Hysteresis in River Discharge Rating Curves Histerésis en las curvas de gasto en ríos (caudal/calado) Madrid, March 25, 2013 Marian Muste and Kyutae Lee IIHR Hydroscience & Engineering The University of Iowa, U.S.A.
Conventional Discharge Rating Curves Rating Curves (RC): Practical solutions to continuously provide stream discharge Option 1: stage discharge (most often) One rating curve Requires continuous stage measurement (pressure sensors, radar, ultrasonic, etc) Option 2: index velocity (emerging with the advent of acoustic and image based instruments) One to three rating curves (Kennedy, 1984) Requires continuous stage & velocity measurements Option 3: slope area (rarely used for continuous, mostly for RC extrapolation) No rating curves (synthetic) Requires cross section and free surface slope measurements
Option 1: Stage discharge Rating Curves 1. Direct discharge measurements over a wide range of flows 2. Build the RC 3. Convert measured stages in discharges using RC h Step 1 Step 2 Underlying assumption: Steady Flow Step 3 USGS 05454200 Coralville, Iowa, 7 years of records RC derived measurements (125,865) direct measurements (237)
Option 2: Index velocity Rating Curves 1. Direct measurements for V index, Q, h, and A 2. Build stage area RC 3. Build velocity index RC 4. Compute instantaneous discharges as Q = V*A Step 2: Stage-Area Rating (h A) Step 1 WMO (2011) Step 3: Index Velocity Rating (V index V) Vmean 1.20 1.00 0.80 0.60 0.40 0.20 0.00-0.50-0.200.00 0.50 1.00 1.50-0.40 V(index) Step 4: Q = V*A
Option 3: Slope area Rating Curves 1. Survey cross section 2. Survey free surface slope (HGL) 3. Compute instantaneous discharges using Manning eqn. Step 2 Step 1 Step 3 1 2 3 1 2 SI units
What is hysteresis? Dependence of a system not only of the present state but also of its past (Wikipedia) Example: Loading and unloading a rubber band
Hysteresis in discharge RCs Conventional assumption for Options 1, 2, and 3: STEADY FLOW STATIC RCs (one to one relationship) Calibration measurements can be randomly acquired over the flow range However, storm runoff conveyed to streams propagates as UNSTEADY TRANSITORY FLOWS HYSTERESIS in RC (dynamic, looped curve) Calibration measurements need to be sampled commensurate with the event time scale Accelerated flow (phase I) Decelerated flow (Phase II) Steady (normal) Focus: Stage discharge (h Q) RCs Adapted from Graf & Qu (2004)
Sample Hysteresis in Stage Discharge RC Measurements with appropriate protocols enable to capture hysteresis 0.8 886 Δh= 10% 885.5 Δh= 26% H(m) 0.7 0.6 ΔQ=18% Stage (ft) 885 884.5 884 883.5 883 ΔQ=27% 0.5 882.5 882 0.4 0.40 0.55 0.70 0.85 1.00 1.15 1.30 1.45 1.60 Q(m 3 /s) Source: Budi Gunawan, 2008 881 1000 2000 3000 4000 5000 6000 7000 Discharge (cfs) Small streams: Blackwater (UK); Gunawan (2010) Medium streams: Chattahoochee (USA); Faye and Cherry (1980) 881.5 Δh= 13% ΔQ=41% Δh= 14 % Large rivers: Mississippi River (USA); Fread (1973) Large rivers: Yantze (China); Herschy (2009)
Hysteresis sensitivity factors Most important factors in welldeveloped hysteresis: Gage setting Event intensity and duration Stage (ft) 706 705 704 703 702 Discharge Q (ft 3 /t) 12000 10000 8000 6000 4000 C3 (Tp=24hr,Tb=24hr) C6 (Tp=24hr,Tb=12hr) C7 (Tp=24hr,Tb=72hr) Depth (ft) 40 35 30 25 20 15 10 C3 (Tp=24hr,Tb=24hr) C6 (Tp=24hr,Tb=12hr) C7 (Tp=24hr,Tb=72hr) 701 Bed Slope = 0.0001 Bed Slope = 0.001 Bed Slope = 0.01 700 0 500 1000 1500 2000 2500 3000 Discharge (cfs) 2000 5 0 0 20 40 60 80 100 120 140 Time (hr) 0 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 110 Discharge Q (ft 3 /t) Discharge Q (ft 3 /t) 25000 20000 15000 10000 C3 (peak = 10000) C8 (peak = 20000) Depth (ft) 70 60 50 40 30 20 C3 (peak = 10000) C8 (peak = 20000) Need for diagnostic protocols (currently under development) 5000 10 0 0 20 40 60 80 100 120 140 Time (hr) 0 0 2000 4000 6000 8000 10000 12000 14000 16000 18000 20000 22 Dischrage Q (ft 3 /t)
