This document summarizes geomorphological aspects of hydrological modeling from 1979 to the present. It discusses how geomorphological information has been incorporated into hydrological models over time. In the 1980s, the availability of digital elevation models allowed for isochrones and width functions to be derived from digital data. This provided a finer representation of geomorphology. In the following decades, models incorporated more geomorphological details like separating hillslope and channel flow velocities. Overall, incorporating geomorphological details improved the ability of models to predict rainfall-runoff responses and event hydrographs.

1. Geomorphological Aspect of Hydrological
modelling in 2020
Riccardo Rigon, Marialaura Bancheri, Wuletawu Abera, Giuseppe Formetta and Alban de Lavenne
Perugia, Giornate dell’idrologia, 7 Ottobre 2015
ParcoQuerini,Ottobre2015,Vicenza

2. !2
What was Geomorphology
in hydrological modelling
before 1979 ?
but this is a post-reinterpretation
that there was geomorphology there
A little of (biased) history
R.B.A.F.L

3. !3
What was geomorphology ?
Other Ancestors
Zoch RT, 1937. On the relation between rainfall and stream flow. Monthly
Review 55: 135–147.
Clark C, 1945. Storage and the unit hydrograph. Transactions
of the American Society of Civil Engineers 110(1): 1419–1446.
Not to go back to the old papers on the:
Mulvany, T. J. (1851) On the use of self-registering rain and flood gauges in making
observations of the relations of rain fall and of flood discharges in a given catchment.
Proceedings of the Institution of Civil Engineers of Ireland 4, 18–33.
Ross, C. N. (1921) The calculation of flood discharge by the use of a time contour plan.
Transactions of the Institution of Engineers, Australia 2, 85–92.
See Also: “Rainfall-runoff modelling, IAHS Benchmark papers in Hydrology, 2010, J. McDonnell (Ed.)
R.B.A.F.L

4. !4
A hybrid
Surkan,A.J.,Wrr,1976What was geomorphology ?
R.B.A.F.L

5. !5
1979
Clearly there is not a unique thread in this history, and possibly Rodriguez-Iturbe and Valdes did
not know, when they wrote their paper, neither Dooge’s nor the other works, and they were more
inspired by other contributions, as those in stochastic hydrology by V. Yevievich
GIUH
R.B.A.F.L

6. !6
Iturbe and Valdés, marked the beginning of a new era in rainfall-
runoff models. The use of geomorphological information to assist
in definining the unit hydrograph (or more general hydrological
response functions such as travel time distributions), and the
conceptualisation of hydrologic response as the convolution of
travel time distributions, was represented with a mathematically
neat method.
It Was really new
GIUH
Q(t) = A
Z t
0
p(t ⌧)Je(⌧)d⌧
p(t) =
X
2
(p 1
⇤ · ⇤ p ⌦
)(t)
R.B.A.F.L

7. !7
Or, really syntethic
Q(t) = A
X
2
p (Je ⇤ p 1
⇤ · ⇤ p ⌦
)(t)
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GIUH

8. !8
Notably there was an effort to get information from the
knowledge of terrain
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GIUH
Rigon et al., ESP&L, 2015

9. !9
BUT
Geomorphological information at that time was limited
NO SRTM
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GIUH

10. !10
Horton Laws
Geomorphology was
daTarboton:www.cuahsi.org
• The bifurcation law … just for illustrative purposes
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11. !11
Somewhat the interpretation given by
researchers was rigid
• Partition was taken according to Horton Laws
• Pdfs were chosen exponential
And obviously the problem of choosing a right
effective rainfall was ubiquitous
R.B.A.F.L
A somewhat limited interpretation

12. WFIUH
Riccardo Rigon, Marialaura Bancheri, Wuletawu Abera, Giuseppe Formetta and Alban de Lavenne
Perugia, Giornate dell’idrologia, 7 th October 2015

13. !13
What was Geomorphology
in hydrological modelling
in eighties ?
The Digital Elevation Models Era
The old concept of isochrones Ross (1921), or, e.g. Beven (2011), could
be derived from the definition of width function (Kirkby, 1976), under
the hypothesis of constant celerities throughout the network , from
digital data.
Figure 2: A basin can be continuously subdivided into strips of terrain at the same distance
from the outlet. These strips are not necessarily continuous.They are physically connected
to the outlet by many channels, but of these channels the physically significant quantity is
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WFIUH

14. !14
A finer view of
some isochrones
At steps of 2.5 km apart
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WFIUH

15. !15
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WFIUH
Width function after, Kirkby 1976

