Methods of Analysis

Lesson

Empirical estimation: in time

The Trace Moment method is widely used in geophysics. We can fiend in literature several studies of rainfall analysis using Trace Moment to determine the UM parameters. We give an example of the estimation of UM using TM for rainfall time series obtained with the help of a tipping bucket rain gauge. The sampling time scale is equal to one hour and the total time sampling corresponds to 1,3 days. This results were obtained by A.Gires (Multifractal data analysis technique and simulation ):

Illustration of the TM technique for the studied rainfall field

We can observe a scaling (scale invariance) up to the resolution ( 11 days). The UM parameter and are estimated as follow:

$C_1 = K^{'}(1) = \frac{K(1.1)- K(0.9)}{0.2} \approx 0.54 \\ \quad\\ \alpha = \frac{K^{''}(1)}{K^{'}(1)} \approx \frac{0.2}{0.1^2} \frac{K(1.1)+ K(0.9)}{K(1.1)- K(0.9)} \approx 0.51$

The above figure shows that the theoretical curve fit the empirical one until the statistical moment . This is due to the maximum observable singularity which can be observed with a finite sample size. The calculation of this moment in the UM framework is given by :

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