Comparing the Three Cornered and Two Cornered Hat Methods for Estimating Error Variances Using Simulated Data Sets with Known Errors

In this paper we compare two different methods of estimating the error variances of two or more independent data sets. One method, called the “three-cornered hat” (3CH) method, requires three data sets. Another method, which we call the “two-cornered hat” (2CH) method, requires only two data sets. Both methods have been used in previous studies to estimate the error variances associated with a number of physical and geophysical data sets. A key assumption in both methods is that the errors of the data sets are not correlated.

We compare the 3CH and 2CH methods using a simple error model to simulate two or three data sets with various error correlations and biases. The model consists of a set of 1460 assumed perfect (truth) vertical profiles of specific humidity based on the annual (2007) mean ERA-Interim reanalysis sounding at the subtropical location of Minamidaitojima, Japan. We create data sets X, Y and Z by superimposing prescribed errors on the true profile for the three data sets. We assume various degrees of correlation of the errors associated with Z and those with X. The errors of Y are assumed to be uncorrelated with the X and Z errors. With this model, we know the exact error variances and covariances, which we use to assess the accuracy of the 3CH and 2CH estimates. We examine the sensitivity of the estimated error variances to the degree of error correlation between two of the data sets, as well as the sample size. We find that the 3CH method is less sensitive to random errors and sample size than the 2CH method and hence is more accurate. We also find that biases in one of the data sets has a minimal effect on the 3CH method, but can produce large errors in the 2CH method. An example for a relatively high correlation of Z and X errors of 0.45 is given in the figure below.

Figure: Vertical profiles of estimated and exact normalized error standard deviations for (a) 3CH method and (b) 2CH method for a correlation coefficient of Z and X errors of 0.45 (represented by an error model parameter a = 0.5). (a) Exact STD errors for X, Y and Z given by blue, red and orange solid lines respectively. Estimated error STD error using the 3CH method given by dashed lines of same color. (b) Exact error STD of Z given by solid black line. Estimated error STD of X using Z or Y and the 2CH method given by blue lines, estimated error STD of Y using X or Z and the 2CH method given by red lines, and estimated error STD of Z using X or Y and the 2CH method given by orange lines. For all colored lines, error terms are neglected. Solid lines indicate estimates from combinations of data sets with uncorrelated errors, dashed lines indicate estimates from combinations of data sets with correlated errors and hence larger error terms. From Rieckh and Anthes, 2018: Evaluating two methods of estimating error variances using simulated data sets with known errors. Atmos. Meas. Tech., 11, 4309-4325,2018 https://doi.org/10.5194/amt-11-4309-2018