Hydrograph ModelingGoal: Simulate the shape of a hydrograph given a known or designed water input (rain or snowmelt)

Hydrograph Modeling: The input signalHyetograph can beA future design eventWhat happens in response to a rainstorm of a hypothetical magnitude and durationSee http://hdsc.nws.noaa.gov/hdsc/pfds/A past stormSimulate what happened in the pastCan serve as a calibration data set

Hydrograph Modeling: The ModelWhat do we do with the input signal?We mathematically manipulate the signal in a way that represents how the watershed actually manipulates the water

Q = f(P, landscape properties)

Hydrograph ModelingWhat is a model?What is the purpose of a model?Types of ModelsPhysicalhttp://uwrl.usu.edu/facilities/hydraulics/projects/projects.htmlAnalogOhms law analogous to Darcys law MathematicalEquations to represent hydrologic process

Types of Mathematical ModelsProcess representationPhysically BasedDerived from equations representing actual physics of processi.e. energy balance snowmelt modelsConceptualShort cuts full physics to capture essential processesLinear reservoir modelEmpirical/Regressioni.e temperature index snowmelt modelStochasticEvaluates historical time series, based on probabilitySpatial representationLumpedDistributed

Hydrograph ModelingPhysically Based, distributedPhysics-based equations for each process in each grid cellSee dhsvm.pdfKelleners et al., 2009Pros and cons?

Hydrologic ModelingSystems ApproachA transfer function represents the lumped processes operating in a watershed

-Transforms numerical inputs through simplified paramters that lump processes to numerical outputs-Modeled is calibrated to obtain proper parameters-Predictions at outlet only-Read 9.5.1PtQtMathematical Transfer Function

*Integrated Hydrologic Models Are Used to Understand and Predict (Quantify) the Movement of WaterHow ? Formalization of hydrologic process equationsLumped ModelDistributed Modele.g: Stanford Watershed Modele.g: ModHMS, PIHM, FIHM, InHMSemi-Distributed Modele.g: HSPF, LASCAMData Requirement:Computational Requirement:Process Representation:Predicted States Resolution:

Transfer Functions2 Basic steps to rainfall-runoff transfer functions1. Estimate losses.W minus losses = effective precipitation (Weff) (eqns 9-43, 9-44)Determines the volume of streamflow response

2. Distribute Weff in timeGives shape to the hydrograph

Recall that Qef = WeffBase FlowEvent flow (Weff)

Transfer FunctionsGeneral ConceptWLossesWeff = QefTaskDraw a line through the hyetograph separating loss and Weff volumes (Figure 9-40)tW?

Loss MethodsMethods to estimate effective precipitationYou have already done it one wayhow?However,

Loss MethodsPhysically-based infiltration equationsChapter 6Green-ampt, Richards equation, DarcyKinematic approximations of infiltration and storageWUniform: Werr(t) = W(t) - constantExponential: Weff(t) = W0e-ctc is unique to each site

Examples of Transfer Function ModelsRational Method (p443)qpk=urCrieffAdNo loss methodDuration of rainfall is the time of concentrationFlood peak onlyUsed for urban watersheds (see table 9-10)SCS Curve NumberEstimates losses by surface propertiesRoutes to stream with empirical equations

SCS Loss MethodSCS curve # (page 445-447)Calculates the VOLUME of effective precipitation based on watershed properties (soils)Assumes that this volume is lost

SCS ConceptsPrecipitation (W) is partitioned into 3 fatesVi = initial abstraction = storage that must be satisfied before event flow can begin

Vr = retention = W that falls after initial abstraction is satisfied but that does not contribute to event flow

Qef = Weff = event flow

Method is based on an assumption that there is a relationship between the runoff ratio and the amount of storage that is filled:Vr/ Vmax. = Weff/(W-Vi)

where Vmax is the maximum storage capacity of the watershed

If Vr = W-Vi-Weff,

SCS ConceptAssuming Vi = 0.2Vmax (??)

Vmax is determined by a Curve Number

Curve NumberThe SCS classified 8500 soils into four hydrologic groups according to their infiltration characteristics

Curve NumberRelated to Land Use

Transfer Function1. Estimate effective precipitationSCS method gives us Weff2. Estimate temporal distributionVolume of effective Precipitation or event flow-What actually gives shape to the hydrograph?

Transfer Function2. Estimate temporal distribution of effective precipitationVarious methods route water to stream channelMany are based on a time of concentration and many other rules

SCS methodAssumes that the runoff hydrograph is a triangleTb=2.67TrQtOn top of base flowTw = duration of effective PTc= time concentrationHow were these equations developed?

Transfer FunctionsTime of concentration equations attempt to relate residence time of water to watershed propertiesThe time it takes water to travel from the hydraulically most distant part of the watershed to the outletEmpically derived, based on watershed propertiesOnce again, consider the assumptions

Transfer Functions2. Temporal distribution of effective precipitationUnit HydrographAn X (1,2,3,) hour unit hydrograph is the characteristic response (hydrograph) of a watershed to a unit volume of effective water input applied at a constant rate for x hours. 1 inch of effective rain in 6 hours produces a 6 hour unit hydrograph

Unit HydrographThe event hydrograph that would result from 1 unit (cm, in,) of effective precipitation (Weff=1)A watershed has a characteristic responseThis characteristic response is the modelMany methods to construct the shapeQeft11

Unit HydrographHow do we Develop the characteristic response for the duration of interest the transfer function ?Empirical page 451Synthetic page 453

How do we Apply the UH?: For a storm of an appropriate duration, simply multiply the y-axis of the unit hydrograph by the depth of the actual storm (this is based convolution integral theory)

Unit HydrographApply: For a storm of an appropriate duration, simply multiply the y-axis of the unit hydrograph by the depth of the actual storm. See spreadsheet exampleAssumes one burst of precipitation during the duration of the stormIn this picture, what duration is 2.5 hours Referring to?

