1. 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

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
A past storm
Simulate what happened in the past
Can serve as a calibration data set
time
Precipitation
time
flow
Hydrologic Model

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)
time
Precipitation
time
flow
Hydrologic Model

4. Hydrograph Modeling What is a model? What is the purpose of a model?
Types of Models
Physical
Analog
Ohm’s law analogous to Darcy’s 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 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

8. 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

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 Q t
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 way…how?
However, … Q t

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)
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

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
Base flow Q t
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?
Tb=2.67Tr t

21. 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…

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 (Weff=1)
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.
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?

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
J=n-i+1

28. 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

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…”
See spreadsheet example

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 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, Highlands Ranch, Colorado, p
“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 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

45. 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

46. 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

47. 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

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