Multifractal

Lesson

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

Singularities

In order to describe the observations, the multifractal formalism introduces the singularity notion defined as :

$\gamma = \log_{\gamma} (\varepsilon_{\gamma})$

The singularity the logarithmic transformation (base ) of the measured observations at the resolution .The transformation allows to deal with one variable in which information from all scales are integrated. It is easy to inverse the relation defined above in order to construct the observations at all scales:

$\varepsilon_{\lambda} = \lambda^{\gamma}$

Principal of transformation of all available observations at all scales in to singularity

The above figure illustrated the transformation procedure of the identified observations at scale into the singularity

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