Estimating Observation and Model Error Variances Using the Three-Cornered Hat Method

We combine multiple data sets, including observations and models, using the “three-cornered hat” (3CH) method to estimate vertical profiles of the errors of each system. Using data from 2007, we estimate the error variances of two versions of radio occultation (RO) retrievals¹, radiosondes, ERA-Interim, and Global Forecast System (GFS) model data sets at four radiosonde locations in the tropics and subtropics. We computed vertical profiles of estimated error variances for four variables (specific humidity q, relative humidity RH, temperature T, and refractivity N) for the five data sets using three linearly independent equations for each data set. A key assumption is the neglect of error correlations among the different data sets, and we examine the consequences of this assumption on the resulting error estimates. Using various combinations of the five data sets, we obtain three different estimates of the error variance for each data set. Ideally, with a very large sample size and zero correlation of errors among the different data sets, all three estimates would be identical. However, a finite sample size and non-zero error correlations among the data sets lead to three slightly different estimates. The differences among the three estimates is a measure of these effects.

Our results show that different combinations of the five data sets yield similar relative and specific humidity, temperature, and refractivity error variance profiles at the four stations, and these estimates are consistent with previous estimates where available. These results thus indicate that the correlations of the errors among all data sets are small and the 3CH method yields realistic error variance profiles. The estimated error variances of the ERA-Interim data set are smallest, a reasonable result considering the excellent model and data assimilation system and assimilation of high-quality observations. For the four locations studied, RO has smaller error variances than radiosondes, in agreement with previous studies. Part of the larger error variance of the radiosondes is associated with representativeness differences because radiosondes are point measurements, while the other data sets represent horizontal averages over scales of approximately 100 km.

¹In the RO-Direct version we computed water vapor from observed RO refractivity using the GFS temperature as ancillary data. In the RO 1D-VAR retrieval, temperature and water vapor are computed from the observed RO refractivity using the ERA-Interim profile as background data.

Figure: Mean of the three estimates of error variance profiles for specific humidity a), relative humidity b), temperature c) and refractivity d) using RO-Direct and RO-1D-VR for each data set at Minamidaitojima, Japan. The standard deviation about the mean is indicated by shaded areas. RO (light and dark blue), radiosonde (red), GFS (gray) and ERA-Interim (purple). From Anthes and Rieckh, 2018: Estimating observation and model error variances using multiple data sets. Atmos. Meas. Tech. 11, 4239-4260,2018 https://doi.org/10.5194/amt-11-4239-2018