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Extreme Events, Heavy Tails, and the Generating Processes:
Examples from Hydrology and Geomorphology
Efi Foufoula-Georgiou
SAFL, NCED
University of Minnesota
E2C2 – GIACS Advanced School on “Extreme Events: Nonlinear Dynamics and Time Series Analysis
Comorova, Romania
September 3-11, 2007
In Hydrology and Geomorphology “Fluctuations” around the mean behavior are of high magnitude.
Understanding their statistical behavior is useful for prediction of extremes and also for understanding spatio-temporal heterogeneities which are hallmarks of the underlying process- generating mechanism.
These fluctuations are often found to exhibit power law tails and scaling
Underlying Theme
PRESENCE OF SCALING
... scaling laws never appear by accident. They always manifest a property of the phenomenon of basic importance … This behavior should be discovered, if it exists, and its absence should also be recognized.”
Barenblatt (2003)
High-resolution temporal rainfall data
(courtesy, Iowa Institute of Hydraulic Research – IIHR)
~ 5 hrs
t = 10s
~ 1 hr
t = 5s
STREAMLAB 2006
Data Available:
-Sediment accumulation series
-Time series of bed elevation
-Laser transects of bed elevation
Pan-1 Pan-2 Pan-3 Pan-4 Pan-5
Bed Elevation
0 100 200 300 400 5000
100
200
300
h(t)
(m
m)
Q = 4300 lps
0 100 200 300 400 5000
100
200
300
time (min)
h(t)
(m
m)
Q = 5500 lps
= 28.64 mm = 9.79 mm
= 184.14 mm = 65.78 mm
Noise-free sediment transport rates
Weigh pan bedload transport rates (Q = 5.5 m3/s) (a) 1 s averaging and 0 point skip (b) 15s averaging time and 6 point skip (from Ramooz and Rennie, 2007)
• Characterize a signal f(x) in terms of its local singularities
( ) ( ) ( )0
0 0
h xf x f x Ce e- + £ ×
Ex: h(x0) = 0.3 implies f(x) is very rough around x0.
h(x0) = 0.7 implies a “smoother” function around xo.
Localized Scaling Analysis: Multifractal Formalism
h=0.3
h=0.7
• Spectrum of singularities D(h)
• D(h) can be estimated from the statistical moments of the fluctuations.
( ) ( ) ( ) ( ), ~ qqM q a f x f x a a t= - +
( ) ( )min 1qqh h qD té ù= - +ë û Legendre Transform
Multifractal Formalism
h
D(h)
H
Multifractal Spectra
• Spectrum of scaling exponents (q) and Spectrum of singlularities D(h)
monofractal
multifractal
h
h
1
2 0
c H
c
=
=
1
2 0
c
c ¹
Multifractal Spectra
Spectrum of scaling exponents
h
Spectrum of singularities
-11 2 3 4 5 6 7
(2)
Slopes (q)
q
D(h)
0q
0q 0q
q
dh
hdDDf
hmin hmax
h
Wavelet-based multifractal formalism(Muzy et al., 1993; Arneodo et al., 1995)
1[ ] ,
u xT f x a f u du
a a
CWT of f(x) :
The local singularity of f(x) at point x0 can be characterized
by the behavior of the wavelet coefficients as they change with scale, provided that the order of the analyzing wavelet
n > h(x0)
Can obtain robust estimates of h(x0) using “maxima lines”
only: Ta(x) i.e. WTMM
0~ 0xh
a axT 0a
It can be shown that
fDqq
a axT ~)( 0
f(x)
T[f](x,a)
WTMMTa(x)
Structure FunctionMoments of
|f(x+l) – f(x)|
Partition FunctionMoments of|T[f](x,a)|
Partition FunctionMoments of |Ta(x)| (access to q < 0)
Cumulant analysisMoments of ln |Ta(x)|
(direct access to statistics of singularities)
Landscape dissection
• Planform topology of channelized and unchannelized paths (branching structure of river networks and hillslope drainage patterns)
• Vertical structure of landform heterogeneity perpendicular to the river paths.
