A reliable method to study the fractal behaviour of hydrological variables

What is it about?

Fractals are mathematical sets that are formed by repeating a (usually simple) pattern recursively at all scales. This produces an object that looks exactly the same at all zoom levels or at all scales. Cauliflowers are an example of fractals (Figure 1). They exhibit this peculiar property which makes smaller parts of it identical to the whole. In other words, flower heads of a cauliflower look exactly like the whole cauliflower except that they are smaller, and each flower head is formed of even smaller flower heads identical to the whole of the cauliflower. This property of being fractal is observed in many objects in nature such as trees, clouds, mountains, etc. Variables in hydrology, like river flows, groundwater levels, rainfall intensity etc., are fractal. By extending the notion of the fractal nature of a cauliflower from geometric similarity in the spatial domain, to a statistical similarity in the time domain, one can explain the fractal nature of hydrological variables. In other words, the fractal nature of hydrological time series is evident when the second statistical moment (variance) of the hydrological time series plotted against the time scale (on logarithmic scales) is a straight line. This means the time series exhibits a power-law relationship against these two axes. This power-law relationship may change at a certain time scale, known as a crossover, and this will give rise to different 'scaling regimes'. How to locate these crossovers in order to identify the different scaling regimes of a time series in a reliable and objective way is the subject of this paper.

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Why is it important?

Well, if we want to study a property and include it in our models we have to be able to quantify it objectively. The method introduced in this paper is a tool that researchers can use to study the fractal behaviour of hydrological time series in a reliable and systematic way.