Extreme Events, Heavy Tails, and

1
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
2
Underlying Theme

• 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

3
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)

4
High-resolution temporal rainfall data

• (courtesy, Iowa Institute of Hydraulic Research
  IIHR)

5
Noyo River basin
6
STREAMLAB 2006

• Data Available
• Sediment accumulation series
• Time series of bed elevation
• Laser transects of bed elevation

Pan-2
Pan-1
Pan-3
Pan-4
Pan-5
7
Experimental setup

• Data Available
• Sediment accumulation series
• Time series of bed elevation
• Laser transects of bed elevation

D50 11.3mm
Discharge controlled here
Channel Width 2.75 m Channel Depth 1.8 m
D5011.3 mm
100
Diameter mm
1.0

• Discharge capacity 8500 lps
• Coarse sediment recirculation system located 55 m
  from upstream end.

8
Bed Elevation
9
Sediment Transport Rates
Accumulated series (Sc(t))
Nearest neighbor differences (S(t))
10
SEDIMENT FLUX VARIABILITY AT ALL SCALES
Sc (t) Accumulated sediment over an interval of
0 to t sec
11
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)
12
Background

• 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
• There is a continuous need for new mathematical
  tools of analysis and new paradigms of modeling
  physical phenomena that exhibit a rich
  statistical structure

13
Localized Scaling Analysis Multifractal Formalism
Characterize a signal f(x) in terms of its
local singularities
h0.3
h0.7
Ex h(x0) 0.3 implies f(x) is very rough around
x0. h(x0) 0.7 implies a smoother function
around xo.
14
Multifractal Formalism
Spectrum of singularities D(h)
D(h)
h
D(h) can be estimated from the statistical
moments of the fluctuations.
Legendre Transform
15
Multifractal Spectra
Spectrum of scaling exponents t(q) and Spectrum
of singlularities D(h)
monofractal
h
multifractal
h
16
Multifractal Spectra

• Spectrum of scaling exponents

Spectrum of singularities
D(h)
Df
h
hmax
hmin
17
Wavelet-based multifractal formalism(Muzy et
al., 1993 Arneodo et al., 1995)
CWT of f(x)

18
f(x)
Structure Function Moments of f(xl) f(x)
T?f(x,a)
Partition Function Moments of T?f(x,a)

• ? Partition Function
• Moments of Ta(x)
• (access to q lt 0)
• ? Cumulant analysis
• Moments of ln Ta(x)
• (direct access to statistics of singularities)

WTMMTa(x)
19
Two Examples

• 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

20
Two Ways of Looking at Landscapes

• Planform Dissection
• Topology of the river network (channelized
  paths) W(x)
• Topology of the unchannelized flow paths A(x)
  W(x)
• Vertical roughness of topography
• Structure of the river corridor width (RCW)
  series

21
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
22
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
23
Walawe River, Sri Lanka (90x90m) A2,000 km2
24
Noyo River Basin, California, USA (10x10m) A143
km2
25
A(x)
W(x)
W(x)
A(x)
A rich multifractal structure is observed which
is different for A(x) and W(x)
26
A(x)
W(x)
A rich multifractal structure is observed which
is different for A(x) and W(x)
27
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
28
c1 0.40 c2 0.05 SR 0.4 - 3.0 km
c1 0.50 c2 0.13 SR 2.4 - 6.0 km
Accurate characterization of higher moments needs
high resolution topography data
29
Noyo River basin, CA (10x10m) A1430 km2
Walawe River, Sri Lanka (90x90m) A2,000 km2
South Fork Eel River, CA (1x1m 10x10m) A154 km2
C10.77 C20.11
C10.40 C20.05
C10.80 C20.05
A(x)
30
Peano Basin
Not comparable to real networks
Shreves random network model
c10.5 c20
Stochastic S-S model with (a , b)(1 , 2)
c10.62 c20
31
(a,b)(1,2) Tk correspond to those of Shreve
model.
Shreve Model H 0.5 S-S tree H 0.62
Yet
ß
Difference reflects the effect of the statistical
distribution of Tk (Tk Deterministic for
Shreves model Poisson distributed for S-S tree)
ß
Randomness in the River Network Topology is
reflected in the statistical properties of the
width function
32

• 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.

