ANALYTICAL REPRODUCTION OF RADAR IMAGES OF RAINFALL
SPATIAL (2D) DISTRIBUTION AS AN INPUT TO GIS BASED
FLOOD MODELLING
B. Dodov1, R. Arsov1, B. Marinov2
University of Architecture, Civil Engineering and Geodesy; Faculty of Hydrotechnics;
1 Chr.Smirnensky Blvd., 1421 Sofia, Bulgaria; Tel./ Fax: +359 2 668 995;
E-mail: dodov_fhe@uacg.acad.bg ; r_arsov_fhe@uacg.acad.bg
2
Bulgarian Academy of Science; Central Laboratory for Parallel Processing;
Acad. G. Bonchev Str., Bl. 25 A; 1113 Sofia, Bulgaria; E-mail: pencho@cantor.bas.bg
1
Abstract
A new approach for radar rainfall images compression, recovering and interpolation
have been suggested, based on an adaptive wavelet transform with fractal basis. It
combines advantages of the wavelet transform (WT) and the adaptive wavelet
approximation (AWA) approaches and offers a possibility for semi-deterministic
reproduction of radar rainfall images by means of restricted number of parameters on
which statistical analyses could be performed. This technology promises to be a
powerful tool for reliable prediction of storms detailed spatial and temporal distribution
with prescribed probability.
Keywords
adaptive wavelet transform, cascade processes, fractals, multifractals, radar rainfall
images, modelling
1. Objectives
Increasingly implementation of the remote sensing technologies for atmosphere
observation and availability of automated radar, satellite and airborne rainfall images in
particular, is a good base for improving the accuracy of the rain fields assessment as an
input to the adequate floods and pollution transfer modelling. Although the information
from these observations is enough detail to support directly the sophisticated
hydrological models, it is huge in quantity and is organised in a manner, which makes
very difficult the extraction of whatsoever general trends with respect to process
development. Regardless of the fact that this information is much more detail than the
conventional one, the resolution of the most popular and accessible radar images is
usually not less than 2x2 km per pixel, with time interval of about 4-5 minutes.
*Published in Flood Issues in Contemporart Water Management, NATO Science Series,
Environmental Security – Vol.71, Kluwer Academic Publishers,1999, The Netherlands
Such a resolution is still too “coarse” for adequate flood modelling, particularly at urban
territories. It is reasonable question to be asked if there is such a technique, which could
allow extraction of the most important features of any sequence of images, describing a
rainfall event with minimum information resources in a manner, that would allow
subsequent restoration of the process to prescribed resolution, preserving the visual
resemblance and statistical properties of the original data. The importance of this
question motivated the reported research and fortunately now we can answer it
positively.
In order to explain better the idea, we find it reasonable to introduce some important
definitions and explanations, concerning the terms and analytical procedures, used in
the paper.
There is not a unique definition of fractals, but they could be described as objects of a
complex structure, consisting of parts, which are at any scale similar to the whole in
some way. An important feature of the most of the fractals is the scale invariance (or
scaling). A fractal function (t) is considered to be scaling with factor  > 0 if the
homogeneity relation (t) = (t) holds for all  > 0 [4].
While the fractal is a set, the multifractal could be considered as a measure or
distribution (of mass, energy, probability, feature, etc.). The Hausdorf dimensions of the
sets, represented in a relevance to their singularity exponents [4], define the multifractal
spectrum of the relevant density (precipitation) distribution. The singularity exponent
measures the “strength” of singularity of the relevant distribution at particular position
x0 in the domain under consideration. The estimated spectrum is known also as
Legendre’s one, because it is calculated trough Legendre transform of the so-called
mass exponents – τ(q), accounting for the scaling lows of the q-th order moments of the
data. For the distribution  this parameter could be defined as follows [20]:
 (q)  lim
log S  (q)
, S  (q)    (C ) q
log 
CG
(1a,b)
where (C) is the total mass of  over the (hyper)cube C with side ;
G - the set of all the non-overlapping cubes, covering the support of .
If the distribution  is scale invariant, the values of S for particular q form a straight
line in a log-log plot (1a), which slope is the corresponding value of τ. Then the
Legendre spectrum could be estimated through the transform:
  d / dq, f ( )  q   (q)
(2a,b)
where  is the singularity exponent, f() - corresponding value of the singularity
spectrum.