Where does 2.4 come from?

What if storm comes in multiple bursts?Application of the Convolution IntegralConvolves an input time series with a transfer function to produce an output time seriesU(t-t) = time distributed Unit Hydrograph

Weff(t)= effective precipitation t =time lag between beginning time series of rainfall excess and the UH

Convolution integral in discrete formJ=n-i+1

Unit HydrographMany ways to manipulate UH for storms of different durations and intensitiesS curve, instantaneousThats for an engineering hydrology classYOU need to know assumptions of the application

Unit HydrographHow do we derive the characteristic response (unit hydrograph)?Empirical

Unit HydrographHow do we derive the characteristic response (unit hydrograph)?Empirical page 451Note: 1. approximately equal durationWhat duration are they talking about?Note: 8. adjust the curve until this area is satisfactorily close to 1unitSee spreadsheet example

Unit HydrographAssumptionsLinear responseConstant time base

Unit HydrographConstruction of characteristic response by synthetic methodsScores of approaches similar to the SCS hydrograph method where points on the unit hydrograph are estimated from empirical relations to watershed properties.SnyderSCSClark

Snyder Synthetic Unit HydrographSince peak flow and time of peak flow are two of the most important parameters characterizing a unit hydrograph, the Snyder method employs factors defining these parameters, which are then used in the synthesis of the unit graph (Snyder, 1938).

The parameters are Cp, the peak flow factor, and Ct, the lag factor.

The basic assumption in this method is that basins which have similar physiographic characteristics are located in the same area will have similar values of Ct and Cp.

Therefore, for ungaged basins, it is preferred that the basin be near or similar to gaged basins for which these coefficients can be determined. The final shape of the Snyder unit hydrograph is controlled by the equations for width at 50% and 75% of the peak of the UHG:

SCS Synthetic Unit HydrographTriangular RepresentationThe 645.33 is the conversion used for delivering 1-inch of runoff (the area under the unit hydrograph) from 1-square mile in 1-hour (3600 seconds).

Synthetic Unit HydrographALL are based on the assumption that runoff is generated by overland flowWhat does this mean with respect to our discussion about old water new water?How can Unit Hydrographs, or any model, possibly work if the underlying concepts are incorrect?

Other ApplicationsWhat to do with storms of different durations?

Other ApplicationsDeriving the 1-hr UH with the S curve approach

Physically-Based Distributed

Hydrologic Similarity ModelsMotivation: How can we retain the theory behind the physically based model while avoiding the computational difficulty? Identify the most important driving features and shortcut the rest.

TOPMODELBeven, K., R. Lamb, P. Quinn, R. Romanowicz and J. Freer, (1995), "TOPMODEL," Chapter 18 in Computer Models of Watershed Hydrology, Edited by V. P. Singh, Water Resources Publications, Highlands Ranch, Colorado, p.627-668.TOPMODEL is not a hydrological modeling package. It is rather a set of conceptual tools that can be used to reproduce the hydrological behaviour of catchments in a distributed or semi-distributed way, in particular the dynamics of surface or subsurface contributing areas.

TOPMODELSurface saturation and soil moisture deficits based on topographySlopeSpecific Catchment AreaTopographic ConvergencePartial contributing area conceptSaturation from below (Dunne) runoff generation mechanism

Saturation in zones of convergent topography

TOPMODELRecognizes that topography is the dominant control on water flowPredicts watershed streamflow by identifying areas that are topographically similar, computing the average subsurface and overland flow for those regions, then adding it all up. It is therefore a quasi-distributed model.

Key Assumptionsfrom Beven, Rainfall-Runoff ModelingThere is a saturated zone in equilibrium with a steady recharge rate over an upslope contributing area a

The water table is almost parallel to the surface such that the effective hydraulic gradient is equal to the local surface slope, tan

The Transmissivity profile may be described by and exponential function of storage deficit, with a value of To whe the soil is just staurated to the surface (zero deficit

Hillslope ElementPqtotal = qsub + q overlandWe need equations based on topography to calculate qsub (9.6) and qoverland (9.5)

Subsurface Flow in TOPMODELqsub = TctanWhat is the origin of this equation?What are the assumptions?How do we obtain tanHow do we obtain T?

Recall that one goal of TOPMODEL is to simplify the data required to run a watershed model. We know that subsurface flow is highly dependent on the vertical distribution of K. We can not easily measure K at depth, but we can measure or estimate K at the surface. We can then incorporate some assumption about how K varies with depth (equation 9.7). From equation 9.7 we can derive an expression for T based on surface K (9.9). Note that z is now the depth to the water table.

z

Transmissivity of Saturated ZoneK at any depth

Transmissivity of a saturated thickness z-D

z

Equations

SubsurfaceSurfaceAssume Subsurface flow = recharge rateTopographic IndexSaturation deficit for similar topography regions

Saturation DeficitElement as a function of local TI

Catchment Average

Element as a function of average

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