33

1. W(x) of channels intersected by a contour of
   equal length x to the outletW(xdx)-W(x)
   of new channels within a strip of dx flow
   distance to outlet of copies of new
   hillslope hydrographs combined in phase and
   delivered to channel.
3. Shreves random topology model c10.5, c20.0,
   for all dx
4. Walawe River Basin c10.5, c20.13 for dx 2
   6 km
5. Deviation from scale invariance (c2 ¹ 0) Þ
   extended S-S within a limited range of scales
6. An intermittent application of in-phase new
   hillslope hydrographs along the river network Þ
   Higher chance of a disproportionately larger of
   in-phase new hydrographs to enter the network at
   smaller than larger distances apart Þ
   Implications for routing (e.g., scale-dependent
   convolution? geomorphologic dispersion?) Þ
   Implications for scaling of hydrographs?

34

• Conjecture Deviation from scale invariance in
  W(x), implies that the variability of the
  in-phase hillslope hydrographs entering the
  network depends on scale
• ÞImplications for routing? scale-dependent
  convolution? geomorphologic dispersion?
• Þ Implications for scaling of hydrographs?

35
River Corridor Width Functions
36
River Corridor Width Function (D5m)
37
SCALING OR NOT?

• Why are scaling laws of such distinguished
  importance?
• The answer is that 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)

38
Specific Questions Area and Width Functions

1. Does the topology of river networks leave a
   unique signature on the scaling of W(x) and A(x)?
   
2. How different are the scaling properties of
   commonly used simulated trees and those of real
   river networks?
3. Are there differences between the scaling
   properties of A(x) and W(x) and what do these
   tell us about the topology of hillslope vs.
   channelized drainage patterns in a river basin?
4. What are the hydrological implications of the
   above?

39

• Need a localized multiscale analysis methodology
  to locally characterize abrupt fluctuations
  (coming about from the underlying branching
  topology)
• Energy associated with the small scales is
  not uniformly distributed over the river network
  characterize the statistical nature of the
  points (flow distances from outlet) at which
  abrupt local changes in W(x) or A(x) exist.
• The multifractal formalism (Parisi and Frisch,
  1985) allows this characterization from the
  statistics of W(x) and A(x) fluctuations.

40
Multifractal Formalism
Spectrum of singularities D(h)
D(h)
h
D(h) can be estimated from the statistical
moments of the fluctuations.
Legendre Transform
41
Multifractal Spectra
Spectrum of scaling exponents t(q) and Spectrum
of singlularities D(h)
monofractal
h
multifractal
h
42
A(x)
W(x)
W(x)
A(x)
A rich multifractal structure is observed which
is different for A(x) and W(x)
43
A(x)
W(x)
A rich multifractal structure is observed which
is different for A(x) and W(x)
44
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
45
c1 0.40 c2 0.05 SR 0.4 - 3.0 km

• Hillslope path dominated
• rougher overall than W(x)
• Hillslope drainage dissection is s-s between
  scales 0.4 km 3.0 km
• Statistics of the density of hillslope drainage
  paths depends on scale

c1 0.50 c2 0.13 SR 2.4 - 6.0 km

• River network path dominated
• Smoother overall than A(x)
• Channel network landscape dissection is s-s
  between scales 2.6 km to 6.0 km
• Much more intermittent (higher moments of pdf of
  channel drainage density has a strong dependence
  on scale)

Accurate characterization of higher moments needs
high resolution topography data
46

• Conjecture Deviation from scale invariance in
  W(x), implies that the variability of the
  in-phase hillslope hydrographs entering the
  network depends on scale
• ÞImplications for routing? scale-dependent
  convolution? geomorphologic dispersion?
• Þ Implications for scaling of hydrographs?

