dependence, manifested also as latitudinal dependence. This is further confirmed when examining the variability of the random uncertainties with wind speed and SST. The AATSR random uncertainty slightly decreases linearly with wind speed, while AMSR-E increases by up to 50% between low and high wind speeds. A non-negligible dependence with SST has been found for both AATSR and drifting buoys with higher random uncertainty for SST in the range 10 degrees-15 degrees C. The AMSR-E random uncertainty decreases linearly with SST, when SST is above about 15 degrees C. Obviously, the dependence of the random uncertainty with the geophysical variable, here SST, violates one of the assumptions behind TC. Nevertheless, this can be mitigated through the application of TC into intervals of the geophysical variable for which the independence from the random uncertainty holds. These practical results unveil the real power of TC which based on the use of three collocated datasets is able to characterise their random uncertainties in a robust way, something extremely difficult to achieve by employing only two datasets without prior knowledge.

DOI:
10.1016/j.rse.2022.112936

ISSN:
1879-0704