Multifractal

Lesson

Introduction
Multifractal theory
- Scale ratio
- Observations and scale in geophysics
- Singularities
Hierarchy of singularities
Multifractal and extremes
Universal multifractal model
Exercises
Conclusion

Observations and scale in geophysics

The problem of scales is very important in the description of complex systems (geophysical, engineering ...etc) for which we have a very fluctuating fields (velocity, temperature, salinity, ....etc) in space and time. These data bases are representative of different overlapping processes which appear at different scales in time and space.

(a) : Temperature time series; (b) : Aggregated time series

In the geophysical field, the data analysis, we faced on measure at unique scale, conditioned by the sampling mode of the sensors (devices). The information at other scales must necessarily be deduced at the basis scale. Several methods exist to aggregate a time/space series at different scales. The most common procedure uses the non-overlapping average.

Lets use as a signal obtained by aggregation at the scale . The transition from any scale characterized by the scale to a scale is obtained as :

$\varepsilon_{\lambda_{2,j}} = \frac{\lambda_{2}}{\lambda_{1}} \sum \limits_{j=1}^{\lambda_1 / \lambda_2} \varepsilon_{\lambda_{1,j}}$

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