47
Noyo River basin, CA (10x10m) A1430 km2
Walawe River, Sri Lanka (90x90m) A2,000 km2
South Fork Eel River, CA (1x1m 10x10m) A154 km2
C10.77 C20.11
C10.40 C20.05
C10.80 C20.05
A(x)
48
Noyo River basin
49
Peano Basin
Not comparable to real networks
Shreves random network model
c10.5 c20
Stochastic S-S model with (a , b)(1 , 2)
c10.62 c20
50
(Tokunaga, 1996, 1978 Peckham, 1995)

average of tributaries of order w that
branch into a stream of order w
51
(a , b) (0.75, 1.894) (1, 2) (1.25, 2.095) (1.5, 2.183) (1.75, 2.266) (1.5, 2.5) (1, 3)
D 2 2 2 2 2 1.76 1.41
order 13 12 11 11 10 10 10
3894 4160 3618 5817 3950 6435 14827
c1 0.65 0.62 0.55 0.55 0.55    
c2 0.00 0.01 0.00 0.01 0.00    
Note as a increases, c1 decreases, i.e., when
branching rate increases A(x) exhibits wilder
fluctuations and becomes more irregular.
52
(a,b)(1,2) Tk correspond to those of Shreve
model.
Shreve Model H 0.5 S-S tree H 0.62
Yet
ß
Difference reflects the effect of the statistical
distribution of Tk (Tk Deterministic for
Shreves model Poisson distributed for S-S tree)
ß
Randomness in the River Network Topology is
reflected in the statistical properties of the
width function
53
A(x) W(x)
Walawe A2,000 km2 (90x90m) c10.37 0.40 c20.05 c10.50 c20.132
South Fork A154 km2 (1x1m 10x10m) c10.80 c20.05
Beaver Creek A622 km2 (30x30m) c10.44 c20.05
Noyo River Basin A 143 km2 (1x1m) c10.77 c20.11 c10.46 c20.11
Lower Noyo River Basin A km2 (1x1m) c1 c2 c1 c2
54

• 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.

55

1. W(x) of channels intersected by a contour of
   equal length x to the outletW(xdx)-W(x)
   of new channels within a strip of dx flow
   distance to outlet of copies of new
   hillslope hydrographs combined in phase and
   delivered to channel.
3. Shreves random topology model c10.5, c20.0,
   for all dx
4. Walawe River Basin c10.5, c20.13 for dx 2
   6 km
5. Deviation from scale invariance (c2 ¹ 0) Þ
   extended S-S within a limited range of scales
6. An intermittent application of in-phase new
   hillslope hydrographs along the river network Þ
   Higher chance of a disproportionately larger of
   in-phase new hydrographs to enter the network at
   smaller than larger distances apart Þ
   Implications for routing (e.g., scale-dependent
   convolution? geomorphologic dispersion?) Þ
   Implications for scaling of hydrographs?

56

• Conjecture Deviation from scale invariance in
  W(x), implies that the variability of the
  in-phase hillslope hydrographs entering the
  network depends on scale
• ÞImplications for routing? scale-dependent
  convolution? geomorphologic dispersion?
• Þ Implications for scaling of hydrographs?

57
South Fork Eel River, CA
Area 351 km2
58
River Corridor Width Function (D5m)
59
Questions

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?

60
SOUTH FORK EEL RIVER, CA Hypsometric Profile
61
River Reach 0-6 Km
62
Reach 20-28 km
Right
Left
63
Motivating questions

• Are statistically-distinct regimes the result of
  physically-distinct valley-forming processes?
• Do differences in mechanistic laws governing
  valley-forming processes leave their signature on
  the statistical properties of valley geometry?
• How can these statistical properties be used for
  modeling and prediction?