The multifractal theory implementation with respect to the atmospheric fields modelling
is founded on the assumption that fluxes of mass (water) and energy in the atmosphere
could be represented by the so called multiplicative cascade processes, successively
transferring these quantities from larger to smaller scales [12,13,28]. The cascade
processes could be considered as eddies of fluxes, broken down into smaller sub-eddies,
receiving a part of the flux of its parent eddy and the amount of which is being
determined by a random factor.
Applying different strategies and “levels” of randomness, as well as some flux
dissipation, an imitation of very complex structures such as rainfall patterns or clouds
could be simulated [28]. The artificial fields produced by cascade processes exhibit
usually good multifractal properties, which could be changed at different stages of
resolution [9]. Regardless of this, the fact that they are stochastically based makes the
chance to fit some real data insignificant. May be the most promising deterministic
technique to produce “naturally looking” fields is the so-called “Fractal/Multifractal”
(FM) approach, which could be considered as a deterministic cascade process
“refracted” through a Fractal Interpolation Functions (FIF) [1, 6, 14-18].
Morlet [10] introduced the wavelet transform (WT) in connection with the analyses of
geophysical signals with very different scales. This transformation basically leads to
decomposition of the signal in terms of some elementary functions b,a, obtained from a
“mother” (basis) function  by dilations and translations, such as:
b,a(x) = a-1/2b,a((x-b)/a),
(3)
where a and b are real numbers.
The wavelet transform applied continuously with real values of a and b is named
continuous wavelet transform (CWT) since it produces continuous function in scalespace domain (called often wavelet space). Later on, the multiresolution analysis
discovered by Mallat [7, 8] and describing any orthogonal wavelet basis as common
multiscale structure, was combined with the compactly supported wavelet basis [2, 3] to
give a fast transform, working on discrete scale-space domain. This transform was
respectively called discrete wavelet transform (DWT). The inherent scaling structure of
the wavelet transform makes it a perfect tool for multifractal analyses, including
continuous [11] and recently - discrete wavelet transform techniques [19, 20].
The so-called Haar scaling and wavelet functions provide the simplest example of an
ortho-normal wavelet basis. The Haar scaling and wavelet coeficients could be
represented as:
Sj,k = 2-1/2 (Sj+1,2k+ Sj+1,2k+1),
Wj,k = 2-1/2 (Sj+1,2k  Sj+1,2k+1).
In the notion of the multiresolution analyses the same expression could be written as
(4)
 S j ,k 
 
 S


  1  j 1,1 j 1, 2  j 1, 2 k ,
W 
2   j 1, 2   j 1,1  S j 1, 2 k 1 
 j ,k 
(5)
where j+1,1  j+1,2  1.
Wavelet decompositions contain considerable information about the oscillations
(singularities) of the recorded signal. The greater the oscillations at particular scale are,
the greater are the WT coefficients at positions at which the basis “fits” best the data. As
has been observed in many contexts, the most wavelet coefficients are very small and
the large ones draw the so-called “maxima” lines in the wavelet space. These features
are the reasons for the image compression and detecting of the singularities (edge
detection in 2D) to be among the main applications of the WT.
In his recent work Riedi [20] introduces the term asymptotically wavelet self-similar
with parameter Hwave for the increments X of process Y if the following limit exist:
H wave 
1 1
 lim
2 2
j 
1
( log 2 var(W j ,k ))
2
(6)
where Wj,k are the wavelet coefficients at the j-th level of transformation.
The parameter Hwave could be used as equivalent to the H parameter of the so-called
long-range dependence (LRD), governing the correlation at large lags. This means that
changing the trend of the wavelet coefficients’ decay, we could control the power low
behavior at prescribed scale intervals and therefore to modell a long-range dependence
observed at real data.
The values of the parameters j,1 and j,2 could be optimized in a way that provides the
lowest possible values of the squares of the wavelet coefficients, generated at the j-th
level of transformation. Such optimization, called adaptive wavelet approximation
(AWA), has a unique and relatively simple solution, which is discussed elsewhere [32].
The support of the wavelet basis could be extended to work over four or eight values of
the approximated function using the so-called Rademacher -matrices [26]. This
extension provides possibilities to apply the Haar WT and the adaptive wavelet
approximation with respect to the surfaces and volumes. More information could be
found in [21-27].
