Hydrograph Modeling Goal: Simulate the shape of a hydrograph given a known or designed water input (rain or snowmelt) time Precipitation time flow Hydrologic Model

Hydrograph Modeling: The input signal Hyetograph can be A future “design” event What happens in response to a rainstorm of a hypothetical magnitude and duration See http://hdsc.nws.noaa.gov/hdsc/pfds/ A past storm Simulate what happened in the past Can serve as a calibration data set time Precipitation time flow Hydrologic Model

Hydrograph Modeling: The Model What 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) time Precipitation time flow Hydrologic Model

Hydrograph Modeling What is a model? What is the purpose of a model? Types of Models Physical http://uwrl.usu.edu/facilities/hydraulics/projects/projects.html Analog Ohm’s law analogous to Darcy’s law Mathematical Equations to represent hydrologic process

Types of Mathematical Models Process representation Physically Based Derived from equations representing actual physics of process i.e. energy balance snowmelt models Conceptual Short cuts full physics to capture essential processes Linear reservoir model Empirical/Regression i.e temperature index snowmelt model Stochastic Evaluates historical time series, based on probability Spatial representation Lumped Distributed

Hydrograph Modeling Physically Based, distributed Physics-based equations for each process in each grid cell See dhsvm.pdf Kelleners et al., 2009 Pros and cons?

Hydrologic Modeling Systems Approach A 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 9.5.1 P Mathematical Transfer Function Q t t

How ? Formalization of hydrologic process equations Integrated Hydrologic Models Are Used to Understand and Predict (Quantify) the Movement of Water How ? Formalization of hydrologic process equations Lumped Model Semi-Distributed Model Distributed Model REW 1 REW 2 REW 3 REW 4 REW 5 REW 6 REW 7 p q e.g: Stanford Watershed Model e.g: HSPF, LASCAM e.g: ModHMS, PIHM, FIHM, InHM Parametric Physics-Based Process Representation: Predicted States Resolution: Coarser Fine Data Requirement: Small Large Computational Requirement: 8

Transfer Functions 2 Basic steps to rainfall-runoff transfer functions 1. Estimate “losses”. W minus losses = effective precipitation (Weff) (eqns 9-43, 9-44) Determines the volume of streamflow response 2. Distribute Weff in time Gives shape to the hydrograph Recall that Qef = Weff Q t Event flow (Weff) Base Flow

Transfer Functions General Concept Task Draw a line through the hyetograph separating loss and Weff volumes (Figure 9-40) W Weff = Qef W ? Losses t

Loss Methods Methods to estimate effective precipitation You have already done it one way…how? However, … Q t

Loss Methods Physically-based infiltration equations Chapter 6 Green-ampt, Richards equation, Darcy… Kinematic approximations of infiltration and storage Exponential: Weff(t) = W0e-ct c is unique to each site W Uniform: Werr(t) = W(t) - constant

Examples of Transfer Function Models Rational Method (p443) qpk=urCrieffAd No loss method Duration of rainfall is the time of concentration Flood peak only Used for urban watersheds (see table 9-10) SCS Curve Number Estimates losses by surface properties Routes to stream with empirical equations

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

SCS Concepts Precipitation (W) is partitioned into 3 fates Vi = 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 Concept Assuming Vi = 0.2Vmax (??) Vmax is determined by a Curve Number

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

Curve Number Related to Land Use

Transfer Function 1. Estimate effective precipitation SCS method gives us Weff 2. Estimate temporal distribution Base flow Q t Volume of effective Precipitation or event flow -What actually gives shape to the hydrograph?

Transfer Function 2. Estimate temporal distribution of effective precipitation Various methods “route” water to stream channel Many are based on a “time of concentration” and many other “rules” SCS method Assumes that the runoff hydrograph is a triangle On top of base flow Tw = duration of effective P Tc= time concentration Q How were these equations developed? Tb=2.67Tr t

Transfer Functions Once again, consider the assumptions… Time of concentration equations attempt to relate residence time of water to watershed properties The time it takes water to travel from the hydraulically most distant part of the watershed to the outlet Empically derived, based on watershed properties Once again, consider the assumptions…

Transfer Functions 2. Temporal distribution of effective precipitation Unit Hydrograph An 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 Hydrograph The event hydrograph that would result from 1 unit (cm, in,…) of effective precipitation (Weff=1) A watershed has a “characteristic” response This characteristic response is the model Many methods to construct the shape 1 Qef 1 t

Unit Hydrograph How do we Develop the “characteristic response” for the duration of interest – the transfer function ? Empirical – page 451 Synthetic – 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 Hydrograph Apply: 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 example Assumes one burst of precipitation during the duration of the storm In 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 Integral Convolves an input time series with a transfer function to produce an output time series U(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 form J=n-i+1

Unit Hydrograph Many ways to manipulate UH for storms of different durations and intensities S curve, instantaneous… That’s for an engineering hydrology class YOU need to know assumptions of the application

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

Unit Hydrograph How do we derive the characteristic response (unit hydrograph)? Empirical page 451 Note: 1. “…approximately equal duration…” What duration are they talking about? Note: 8. “…adjust the curve until this area is satisfactorily close to 1unit…” See spreadsheet example

Unit Hydrograph Assumptions Linear response Constant time base

Unit Hydrograph Construction of characteristic response by synthetic methods Scores of approaches similar to the SCS hydrograph method where points on the unit hydrograph are estimated from empirical relations to watershed properties. Snyder SCS Clark

Snyder Synthetic Unit Hydrograph Since 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 Hydrograph Triangular Representation The 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 Hydrograph ALL are based on the assumption that runoff is generated by overland flow What 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 Applications What to do with storms of different durations?

Other Applications Deriving the 1-hr UH with the S curve approach

Physically-Based Distributed

Hydrologic Similarity Models Motivation: 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.

TOPMODEL Beven, 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.”

TOPMODEL Surface saturation and soil moisture deficits based on topography Slope Specific Catchment Area Topographic Convergence Partial contributing area concept Saturation from below (Dunne) runoff generation mechanism

Saturation in zones of convergent topography

TOPMODEL Recognizes that topography is the dominant control on water flow Predicts 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 Assumptions from Beven, Rainfall-Runoff Modeling There 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, tanβ 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 Element P a c asat qoverland β qsubsurface We need equations based on topography to calculate qsub (9.6) and qoverland (9.5) qtotal = qsub + q overland

Subsurface Flow in TOPMODEL qsub = Tctanβ What is the origin of this equation? What are the assumptions? How do we obtain tanβ How do we obtain T? a β asat qoverland qsubsurface c

a z c asat qoverland β qsubsurface 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. a β asat qoverland qsubsurface c   z

Transmissivity of Saturated Zone K at any depth Transmissivity of a saturated thickness z-D   D a β asat qoverland qsubsurface c   z  

Equations Subsurface Assume Subsurface flow = recharge rate           Saturation deficit for similar topography regions Surface Topographic Index  

Saturation Deficit Element as a function of local TI Catchment Average Element as a function of average