Determining confidence intervals in analyzing climatic series

In the context of the problem of global warming, the question of how much the observed increase in the mean air temperature over the last decades is likely due only to natural variability is studied. It is assumed that an observed temperature-data series of length N is a segment of a stationary random sequence with a spectrum known on an interval that does not include the lowest frequencies. The variance σ2(N) of the sample mean m* and the standard deviation σ(N), which determines the width of the confidence interval of m*, are calculated for different continuations of the spectrum to the low-frequency region. For a continuation of the spectrum that boundlessly increases, as does ω−2α (0 < α < 1/2) when the frequency ω tends to zero (red-noise spectrum), it is shown that, the closer the parameter α is to 1/2, the more slowly σ(N) tends to zero at N → ∞. Using an empirical spectrum of a global mean annual temperature series as an example, it is shown that the standard deviation significantly depends on the form of the spectrum in the lowest frequency region. An attempt has been made to assess the measure of standard-deviation uncertainty due to a lack of exact information on the spectrum in the low-frequency region. Important series characteristics—the equivalent number of independent observations Neq(N) and the time scale of correlation T1(N) which are determined through the variance σ2(N) of m*—also depend on the form of the spectrum in the low-frequency region. For the red-noise spectrum, Neq(N) increases with an increase in N proportionally to N1–2α (but not proportionally to N as in the case of bounded spectrum); the correlation scale T1(N) is no longer constant (as in the case of bounded spectrum) and increases with an increase in N proportionally to N2α.