1. Ricardo Mantilla 1, Vijay Gupta 1 and Oscar Mesa 2 1 CIRES, University of Colorado at Boulder 2 PARH, Universidad Nacional de Colombia Hydrofractals ’03, Fractals in Hydrosciences 24th - 29th August 2003 Centro Stefano Franscini Monte Verità, Ascona, Switzerland Testing Physical Hypotheses on Channel Networks Using Flood Scaling Exponents

2. Contact Information: gupta@cires.colorado.edu ricardo@cires.colorado.edu ojmesa@perseus.unalmed.edu.co

3. 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

4. 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

5. Our Approach – Scaling Framework Theory DataSimulations 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

6. Pioneering Work – Background Gupta et al. 1996Menabde 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)

7. Pioneering Work – Results Gupta et al. 1996Menabde 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.

8. Pioneering Work – Results Gupta et al. 1996Menabde 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.

9. The Scaling Framework in Real Basins – Walnut Gulch, AZ SCALING EXPONENT OF THE WIDTH FUNCTION MAXIMA FOR WALNUT GULCH BASIN CHANNEL NETWORKS EXHIBIT SELF-SIMILARITY Mean # of Links at Maxima Mean Magnitude Width Function at the Walnut Gulch Basin Outlet This finding shows that Gupta et al result for Peano Network generalizes to real networks

10. Data Analysis – Data Distribution (discharge)

11. Result for 20 events:

12. Simulation Environment for Scaling Work - HidroSig Network extraction & Analysis 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)

13. Tools - HidroSig Hydrographs at every spatial scale Scaling Analysis of Peak flows Network extraction & Analysis Stream flow simulation (Hillslope – Link system)

14. 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?

15. Test of Hypothesis - Friction Hydraulic Geometry in a Nested Basin Routing on Real Networks: [Variable Chezy Coef.] Introducing Downstream Hydraulic Geometry for Channel Friction in a nested basin from channel hydraulics

16. 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.

17. 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.

18. Testing Physical Hypotheses on Channel Networks Using Flood Scaling Exponents Contact Information: gupta@cires.colorado.edu ricardo@cires.colorado.edu ojmesa@perseus.unalmed.edu.co