River bedform morphodynamics and sediment transport rates
Two Examples
Width and Area Functions of a River Network
( )W x =
( )A x =
( )W x ®
( )A x ®
# of channels intersected by a contour of equal flow length to the outlet
# of pixels of equal flow length to the outlet
Topology of river network
Topology of the hillslope drainage paths and topology of river network
Width and Area Functions of a River Network
( )W x =
( )A x =
( )W x ®
( )A x ®
# of channels intersected by a contour of equal flow length to the outlet
# of pixels of equal flow length to the outlet
Topology of river network
Topology of the hillslope drainage paths and topology of river network
Walawe River Basin: deviation from simple scaling
A rich multifractal structure is observed which is different for A(x) and W(x)
A(x)
A(x)
W(x)W(x)
Noyo River Basin: deviation from simple scaling
A rich multifractal structure is observed which is different for A(x) and W(x)
A(x)
W(x)
Noyo River Basin (10x10m; A143 km2)
c1 0.77 c2 0.11 SR = 0.07 - 0.43 km
-“Hillslope” path dominated
-“smoother” overall than W(x)
-Hillslope drainage dissection is s-s between scales 0.1 km – 0.5km
-Statistics of the density of hillslope drainage paths strongly depend on scale
c1 0.46 c2 0.10 SR = 0.13 – 0.70 km
-River network path dominated
-“Rougher “overall” than A(x)
-Channel network landscape dissection is s-s between scales 0.1 km to 0.7 km
-Strong inntermittency (higher moments of pdf of channel drainage density has a strong dependence on scale)
Pay attention not only to the average properties of landscape dissection but to higher moments
Real River Networks
Walawe River, Sri Lanka
(90x90m)
A2,000 km2
South Fork Eel River, CA
(1x1m & 10x10m)
A154 km2
Noyo River basin, CA
(10x10m)
A1430 km2
C1=0.40C2=0.05
C1=0.80C2=0.05
C1=0.77C2=0.11A(x)
Area Functions of Simulated Trees
Peano Basin
Shreve’s randomnetwork model
Stochastic S-S model with ()=(1 , 2)
c1=0.5c2=0
c1=0.62c2=0
Not comparable to real networks
Conclusions on topology of drainage paths
Simulated river networks show different multifractal properties than real river networks. [s-s trees are monofractal with H = 0.5 – 0.65 while real networks are multifractal with H 0.4 – 0.8].
Differences between scaling properties of A(x) and W(x) depict differences in the branching topology of channelized vs. unchannelized drainage paths.
Deviation from monoscaling stresses the importance of the dependence on scale of higher order statistics of the branching structure.
Implications for Network Hydrology?
Conjecture: Deviation from scale invariance in W(x), implies that the variability of the in-phase hillslope hydrographs entering the network depends on “scale”
Implications for routing? scale-dependent convolution? geomorphologic dispersion?
Implications for scaling of hydrographs?
1. What is the statistical structure of RCW(x)?
2. Do physically distinct regimes exhibit statistically distinct signatures?
3. How can the statistical structure be used in modeling and prediction of hydrographs, sedimentographs and pollutographs across scales?
Questions
89 tributaries: (1 km2 – 150 km2)
River Corridor Width Function: South Fork Eel River
6 km 14 km 20 km 28 km 35 km
INTERPRETATION OF RESULTS
More localized NL transport mechanism?
More localized on L than R side?
Smoother overall valleys?
Presence of more terraces in R than L?
Conclusions and Open Questions…
• Hillslope “roughness” seems to carry the signature of valley forming processes; need to provide a complete hierarchical characterization. Do hillslope evolution models reproduce this structure? What is the effect on hillslope sediment variability of the higher order statistics of travel paths to streams?
Experimental setup
Data Available:
-Sediment accumulation series
-Time series of bed elevation
-Laser transects of bed elevation
Pan-1 Pan-2 Pan-3 Pan-4 Pan-5D50 = 11.3mm
StreamLab06, Bed Material Grainsize Distributions
0
10
20
30
40
50
60
70
80
90
100
1.0 10.0 100.0
diameter (mm)
Cu
mm
ula
tiv
e P
erc
en
t F
ine
r
1.0 100Diameter [mm]
Discharge controlled here Channel Width = 2.75 m
Channel Depth = 1.8 m
• Discharge capacity: 8500 lps• Coarse sediment recirculation system located 55 m from upstream end.
D50=11.3 mm
QUESTIONS
1. Do the statistics of sediment transport rates depend on “scale” (sampling interval or time interval of averaging) and how?
2. Does this statistical scale-dependence depend on flow rate, bed shear stress, and bedload size distribution (e.g., gravel vs. sand, etc.)
