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Hydrograph Modeling

- Goal Simulate the shape of a hydrograph given a

known or designed water input (rain or snowmelt)

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

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)

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
- Ohms law analogous to Darcys 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 ModelingSystems 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

Integrated Hydrologic Models Are Used to

Understand and Predict (Quantify) the Movement of

Water

How ? Formalization of hydrologic process

equations

Distributed Model

Semi-Distributed Model

Lumped Model

e.g Stanford Watershed Model

e.g ModHMS, PIHM, FIHM, InHM

e.g HSPF, LASCAM

Process Representation

Predicted States Resolution

Data Requirement

Computational Requirement

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

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 wayhow?
- However,

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)
- qpkurCrieffAd
- 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

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?

Tb2.67Tr

t

Transfer Functions

- 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

(Weff1) - 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

Jn-i1

Unit Hydrograph

- Many ways to manipulate UH for storms of

different durations and intensities - S curve, instantaneous
- Thats 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 Assumptionsfrom 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, tanß - 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

We need equations based on topography to

calculate qsub (9.6) and qoverland (9.5)

qtotal qsub q overland

Subsurface Flow in TOPMODEL

- qsub Tctanß
- What is the origin of this equation?
- What are the assumptions?
- How do we obtain tanß
- How 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 Zone

- K at any depth
- Transmissivity of a saturated thickness z-D

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