Distribution of natural trends in long-term correlated records: A scaling approach

We estimate the exceedance probability W(x,α;L) that, in a long-term correlated Gaussian-distributed (sub) record of length L characterized by a fluctuation exponent α between 0.5 and 1.5, a relative increase Δ/σₜ of size larger than x occurs, where Δ is the total observed increase measured by linear regression and σₜ is the standard deviation around the regression line. We consider L between 500 and 2000, which is the typical length scale of monthly local and reconstructed annual global temperature records. We use scaling theory to obtain an analytical expression for W(x,α;L). From this expression, we can determine analytically, for a given confidence probability Q, the boundaries ±xQ(α,L) of the confidence interval. In the presence of an external linear trend, the total observed increase is the sum of the natural and the external increase. An observed relative increase Δ/σₜ is considered unnatural when it is above xQ(α,L). In this case, the size of the external relative increase is bounded by Δ/σₜ±xQ(α,L). We apply this approach to various global and local climate data and discuss the different results for the significance of the observed trends.