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**University of Minnesota**

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

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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

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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)

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**High-resolution temporal rainfall data**

(courtesy, Iowa Institute of Hydraulic Research – IIHR) ~ 5 hrs t = 10s ~ 1 hr t = 5s If easy we could add inside the red box approx 5 hours and underneath Dt= 10 secs and in the green box approx 1 hour and underneath Dt=5 secs DONE

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Noyo River basin

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**Sediment accumulation series Time series of bed elevation **

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

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**Experimental setup Data Available: Sediment accumulation series**

Time series of bed elevation Laser transects of bed elevation D50 = 11.3mm Pan-1 Pan-2 Pan-3 Pan-4 Pan-5 Discharge controlled here Channel Width = 2.75 m Channel Depth = 1.8 m D50=11.3 mm 1.0 Diameter [mm] 100 Discharge capacity: 8500 lps Coarse sediment recirculation system located 55 m from upstream end.

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Bed Elevation

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**Sediment Transport Rates**

Accumulated series (Sc(t)) Nearest neighbor differences (S(t))

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**SEDIMENT FLUX: VARIABILITY AT ALL SCALES**

Sc (t) = Accumulated sediment over an interval of 0 to t sec

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**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)

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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

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**Localized Scaling Analysis: Multifractal Formalism**

• Characterize a signal f(x) in terms of its local singularities h=0.3 h=0.7 Ex: h(x0) = 0.3 implies f(x) is very rough around x0. h(x0) = 0.7 implies a “smoother” function around xo.

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**Multifractal Formalism**

• Spectrum of singularities D(h) D(h) h • D(h) can be estimated from the statistical moments of the fluctuations. Legendre Transform

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Multifractal Spectra • Spectrum of scaling exponents t(q) and Spectrum of singlularities D(h) monofractal h multifractal h

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**Spectrum of singularities**

Multifractal Spectra Spectrum of scaling exponents Spectrum of singularities D(h) -1 1 2 3 4 5 6 7 (2) Slopes (q) q Df Please add the symbol D_f, <h> horizontal axis h, hmin/hmax and possibly an arrow to show that the derivative is taken at that point (as in paper) DONE h hmin hmax

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**Wavelet-based multifractal formalism (Muzy et al., 1993; Arneodo et al., 1995)**

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 In definition of wavelet either correct f(u) inside the integral or replace location with x_o and have the running variable as x This is strange.. I thought I fixed this the very first time I sent you. Same thing here, I fixed it, and now the font changes. I suspect Roshan uses a different equation editor (probably a better one) than I do. It can be shown that

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**(direct access to statistics of singularities)**

f(x) Structure Function Moments of |f(x+l) – f(x)| T[f](x,a) Partition Function Moments of |T[f](x,a)| Partition Function Moments of |Ta(x)| (access to q < 0) Cumulant analysis Moments of ln |Ta(x)| (direct access to statistics of singularities) WTMM Ta(x)

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**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

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**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

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**Width and Area Functions of a River Network**

# 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

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**Width and Area Functions of a River Network**

# 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

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**Area and Width Functions**

Walawe River, Sri Lanka (90x90m) A;2,000 km2

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**Area and Width Functions**

Noyo River Basin, California, USA (10x10m) A;143 km2

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**Walawe River Basin: deviation from simple scaling**

A(x) W(x) W(x) A(x) A rich multifractal structure is observed which is different for A(x) and W(x)

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**Noyo River Basin: deviation from simple scaling**

A(x) W(x) A rich multifractal structure is observed which is different for A(x) and W(x)

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**Noyo River Basin (10x10m; A;143 km2)**

c1 ; c2 = SR = 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 ; c2 ; 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

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**Walawe River Basin (90x90m; A;2,000 km2)**

c1 ; c2 = SR = km c1 ; c2 ; SR = km Accurate characterization of higher moments needs high resolution topography data

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**Real River Networks C1=0.77 C2=0.11 C1=0.40 C2=0.05 C1=0.80 C2=0.05**

Noyo River basin, CA (10x10m) A;1430 km2 Walawe River, Sri Lanka (90x90m) A;2,000 km2 South Fork Eel River, CA (1x1m & 10x10m) A;154 km2 C1=0.77 C2=0.11 C1=0.40 C2=0.05 C1=0.80 C2=0.05 A(x)

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**Area Functions of Simulated Trees**

Peano Basin Not comparable to real networks Shreve’s random network model c1=0.5 c2=0 Stochastic S-S model with (a , b)=(1 , 2) c1=0.62 c2=0

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**Important Observation: The effect of higher order statistical moments**

(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 Shreve’s model; Poisson distributed for S-S tree) ß Randomness in the River Network Topology is reflected in the statistical properties of the width function

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**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.

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**Implications for Network Hydrology?**

W(x) = # of channels intersected by a contour of equal length x to the outlet |W(x+dx)-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. Shreve’s random topology model: c1=0.5, c2=0.0, for all dx Walawe River Basin: c1=0.5, c2=0.13 for dx =2 – 6 km Deviation from scale invariance (c2 ¹ 0) Þ extended S-S within a limited range of scales 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?