64
River Corridor Width Function South Fork Eel
River
6 km
14 km
20 km
28 km
35 km
89 tributaries (1 km2 150 km2)
65
River Reach 0-6 km
66
River Reach 20-28 km
67
Recall the interpretation of multifractal
parameters

• C1 a larger value means a smoother function
  (more smoothing than roughening mechanisms are
  responsible for the formation of this surface)
• C2 a larger value means more intermittency
  (localized very large fluctuations or bursts
  are present signaling nonlinear and localized
  transport mechanisms)

68
SUMMARY OF RESULTS
Right-Left asymmetry
69
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?
70
Conclusions on RCW Series

1. RCW fluctuations exhibit a deviation from
   scale-invariance
2. As we move from the bedrock to more alluvial
   valleys, the statistical intermittency increases
   indicating an increased presence of very
   localized abrupt fluctuations probably due to
   increasingly localized transport mechanisms.
3. There is a significant left-right asymmetry in
   the statistical structure of RCWs reflecting
   different underlying processes in each side of
   the river.

71
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?

72
Some Words of Caution
There exist multiple ways by which an emergent
pattern can manifest itself from mechanistic or
physical processes
Ex. 1
Omittance of floodplain and two distinct rainfall
regimes -or- channel-floodplain interactions with
a single rainfall regime ß Both can give a
scaling break in floods Þ need enough underlying
observations to pose the right hypotheses which
might be region-dependent
Peano basins have been used in modeling studies
to relate the scaling of hydrograph peaks to the
scaling of the peaks of the width functions and
several runoff production mechanisms. But scaling
of Peano basin W(x) ¹ scaling of real network
W(x) Þ Implications?
Ex. 2
73
References
Gangodagamage, C., E. Barnes, and E.
Foufoula-Georgiou, Scaling in river corridor
widths depicts organization in valley morphology,
Geomorphology, doi10.1016/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, doi10.1029/2006WR005329,
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,
doi10.1029/2006WR005035, 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.
74
Experimental setup

• Data Available
• Sediment accumulation series
• Time series of bed elevation
• Laser transects of bed elevation

D50 11.3mm
Discharge controlled here
Channel Width 2.75 m Channel Depth 1.8 m
D5011.3 mm
100
Diameter mm
1.0

• Discharge capacity 8500 lps
• Coarse sediment recirculation system located 55 m
  from upstream end.

75
STREAMLAB 2006

• Data Available
• Sediment accumulation series
• Time series of bed elevation
• Laser transects of bed elevation

Pan-2
Pan-1
Pan-3
Pan-4
Pan-5
76
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?

77
Sediment Transport Rates
Accumulated series (Sc(t))
Nearest neighbor differences (S(t))
78
VARIABILITY AT ALL SCALES
Sc (t) Accumulated sediment over an interval of
0 to t sec
79
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)
80
LOCAL ROUGHNESS OF A SIGNAL
Characterize a signal f(x) in terms of its local
singularities
Ex h(x0) 0.3 implies f(x) is very rough around
x0. h(x0) 0.7 implies a smoother function
around xo.
81
MULTIFRACTAL FORMALISM
Spectrum of singularities D(h)
D(h)
h
D(h) can be estimated from the statistical
moments of the fluctuations.
Legendre Transform
82
MULTIFRACTAL FORMALISM
Spectrum of scaling exponents t(q)
monofractal
h
multifractal
h
83
ANALYSIS METHODOLOGY ADVANTAGES

• Local analysis (as opposed to global, e.g.,
  spectral analysis)
• Can characterize the statistical structure of
  localized abrupt fluctuations over a range of
  scales
• Wavelet-based multifractal formalism -- uses
  generalized fluctuations instead of standard
  differences (f(x) f(xdx))
• 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

84
SEDIMENT TRANSPORT RATES Q 5500 lps
log2
Noise
Variability levels off
Scaling range
C11.10 C20.10
15 min
1 min
85
Q 4300 lps
log2
Noise
C10.55 C20.15
Statistical Variability regime changes
Scaling range
1 min
10 min
86
SEDIMENT TRANSPORT RATES SUMMARY TABLE OF c1, c2
          Polynomial approx. Polynomial approx.
Q (lps) Pan Scaling Range (min) Shield stress t(2) 2 t(1) c1 c2 c1 (cumulants)
               
  2 1 10 0.085 -0.26 0.40 0.15 0.53
4300 3 1 10   -0.20 0.56 0.14 0.57
  4 1 -- 3   -0.10 0.52 0.05 0.53
               
