Simulating Temperature and Evapotranspiration using a Universal Multifractal approach

Abstract

Temporal structure functions are usually defined as the q-th order statistical moment of the absolute fluctuation in a time series over a temporal lag at a given resolution. However, applying this in analyzing a temperature time series results in the possibility of simulating only a similar fluctuation over a temporal lag at a resolution and not the temperature directly. Since the aim is to simulate a temperature time series this simulated fluctuation series can be added to an assumed mean temperature to obtain a temperature time series. However, proceeding this way seems to necessitate some ad-hoc moving average technique that seems difficult to be physically reasoned. Secondly but more importantly both diurnal and seasonal periodicity have to be forcibly introduced once again in a non-rigorous manner. A drastic yet reasonably useful alternative would be to modify the definition of the structure-function instead. For order of statistical moment q  the modified structure function is now defined here as

S_q(Δt)=⟨ΙT_λ - T_λ/2,2Ι^q⟩

Where the scale ratio λ∝1/ΙΔtΙ; 2^m/2^m=1≤λ≤Λ=2^m/2⁰ and ΙΔtΙ is the time lag, whereas 2^m is the largest possible scale out of the scales analyzed that can be represented as a power of 2. While T_λ is the temperature at scale ratio λ or scale l, T_λ/2,2 is the upscaled (by a scale ratio of 2) temperature at scale ratio λ/2 or scale 2l, and the subscript ‘2’ indicates that each element of  T_λ/2 (upscaled temperature) is repeated twice consecutively. It should be noted that T_λ/2,2 is not the same as T_λ because the former is an upscaled series, twice repeated (consecutively) of the latter. The largest scale ratio considered in the analysis is Λ. By defining the structure-function in this way temperature at a larger scale after being repeated a sufficient number of times can be directly added to the fluctuation at a smaller scale to result in the temperature at a smaller scale. The universal multifractal parameters obtained from the modified structure-function analysis are not necessarily equal to those obtained from the usual structure-function analysis (i.e. the two different structure functions follow two different scaling laws). An iterative curve fitting technique is used to estimate the values of Universal Multifractal (UM) parameters C₁, H, and a  while the value of α  is estimated using a normalized form of the modified structure function along with the un-normalized one. A simulation procedure that utilizes the aforementioned modified structure function definition is proposed here to generate temperature scenarios. Finally, reference evapotranspiration is estimated based on the simulated temperature using a simple empirical power law function. The actual evapotranspiration is estimated using the reference evapotranspiration and water content via a different, simpler empirical function. The tentative methodology proposed here when used along with simulated reference rainfall scenarios could help design zero-emission green roof solutions.

 

Keywords

Multifractals, Non-linear geophysical systems, Cascade dynamics, Scaling, Hydrology, Meteorology.