16. !16
f(t|x) =
x
p
4⇡Dt3
exp

(x uct)2
4Dt
Q(t) =
Z t
0
p(t ti)Je(ti)dti
WFIUH
pw(t ti) =
Z L
0
w(x)f(t ti|x)dx
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WFIUH
Rinaldo et al., 1991; Mesa and Mifflin, 1986

17. !17
In case channel celerity and hillslope celerity were
separated
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WFIUH

18. !18
x0
= xc + r xh
r :=
uh
uc
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Rescaled width function
Rinaldo et al, 1995; D’Odorico and Rigon, 2003; Rinaldo and Botter, 2003

19. !19
Achievements
• It was possible to assess the role of geometry/topology and water
dynamics by showing that geometry counts more than dynamics
(geomorphological dispersion, Rinaldo et al., 1991)
• It was possible to assess the role of hillslope and channels, by
showing that hillslopes count more than channels, e.g. D’Odorico
and Rigon, 2003.
• It was possible to give semi-analytical formula for the peak flows
(Rigon et al., 2011)
R.B.A.F.L
The good

20. !20
R
Rn
Q
Total flowpath
travel time
Baseflow
travel time
(a) 1F1U1P
R
Rn
Q
Network
travel time
Hillslope
travel time
Baseflow
travel time
(b) 1F2U2P
R
Rn
Q
Total flowpath
slow travel time
Total flowpath
fast travel time
Baseflow
travel time
(c) 2F2U3P
R
Rn
Q
Hillslope
slow travel time
Network
travel time
Hillslope
fast travel time
Baseflow
travel time
(d) 2F3U4P
Figure 3: Graphical description of the di↵erent model’s structure.
2.2.2 Production function
In order to be able to implement those transfer functions, a production function and a baseflow
Various complications were tried
R.B.A.F.L
Various complications
de Lavenne et al., in preparation, 2015

21. !21
and is only use to describe the discharge between the events which are, as for them, modelled by
the transfer function. For Co¨et-Dan at Naizin basin, the baseflow function can even be totally
removed from the global structure of the model (↵1 closed to 0) which underline that a simple
transfer function which is considering dispersion could be enough to describe the basin’s response.
In that case, model’s structure would be greatly simplified.
0.0
0.1
0.2
0.3
01 Nov
2007
05 Nov
2007
12 Nov
2007
19 Nov
2007
26 Nov
2007
03 Dec
2007
10 Dec
2007
17 Dec
2007
24 Dec
2007
31 Dec
2007
Runoff simulations on Fremeur at Plumeliau
Discharge[m³/s]
KGE
0.72
0.72
0.5
0.77
0.77
0.79
0.66
0.66
0.53
0.75
0.75
0.74
Models
1F1U1P
1Frv1U1P
1Fd1U2P
1F2U2P
1Frv2U2P
1Fd2U3P
2F2U3P
2Frv2U3P
2Fd2U4P
2F3U4P
2Frv3U4P
2Fd3U5P
Observation
Figure 11: Runo↵ simulation on Fremeur at Plumeliau from 2007-11-01 to 2007-01-01 using
the di↵erent transfer function.
Performances
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Good Performances in events predictions
de Lavenne et al., in preparation, 2015