  2 2 15 0.111 -0.12 1.30 0.11 1.34
4900 3 2 15   -0.14 1.33 0.10 1.33
  4 2 15   -0.09 1.24 0.08 1.24
               
  2 1 15 0.196 -0.13 1.09 0.09 1.17
5500   3 1 15   -0.15 1.07 0.09 1.18
5500   4 1 -- 15   -0.15 1.15 0.10 1.25
87
RECALL

• Higher c1 means a smoother signal
• Higher c2 means a stronger dependence of the
  higher moments on scale, spatially inhomogeneous
  distribution of extreme bursts, more likelihood
  of extreme bursts at very small scales

c20 monofractal t(2)2t(1) all moments can
be scaled with one parameter c1H only CV is
constant with scale
c2¹0 multifractalfractal t(2)lt2t(1) need 2
parameters c1, c2 to scale pdfs CV decreases
with increase in scale
88
INTERPETATION AND PRACTICAL IMPLICATIONS
Low flows

• The sedimentation rate is a fractal (s-s)
  process
• The longer the time interval, the lower the
  average sedimentation rate (in double the time,
  sedimentation rate decreases by a factor of 0.7
  2(-0.4))
• The smaller the time interval, the higher the
  chance to encounter an extreme localized rate

High Flows

• The change in sedimentation rate is a fractal
  (s-s) process
• The longer the time interval, the higher the
  expected change in sedimentation rate (in double
  the time, the gradient of sedimentation rate
  changes by a factor of 1.1 2(0.2))

89
Bed Elevation
90
Bed elevation data
Note these series are plotted till 3000 data
points to show the same scale
91
BED ELEVATION TEMPORAL SERIES Q 5500 lps
Scaling range
C10.70 C20.11
0.5 min
8 min
92
BED ELEVATION TEMPORAL SERIES Q 4300 lps
Scaling range
C10.55 C20.05
12 min
1 min
93
Inferences on Nonlinearity
Basu and Foufoula-Georgiou, Detection of
nonlinearity and chaoticity in time series using
the transportation distance function, Phys.
Letters A, 2002.
94
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 d to grow by
  a factor r (equals to v2 in this work)
• measure the typical rate of exponential
  divergence of nearby trajectory
• d(nr) size of the perturbation at the time nr at
  which this perturbation first exceeds (or becomes
  equal to) the size rd
• For an initial error d and a given tolerance ?
  rd, the average predictability time

Basu et al., Predictability of atmospherci
boundary layer flows as a function of scale,
Geophys. Res. Letters, 2002.
95
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96
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97
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

98
References
Gangodagamage, C., E. Barnes, and E.
Foufoula-Georgiou, Scaling in river corridor
widths depicts organization in valley morphology,
Geomorphology, doi10.1016/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, doi10.1029/2006WR005329,
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,
doi10.1029/2006WR005035, 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.
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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 gt 0 amount of material
  shipped from box Bi to box Bj
• dij taxi cab metric
  normalized to the embedding dimension between
  the centres of Bi and Bj

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Bed elevation Summary
Table 2
Polynomial approx. Polynomial approx.
Q (lps) Probe Scaling Range (min) Shield stress t(2) -2 t(1) c1 c2 c1 (cumul)
4300 4 1-12 0.085 -0.0396 0.5686 0.0488 0.5546
4900 3 0.5-20 0.111 -0.0404 0.6187 0.0658 0.5784
5500 3 0.5-8 0.196 -0.1567 0.7054 0.1169 0.7523
RESULT The higher the flow, the smoother the bed
elevation fluctuations overall (larger c1) but
the higher chance to find localized rough spots
inhomogeneously distributed over time (c2
higher)
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RECALL
1.
c20 monofractal t(2)2t(1) all moments can
be scaled with one parameter c1H only CV is
constant with scale
2.
c2¹0 multifractalfractal t(2)lt2t(1) need 2
parameters c1, c2 to scale pdfs CV decreases
with increase in scale
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High-resolution temporal rainfall data

• (courtesy, Iowa Institute of Hydraulic Research
  IIHR)