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

Abstract

We estimate the exceedance probability $W(x,\ensuremath{\alpha};L)$ that, in a long-term correlated Gaussian-distributed (sub) record of length $L$ characterized by a fluctuation exponent $\ensuremath{\alpha}$ between 0.5 and 1.5, a relative increase $\ensuremath{\Delta}/{\ensuremath{\sigma}}_{t}$ of size larger than $x$ occurs, where $\ensuremath{\Delta}$ is the total observed increase measured by linear regression and ${\ensuremath{\sigma}}_{t}$ 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 $W(x,\ensuremath{\alpha};L)$. From this expression, we can determine analytically, for a given confidence probability $Q$, the boundaries $\ifmmode\pm\else\textpm\fi{}{x}_{Q}(\ensuremath{\alpha},L)$ 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 $\ensuremath{\Delta}/{\ensuremath{\sigma}}_{t}$ is considered unnatural when it is above ${x}_{Q}(\ensuremath{\alpha},L)$. In this case, the size of the external relative increase is bounded by $\ensuremath{\Delta}/{\ensuremath{\sigma}}_{t}\ifmmode\pm\else\textpm\fi{}{x}_{Q}(\ensuremath{\alpha},L)$. We apply this approach to various global and local climate data and discuss the different results for the significance of the observed trends.