The subject of this paper is a new approach for the radar rainfall images compression,
recovering and interpolation, based on the Haar WT and the AWA. Since the AWA’s
basis (mother) function is generically multiplicative, it is very suitable tool for
representation and analyses of multiplicative (cascade) processes, such as the rainfall
distributions are expected to be because of the turbulent environment of their genesis.
2. Results and Discussion
Since the wavelet transform (WT) is a
1
linear operation, it could not be
expected its product – the wavelet
transform coefficients (WTC), their
0.8
absolute values in particular, to have
different type of distribution then the
0.6
original signal. Instead, the parameters
of the distribution of the WTC will
0.4
change at any successive level of the
WT. As the processes under
0.2
consideration are expected to be
multiplicative cascades, log-normal
distribution of the wavelet coefficients
0
0
10
20
30
40
50
could be observed. This means that
Reflectivity, Z,[dBz]
exploiting (6), a linear correlation
could be expected in regard to the
Figure 1. Z – reflectivity distribution
parameters
of
the
normalized
distribution of the WTC, produced at
any successive level of the WT. If confirmed, this would allow adequate interpolation of
data, resulted from multiplicative cascade process.
To confirm the above statement we processed 20 images, representing values of the
radar reflectivity (so-called dBz-reflectivity - Z), related to the physical reflectivity - z,
[mm6/m3] according to the expression 10log10z = Z. As it is shown in Fig. 1, a normal
statistical distribution of the relevant data Z has been obtained. Therefore, the statistical
distribution of the radar reflectivity z itself is obviously a log-normal.
In the term of the multiplicative cascades, the application of the adaptive approximation
means that for a deterministic binomial process the mean and the variance of the
wavelet coefficients at any successive level of transformation will be practically zero.
Arising the randomness of the process, the mean and the variance of the coefficients
arise up to the level, where “absence of adaptivity” is observed and measured by the
corresponding Haar transform. Figures 2a and 2b represent comparison between Haar
WT and AWA in regards to mean and standard deviation of the first level wavelet
coefficients of binomial cascades with base 0.1 and 0.4, respectively for arising
tolerance of randomness from 0 to 100 % (from 0 to 0.1 and from 0 to 0.4,
respectively). Obviously the AWA insures higher compression of the signal than the
conventional Haar WT . The values of the ratios  j1 /( j1+ j2) for cascade with base 0.4
are shown in Fig. 3.
As it could be seen in the Fig. 3, irrespective of the degree of randomness, the adaptive
wavelet approximation detects well the base of the underlying cascade. The base at any
level can be calculated for data with length 2SN through the following expression:
bj,i = j,i/(j,1+…+j, 2s ),
(7)
where j = 1,…,N; S = 1,2,3; i = 1,…,2S and N represents the index of the highest level
of resolution. It can be seen from Fig.3, that the higher the randomness is, the higher is
the variance of the -s.
Mean
Mean
150
60
100
40
50
20
0
0
20
40
60
Standard deviation
80
100
0
1500
150
1000
100
500
50
0
0
20
40
60
tolerance, %
80
100
0
Haar WT
AWA
0
20
40
60
Standard deviation
80
100
0
20
40
60
tolerance, %
80
100
Figure 2. Comparison between Haar WT and AWA - mean and standard deviation
of the first level wavelet coefficients of binomial cascades with base 0.1 (a) and 0.4(b).
Fig. 4 represents the ratios  j1 /( j1+ j2) = bj1 of a binomial cascade with base 0.4 and
length of 64, 128, 256, 512, 1024 and 2048, respectively with 100 % tolerance of
randomness. It can be seen that the values bi1 (and respectively, the base vectors)
alternatively change their directions, decaying to bN1. This corresponds to the eddies’
direction and size changing at the successive scales of the turbulence, which at the finest
resolution must disappear. Therefore the AWA application allows revealing of the
availability of cascade behavior of given process (signal) and could be denominated
also as inverse cascade approximation (ICA).
