Hurst-Kolmogorov dynamics and uncertainty

Perhaps the most significant contribution of the intensifying climatic research is the accumulation of evidence that climate has never been static. Rather, it has been ever changing at all time scales. The changing character of climate, verified for the past or predicted for the future, has been sometimes described by the term nonstationarity. However, revisiting the notions of stationarity and nonstationarity, which are defined within stochastics, we may understand that the terms are often abused. Literally, claims of nonstationarity cannot stand unless the evolution in time of the statistical characteristics of the process is known in deterministic terms not only for the past, but also for the future in particular. This however can hardly be the case, because, as is known, deterministic predictions are difficult, especially of the future.

Change is not synonymous to nonstationarity, since even an ideal stationary white noise process involves change, which however becomes less and less distinct as the time scale of viewing the process (e.g., time scale of averaging) increases. However, the climatic and all geophysical processes demonstrate more prominent change at large scales in comparison to white noise or even to typical stochastic models such as Markovian. This does not reflect nonstationarity. Rather it warns us to change our perception of natural processes as resembling these simple idealized mathematical processes and to move towards a new type of stochastic dynamics.

The “new” description does not depart from the 60- to 70-year old pioneering works of Hurst on natural processes and of Kolmogorov on turbulence. Essentially, Hurst’s discovery in 1950 of the behaviour named after him and the model that had been proposed by Kolmogorov 10 years earlier recognize the multi-scale fluctuation of natural processes and describe it in stationary terms.

Contrasting stationary with nonstationary descriptions is not just a matter of semantics and of rigorous use of scientific terminology. Rather it has important implications in engineering and management. Because nonstationarity uses deterministic functions of time, it explains part of the variability and thus reduces future uncertainty. This is consistent with reality only if the produced deterministic functions are indeed deterministic, i.e., exact and applicable in future times. As this is hardly the case as far as future applicability is concerned, the uncertainty reduction is a delusion and results in a misleading perception and underestimation of risk.

Apparently, the stationary description with Hurst-Kolmogorov stochastic dynamics results in higher uncertainty in comparison to either nonstationary descriptions or to typical stationary stochastic processes. In particular, the uncertainty under Hurst-Kolmogorov dynamics is dramatically increased at large scales, i.e., time scales comparable to those used to define climate, to lifetimes of engineering projects, and to horizons of management strategies. In addition, as far as typical statistical estimation is concerned, the Hurst-Kolmogorov dynamics implies dramatically higher intervals in the estimation of location statistical parameters (e.g., mean) and highly negative bias in the estimation of dispersion parameters (e.g., standard deviation). It is important that the Hurst-Kolmogorov framework allows calculating the statistical measures of bias and uncertainty, either of statistical parameters or of future predictions, which are theoretically consistent and also consistent with empirically observed natural behaviours.

The above framework is illustrated using several examples of hydrometeorological time series, which initially show the consistency of the framework with reality and illustrate the implications for uncertainty. A final example demonstrates how this framework was implemented in the planning and strategic management of the water supply system of Athens, Greece, which comprises four reservoirs and several aquifers. After a long persistent drought (7 years) that shocked Athens in the beginning of the 1990s, the strategic water management became a crucial task, with amplified importance in the phase of preparation of the 2004 Olympics. The demonstration of the methodological framework also includes comparison with alternative nonstationarity modelling approaches, including a trend-based approach that yields absurd results, and a climate-model-based approach that substantially underestimates uncertainty and risk.