alpha - the index of multifractality; it varies
from 0 to its maximum value 2, and it describes how rapidly the fractal dimensions
vary as we leave the mean. See Universality in the Introduction, and the Multifractals
section for a graphical display.
anisotropic - the most general multifractal
field - differently stratified and/or rotated in (possibly nonorthogonal) directions.
See GSI below
chaos - chaos often comes up in geophysical contexts.
but, which chaos is relevant? see the slides:
Deterministic chaos?
eg
Nonlinear Mappings: Discrete time (=n) evolution
of a few variables (x):
(Z, C are complex numbers)
…the Mandelbrot set
Flows:
Few degrees of freedom… few applications
Or stochastic chaos...
Nonlinear PDE ’s: Fields/spatial structures evolving
in time
Example: Navier-Stokes Equations:
where v = velocity, t = time, p = pressure, r = density,
n = viscosity, f = body forces (e.g. stirring,
gravity).
1 second of wind data
C1 - the sparseness of the level of activity which
gives the dominant contribution to the mean field. See Universality in the Introduction,
and the Multifractals section for a graphical display.
codimension - codimensions tend to be more
useful than traditional fractal dimensions. codimension = D - d where D is the
dimension of the embedding space, d is the fractal dimension. The table below
shows some details about the c(g) curve for universal
multifractals.
c(g) curve:
C1 is the mean inhomogeniety order and codimension
of the mean singularities of the corresponding conservative flux
convexity of c(g):
local characterization, but global for universal processes
fractal - Fractals are geometric sets of points
which are invariant under "zooming". See the Introduction
Generalized Scale Invariance (GSI) - a general scheme
for parameterizing anisotropic universal multifractals. See GSI in the Introduction
and GSI in the Multifractals section
The basic elements of GSI:
1) unit ball
The unit ball B1 which defines all the unit vectors. In general, B1
will be defined by an implicit equation:
where delB1 is the "frontier" of the unit ball, and ||x|| is a function
of position.
2) The scale changing operator
Tl which transforms the scale of vectors
by scale ratio l. Tl
is the rule relating the statistical properties at one scale
to another and involves only the scale ratio l
(there is no characteristic "size"). This implies that Tl
has certain properties. In particular, if and only if ,
then
i.e., Tl has the group properties:
Hence:
and Tl is therefore a one parameter multiplicative
Lie group:
where Gop is an operator
called the "infinitesimal generator" of the group.
3) Anisotropic Hausdorff measures
Scale functions
Infinitesimal transformations
Linear GSI
Diagonal G: self-affinity
Using the horizontal x direction to define the scale, we have the following
generator and convenient scale function (Hz>0 any complex h.
Isolines of the scale function with Hz=3, h=3,
ls=1 (left), with h=1 (right)
Nondiagonal linear GSI
If the eigenvalues Hx, Hz are real we can also obtain analytical scale
functions using the scale function derived from the diagonal scale function
by:
H - the order of the integration (H>0) or differentiation
(H<0) needed to obtain the observed field from a (direct) multifractal cascade
process. See Universality in the Introduction, and the Multifractals section
for a graphical display.
Fractionally Integrated Flux:
Observable fields are generally not the direct result of conservative
cascade processes; they are related to them via fractional integration
of order H; the Fractionally Integrated Flux model:
D is the dimension of space-time, the “*” means “convolution”. If H=1/3,
v the velocity, e the turbulent energy flux,
then we have the famous Kolmogorov law in turbulence.
isotropic - the simplest case for a multifractal
field - the same in all directions.
multifractal - A multifractal is the generalization
of a fractal required to deal with a field that has values at each point. See
the Introduction
self-affine - the simplest type of anisotropic
field - differentially stratified along (possibly nonorthogonal) directions.
self-similar - see isotropic
universality - See Universality in the Introduction.
K(q) curve:
Multifractals have multiple scaling of their moments:
.gif)
,
then
where Gop is an operator
called the "infinitesimal generator" of the group.
is the Mellin
transform of 
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2.gif)
2.gif)
curve.gif)
graph.gif)