3. Do the statistics of sediment transport relate to the statistics of bedform morphodynamics and how?
4. What are the practical implications of all these?
0 5 10
x 104
0
100
200
300Q=4300 lps
time(sec)
Sc(t
) kg
)
0 5 10
x 104
-1
0
1 Q=4300 lps
time(sec)
S(t
)(kg
)0 5 10
x 104
0
5000 Q=4900lps
time(sec)
Sc(t
)(kg
)
0 5 10
x 104
-2
0
2 Q=4900 lps
time(sec)S
(t)(
kg)
0 5 10
x 104
0
1
2
3x 10
4
Q=5500 lps
time(sec)
Sc(t
)(kg
)
0 5 10
x 104
-5
0
5 Q=5500 lps
time(sec)
S(t
)(kg
)
Sediment Transport Rates
Accumulated series (Sc(t)) Nearest neighbor differences (S(t))
0 2 4 6 8 10
x 104
0
0.5
1
1.5
2
2.5
3x 10
4
Q=5500 lps
time(sec)
Sc(t
)(kg
)
0 20 40 60 80 100 1201.2736
1.2738
1.274
1.2742
1.2744
1.2746
1.2748
1.275x 10
4
Q=5500 lps
time(sec)
Sc(t
)(kg
)
0 200 400 600 800 1000 12001.265
1.27
1.275
1.28
1.285
1.29
1.295
1.3
1.305x 10
4 Q=5500 lps
time(sec)
Sc(t
)(kg
)
VARIABILITY AT ALL SCALES
Sc (t) = Accumulated sediment over an interval of 0 to t sec
ANALYSIS METHODOLOGY: ADVANTAGES
1. Local analysis (as opposed to global, e.g., spectral analysis)
2. Can characterize the statistical structure of localized abrupt fluctuations over a range of scales
3. Wavelet-based multifractal formalism -- uses generalized fluctuations instead of standard differences (f(x) – f(x+dx))
Can automatically remove non-stationarities in the signal both in terms of overall trends and in terms of low-frequency oscillations coming from dune or ripple effects
Can automatically remove noise in the signals and point to the minimum scale that can be safely interpreted
Can characterize effectively how pdfs change with scale with only one or two parameters
Noise Variability levels off
1 min 15 min
4 6 8 10 12 140
5
10
15
20
25
30
35
40
45Q = 5500 lps
log2(a) (sec)
S(q
,a)
q = 0.5
q = 1.0
q = 1.5
q = 2.0
q = 2.5
q = 3.0
Scaling range
0 1 2 3 40
0.5
1
1.5
2
2.5
3
3.5
4
(q
)
Q = 5500 lps
q
SEDIMENT TRANSPORT RATES: Q = 5500 lpslo
g 2
C1=1.10C2=0.10
Noise Statistical Variability regime changes
1 min 10 min
Scaling range
0 0.5 1 1.5 2 2.5 3 3.50
0.5
1
1.5
2
2.5
3
3.5
4
(q)
Q=4300 lps
q4 6 8 10 12 14
0
5
10
15
20
25
30
35
40
45Q = 4300 lps
log2(a) (sec)
S(q
,a)
q =3.0
q =2.5
q =2.0
q =1.5
q =1.0
q =0.5
Q = 4300 lpslo
g 2
C1=0.55C2=0.15
Bed Elevation
0 100 200 300 400 5000
100
200
300
h(t)
(m
m)
Q = 4300 lps
0 100 200 300 400 5000
100
200
300
time (min)
h(t)
(m
m)
Q = 5500 lps
= 28.64 mm = 9.79 mm
= 184.14 mm = 65.78 mm
BED ELEVATION TEMPORAL SERIES: Q = 5500 lps
0 1 2 3 40
0.5
1
1.5
2
2.5
3
3.5
4
(q)
Q = 5500 lps
q1 2 3 4 5 6 7 8
0
5
10
15
20
25
30
35
40
45Q = 5500 lps
log2(a) (sec)
log
2 S
(q,a
)
q = 3.0
q = 2.5
q = 2.0
q = 1.5
q = 1.0
q = 0.5
Scaling range
0.5 min 8 min C1=0.70C2=0.11
BED ELEVATION TEMPORAL SERIES: Q = 4300 lps
0 1 2 3 40
0.5
1
1.5
2
2.5
3
3.5
4
(q)
Q = 4300 lps
q2 3 4 5 6 7 8
5
10
15
20
25
30
35
40
45Q = 4300 lps
log2(a) (sec)
log
2 S
(q,a
)
q = 3.0
q = 2.5
q = 2.0
q = 1.5
q = 1.0
q = 0.5
Scaling range
12 min1 min C1=0.55C2=0.05
Inferences on Nonlinearity
0.4 0.5 0.6 0.7 0.80
2
4
6
8
10
12Q=4300 lps
Rel
ativ
e fre
quen
cy
Transportation Distance
elevationssurrogates
4 6 8 10 120
0.2
0.4
0.6
0.8
1
1.2
1.4Q=5500 lps
Rel
ativ
e fre
quen
cy
Transportation Distance
elevationssurrogates
Basu and Foufoula-Georgiou, Detection of nonlinearity and chaoticity in time series using the transportation distance function, Phys. Letters A, 2002.