0.6
0.8
tolerance 0%
tolerance 20%
tolerance 40%
tolerance 60%
tolerance 80%
tolerance 100%
0.55
0.5
0.6
0.4
1/(12)
1/(12)
0.45
64
128
256
512
1024
2048
0.7
0.35
0.3
0.25
0.5
0.4
0.3
0.2
0.2
0.15
0.1
1
2
3
4
5
6
Resolution, 2x
7
8
9
Figure 3. Values of the ratios  j1 /( j1+ j2)
for cascades with base 0.4 and different
tolerances of randomness
10
0.1
1
2
3
4
5
6
Resolution, 2x
7
8
9
Figure 4. Ratios  j1 /( j1+ j2) of a binomial
cascade with base 0.4 and different lengths with
100 % tolerance of randomness
10
Some of our preliminary analyses of real data indicate that the values of N,i tend to one.
This could be expected, since any other value means that the energy and the mass will
never dissipate in the turbulent processes. In this sense the AWA is a good tool for
verification of different types of turbulent processes’ models. The first model which we
tested and which exhibited good properties in this sense is the FM one.
The analyses carried out so far show that the adaptive wavelet approximation exhibits
high sensitivity with respect to the existence of a “cascadeness” in the analyzed signal.
This sensitivity reflects in the possibility to extract the main features of the signal within
minimum amount of information, or in other words – with higher compression in
comparison to the corresponding Haar transform. The rate of the compression is strictly
related to the base of the underlying cascade and the “level of randomness” implanted in
the signal. The closer the underlying cascade is to the Lebesque measure (equal bases)
and the higher the degree of randomness is, the less is the possibility for additional
compression, compared to the Haar WT. In this sense the practical aspects of
application of the adaptive wavelet approximation are related mainly to the possibilities
for higher degree of compression of turbulence-related data and respectively, the
possibilities for minimizing of the randomness, implied in the restored (modelled) ones.
An important feature of the WT is the possibility for the data interpolation based on (6),
which allows more detailed representation of the relevant process development in space
and time, preserving its inherent multifractal properties. Interpolation of the data must
be performed by the Haar transform or by AWA if the last three -sets are relatively
equal and the trend of the WTC is captured. If the last condition is not fulfilled the
AWA is not applicable, because the -s distortions the WTC distribution and (6) will
not be valid.
The practical conclusions from the above notions in the context of the radar data
analyses are related to the possibilities for:
 storage and analyses of the -coefficients, the core wavelet coefficients marking the
general maxima lines and the trends of decay (or distributions at any level of
transformation) of these coefficients;
 analyses of the obtained information in the multiparametric probability space to
obtain the trend in a particular process, or to find the position of this process in the
probability space;
 restoration, interpolation and modelling of radar images, based on the derived
coefficients, filling in the gaps of the missing coefficients with random ones (but
derived from distribution, inherent for the data) and finally - permutation of the
random coefficients to provide positive values for the recovered data;
 governing the scaling behavior of the model through control over the rates of decay
of the wavelet coefficients at any level of the transform and therefore to match the
bifurcation points (or zones) of the natural turbulent process. This is essential for
the successful interpolation of this process in space and time.
Example of 1D interpolation of data, produced by means of the FM approach is shown
in Fig. 5. The corresponding τ(q) and f() are given in Fig.6 and the power spectrum is
given in Fig. 7. A similar example but in 2D is represented in Fig. 8a and Fig. 8b.
Interpolation of real data of a
radar image is illustrated in
Fig.9.
The
interpolated
image is not an exact copy
of the natural one in visual
aspect (as it is at the
examples, illustrated in Fig.
5 and Fig. 8 a, b), because of
missing of detailed data of
the registered radar signal.
The low intensity signals are
cut automatically by the
radar being used (as could be
seen in Fig. 1), since they
are currently considered to
be of a little practical
importance. Taking into
account as many details as
possible however is of
crucial importance for the
1500
1000
500
0
0
500
1000
1500
2000
0
500
1000
1500
2000
1500
1000
500
0
Figure 5. Comparison between signals produced by FM
procedure with length 2048 (up) and the same signal with
length 128 interpolated through WT to 2048 (down)
50
tau(q)
10
5
0
-50
-100
-30
Slope-1.2142
FM output - 2048 intervals
WT interpolation from 128 to 2048
-20
-10
0
q
10
20
30
10
-2
-1
10
10
0
1
f(alfa)
10
5
0.5
Slope-1.1818
0
0.6
0.8
1
1.2
1.4
1.6
1.8
2
alfa
Figure 6. Comparison between τ(q) and
singularity spectrums f() of the above signals
10
-2
-1
10
Frequency
10
Figure 7. Comparison between power
spectrums of the of the above signals
adequate modelling of the multiplicative cascade processes [18]. Missing of low
intensity signals data changes the pattern of “cascadnes” of the turbulent process (and
respectively, the distribution of the wavelet coefficients) and does not allow an adequate
interpolation. This reflects the accuracy of the modelling, as well.