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**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” ÞImplications for routing? scale-dependent convolution? geomorphologic dispersion? Þ Implications for scaling of hydrographs?

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**River Corridor Width Functions**

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**River Corridor Width Function (D=5m)**

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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)

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**Specific Questions: Area and Width Functions**

Does the topology of river networks leave a unique signature on the scaling of W(x) and A(x)? How different are the scaling properties of commonly used simulated trees and those of real river networks? 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? What are the hydrological implications of the above?

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**Statistical multiscaling: beyond global analysis**

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.”

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**Multifractal Formalism**

• Spectrum of singularities D(h) D(h) h • D(h) can be estimated from the statistical moments of the fluctuations. Legendre Transform

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Multifractal Spectra • Spectrum of scaling exponents t(q) and Spectrum of singlularities D(h) monofractal h multifractal h

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**Walawe River Basin: deviation from simple scaling**

A(x) W(x) W(x) A(x) A rich multifractal structure is observed which is different for A(x) and W(x)

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**Noyo River Basin: deviation from simple scaling**

A(x) W(x) A rich multifractal structure is observed which is different for A(x) and W(x)

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**Noyo River Basin (10x10m; A;143 km2)**

c1 ; c2 = SR = 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 ; c2 ; 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

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**Walawe River Basin (90x90m; A;2,000 km2)**

c1 ; c2 = SR = 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 ; c2 ; SR = 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

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**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” ÞImplications for routing? scale-dependent convolution? geomorphologic dispersion? Þ Implications for scaling of hydrographs?

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**Real River Networks C1=0.77 C2=0.11 C1=0.40 C2=0.05 C1=0.80 C2=0.05**

Noyo River basin, CA (10x10m) A;1430 km2 Walawe River, Sri Lanka (90x90m) A;2,000 km2 South Fork Eel River, CA (1x1m & 10x10m) A;154 km2 C1=0.77 C2=0.11 C1=0.40 C2=0.05 C1=0.80 C2=0.05 A(x)

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Noyo River basin

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**Area Functions of Simulated Trees**

Peano Basin Not comparable to real networks Shreve’s random network model c1=0.5 c2=0 Stochastic S-S model with (a , b)=(1 , 2) c1=0.62 c2=0

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Stochastic S-S Trees (Tokunaga, 1996, 1978; Peckham, 1995) • = average # of tributaries of order w¢ that branch into a stream of order w

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**Stochastic S-S Network Model: Area Function**

(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 1.76 1.41 order 13 12 11 10 3894 4160 3618 5817 3950 6435 14827 c1 0.65 0.62 0.55 c2 0.00 0.01 Note: as a increases, c1 decreases, i.e., when “branching rate” increases; A(x) exhibits wilder fluctuations and becomes more irregular.

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**Important Observation: The effect of higher order statistical moments**

(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 Shreve’s model; Poisson distributed for S-S tree) ß Randomness in the River Network Topology is reflected in the statistical properties of the width function

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**A(x) W(x) Walawe: A;2,000 km2 (90x90m) c1=0.37 – 0.40 c2=0.05 c1;0.50**

South Fork: A;154 km2 (1x1m & 10x10m) c1=0.80 Beaver Creek: A;622 km2 (30x30m) c1=0.44 Noyo River Basin: A; 143 km2 (1x1m) c1=0.77 c2=0.11 c1=0.46 Lower Noyo River Basin: A; km2 (1x1m) c1= c2=

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**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.

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**Implications for Network Hydrology?**

W(x) = # of channels intersected by a contour of equal length x to the outlet |W(x+dx)-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. Shreve’s random topology model: c1=0.5, c2=0.0, for all dx Walawe River Basin: c1=0.5, c2=0.13 for dx =2 – 6 km Deviation from scale invariance (c2 ¹ 0) Þ extended S-S within a limited range of scales 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?

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**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” ÞImplications for routing? scale-dependent convolution? geomorphologic dispersion? Þ Implications for scaling of hydrographs?

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South Fork Eel River, CA Area = 351 km2

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**River Corridor Width Function (D=5m)**

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**Questions What is the statistical structure of RCW(x)?**

Do physically distinct regimes exhibit statistically distinct signatures? How can the statistical structure be used in modeling and prediction of hydrographs, sedimentographs and pollutographs across scales?

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**SOUTH FORK EEL RIVER, CA: Hypsometric Profile**

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River Reach: 0-6 Km

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Reach: km Right Left

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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?

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**River Corridor Width Function: South Fork Eel River**

6 km 14 km 20 km 28 km 35 km 89 tributaries: (1 km2 – 150 km2)

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River Reach: 0-6 km

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River Reach: km

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**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)

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SUMMARY OF RESULTS Right-Left asymmetry

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**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?

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**Conclusions on RCW Series**

RCW fluctuations exhibit a deviation from scale-invariance 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. There is a significant left-right asymmetry in the statistical structure of RCWs reflecting different underlying processes in each side of the river.

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**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?

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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 Ex. 2: 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?