How to capture hysteresis? A) Direct discharge measurements (using event based, high temporal frequency sampling protocols) EXPENSIVE, NO PROTOCOLS, INCREASINGLY TESTED B) Analytical investigation using simplified approaches (1D) INEXPENSIVE, MANY PROTOCOLS, SCARSELY VALIDATED C) Numerical modeling using physically based modeling (2D, 3D) EXPENSIVE, MANY MOELS, SCARSELY VALIDATED
How to capture hysteresis? A) Direct discharge measurements (using event based, high temporal frequency sampling protocols) B) Analytical investigation using simplified approaches C) Numerical modeling using physically based modeling
Hysteresis: Direct measurements Our attempts to capture hysteresis (2011 13) Measurement Site: Clear Creek, Oxford, IA (USGS 05454220)
How to capture hysteresis? A) Direct discharge measurements B) Analytical investigation using simplified approaches (1D corrections formulae) C) Numerical modeling using physically based modeling
Hysteresis correction methods Abundant choices, few validations or recommendations for implementation Method Data required Flood Routing 0 1 1 1 0 0 0 1 Jones Q o, B, S o,( y/ t), ( Q o / z) Kinematic approximation 2 Henderson Q o, S o,( y/ t), ( y/ x) Parabolic approximation 3 Di Silvio Q b, Q p, A, S o, F r, R, T r, T f, A p, R p, A m, ( C/ A) 4 Fread S o, A, B,,( B/ y), ( z/ t), ( U/ t), Q p, Q b, T r, h p, h b, A m, Triangular approximation Parabolic approximation Q n normal flow kinematic wave: term a diffusion wave: terms a and b full dynamic wave: terms a, b, and c 5 Marchi Q s, B, S o, A,,( B/ y), ( A/ t) Kinematic approximation 6 Faye & Cherry K, A, y (t± t), y t, R, U t, ( Q o / z), S o, U (t± t), n Kinematic approximation 7 Fenton Q s, A, K, U, S o, Q o, B, ( Q o / z), ( y/ t), ( 2 y/ t 2 ), ( 3 y/ t 3 ) Kinematic approximation Our option: Fread (1975) full dynamic wave stage measurements at one station 8 Perumal Q s, B, S o, ( Q o / z), ( y/ t), F r, P, ( R/ y), ( A/ y), ( 2 y/ t 2 ) Approximate convection diffusion 9 Boyer Plots of Q m vs. z, z/ t Kinematic approximation 10 Lewis Qm, z/ t, Plots of Q m vs. z, J Kinematic approximation 11 Wiggins Plots of R vs. V m,, n, Classification of bed surface, z/ t, Q m 12 Peterson- Overleir z/ t, BFGS algorithm and its parameters No convective and local acceleration term Kinematic approximation
Fread s formula Fread (1973 & 1975) 1. Inputs: Hydraulic depth, width, bed slope, Manning s roughness, rate of changes of depth (dh/dt), initial discharge (randomly selected), time step for output 1. Output: looped rating curve
Fread s formula Modified Fread method for small stream channels (iterative solution) Energy slope, S f Wave celerity coefficient, K Implementation case studies Case 1 Case 2 Case 3 One event, Clear Creek, USGS 05454220 Oxford, Iowa (USA) One event, Ebro River (Spain) Multiple events, Clear Creek, USGS 05454220 Oxford, Iowa (USA)
Fread s formula implementation case 1: one event USGS 05454220, Oxford Iowa (processed data) 700 600 4 Evaluation of Saint-Vernant equation Steady-state Fread (1975) Points Discharge (cfs) 500 400 3 5 300 6 200 2 1 706 Stage-discharge rating curve comparisons 12 100 14-Apr-2012 15-Apr-2012 16-Apr-2012 17-Apr-2012 18-Apr-2012 Time Series Evaluation of the uncertainty in Prediction of Q Stage (ft) 705 704 703 702 701 Modified Fread RC USGS Steady RC 700 100 200 300 400 500 600 700 Discharge (cfs) Relative uncertainty in prediction of Q (%) 10 8 6 4 2 0-2 -4 14-Apr-2012 15-Apr-2012 16-Apr-2012 17-Apr-2012 17-Apr-2012 18-Apr-2012 Time Modified Fread vs. USGS steady RC 4% to 10.5%