Finite Size Lyapunov Exponent (FSLE)
• FSLE is based on the idea of error growing time (Tr(d)), which is the time it takes for a perturbation of initial size δ to grow by a factor r (equals to √2 in this work)• measure the typical rate of exponential divergence of nearby trajectory
δ(nr) size of the perturbation at the time nr at which this perturbation first exceeds (or becomes equal to) the size rδ
For an initial error δ and a given tolerance ∆ = rδ, the average predictability time
)(
lndTP
)(
ln1
)( r
r
n
n
Basu et al., Predictability of atmospherci boundary layer flows as a function of scale, Geophys. Res. Letters, 2002.
CONCLUDING REMARKS
• Documented a clear dependence of sediment transport rates and of the corresponding bed elevation series on “scale”
• Need to explore more rigorously the dependence on flow rate, grain size distribution, etc. and how the self-organized structure of the bed elevation reflects itself in the statistics of the sediment transport rate
• Must think about the implications of scaling for sampling and also for the development of sediment transport equations
References
Gangodagamage, C., E. Barnes, and E. Foufoula-Georgiou, Scaling in river corridor widths depicts organization in valley morphology, Geomorphology, doi:10.1016/j.geomorph.2007.04.414, 2007.
Lashermes, B. and E. Foufoula-Georgiou, Area and width functions of river networks: new results on multifractal properties, Water Resources Research, doi:10.1029/2006WR005329, 2007
Lashermes, B., E. Foufoula-Georgiou, and W. Dietrich, Channel network extraction from high resolution topograhy using wavelets, Geophysical Research Letters, in press, 2007.
Sklar L. S., W. E. Dietrich, E. Foufoula-Georgiou, B. Lashermes, D. Bellugi, Do gravel bed river size distributions record channel network structure?, Water Resources Research, 42, W06D18, doi:10.1029/2006WR005035, 2006.
Barnes, E. M.E. Power, E. Foufoula-Georgiou, M. Hondzo, and W.E. Dietrich, Scaling Nostic biomass in a gravel-bedrock river: Combining local dimensional analysis with hydrogeomorphic scaling laws, Geophysical Research Letters, under review.
0.2 0.25 0.3 0.35 0.4 0.45 0.50
5
10
15
20
25 H = 0.5
Rel
ativ
e fre
quen
cy
Transportation Distance
fbmsurrogates
10-2
10-1
100
101
10-4
10-3
10-2
10-1
100
101
( )
H = 0.5
slope = -2.0
0.4 0.6 0.8 1 1.2 1.4 1.6 1.8
x 10-3
0
2000
4000
6000
8000
10000
c1=0.7 c
2=0.2
Re
lativ
e fr
eq
ue
ncy
Transportation Distance
RWCsurrogates
10-5
10-4
10-3
10-2
10-1
10-4
10-3
10-2
10-1
100
( )
c1=0.7 c
2=0.2
slope= -2
Transportation Distance
• based on both the geometric and probabilistic aspects of point distributions • provide a measure of long term qualitative differences between any
two time series (x and y).
μij > 0 amount of material shipped from box Bi to box Bj
δij taxi cab metric normalized to the embedding dimension between the centres of Bi and Bj
b
jiijijqpMqpd
1,),(inf),(
RECALL
( ) ( ) ( ) ( )~q q
c ct k t t k t tx x+ ×D - ×D
( )2
1 2 2q
q c q ct = × - ×
• c2=0 monofractal (2)=2(1)
• all moments can be scaled with one parameter c1=H only
• CV is constant with scale
• c20 multifractalfractal (2)<2(1)
• need 2 parameters c1, c2 to scale pdfs
• CV decreases with increase in scale
{
{
1.
2.
(k t) =Sediment transported during a time period k t
t =Sampling interval
~
~
~
( )E k txD ×D
( )2
E k txD ×D
( )CV k txD ×D
( ) ( )1k t t×D
( ) ( )2k t t×D
( ) ( ) ( )2 2 1k t t t-×D