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References Gangodagamage, C., E. Barnes, and E. Foufoula-Georgiou, Scaling in river corridor widths depicts organization in valley morphology, Geomorphology, doi: /j.geomorph , 2007. Lashermes, B. and E. Foufoula-Georgiou, Area and width functions of river networks: new results on multifractal properties, Water Resources Research, doi: /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: /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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**Experimental setup Data Available: Sediment accumulation series**

Time series of bed elevation Laser transects of bed elevation D50 = 11.3mm Pan-1 Pan-2 Pan-3 Pan-4 Pan-5 Discharge controlled here Channel Width = 2.75 m Channel Depth = 1.8 m D50=11.3 mm 1.0 Diameter [mm] 100 Discharge capacity: 8500 lps Coarse sediment recirculation system located 55 m from upstream end.

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**Sediment accumulation series Time series of bed elevation **

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

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QUESTIONS Do the statistics of sediment transport rates depend on “scale” (sampling interval or time interval of averaging) and how? Does this statistical scale-dependence depend on flow rate, bed shear stress, and bedload size distribution (e.g., gravel vs. sand, etc.) Do the statistics of sediment transport relate to the statistics of bedform morphodynamics and how? What are the practical implications of all these?

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**Sediment Transport Rates**

Accumulated series (Sc(t)) Nearest neighbor differences (S(t))

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**VARIABILITY AT ALL SCALES**

Sc (t) = Accumulated sediment over an interval of 0 to t sec

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**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)

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**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.

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**MULTIFRACTAL FORMALISM**

• Spectrum of singularities D(h) D(h) h • D(h) can be estimated from the statistical moments of the fluctuations. Legendre Transform

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**MULTIFRACTAL FORMALISM**

• Spectrum of scaling exponents t(q) monofractal h multifractal h

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**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(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

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**SEDIMENT TRANSPORT RATES: Q = 5500 lps**

log2 Noise Variability levels off Scaling range C1=1.10 C2=0.10 15 min 1 min

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**Q = 4300 lps Noise C1=0.55 C2=0.15 Scaling range**

log2 Noise C1=0.55 C2=0.15 Scaling range Statistical Variability regime changes 1 min 10 min

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**SEDIMENT TRANSPORT RATES: SUMMARY TABLE OF c1, c2**

Polynomial approx. Q (lps) Pan Scaling Range (min) Shield stress τ(2) – 2 τ(1) c1 c2 c1 (cumulants) 2 1 – 10 0.085 -0.26 0.40 0.15 0.53 4300 3 -0.20 0.56 0.14 0.57 4 1 -- 3 -0.10 0.52 0.05 2 – 15 0.111 -0.12 1.30 0.11 1.34 4900 -0.14 1.33 0.10 -0.09 1.24 0.08 1 – 15 0.196 -0.13 1.09 0.09 1.17 5500 -0.15 1.07 1.18 1.15 1.25

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**Higher c1 means a “smoother” signal **

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 • c2=0 monofractal ® t(2)=2t(1) • all moments can be scaled with one parameter c1=H only • CV is constant with scale • c2¹0 multifractalfractal ® t(2)<2t(1) • need 2 parameters c1, c2 to scale pdfs • CV decreases with increase in scale

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**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))

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Bed Elevation

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Bed elevation data Note: these series are plotted till 3000 data points to show the same scale

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**BED ELEVATION TEMPORAL SERIES: Q = 5500 lps**

Scaling range C1=0.70 C2=0.11 0.5 min 8 min

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**BED ELEVATION TEMPORAL SERIES: Q = 4300 lps**

Scaling range C1=0.55 C2=0.05 1 min 12 min

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**Inferences on Nonlinearity**

Basu and Foufoula-Georgiou, Detection of nonlinearity and chaoticity in time series using the transportation distance function, Phys. Letters A, 2002.

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

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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

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References Gangodagamage, C., E. Barnes, and E. Foufoula-Georgiou, Scaling in river corridor widths depicts organization in valley morphology, Geomorphology, doi: /j.geomorph , 2007. Lashermes, B. and E. Foufoula-Georgiou, Area and width functions of river networks: new results on multifractal properties, Water Resources Research, doi: /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: /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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THE END

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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 > 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

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**4300 4900 5500 Bed elevation Summary Table 2 4 1-12 0.085 -0.0396**

Polynomial approx. Q (lps) Probe Scaling Range (min) Shield stress τ(2) -2* τ(1) c1 c2 c1 (cumul) 4300 4 1-12 0.085 0.5686 0.0488 0.5546 4900 3 0.5-20 0.111 0.6187 0.0658 0.5784 5500 0.5-8 0.196 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. Dx(k ×Dt) = Sediment transported during a time period k ×Dt Dt = Sampling interval ~ • c2=0 monofractal ® t(2)=2t(1) • all moments can be scaled with one parameter c1=H only • CV is constant with scale 2. • c2¹0 multifractalfractal ® t(2)<2t(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) ~ 5 hrs t = 10s ~ 1 hr t = 5s If easy we could add inside the red box approx 5 hours and underneath Dt= 10 secs and in the green box approx 1 hour and underneath Dt=5 secs DONE

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