Fractal and multifractal analysis of the rise of oxygen in Earth's early atmosphere

The rise of oxygen in Earth's atmosphere that occurred 2.4-2.2 billion years ago is known as the Earth's Great Oxidation, and its impact on the development of life on Earth has been profound. Thereafter, the increase in Earth's oxygen level persisted, though at a more gradual pace. The proposed underlying mathematical models for these processes are based on physical parameters whose values are currently not well-established owing to uncertainties in geological and biological data. In this paper, a previously developed model of Earth's atmosphere is modified by adding different strengths of noise to account for the parameters' uncertainties. The effects of the noise on the time variations of oxygen, carbon and methane for the early Earth are investigated by using fractal and multifractal analysis. We show that the time variations following the Great Oxidation cannot properly be described by a single fractal dimension because they exhibit multifractal characteristics. The obtained results demonstrate that the time series as obtained exhibit multifractality caused by long-range time correlations.