0
Figure 8 a. FM-output image
128x128 (left) interpolated to
512x512 by WT (up)
Figure 8 b. FM image 512x512 produced by
means of the same FM parameters as for the one,
given in Fig. 8a (left)
3. Conclusions
The adequate assessment and modelling of the rainfall processes have been recognized
as a key issue in the successive flood prediction and mitigation. The rainfall/runoff
processes modelling in the current practice emphasize mainly on the developments on
the ground. Even in this cases the possibilities, offered by the GIS have not been used in
the advanced models in the way, which allows an accurate accounting for the topology
and land use on the runoff development. Higher space and time resolution can be
implemented for more adequate modelling both of rainfall and runoff processes in a GIS
environment, than the currently used approaches offer.
Since many of the existing representations
are limited by their analytical tractability
and because there has been a recognition
of chaotic effects in atmospheric
precipitation, a new class of models, based
on Fractal/Multifractal (FM) approach,
wavelet transform (WT) and adaptive
wavelet approximation (AWA) have been
considered in the rainfall pattern
description.
Figure 9. Radar image in it’s
original size 100x100 pixels or
400x400 km (left) and its
interpolation (zoom) of 16 times
(up)
The multifractal theory implementation
with respect to the atmospheric fields is
founded on the assumption that fluxes of
water and energy in the atmosphere could
be represented by multiplicative cascade
processes, successively transferring these
quantities from larger to smaller scales.
For this considering of the smallest details
of the turbulent process is of crucial
importance.
The new technology discussed in this paper is based on the advantages of the WT and
AWA approaches and allows analyses, modelling and detailed interpolation of
turbulence-related processes, such as rainfalls. It offers a possibility for semideterministic reproduction of radar rainfall images by means of restricted number of
parameters and allows statistical analyses to be performed on them. This technology
promises to be a powerful tool for reliable prediction of storms detailed spatial and
temporal distribution with prescribed probability.
4. Recommendation
It is reasonable a catalogue to be created with optimal number of the coefficients, which
for prescribed size of radar images, could allow detailed restoration of the data in
statistical and visual sense. Fortunately the numerical aspects of the notion “similarity in
statistical and visual sense” has it’s mathematical analogue, such as: conventional
statistics and multifractal analyses for statistical similarity, as well as integrated
Hausdorf distance [27] for visual similarity.
In conclusion, the stages of implementation of the Haar WT and adaptive (extended
Haar) wavelet transforms (AWA) in radar data analysis and modelling for a given
region and resolution (image size) could be summarized as follows:
 Determination of the optimal data quantity and quality, which to be stored in the
catalogue;
 Filling in the catalogue in a two-step procedure. First, extracting the information
per image for all the images, scanned during particular rainfall - 2D transform.
Second, extracting the information for the same rainfall as 3D representation
(images as slides in time step) – 3D transform. If there are missing slides, they
could be obtained through interpolation in time in 3D. This technique is also known
as “morphing”;
 Analyzing of the data by means of the Fuzzy Set Theory and the Cluster Analysis
for revealing of the trend of the parameters development in respect of an individual
rainfall 2D image. Another possibility is analyzing the probabilistic distribution of
the 3D data in the multiparametric space, to define what “likely” is a particular rain
to happen, or to “construct” the parameters of a rainfall with prescribed probability;
 Reproduction through inverse adaptive wavelet transform of the rainfall images
from the chosen set of parameters in “standard” size;
 Interpolation, if necessary, to prescribed resolution in space and time through
inverse Haar transform. Anisotropy in scaling behavior could be achieved through
control over the decay of the wavelet coefficients at any stage of the interpolation.
 Implementation of the produced grid (image) in the modelling of the processes
through the conventional GIS tools.
5. References
1.
Barnsley, M.F., (1988) Fractals Everywhere, Academic Press, New York.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
22.
23.
24.
25.
26.
27.
28.
Daubechies, I., Grossmann, A. and Meyer, Y. (1986) Painless nonorthogonal expressions, J. Math.
Phys., 27, 1271-1283.
