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Ambiguities in Determination Of Self-Affinity in the AE-index Time Series

The interaction between the Earth's magnetic field and the solar wind plasma results in a natural plasma confinement system which stores energy. Dissipation of this energy through Joule heating in the ionosphere can be studied via the Auroral Electrojet (AE) index. The apparent broken power law form of the frequency spectrum of this index has motivated investigation of whether it can be described as fractal coloured noise. One frequently-applied test for self-affinity is to demonstrate linear scaling of the logarithm of the structure function of a time series with the logarithm of the dilation factor $λ$. We point out that, while this is conclusive when applied to signals that are self-affine over many decades in $λ$, such as Brownian motion, the slope deviates from exact linearity and the conclusions become ambiguous when the test is used over shorter ranges of $λ$. We demonstrate that non self-affine time series made up of random pulses can show near-linear scaling over a finite dynamic range such that they could be misinterpreted as being self-affine. In particular we show that pulses with functional forms such as those identified by Weimer within the $AL$ index, from which $AE$ is partly derived, will exhibit nearly linear scaling over ranges similar to those previously shown for $AE$ and $AL$. The value of the slope, related to the Hurst exponent for a self-affine fractal, seems to be a more robust discriminator for fractality, if other information is available.