22. !22
Performances
2006−10to2010−10
2006−11to2007−01
2007−05to2007−07
2007−11to2008−01
2008−05to2008−07
2008−11to2009−01
2009−05to2009−07
2009−11to2010−01
2010−05to2010−07
2010−11to2011−01
2011−05to2011−07
2011−11to2012−01
2012−05to2012−07
1F1U−d
2F2U−d
1F1U−rv
(a) Calibration periods, colorized by columns
2006−10to2010−10
2006−11to2007−01
(b) Calibra
2007−01to2007−03
2007−07to2007−09
2008−01to2008−03
2008−07to2008−09
2009−01to2009−03
2009−07to2009−09
2010−01to2010−03
2010−07to2010−09
2010−10to2012−10
2011−01to2011−03
2011−07to2011−09
2012−01to2012−03
2012−07to2012−09
2F3U
2F3U−rv
2F3U−d
1F2U−d
1F1U−d
2F2U−d
2F2U−rv
2F2U
1F2U−rv
1F2U
1F1U−rv
1F1U
(c) Validation periods, colorized by columns
2007−01to2007−03
2007−07to2007−09
(d) Validat
Figure 10: Heatmap of mean models’ simulations e ciency est
tion and validation periods over the six studied basin
models and sub-figure 10b is comparing periods of s
relatively poorer performance, and darker color mean
Heatmaps of figure 11 summarize model e ciency accordin
average for all studied periods. Even if the description of each
(because models are not compared according to simulation perio
2006−10to2010−10
2006−11to2007−01
2007−05to2007−07
2007−11to2008−01
2008−05to2008−07
2008−11to2009−01
2009−05to2009−07
2009−11to2010−01
2010−05to2010−07
2010−11to2011−01
2011−05to2011−07
2011−11to2012−01
2012−05to2012−07 1F1U−d
2F2U−d
1F1U−rv
1F1U
2F2U−rv
2F2U
2F3U
2F3U−rv
2F3U−d
1F2U−d
1F2U
1F2U−rv
(a) Calibration periods, colorized by columns
2006−10to2010−10
2006−11to2007−01
2007−05to2007−07
2007−11to2008−01
2008−05to2008−07
2008−11to2009−01
2009−05to2009−07
2009−11to2010−01
2010−05to2010−07
2010−11to2011−01
2011−05to2011−07
2011−11to2012−01
2012−05to2012−07
1F1U−d
2F2U−d
1F1U−rv
1F1U
2F2U−rv
2F2U
2F3U
2F3U−rv
2F3U−d
1F2U−d
1F2U
1F2U−rv
(b) Calibration periods, colorized by rows
1F2U−d
1F1U−d
2F2U−d
2F2U−rv
2F2U
1F2U−rv
1F2U
1F1U−rv
1F1U
1F2U−d
1F1U−d
2F2U−d
2F2U−rv
2F2U
1F2U−rv
1F2U
1F1U−rv
1F1U
The darker, the better
R.B.A.F.L
Well … Custer heat maps
de Lavenne et al., in preparation, 2015

23. !23
Issues
• It is still an event-based model
• Does not include evapotranspiration (the World has gone
elsewhere)
• Tracers tell another story (“The old water paradox”)
R.B.A.F.L
The idea that something is missing

24. The business as usual
Riccardo Rigon, Marialaura Bancheri, Wuletawu Abera and Giuseppe Formetta
Already used in Padova, 23-24 September 2015

25. !25
Each HRU is a control volume
• No lateral fluxes
• No deep losses and
recharge terms supplying
deep groundwater
S(t) : Water storage in the
control volume V
M(t) : Solute storage in the
control volume V
Figure From Catchment travel times distributions
and water flow in soils, Rinaldo et al. (2011)
HRUs level example
R.B.A.F.L

26. !26
Water Budget
dS(t)
dt
= J(t) Q(t) AET (t)
HRUs level example
R.B.A.F.L

27. !27
Water Budget
Volume of water in the
control volume
Total precipitation =
rainfall + snow melting
Discharge
Actual
Evapotranspiration
dS(t)
dt
= J(t) Q(t) AET (t)
HRUs level example
R.B.A.F.L

28. !28
The business as usual
dS(t)
dt
= J(t) Q(t) AET (t)
AET(t) =
S(t)
Smax
ET (t)
where ET(t) is potential evapotranspiration (maybe space-averaged) and a,b,Smax
are parameters (in principle different for any HRU)
Q(t) = k S(t)b
HRUs level example
R.B.A.F.L

29. !29
In this case:
Let for a moment b=1, then the equation is linear and has a solution
dS(t)
dt
= J(t) kS(t)b S(t)
Smax
ET (t)
S(t) = e ( t
k + 1
Smax
R t
0
ET (t0
)dt0
)
Z t
0
e(s
k + 1
Smax
R s
0
ET (t0
)dt0
)J(s)ds
if S(0) = 0 which is known, as soon as, ET(t) and J(t) are known
HRUs level example
R.B.A.F.L

30. !30
If we define
S(t) :=
Z t
0
S(t, ⌧)d⌧
Storage at time t
generated by precipitation
at time
Z t
0
S(t, s)ds =
Z t
0
e (t s
k + 1
Smax
R t s
0
ET (t0
)dt0
)J(s)ds
we have
S(t, s) = e (t s
k + 1
Smax
R t s
0
ET (t0
)dt0
)J(s)
HRUs level example
R.B.A.F.L

31. !31
Q(t) :=
Z t
0
Q(t, ⌧)d⌧
AET (t) :=
Z t
0
AET (t, ⌧)d⌧
Discharge at time t
generated by
precipitation at time
Actual
evapotranspiration
generated by
precipitation at time
We can also define
HRUs level example
R.B.A.F.L

