1. Residence Time

2. Residence Time
Mean Water Residence Time (aka: turnover time, age of water leaving a system, exit age, mean transit time, travel time, hydraulic age, flushing time, or kinematic age)
T = V / Q = turnover time or age of water leaving a system
For a 10 L capped bucket with a steady state flow through of 2 L/hr, T = 5 hours
Assumes all water is mobile
Assumes complete mixing
For watersheds, we don’t know V or Q
Mean Tracer Residence Time (MRT) considers variations in flow path length and mobile and immobile flow

3. Residence and Geomorphology
Geomorphology controls fait of water molecule
Soils
Type
Depth
Bedrock
Permeability
Fracturing
Slope
Elevation

4. MRT estimated using Transfer Function Models

5. Transfer Function Models
Signal processing technique common in
Electronics
Seismology
Anything with waves
Hydrology

6. Transfer Function Models
Brief reminder of transfer function HYDROGRAPH model before returning to

7. 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

8. 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

9. 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

10. 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

11. 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

12. 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: 12

13. 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?

14. 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.

15. 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.”

16. 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

17. Saturation in zones of convergent topography

18. 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.

19. 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

20. 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

21. 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

22. 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

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

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

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

26. 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

27. 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

28. 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

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

30. 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

31. 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

32. 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”

33. 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,

34. SCS Concept Assuming Vi = 0.2Vmax (??)
Vmax is determined by a Curve Number

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

36. Curve Number
Related to Land Use

37. 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?

38. 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

39. 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…

40. 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

41. 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

42. 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)

43. 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?

44. 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

45. Convolution Convolution is a mathematical operation
Addition, subtraction, multiplication, convolution…
Whereas addition takes two numbers to make a third number, convolution takes two functions to make a third function
𝑥 𝑡 ∗𝑈 𝑡 =𝑦(𝑡)≝ −∞ ∞ 𝑥 𝜏 𝑈 𝑡−𝜏 𝑑𝜏
x(t)
U(t)
𝑥 𝑡 ∗𝑈 𝑡 =𝑦(𝑡)≝ −∞ ∞ 𝑥 𝑡−𝜏 𝑈 𝜏 𝑑𝜏
y(t)
x(t) = input function
U(t) = system response function
τ = dummy variable of integration

46. Convolution Watch these: http://www.youtube.com/watch?v=SNdNf3mprrU

47. Convolution Convolution is a mathematical operation
Addition, subtraction, multiplication, convolution…
Whereas addition takes two numbers to make a third number, convolution takes two functions to make a third function
𝑥 𝑡 ∗𝑈 𝑡 =𝑦(𝑡)≝ −∞ ∞ 𝑥 𝜏 𝑈 𝑡−𝜏 𝑑𝜏
x(t)
U(t)
𝑥 𝑡 ∗𝑈 𝑡 =𝑦(𝑡)≝ −∞ ∞ 𝑥 𝑡−𝜏 𝑈 𝜏 𝑑𝜏
y(t)
x(t) = input function
U(t) = system response function
τ = dummy variable of integration

48. Unit Hydrograph Convolution integral in discrete form
𝑥 𝑡 ∗𝑈 𝑡 =𝑦(𝑡)≝ −∞ ∞ 𝑥 𝑡−𝜏 𝑈 𝜏 𝑑𝜏
𝑦(𝑡)≝ 𝜏=−∞ ∞ 𝑥 𝑡−𝜏 𝑈(𝜏)
For Unit Hydrograph (see pdf notes)
J=n-i+1

49. Catchment Scale Mean Residence Time: An Example from Wimbachtal, Germany

50. Wimbach Watershed Drainage area = 33.4 km2
Streamflow Gaging Station
Major Spring Discharge
Precipitation Station
Drainage area = 33.4 km2
Mean annual precipitation = 250 cm
Absent of streams in most areas
Mean annual runoff (subsurface discharge to the topographic low) = 167 cm
Maloszewski et. al. (1992)

