The uncertainty of break positions detected by homogenization algorithms in climate records

Long instrumental climate records suffer from inhomogeneities due to, e.g. relocations of the stations or changes in instrumentation, which may introduce sudden jumps into the time series. These inhomogeneities may mask or strengthen true trends. Relative homogenization algorithms use the difference time series of a candidate station with neighboring stations to identify such breaks (changepoints). Modern multiple breakpoint methods search for the optimum segmentation, which is characterized by minimum internal variance within the segments and maximum external variance between the segment means. We analyse the accuracy of these homogenization methods and concentrate on the uncertainty in the position of the break. Due to unavoidable random noise in the difference time series, the segmentation method may find a shifted break position, which attains a higher external variance than the true one. Different lengths of potentially exchanged subsegments are considered; that one providing the largest external variance will be chosen as possibly erroneous optimum. We will show that the variances of shifted segmentations can be approximated by a Brownian motion with drift, where the signal-to-noise ratio (SNR) defines the drift size. Available formulae for one-sided and continuous Brownian motion with drift are expanded to two-sided and discrete processes as they occur in praxis. The error probability increases strongly for SNRs lower than 1. Thus, when the internal variance is larger than the variance introduced by the breaks, the probability of finding the right break position is small.