The N-Cornered Hat method for estimating error variances between multiple data sets: theoretical considerations and comparisons with the two-cornered hat method
Presenter:
Jeremiah P. Sjoberg
UCAR COSMIC Program
UCAR COSMIC Program
Poster
Variations of the three-cornered hat (3CH) method for estimating random error variances associated with three or more data sets have been reported in the literature for studying geophysical data sets, including sea-surface winds, sea-surface temperatures, precipitation, leaf area index and soil moisture. Anthes and Rieckh (2018) and Rieckh and Anthes (2018) used the 3CH method to estimate the error variances of multiple atmospheric sounding data sets. However the methods used often contain subtle variations and make different assumptions. Here we derive the full 3CH equations that relate the error variance of three or more observations to the mean square of the differences between the data sets, the bias errors, and the error covariances among the different data sets.
We then show that the 3CH method can be generalized to use N different data sets, where N is any integer ≥ 3. This so-called N-cornered hat method produces a single estimate of error variance for each individual data set. With the N data sets it is also possible to compute (N-1)(N-2)/2 estimates of error variances for each individual data set using the 3CH method. In general the multiple 3CH estimates do not agree exactly because of correlations of errors among the different data sets and other factors related to sample size and the co-location process. We show that the single estimate of error variance using the N-cornered hat method is equal to the mean of the (N-1)(N-2)/2 estimates of error variances for each individual data set using the 3CH method.
Finally, we show that the 3CH method actually contains as a subset three estimates of the error variances using only two of the data sets at a time. We call these subsets the two-cornered hat (2CH) method. Several studies in the literature have also used this method, sometimes called “triple co-location method,” but its relationship to the more accurate and stable 3CH method has not been shown before to our knowledge.
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