Daubechies, I. (1992) Ten Lectures on Wavelets (S.I.A.M., Philadelphie).
Feder, J. (1988) Fractals, Plenum Press, New York.
Goncalves, P., Riedi, R. H. and Baraniuk, R. G., (1998) Simple Statistical Analysis of Wavelet-based
Multifractal Spectrum Estimation, Proceedings of the 32nd Conference on `Signals, Systems and
Computers', Asilomar, 7.
Lutton, E., Levy-Vehel, J., Cretin, C., Glevarec, P. and Roll, C., (1995) Mixed IFS: resolution of the
inverse problem using Genetic Programming, Complex Systems, 9, (5), October and Inria research
report No. 2631, August.
Mallat, S.G. (1989) A Theory for multiresolution signal decomposition: The wavelet representation,
IEEE Trans. On Pattern Analysis and Machine Intelligence, 11, 674-693.
Mallat, S.G. (1989) Multiresolution approximations and wavelet orthogonal bases of L2(R), Trans. of the
Am. Math. Soc., 315, 69-87.
Marshak, A., Davis, A., Cahalan, R. F. and Wiscombe, W. J. (1994) Bounded Cascade Models as
Nonstationary Multifractals, Physical Review E, 49, 55-69.
Morlet, J. (1983) Sampling theory and wave propagations, NATO ASI series FI (Springer, Berlin), 233261.
Muzy, J. F., Bacry, E. and Arneodo, A. (1994) The multifractal formalism revisited with wavelets,
International Journal of Bifurcation and Chaos, 4(2), 245-302.
Olsson, J., Niemczynowicz, J. & Berndtsson, R. (1993) Fractal of high-resolution rainfall time series,” J.
Geophys. Res., 98 (D12), 23265-23274.
Olsson, J. & Niemczynowicz, J. (1994) Multifractal relations in rainfall data, Nordic Seminar on Spatial
and Temporal Variability and Interdependancies Among Hydrological Processes, Kirkkonummi,
Finland, 14-16 Sept.
Puente, C. & Klebanoff, A. (1994) Gaussians everywhere, Fractals, 2 (1), 65-79.
Puente, C. (1994) Deterministic fractal geometry and probability, International Journal of Bifurcation
and Chaos, 4 (6), 1613-1629.
Puente, C. & Obregon, N. (1996) A deterministic geometric representation of temporal rainfall: Results
for a storm in Boston, Water Resources Research, 32 (9), 2825-2839.
Puente, C., Lopez, M., Pinzon, J. & Angulo, J. (1996) The gaussian distribution revisited, Adv. Appl.
Prob., 28 (2), 500-524.
Puente, C. (1996) A new approach to hydrologic modelling: derived distributions revisited, Journal of
Hydrology, 187, 65-80.
Riedi, R. H., Crouse, M. S., Ribeiro, V. J., and Baraniuk, R. G., A Multifractal Wavelet Model with
Application to Network Traffic, IEEE Special Issue on Information Theory, 45 (4), 992-1018.
Riedi, R. H., Multifractal Processes, (submitted) Stoch. Proc. and Appl..
Sendov, Bl. (1997) Multiresolution Analysis of Functions Defined on the Dyadic Topological group,
East Journal of Approximations, 3 (2) 225-239.
Sendov, Bl and Marinov P. (1997) Orthonormal Systems of Fractal Functions, Mathematica Balkanica,
New Series, 11 (1-2), 169-200.
Sendov, B. & Marinov, P. (1998) Binary exponential fractal functions, Fractal Calculas & Applied
Analysis, 1, 23-48.
Sendov, Bl. (1998) Adaptive approximation and compression, Approximation Theory IX, (eds. Charles
K. Chui and Larry L. Schumaker), 1-8.
Approximation Theory, Vanderbild Univ. Nashville, Tennesse, USA, January 3-6, p. 8.
Sendov, Bl. (1999) Adaptive Multiresolution Analysis on the Dyadic Topological Group, Journal of
Approximation Theory 96, 258-280.
Sendov, Bl. (1999) Walsh-Similar Functions, East Journal of Approximations, 5 (1),1-65.
Tessier, Y., Lovejoy & Schertzer, D. (1993) Universal multifractals: Theory and observations for rain
and clouds, J. Appl. Meteorol., 32 (2), 223-250.