Simulating Temperature and Evapotranspiration using a Universal Multifractal approach
Authors: Ramanathan Arun , Versini Pierre-Antoine , Schertzer Daniel , Tchiguirinskaia Ioulia , Perrin Remi , Sindt Lionel 
Published: March 2024

AbstractTemporal 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&#160; the modified structure function is now defined here asSq(&#916;t)=&#10216;&#921;T&#955; - T&#955;/2,2&#921;q&#10217;Where the scale ratio &#955;&#8733;1/&#921;&#916;t&#921;; 2m/2m=1&#8804;&#955;&#8804;&#923;=2m/20 and &#921;&#916;t&#921; is the time lag, whereas 2m is the largest possible scale out of the scales analyzed that can be represented as a power of 2. While T&#955; is the temperature at scale ratio &#955; or scale l, T&#955;/2,2 is the upscaled (by a scale ratio of 2) temperature at scale ratio &#955;/2 or scale 2l, and the subscript &#8216;2&#8217; indicates that each element of &#160;T&#955;/2 (upscaled temperature) is repeated twice consecutively. It should be noted that T&#955;/2,2 is not the same as T&#955; because the former is an upscaled series, twice repeated (consecutively) of the latter. The largest scale ratio considered in the analysis is &#923;. 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&#160; while the value of &#945; &#160;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.&#160;KeywordsMultifractals, Non-linear geophysical systems, Cascade dynamics, Scaling, Hydrology, Meteorology.