32. !32
Is also
Q(t, s) = ke (t s
k + 1
Smax
R t s
0
ET (t0
)dt0
)J(s)
AET (t, s) = S 1
max
h
e (t s
k + 1
Smax
R t s
0
ET (t0
)dt0
)J(s)
i
ET (t)
Given
S(t, s) = e (t s
k + 1
Smax
R t s
0
ET (t0
)dt0
)J(s)
HRUs level example
R.B.A.F.L

33. Introducing residence/travel times
Riccardo Rigon, Marialaura Bancheri
October, 2015
ParcoQuerini,Ottobre2015,Vicenza

34. !34
Definitions
Introduction
Let’s define
⌧
⌦
the injection time
the exit time
R.B.A.F.L

35. !35
Storage is usually defined as S. It is varying with time, so:
S = S(t)
of this storage, we can think to distinguish, the part that was injected at time tau
and, further, the part that is expected to exit at time Omega. This part is:
that integrated gives:
S(⌦, t, ⌧)
S(⌦, t) =
Z 1
0
S(⌦, t, ⌧)d⌧
S(t, ⌧) =
Z 1
0
S(⌦, t, ⌧)d⌦
R.B.A.F.L
The Kinematics of travel time distributions

36. !36
Obviously:
S(t) =
Z 1
0
Z 1
0
S(⌦, t, ⌧)d⌦d⌧
Observe also that we can define such probabilities that *:
p (⌧|t) :=
S(t, ⌧)
S(t)
!p (⌦|t) :=
S(⌦, t)
S(t)
* One obtains normalisation by integrating over
R.B.A.F.L
The Kinematics of travel time distributions
⌧

37. !37
Then:
Define, analogously
J(t, ⌧)
Q(t, ⌧)
AET (t, ⌧)
S(t, ⌧) = S(t) p (⌧|t)
S(⌦, t) = S(t)!p (t|⌦)
R.B.A.F.L
The Kinematics of travel time distributions

38. !38
When we are dealing with hydrology, we are doing water budgets. This can be
presented in integrated form:
dS(t)
dt
= J(t) Q(t) AEt(t)
However, we can do it by any of the sub volumes of S:
dS(t, ⌧)
dt
= J(t) Q(t, ⌧) AEt(t, ⌧)
dS(⌦, t)
dt
= J(⌦, t) Q(t) AEt(t)
R.B.A.F.L
The Kinematics of travel time distributions

39. !39
Therefore, making the appropriate substitutions
backward equation
forward equation
d
dt
S(t) p (⌧|t) = J(⌧) Q(t) p Q(⌧|t) AEt(t) p E(t, ⌧)
d
dt
S(t)!p (⌦|t) = J(t, ⌦)!p J (⌦|t) Q(t) AEt(t)
Note: both or them are probability conditional to the actual time, t.
However, backward and forward refers to the fact that the free variable is
before or after the actual time.
R.B.A.F.LR.B.A.F.L
The Kinematics of travel time distributions

40. !40
The above equations, once known S(t) are differential equations in p, but, to be
solved, some trick must be made to express pQ and pAET as function of p.
This can be done, introducing some StorAge Selection functions (SAS), each,
for each output.
p Q(⌧|t) := !(⌧, t) p (⌧|t)
p AET
(⌧|t) := !AET
(⌧, t) p (⌧|t)
!p AET
(⌦|t) := !AET
(⌦, t)!p (⌦|t)
For the backward equation:
For the forward equation:
!p Q(⌦|t) := !Q(⌦, t)!p (⌦|t)
R.B.A.F.L
The Kinematics of travel time distributions

41. !41
Hence, the backward equation reads:
d
dt
S(t) p (⌧|t) = J(⌧) !Q(t, ⌧)Q(t) p (⌧|t) !AET
(t, ⌧)AET (⌧|t) p E(t, ⌧)
R.B.A.F.L
And can be solved as:
The Kinematics of travel time distributions
Notably from this distribution we can derive the
average age of water and connect, for instance,
tracers experiments with the model that gives Q and
ET

42. !42
The Kinematics of travel time distributions
R.B.A.F.L
Botter, G. (2012). Catchment mixing processes and travel time distributions. Water
Resources Research, 48(5), http://doi.org/10.1029/2011WR011160
Botter, G., Bertuzzo, E., & Rinaldo, A. (2010). Transport in the hydrologic
response: Travel time distributions, soil moisture dynamics, and the old water
paradox, 46(3). http://doi.org/10.1029/2009WR008371
Botter, G., Bertuzzo, E., & Rinaldo, A. (2011). Catchment residence and travel time
distributions: The master equation. Geophysical Research Letters, 38(11). http://
doi.org/10.1029/2011GL047666
Benettin, P., Catchment transport and travel time distributions: theoretical developments and
applications, Ph.D. dissertation (A. Rinaldo & G. Botter, Supervisors.).
Details on

