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The stochastic analysis in the scale domain (instead of the traditional lag or frequency domains) is introduced as a robust means to identify, model and simulate the Hurst–Kolmogorov (HK) dynamics, ranging from small (fractal) to large scales exhibiting the clustering behavior (else known as the Hurst phenomenon or long-range dependence). The HK clustering is an attribute of a multidimensional (1D, 2D, etc.) spatio-temporal stationary stochastic process with an arbitrary marginal distribution function, and a fractal behavior on small spatio-temporal scales of the dependence structure and a power-type on large scales, yielding a high probability of low- or high-magnitude events to group together in space and time. This behavior is preferably analyzed through the second-order statistics, and in the scale domain, by the stochastic metric of the climacogram, i.e., the variance of the averaged spatio-temporal process vs. spatio-temporal scale.
Clustering in nature has been first identified by H.E. Hurst (1951)  (Figure 1a) while studying the long-term behaviour in a variety of scales of the discharge timeseries of the Nile River in the framework of developing engineering projects in its basin.
Particularly, H.E. Hurst discovered a tendency of high-discharge years to cluster into high-flow periods, and low-discharge years to cluster into low-flow periods. This behaviour, also known as the Hurst phenomenon or Joseph effect (Mandelbrot, 1977) , has been verified in a variety of hydrological , hydrometeorological and turbulent processes   and in other geophysical and alternate fields such as finance, medicine , and art   .
All these processes are characterized by long-term persistence (LTP), which leads to high unpredictability in long-term scales due to the clustering of events as compared to the purely random process, i.e. white-noise (e.g. as in a fair dice game ), or other short-range dependence models (e.g., Markov).
Figure 1: (a) In 1951 H.E. Hurst discovered the clustering behaviour in nature (b) A.N.Kolmogorov proposed a decade before a stochastic process that describes this clustering behaviour.
The mathematical description of the Hurst phenomenon is attributed to A.N. Kolmogorov (Figure 1b) who developed it while studying turbulence in 1940  (Figure 1b), inspiring D. Koutsoyiannis  to name the general behaviour of the Hurst phenomenon as Hurst-Kolmogorov (HK) dynamics (Figure 2), to give credit to both contributing scientists and to distinguish it from the Gaussian LTP processes (e.g., fractional-Gaussian-noise ), and to incorporate alternate short-range dependence (e.g., Markov-behaviour ).
Figure 2. Hurst-Kolmogorov (HK) dynamics and the perpetual change of Earth’s climate
The HK dynamics has been recently also linked to the entropy maximization principle, and thus, to robust physical justification . The stochastic simulation of the HK dynamics has been a mathematical challenge since it requires the explicit preservation of high-order moments in a vast range of scales, affecting both the intermittent behaviour in small scales  and the dependence in extremes  as well as the trends often appearing in geophysical timeseries .