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Applications of Scaling to Regional Flood Analysis Brent M. Troutman U.S. Geological Survey

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Introduction: Flood frequency analysis Objective Objective Estimate magnitude of flow which is exceeded on the average once every T-years at a site (T-year flow) Problem Problem Limited flow data! Approaches Approaches Regional flood analysis: Data from nearby sites Rainfall-runoff models: Process knowledge Scaling: Connections

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Regional flood analysis methods Regional regression Regional regression where q T = T-year flow; B, C = basin, climatic characteristics where q T = T-year flow; B, C = basin, climatic characteristics Index-flood method Index-flood method Q/ has same distribution for all sites where Q = annual peak flow, = at-site mean, also often related to B, C by regression where Q = annual peak flow, = at-site mean, also often related to B, C by regression

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Scaling invariance A type of symmetry such that small systems are “similar” in geometry and/or function to large systems A type of symmetry such that small systems are “similar” in geometry and/or function to large systems How will scaling help in regional flood analysis? How will scaling help in regional flood analysis? A framework for revealing connections with rainfall-runoff processes Predictions of coefficients in regional regressions Indications of appropriate form and assumptions for statistical models

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The role of area Analysis of scaling invariance involves looking at changes with respect to a scale parameter Analysis of scaling invariance involves looking at changes with respect to a scale parameter Drainage area A is a logical choice in regional flood analysis: it is often the only or most significant statistical predictor in regressions: Drainage area A is a logical choice in regional flood analysis: it is often the only or most significant statistical predictor in regressions: Specific focus of this work: Specific focus of this work: How do scaling ideas help in understanding peak flow dependence on A?

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New Mexico scaling exponents T (years) T (years) NE plains N mtns. NW plateau SE plains 20.560.910.520.67 50.550.920.470.59 100.550.920.440.55 250.550.930.410.50 500.550.930.390.47 1000.560.940.370.44 5000.580.940.360.41

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The framework of scaling invariance has been used to look at many of the characteristics known to influence flows … River basin geometry River basin geometry Channel network Channel sinuosity Downstream hydraulic geometry Landscape roughness Longitudinal profiles Rainfall Spatial variability Temporal variability IDF curves Soils Pore structure Flow pathways

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Channel networks: The width function L(x) L(x) = number of links at distance x from the outlet

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The width function & flow Under an idealized scenario, flow Q(t) at the outlet has the same shape as the width function L(x) Under an idealized scenario, flow Q(t) at the outlet has the same shape as the width function L(x) Spatial rainfall pattern: Uniform Temporal rainfall pattern: Instantaneous burst of rain all deposited into network Channel flow: Translation routing at uniform velocity v c

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RRT network model: Peano

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Peano network width function

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Peano width function maximum Straightforward geometric arguments show max width L max and area A are related by Straightforward geometric arguments show max width L max and area A are related by Implication: Peak flows under “idealized scenario” scale as: Implication: Peak flows under “idealized scenario” scale as:

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Flint River, GA Drainage area: Drainage area: 6380 sq km 6380 sq km Number of links: Number of links: 22,959 22,959 25 km

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Flint River width function

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Flint R. width function maximum

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Flint River generators Actual networks: generators vary randomly Extract generators and analyze distribution of no. links per generator Same for different replacement levels

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Goodwin Creek, MS

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Goodwin Creek width function

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Goodwin Cr. width function maximum

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Peak flow scaling, Goodwin Cr. Observed features: Average slope for 329 events is 0.79 Decrease in peak flow variability as A increases Explanation: hillslope processes Travel time Spatial variability of runoff generation

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Travel time: channel and hillslope Width function Width function L(x) = no. links at channel distance x Generalized width function Generalized width function M(x,y) = no. of pixels at channel distance x and hillslope distance y Assume velocities v h and v c such that travel time to outlet is Consider flow again with spatially uniform, instantaneous rainfall

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Peak flow scaling exponent vs. v h /v c This curve is ob- tained using only the function M v h /v c large yields flow proportional to width function, = net = 0.79 corresponds to

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Spatial variability of runoff generation Assumption: Peak flow is sum of flow contributions from a set of links in the basin, and runoff generation from these links (hillslopes) is spatially variable Assumption: Peak flow is sum of flow contributions from a set of links in the basin, and runoff generation from these links (hillslopes) is spatially variable Implication: Implication: Yields a statistical model for log of Q Yields a statistical model for log of Q

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Statistical model for log of peaks , , storm dependent, Z basin effect, e error

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Parameter estimates

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Pooled variability of residuals, Goodwin Creek

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Conclusions: How will scaling help in regional flood analysis? Predictions of coefficients in regional regression relations based on network geometry, basin & storm properties, etc. Predictions of coefficients in regional regression relations based on network geometry, basin & storm properties, etc. Indications of appropriate form and assumptions for statistical models Indications of appropriate form and assumptions for statistical models More generally, a framework for revealing connections between regional flood analysis and rainfall-runoff processes More generally, a framework for revealing connections between regional flood analysis and rainfall-runoff processes

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