Hydrograph Modeling
flow
Precipitation
• Goal: Simulate the shape of a hydrograph
given a known or designed water input
(rain or snowmelt)
time
Hydrologic
Model
time
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
Hydrologic
Model
time
flow
Precipitation
• Simulate what happened in the past
• Can serve as a calibration data set
time
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
Hydrologic
Model
time
flow
Precipitation
• Q = f(P, landscape properties)
time
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
t
Q
t
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
REW 2
p

 .(U )  .( )  Qss
t
REW 3
REW 4
REW 1

 pq
t
Distributed Model
REW 5
REW 7
REW 6
q
e.g: Stanford Watershed Model
Process Representation:
e.g: HSPF, LASCAM
Parametric
Predicted States Resolution: Coarser
Data Requirement:
Small
e.g: ModHMS, PIHM, FIHM, InHM
Physics-Based
Fine
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
Event flow (Weff)
Q
Base Flow
t
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,
(W  Vi ) 2
Weff 
W  Vi  Vmax
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
Q
Base flow
t
-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
Tb=2.67Tr
t
How were these
equations developed?
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 (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
1. How do we Develop the “characteristic
response” for the duration of interest – the
transfer function ?
•
•
Empirical – page 451
Synthetic – page 453
2. 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
t
Q(t )   Weff  U t   d
0
U(t-) = time distributed Unit Hydrograph
Weff()= effective precipitation
 =time lag between beginning time series of
rainfall excess and the UH
• Convolution integral in discrete form
Q(t )  i 1W (i )U (t  i  1)
t
Q(t )  WtU1  Wt 1U2  Wt 2U3  ... W1U j
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.
t LAG  Ct ( L  Lca ) 0.3
tbase  3 
tduration 
q peak 
t LAG
8
t LAG
5.5
640AC p
t LAG
t alt .lag  t LAG  0.25(t alt . duration  t duration )
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
D
SCS Dimensionless UHG & Triangular Representation
Triangular
Representation
Excess
Precipitation
1.2
Tb  2.67 x Tp
Tlag
1
Tr  Tb - Tp  1.67 x Tp
0.8
2
+
qpT r
2
=
qp
2
Flow ratios
( T p +T r )
Cum. Mass
Q/Qpeak
Q=
qpT p
Triangular
0.6
qp=
2Q
T p +T r
Point of
Inflection
Tc
0.4
qp=
654.33x 2 x A x Q
T p +T r
qp=
0.2
484 A Q
0
Tp
0.0
Tp
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).
1.0
2.0
Tb
3.0
T/Tpeak
4.0
5.0
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 quasidistributed 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
qtotal = qsub + q overland
We need equations based on
topography to calculate qsub (9.6)
and qoverland (9.5)
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?
c
a
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
z
c
asat
qoverland
β
qsubsurface
Transmissivity of Saturated Zone
• K at any depth
• Transmissivity of a saturated thickness z-D
a
z
c
qoverland
asat
D
β
qsubsurface
Equations
Subsurface
Surface
Assume Subsurface flow = recharge rate
Saturation deficit for
similar topography
regions
Topographic Index
Saturation Deficit
• Element as a function of local TI
• Catchment Average
• Element as a function of average