43. Where is geomorphology here ?
Riccardo Rigon, Marialaura Bancheri, Wuletawu Abera and Giuseppe Formetta
Already used Padova, 23-24 September 2015
A.Bonomi
Sketches of a theory

44. !44
Even in the new formalism we can think the single sub-
catchments as part of system
R.B.A.F.L
The new theory

45. !45
And we can apply the convolution formulas
Q(t) = A
X
2
p (Je ⇤ p 1
⇤ · ⇤ p ⌦
)(t)
but this is kind of a trivial
application of the theory
R.B.A.F.L
The new theory

46. !46
Even if we applied it to a system complex
like Adige River
R.B.A.F.L
Real applications

47. !47
To really introduce geomorphology
we have to work out the storage problem for each hillslope, and make it
dependent on some space variable. So that, for instance, we spatially
distribute the input, the storage becomes
where x is some space variable, for instance the distance to outlet. One
possibility is taking:
and, then:
Q(t, ⌧) =
Z L
0
J(x, ⌧)w(x)f(t ⌧|x)dx
S(x, t, ⌧)
for some appropriate probability distribution function f( )
S(x, t, ⌧) := J(x, ⌧) J(x, ⌧)w(x)f(t ⌧|x) AET (x, t, ⌧)
R.B.A.F.L
Storage is the key

48. !48
Integrating over the space variable
S(t, ⌧) =
Z L
0
J(x, ⌧) J(x, ⌧)w(x)f(t ⌧|x) AET (x, t, ⌧)dx
and integrating over the injection time
S(t) =
Z t
0
Z L
0
J(x, ⌧) J(x, ⌧)w(x)f(t ⌧|x) AET (x, t, ⌧)dxd⌧
We can reconstruct all the relevant probabilities
R.B.A.F.L
Storage is the key

49. !49
We can do more …
This scheme, in fact, correspond to a unique control volume
R.B.A.F.L
Too simple, too bad

50. !50
We would like to split it in more parts
Figure 4: A HRU is “vertically” split into a surface and a subsurface domain. The overall
response (to channel flow) is obtained just by the sum of the two contributions.
In the same study, notwithstanding the generally optimal response of the WFIUH the-
ory, an accurate analysis of the results over several events by means of ”Cluster Heat
Maps” (Wilkinson and Friendly, 2009) clearly showed that there were basins and events
say, two, for simplicity.
R.B.A.F.L
Too simple, too bad

51. !51
The Hymod-like way
This is giving a distribution function for storages, for instance:
F(C) = 1
✓
1
C
Cmax
◆b
C is the storage
R.B.A.F.L
Partitioning the control volume
After Moore, 1985

52. !52
We can assume
That (by saturation excess) those points where the capacity C is exceeded,
goes into surface runoff, generated, for instance at any distance x.
Surprise!
You thought, by analogy, that the previous scheme, with one storage, was a
kind of surface flow. Instead now it become subsurface storm flow.
Runoff can be also routed through a geomorphologic
scheme but also in a different way.
R.B.A.F.L
Partitioning the control volume

53. !53
I got you tired so I do not proceed further
but clearly one can add layers at will, until necessary
0
100
200
300
2012−01−01 2012−02−01 2012−03−01 2012−04−01 2012−05−01 2012−06−01 2012−07−01 2012−08−01 2012−09−01 2012−10−01 2012−11−01 2012−12−01
monthly
J(mm)
−100
0
100
200
300
10-2011
11-2011
12-2011
01-2012
02-2012
03-2012
04-2012
05-2012
06-2012
07-2012
08-2012
09-2012
Months
Watercomponent:Q,ET,S(mm)
Q
ET
S
components
R.B.A.F.L
It is too late. I close it here

54. !54
A few new things here
R.B.A.F.L
There is no need for finding the effective rainfall
Evapotranspiration is included
Snow modelling can be easily includes (it is just an added storage)
Tracers and temperature dynamics is around the corner
1
2
3
4
Top storages can be set to describe the root zone5
6 Top storages can be set to be coupled with satellite information
about soil moisture
7 …..

55. !55
Find this presentation at
Ulrici,2000?
Other material at
Questions ?
R. Rigon
Mostly from Rigon et al., ESP&L, 2015
http://abouthydrology.blogspot.it/2015/10/geomorphological-modelling-in-2020.html
http://abouthydrology.blogspot.it/search/label/Residence%20time