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Ricardo Mantilla1, Vijay Gupta1 and Oscar Mesa2
1 CIRES, University of Colorado at Boulder2 PARH, Universidad Nacional de Colombia
Hydrofractals ’03, Fractals in Hydrosciences
24th - 29th August 2003
Centro Stefano FransciniMonte Verità, Ascona, Switzerland
Testing Physical Hypotheses on Channel Networks Using Flood Scaling Exponents
Testing Physical Hypotheses on Channel Networks Using Flood Scaling Exponents
Contact Information:
The Problem
• Lack of testability of physical hypothesis on hydrologic models (lumped and distributed).
• Example: Models with disjoint hypothesis can get a good fit of the hydrograph, leaving physical interpretation with no ground.
• Calibration of simple and complex models make interpretation of parameters impossible.
• Statistical models of regionalization are not tide to physical processes
• How the statistical approaches can be linked to physical processes is a long standing question in hydrology
The Problem
• Lack of testability of physical hypothesis on hydrologic models (lumped and distributed).
• Example: Models with disjoint hypothesis can get a good fit of the hydrograph, leaving physical interpretation with no ground.
• Calibration of simple and complex models make interpretation of parameters impossible.
• Statistical models of regionalization are not tied to physical processes
• How the statistical approaches can be linked to physical processes is a long standing question in hydrology
Our Approach – Scaling Framework
Theory
Data Simulations
EMERGENCE OF STATISTICAL SCALE INVARIANCE FROM
DYNAMICS
FROM REGIONALIZATION TO EVENT BASED EMBEDED DATA
HIDROSIG:A NETWORK BASED
HYDROLOGIC MODEL
STATISTICAL SCALING IS A
FRAMEWORK TO TEST PHYSICAL
HYPOTESIS
Pioneering Work – Background
Gupta et al. 1996 Menabde et al. 2001a & 2001b
Different topologies and hypothesis about flow in channels
• Simplest type of routing – All water moves out of the link in t.
• A more realistic type of routing (Constant Velocity in 2001a & Chezy type equation for Velocity in 2001b)
Pioneering Work – Results
Gupta et al. 1996 Menabde et al. 2001a & 2001b
Numerical Result: The scaling exponent 0.49 is smaller than the scaling exponent of the width function for Man-Vis tree log(2)/log(3) = 0.63
Analytical Result: Under this hypothesis of flow routing the scaling exponent of Peak Flows is equal to the scaling exponent of the maxima of the width function.
Scaling of Peak Flow vs. Drainage Area under the “Constant Velocity” assumption.
Pioneering Work – Results
Gupta et al. 1996 Menabde et al. 2001a & 2001b
Numerical Result: The scaling exponent 0.39 is smaller than the scaling exponent of the width function, and smaller than the observed for Constant Velocity
Analytical Result: Under this hypothesis of flow routing the scaling exponent of Peak Flows is equal to the scaling exponent of the maxima of the width function.
Scaling of Peak Flow vs. Drainage Area under the “Nonlinear Velocity” assumption.
* These two results suggest that the scaling exponent for the maxima of the width function is an upper limit for the scaling exponent of the peak discharge.
The Scaling Framework in Real Basins – Walnut Gulch, AZ
47.0
SCALING EXPONENT OF THE WIDTH FUNCTION MAXIMA FOR WALNUT GULCH BASIN
CHANNEL NETWORKS EXHIBIT SELF-SIMILARITY
Mea
n #
of
Lin
ks a
t Max
ima
Mean MagnitudeWidth Function at the Walnut Gulch Basin Outlet
This finding shows that Gupta et al result for Peano Network generalizes to real networks
Data Analysis – Data Distribution (discharge)
Data Analysis – Data Distribution (discharge)
Result for 20 events:
Simulation Environment for Scaling Work - HidroSig
• Network extraction & Analysis
S2
S1
PE
B a se F low(B F )
S atu r a te d O v e rla n d F lowG W R e c h a r g e
H o r to n ia n O v e rla n d F lo w (H O F )
In filtra tio n
• An schematic representation of the dynamical system
Link Based Mass conservation equation (Gupta & Waymire, 1998), and Momentum conservation equation (Regianni et al, 2001)
• Stream flow simulation(Hillslope – Link system)
Tools - HidroSig
Hydrographs at every spatial scale
Scaling Analysis of Peak flows
• Network extraction & Analysis
• Stream flow simulation(Hillslope – Link system)
First Result
Do Menabde et al results apply to Walnut Gulch network?
Result: The observed peak flow scaling exponent cannot be explained by Menabde et al set of assumptions.
Question: What physical processes can explain the observed Flood Scaling Exponents?
Test of Hypothesis - Friction Hydraulic Geometry in a Nested Basin
Routing on Real Networks: [Variable Chezy Coef.]
6/16/16/1
6/1
6/1
6/150
50
1.1449.121
/
1923] ,[Strickler 21
1936] [Shields, 823.10
dd
C
Cdn
dn
dd
Introducing Downstream Hydraulic Geometry for Channel Frictionin a nested basin from channel hydraulics
Aggregation vs. Attenuation
Results:Flood Scaling Exponent(i) is larger than the maxima of WF scaling exponent for variable Chezy(ii) is smaller than WF scaling exponent for constant Chezy
Conclusion:Scaling parameters provide a new way to test hypotheses about the physical processes governing floods without requiring calibration.
Conclusions
•Role of Aggregation via the Width Function and Attenuation via channel friction influence the peak flow scaling exponent.
•The example of how constant velocity vs. nonlinear velocity with and without spatially variable Chezy coefficient determine peak flow scaling was presented here.
•Need to extend the mathematical framework to understand the role of rainfall duration and space time variability on peak flow scaling is an important open problem.
•Walnut Gulch data shows that spatially variable infiltration is a major factor in determine peak flow scaling. This is an important open problem.
Testing Physical Hypotheses on Channel Networks Using Flood Scaling Exponents
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