Theoretical estimation
The different methods to estimation the UM parameters described above can be applied to a conservative field or flux (see Multiscale Analysis : Multfractal courses). However, directly observable quantities are rather the (vector) velocity field of temperature field which are a non-conservative quantities. Kolmogorov in [ Kolmogorov, 1941[1]] proposed to characterize these processes by the analysis of their spatial/ or temporal increments. The statistical moments (in space) of these increments are the structure functions, defined by:


Where the scalar field
can be a passive scalar field (temperature), or one component of the velocity field.
is the smallest resolution
,
is the scaling exponent of the structure function. Using the dimensional analysis the increments of the field
has been related to those of related flux density
:

Using the above relation, the structure function scaling exponent can be expressed in the framework of UM as :

The UM parameters can be calculated as follow :
