On the detection of trends in long-term correlated records

We use the Detrended Fluctuation Analysis (DFA) to quantify underlying trends in long-term correlated records. Our approach is based on the fact that different orders of DFA are affected differently by trends. For a given instrumental record of length N, we compare the fluctuation exponent α0 of DFA0 where trends are not being eliminated, with the fluctuation exponent α2 of DFA2 where possible linear trends in the instrumental record are being eliminated. From this we deduce numerically the probability density p(A) that in the considered long-term correlated record, a linear trend with a slope between A and occurs. Without loss of generality we focus on Gaussian distributed data. As an example, we apply our analysis to several long temperature records (Melbourne, Oxford, Prague, Pusan, Uppsala, and Vienna), where we discuss the trends within the last 90 years, which may originate from both, urban and global warming.