Superdiffusive Fractional Brownian Motion Memory of Annual Mean Global Temperature Anomaly

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Karl Patrick Casas
Allan Roy Elnar
Gibson Maglasang


The NASA’s GISTEMP Team temperature anomaly data from year 1880-2015 has undergone two different data treatments: 5-year moving average and second-generation data. Together with the untreated data, the log-log plot of mean square displacement (MSD) of temperature anomaly versus time was derived. Since it showed a linear response, fractional Brownian motion (fBM), which was derived using white noise analysis was shown. The fBM with Hurst exponent, H = 1 was then used as a fit to the MSD of temperature anomaly which implied super-diffusion. It must be pointed out that the upper bound of the H index is the condition necessary for the chosen memory function rendering the best fit to the temperature anomaly. Considering that global temperature follows a natural cycle when unperturbed, this anomalous behavior can be attributed to a driving force in the time evolution of the temperature anomaly. These ‘forces’ may be attributed to both the natural heating cycle in the Earth’s surface and human activities particularly on technological changes and innovations which already begun even before the 1880s.

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