51. Geology of Wimbach 3 aquifer types – Porous, Karstic, Fractured
Many springs discharge at the base of the Limestone unit
Maloszewski, Rauert, Trimborn, Herrmann, Rau (1992)
3 aquifer types – Porous, Karstic, Fractured
300 meter thick Pleistocene glacial deposits with Holocene alluvial fans above
Fractured Triassic Limestone and Karstic Triassic Dolomite

52. d18O in Precipitation and Springflow
Seasonal variation of 18O in precipitation and springflow
Variation becomes progressively more muted as residence time increases
These variations generally fit a model that incorporates assumptions about subsurface water flow

53. Modeling Approach Lumped-parameter models (black-box models): Filter/
Origanilly adopted from linear systems and signal processing theory and involves a convolution or filtering
System is treated as a whole & flow pattern is assumed constant over the modeling period (can have many system too)
Filter/
Transfer
Function
Watershed/Aquifer Processes 1
Weight
Normalized Time

54. Modeling by Convolution
A convolution is an integral which expresses the amount of overlap of one function g as it is shifted over another function Cin. It therefore "blends" one function with another
where
C(t) = output signature
Cin(t) = input signature
t = exit time from system
t = integration variable that describes the entry time into the system
g(t-t) = travel time probability distribution for tracer molecules in the system
It’s a frequency filter, i.e., it attenuates specific frequencies of the input to produce the result

55. Convolution Illustration
Cin(t) t
g(t) = e -at
Step
g(-t) t
e -(-at)
Folding 1
g(t-t) t
e -a(t-t) 2
Displacement
Cin(t)g(t-t) t
Multiplication 3
C(t) t
Shaded area
Integration 4

56. Transfer Functions - Piston Flow (PFM)
Assumes all flow paths have same residence time
All water moves with advection (no dispersion or diffusion)
Represented by a delta function
This means the output signal at a given time is equal to the input concentration at the mean residence time T earlier.
Maloszewski and Zuber
PFM
PFM

57. Transfer Functions - Exponential (EM)
Assumes contribution from all flow paths lengths and heavy weighting of young portion.
Similar to the concept of a “well-mixed” system in a linear reservoir model DM EM EM
EPM EM
Maloszewski and Zuber

58. Exponential-piston Flow (EPM)
Combination of exponential and piston flow to allow for a delay of shortest flow paths
This model is somewhat more realistic than the exponential model because it allows for the existence of a delay DM
Maloszewski and Zuber

59. Dispersion (DM)
Assumes that flow paths are effected by hydrodynamic dispersion or geomorphological dispersion
Geomorphological dispersion is a measure of the dispersion of a disturbance by the drainage network structure DM
Maloszewski and Zuber
(White et al. 2004)

60. Input Function
We must represent precipitation tracer flux to what actually goes into the soil and groundwater
Weighting functions are used to “amount-weight” the tracer values according recharge: mass balance
where
Pi = the monthly depth of precipitation
N = number of months with observations
= summer/winter infiltration coefficient
Cout = mean output 18O composition (mean infiltration composition)

61. Infiltration Coefficient
a was calculated using 18O data from precipitation and springflow following Grabczak et al., 1984
Application of this equation yielded an a value of 0.2, which means that winter infiltration exceeds summer infiltration by five times
where
Cout ( ) = o/oo (spring water)
Mean Weighted Precipitation ( ) = -8.90o/oo and o/oo, for summer and winter, respectively
Grabczak, J., Maloszewski, P., Rozanski, K. ans Zuber, A., Estimation of the tritium input function with the aid of stable isotopes. Catena, 11:

62. Input Function
Convolution using FLOWPC

63. Application of FLOWPC to estimate MRT for the Wimbach Spring
Maloszewski, P., and Zuber, A., Lumped parameter models for interpretation of environmental tracer data. Manual on Mathematical Models in Isotope Hydrogeology, IAEA:9-58

64. Convolution Summation in EXcel
Work in progress
Your Task:
Evaluate my spreadsheet. Figure out if I’m doing it right
Get FlowPC to work
Reproduce Wimbachtal results
Run FlowPC or Excel for Dry Creek.