Fread s formula implementation case 2: one event Asco station, Ebro River, Spain (Ferrer, Moreno, Sanchez, 2013) Discharge (cms) 1200 1100 1000 900 800 700 600 Evaluation of Saint-Vernant equation Steady-state Modified Fread ADCP Artificial flood event for vegetation removal (June 2012) Not all the needed data available 500 400 300 200 20-Jun-2012 20-Jun-2012 20-Jun-2012 20-Jun-2012 21-Jun-2012 Time Series 5.5 5 4.5 Stage-discharge rating curve comparisons 4 Stage (m) 3.5 3 2.5 2 Steady RC Modified Fread ADCP 1.5 200 300 400 500 600 700 800 900 1000 1100 1200 Discharge (cms)
Fread s formula implementation case 3: event series USGS 05454220, Oxford Iowa (provisional data similar with the info available during floods) Event 1 Event 2 Event 3 Series of rainfalls on frozen ground (good cases for hysteresis) (February March, 2013)
Fread s formula implementation case 3: event series Event 3: most violent rainfall (March 10, 2013) 712.00 710.00 710.67ft (2,340cfs at 11:30am, Mar 10) 709.18ft (1,330cfsat 5:15pm, Mar10) 708.00 706.00 705.63ft (667cfs at 10:00am, Mar 11) 704.00 702.00 700.00 700.22ft (66cfs at 10:00am, Mar 12) 698.00 696.00 0.00 20.00 40.00 60.00 80.00 100.00 120.00 140.00 160.00
Fread s formula implementation case 3: event series USGS 05454220, Oxford Iowa (provisional data) Stage (ft) Event 3: most violent rainfall of the series (March 10, 2013) 710 709 708 707 706 705 704 703 6 3 1 2 5 4 Discharge (cfs) 3500 3000 2500 2000 1500 1000 500 1 2 4 5 3 USGS Hydrograph Modified Fread Points Overbank flow 0 09-Mar-2013 10-Mar-2013 11-Mar-2013 11-Mar-2013 12-Mar-2013 Time Series 6 702 701 USGS Steady RC Modified Fread Points 700 0 200 400 600 800 1000 1200 1400 1600 1800 Discharge (cfs)
Fread s formula implementation case 3: event series Event 1 Event 2 Event 3 USGS 05454220, Oxford Iowa (provisional data) 710 708 Stage-discharge rating curve comparisons ΔH=706.5ft±0.5ft (5%) ΔQ=800cfs±100cfs (12.5%) 706 Uncertainty bounds due to unsteady flows Stage (ft) 704 702 Event1 on Feb 7-9, 2013 Event1 on Feb 7-9, 2013 700 Event2 on Feb 10-12, 2013 Event2 USGS on Steady Feb 10-12, RC 2013 Event3 on Mar 9-12, 2013 USGS Steady RC USGS Steady RC 698 0 200 400 600 800 1000 1200 1400 1600 1800 Discharge (cfs)
How to capture hysteresis? A) Direct discharge measurements B) Analytical investigation using simplified approaches C) Numerical modeling using physically based modeling (2D, 3D)
Hysteresis: numerical simulations Clear Creek watershed including USGS 05454220 Clear Creek, Oxford, Iowa HEC HMS model HEC RAS model Watershed description Size: approximately 103 mi 2 Land use: farm land combined urban areas (Oxford, Tiffin, Coralville, and Iowa City) Length of modeled reach: 24.1km (HEC RAS) and 4.3km (HEC HMS)
Hysteresis: numerical simulations HEC HMS model setup Validations for alternative HEC HMS simulations a) peak weighted RMS error function HEC HMS model setup 6 sub basins, 3 sub reaches, 4 junctions HEC HMS model components Basin model, meteorologic model, control specifications, and time series data b) percent error volume
