MULTIFRACTAL APPROACH TO RIVER RUNOFF AND DRAINAGE AREAS IN POORLY GAUGED BASINS

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MULTIFRACTAL APPROACH TO RIVER RUNOFF AND
DRAINAGE AREAS IN POORLY GAUGED BASINS
Tchiguirinskaia, I. (1) Schertzer, D. (2)
Hubert, P. (3) Bendjoudi, H. (1) Lovejoy, S.(4)
(1) Sisyphe-Univ. Paris VI, (2) ENPC and
Météo-France, (3) Sisyphe-ENMP, (4) McGill Univ.
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Two floods in France
Spring 2001 Floods in Abbeville two years of
rainfall fell in three months. Water gushed from
forgotten springs which had dried up many years
before From press
Abbeville station
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Two floods in France
September 2002 Flash floods in Gard, Herault
and Vaucluse departments six months rain fell in
a few hours From press
Pont de Russan
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Prediction in Gaged Basins
Model parameters and runoff prediction depend
heavily on gaged flow
River flow
River flow
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Peak flows are rarely gaged!
September 8-9, 2002, precipitation of 687mm in 24
hour was observed in Southern France, and Le
Gard, a tributary of Rhone had historical flood
since 1958. Only a few discharge gages could
observe the water level of the peak flood river
over-flew the bridges where gages were set.
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How gaged is our planet?
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Prediction in Ungaged Basins
Clouds
Atmospheric turbulence
Rain
Mountains
Landscapes
PUB
Permeability
River flow
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  Kick-Off Workshop on IAHS DECADE OF PREDICTION
IN UNGAGED BASINS (PUB) - Hydrological Sciences
on Mission -  
20-22 November 2002 Brasilia, Brazil
Organized by IAHS and Brasilia University (UNB),
  Sponsored by IAHS, MEXT Japan, IAEA, IHP-NC
France, IHP-NC UK CGEE, CNPq, CT-Hydro,
ABRH Supported by UNESCO, WMO, CEOP, GEWEX,
HELP, FRIEND, CAUHSI, CHASM, NASA,
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Extreme variability
Problematic the precipitation / discharge
system
a multifractal source (precipitation)
influenced by a multifractal environment (basin
geomorphology)
produce multifractal ouput (river discharge)
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Strategies

• time series analysis
• multifractal behaviour (Tessier et al. 1993,
  Pandey et al. 1996..)
• River network/channel approach
• many known scaling laws,
• channels may be obtained from RS data (Rinaldo
  and coll.)
• ? how to go from geometry to flows?
• (see however, Gupta and Waymire 1998)
• space-time multifractal field approach
• not only channel, but hill slope contributions,
• ? (quasi-)translation invariance?

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Multifractal understanding of phenomenon
The variability over a wide range of scales
The topography a1.8, C10.12, H0.6
Gagnon et al, EPL 2003
lg3
lg2
lg1

• stochastic formalism
• scaling of the pdf and moments with respect to
  the scale ratio l L/l

Pr el ³ lg l-c(g) Û lt (el)q gt lK(q)
add-g t(q) q(d-1) -K(q) fd(ad)d-c(g)

relation with deteminism formalism (gltd)
20000 km
50cm
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Understanding of extreme phenomenon
A multiplicative cascade down to scale ratio L
is equivalent to

• - a bare cascade constructed over ratio l,
• - and multiplied by a hidden cascade obtained by
  re-scaling by factor l a cascade constructed from
  1 to L / l

Averaged, this factor can be scaling therefore
divergent with L

• exists a finite critical order qD
• Pr (elgts) s-qD
  for any q gt qD lt elqgt
• it gives a power-law tail (quite distinct from
  the exponential behavior!)
• a linear codimension c(g) qD (g -g D) c(g
  D) for g gt g D

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Multifractal Universality

• Nonlinearly interracting i.i.d multrifractal
  processes converge towards Universal
  Multifractals
• 3 fundamental exponents
• H mean deviation from conservation lt el gt l-
  H
• C1  mean fractality
• (codimension of the mean field)
• a  (0 a 2 ) index of multi-fractality
  (convexity of c(g))

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From multifractal precipitation to discharge (ex.
Abbeville)
Physical space
Fourier space
Monthly precipitation from January 1963 to April
2001

To filter by k(-H)
Somme monthly discharges(1963-2001)
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Comparison of probability distributions
(precipitation, synthetic discharges and
discharges)
Pr ( Fl gt s ) s-qD
qD of the precipitation and of discharge data are
not the same!..
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Long time rainfall series
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Long time discharge series
qD universel ?
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Towards multifractal flood forecast
To consider discharges Q as a space-time
multifractal field, i.e. defined in the
neighborhood of every space-time point gt
small scale singular behavior
Which data to use discharge Q (output),
drainage area A (characteristic of the
basins) specific discharge QsQ /A and (if
possible) rain R,
Conjectures I multifractal behavior of Qs is
similar to R II statistical independence
between A and Qs gt factorization of the
basin variability and the rain variability
To be able to use the data from inhomogeneously
distributed gauges
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Networks of gauges
From UNESCO database France represented
by Beaucaire, Blois, Gien, Givors, Mas-d'Agenais,
Montjean, Paris
PRECIP Data base (Météo-France)
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database
3500 river runoff gauges with time scales 10
years - 1.5 centuries
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Network of runoff gauges
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Fractal tests with R-ArcticNet Data
Results of box counting
Pr ( Fl gt s ) s-qD
Probability distributions for Ob zone (from left
to right) - specific discharges (12 months of
1980), - discharges (12 months of 1980), -
drainage areas.
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Intersection theorem
Multifractal Field x Gauging Network Measured
Field
Universal multifractals ( two parameters for
K(q) ) C1- mean fractality a  (0a2)-
multifractality index
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Variability factorization
statistical independence
K(q) functions without the network correction
K(q) functions after the network correction -
Discharges (circles, a0.75, C11.35) - Drainage
areas (dashed line, a0.66, C11.11) - Specific
discharges (crosses, a0.39, C10.76), - The sum
of functions of drainage areas and specific
discharges (continues line)
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Relation Qs - R
Multifractal parameters for specific discharge
Q - black, A -gray, Qs-white
Multifractal parameters for rain (PRECIP data
base)
Biaou et al., 2002
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Conclusions (1)

• Statistics of the extremes
• Gumbel law underestimates the return period
• The law of Fréchet (Log-Gumbel) is better
  adapted than Gumbel law
• But we are in the case of long range correlation
  data
• (statistical) properties multifractal process
  mixture? (Loynes, 1965, Leadbetter, 1988,..)

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Conclusions (2)

• Q , A , Qs multifractals, i.e. defined in the
  neighborhood of every space-time point small
  scale singular behavior,
• possibility to use in-situ data measured with the
  help of a fractal network of measurements,
• multifractal behavior of Qs is similar to R,
• rather statistical independence between A and Qs
• gt factorization of the drainage area
  variability and rain variability,
• multiple consequences to explore. (e.g.,
  space-time),
• but already, Qs has more universal behavior than
  Q.