Conservative and non-conservative quantities
A conservative field is defined as :
. This means that when we aggregate the data the average of a signal is the same at all scales. This propriety is implicitly admitted when we aggregate the data according to the cascade process. The obtained
and
functions will therefore present bias, the conservation hypothesis should be checked before the construction of
and
functions.
The study of non-conservative field, the variation of the average of the field according to the scale is given by :
Then, the process is conservative for
. A preliminary estimation of
is necessary in order for the knowledge of the conservativeness of the process. Their exist a method to estimate
base on Fourier analysis, under the scale invariance assumption,
is linked to spectra exponent
as well as the scaling moment function of order two
:
Various techniques are used to isolate the conservative part from a non-conservative field. The most known techniques are the fractional differentiation, which decreases the value of
and the inverse operation, the fractional integration which increases the value of
.