Hysteresis: numerical simulations HEC RAS model setup River system Boundary conditions S1: Discharge hydrographs S4: Normal depth (friction slope: 0.00075) Monitoring locations S2: USGS 05454220 Oxford Clear Creek S3: USGS 05454500 Coralville Clear Creek Geometry setup Reach length: 24.1km Cross sections: 192 (approx 130m interval) Bridges: 10 Roughness coefficient: 0.035 (in bank), LCD (floodplain) Obstructions (buildings) included
Hysteresis: numerical simulations HEC RAS results Scenario 2: typical event December 04, 2011, Q peak_s2 = 3.2m 3 Flow (m3/s) 3.5 3.0 2.5 2.0 1.5 1.0 0.5 b) River: Clear_Cr Reach: Clear_Cr RS: 24131.31 0.0 240006001200180024000600120018002400060012001800 03Dec2011 04Dec2011 05Dec2011 Date Input hydrograph at S1 Legend Flow Scenario 1: large event (June 02, 2008) Q peak_s1 = 68m 3 Flow (m3/s) a) River: Clear_Cr Reach: Clear_Cr RS: 24131.31 70 Legend 60 Flow 50 40 30 20 10 0 2400 0600 1200 1800 2400 0600 1200 1800 03Jun2008 04Jun2008 Date Input hydrograph at S1 Stage (m) Stage (m) Plan: 15 River: Clear_Cr Reach: Clear_Cr RS: 19839.50 214.5 Legend 214.4 RC 214.3 214.2 214.1 214.0 213.9 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 198.9 198.8 198.7 198.6 Flow(m3/s) Max thickness: about 1cm at S2 Plan: 15 River: Clear_Cr Reach: Clear_Cr RS: 1600.056 199.0 Legend 198.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 Flow(m3/s) Max thickness: about 4cm at S3 RC Stage (m) Stage (m) Plan: 1 River: Clear_Cr Reach: Clear_Cr RS: 19839.50 217.0 Legend 216.5 RC 216.0 215.5 215.0 214.5 214.0 0 10 20 30 40 50 60 200.5 200.0 199.5 199.0 Flow(m3/s) Max thickness: about 10cm at S2 Plan: 1 River: Clear_Cr Reach: Clear_Cr RS: 1600.056 201.0 Legend 198.5 0 10 20 30 40 50 60 Flow(m3/s) Max thickness: about 15cm at S3 RC
Hysteresis: numerical simulations HEC RAS: Sensitivity analysis Peak discharge timing Summary of the results Input hydrograph at S1 Simulated RCs at S1 S1 (m) % wrt depth changes S2 (m) % wrt depth changes 2008 Large event 0.1 4.0% 0.15 6.9% 2011 Typical event 0.01 2.2% 0.04 10.3% Peak discharge 0.06 3.8% 0.09 7.1% (low to high) 0.1 4.7% 0.14 8.0% Duration 0.07 3.3% 0.14 8.0% (high to low) 0.18 9.7% 0.18 10.8% Peak timing 0.03 1.9% 0.06 4.8% (slow to fast) 0.13 8.4% 0.15 12.0% Event duration and peak discharge timing most important parameters (max error: 12%) Simulated RCs at S2
Hysteresis practical implications For high, unsteady flows RC uncertainties are considerable increased. The top contributing uncertainties are: measurement uncertainty extrapolation of the rating change in cross section (overbank flow) neglecting the hysteresis effect Hysteresis induced uncertainty is generally small Important for stream reaches on mild slopes, under channel control, and major storm events (during floods when RC accuracy is most important) Selected hysteresis induced uncertainty estimates: 2ft difference from RC in Chatttahooche and Ohio Rivers (Petersen Overleyer, 2006) 5 ft difference from RC in Mississippi River (Fread, 1975) These differences are typically lower then the steady RC reading (occur on the rising limb) important for flood intervention
How can be hysteresis used in practical applications? Uncertainty estimator for steady RCs during storms (based on previous data records ) Predictor for actual discharge during storms using steady RC as basis (based on an initial steady RC data) Stage Stage
How can be hysteresis used in practical applications? Measurements and models embedded in an integrated system for uncertainty assessment and/or forecasting h Q RC Slope area RC Tentative research
How can be hysteresis used in practical applications? Floods Better planning during floods by predicting more accurate flood stages and their timing!
Questions?