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## Hydrograph Modeling

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

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Title: Hydrograph Modeling

1
Hydrograph Modeling
• Goal Simulate the shape of a hydrograph given a
known or designed water input (rain or snowmelt)

2
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

3
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)

4
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

5
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

6
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?
7
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
8
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
9
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
10
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
11
Loss Methods
• Methods to estimate effective precipitation
• You have already done it one wayhow?
• However,

12
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
13
Examples of Transfer Function Models
• Rational Method (p443)
• 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

14
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

15
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,

16
SCS Concept
• Assuming Vi 0.2Vmax (??)
• Vmax is determined by a Curve Number

17
Curve Number
The SCS classified 8500 soils into four
hydrologic groups according to their infiltration
characteristics
18
Curve Number
• Related to Land Use

19
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?
20
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
21
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
22
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

23
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
24
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)

25
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.
• 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?
26
• 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
27
• Convolution integral in discrete form

Jn-i1
28
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

29
Unit Hydrograph
• How do we derive the characteristic response
(unit hydrograph)?
• Empirical

30
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

31
Unit Hydrograph
• Assumptions
• Linear response
• Constant time base

32
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

33
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
34
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).
35
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?

36
Other Applications
• What to do with storms of different durations?

37
Other Applications
• Deriving the 1-hr UH with the S curve approach

38
Physically-Based Distributed
39
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.

40
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,
• 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.

41
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

42
Saturation in zones of convergent topography
43
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.

44
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

45
Hillslope Element
P
We need equations based on topography to
calculate qsub (9.6) and qoverland (9.5)
qtotal qsub q overland
46
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?

47
• 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
48
Transmissivity of Saturated Zone
• K at any depth
• Transmissivity of a saturated thickness z-D

z

49
Equations
Subsurface
Assume Subsurface flow recharge rate

Saturation deficit for similar topography regions
Surface
Topographic Index

50
Saturation Deficit
• Element as a function of local TI
• Catchment Average
• Element as a function of average