Estimation of Break and Noise Variance and the Maximum Distance of Climate Stations Allowed in Relative Homogenisation of Annual Temperature Anomalies

ABSTRACT Relative homogenisation uses the difference time series of neighbouring climate stations in order to reduce the dominating variance of the climate signal. Larger distances between the stations reduce the signal‐to‐noise ratio (SNR) because the noise variance increases while the break variance remains constant. For different continents, we derive the distance where noise and break variance become equal (SNR = 1). This is an estimate for the maximum allowed distance because SNRs below 1 bear the risk that the homogenisation produces nonsense results. Technically, the break variance is derived by extrapolating the autocovariance to zero temporal distance. An analogous treatment of the variance provides an estimate of the sum of break and noise variance, so that the SNR can be concluded. However, the unavoidable usage of anomalies and limited time series shifts and bends the commonly assumed simple functions (decreasing exponential for the covariance and constant for the variance) in a complex manner. The actually occurring functions are theoretically derived and then fitted to observational data. Above that, we show that also in North America, the inhomogeneity levels themselves and not the jumps between them can be described as a random variable. Consequently, no Brownian‐motion‐type inhomogeneities seem to exist, as it was suggested in earlier studies.