Estimating sampling errors in large-scale temperature averages

A method is developed for estimating the uncertainty (standard error) of observed regional, hemispheric, and global-mean surface temperature series due to incomplete spatial sampling. Standard errors estimated at the grid-box level [SE2 = S-2(1 - (r) over bar)(1 + (n - 1)(r) over bar)] depend upon three parameters: the number of site records (n) within each box, the average interrecord correlation ((r) over bar) between these sites, and the temporal variability (S-2) of each grid-box temperature time series. For boxes without data (n = 0), estimates are made using values of S-2 interpolated from neighboring grid boxes. Due to spatial correlation, large-scale standard errors in a regional-mean time series are not simply the average of the grid-box standard errors, but depend upon the effective number of independent sites (N-eff) over the region. A number of assumptions must be made in estimating the various parameters, and these are tested with observational data and complementary results from multicentury control integrations of three coupled general circulation models (GCMs). The globally complete GCMs enable some assumptions to be tested in a situation where there are no missing data; comparison of parameters computed from the observed and model datasets are also useful for assessing the performance of GCMs. As most of the parameters are timescale dependent, the resulting errors are likewise timescale dependent and must be calculated for each timescale of interest. The length of the observed record enables uncertainties to be estimated on the interannual and interdecadal timescales, with the longer GCM runs providing inferences about longer timescales. For mean annual observed data on the interannual timescale, the 95% confidence interval for estimates of the global-mean surface temperature since 1951 is +/-0.12 degrees C. Prior to 1900, the confidence interval widens to +/-0.18 degrees C. Equivalent values on the decadal timescale are smaller: +/-0.10 degrees C (1951-95) and +/-0.16 degrees C (1851-1900).