Ezio Todin symposium talk

About the compromise
among conceptual, mathematical and numerical
tractability in some hydrological models
The&NoAward2-Cover
Riccardo Rigon
Wednesday, June 5, 13

We discover ... that all our laws can be
written in mathematical form; and that
this has a certain simplicity and beauty
about it. So, ultimately, in order to
understand nature it may be necessary to
have a deeper understanding of
mathematical relationships*
R. Feynman
*i.e. equations, and differential equations

3
Extract from the Abstract
We all either try to formulate laws at one scale by
guessing them, using the available knowledge, or try to
deduce them by a mix of algebraic treatment of the
basic laws of mass, energy and momentum
conservation, and educated simplifications.
It is important to have the true equations !
mass, momentum and energy conservation
are
the most “true” equations we know
I liked my abstract ... maybe because it was so obvious!
R. Rigon

4
This is what we did with the GEOtop model
Rigon et al. 2006
Process based models
R. Rigon

5
Picasso,DoraMaar
But since the perfect model does not exist
•shallow water equations for surface flow
•Richards’ equation for subsurface flow
indeed we adopt:
Freeze and Harlan, 1969 (but with much better numerics)
R. Rigon

6
Actually our statement on Richards’ equation
is that what it is true is this
Mass conservation (no nuclear reactions) !
but actually true if the continuum (a.k.a. Darcy) hypothesis is valid
R. Rigon

Not necessarily this:
7
Se = [1 + ( ⇥)m
)]
n
Se :=
w r
⇥s r
C(⇥)
⇤⇥
⇤t
= ⇥ · K( w) ⇥ (z + ⇥)
⇥
K( w) = Ks
⇧
Se
⇤
1 (1 Se)1/m
⇥m⌅2
SWRC +
Darcy-Buckingham
Parametric
Mualem
Parametric
van Genuchten
C(⇥) :=
⇤ w()
⇤⇥
R. Rigon

8
The last representation of mass conservation
is just matter of convenience
habits, and ignorance of some phenomena
I bet that others talked about some other phenomena, but, in the
previous slide, we were also missing:
•variable and changing temperature
•soil freezing
•transition to saturation
R. Rigon

Richards equation is “wrong” !
9
Sure. But then, what else I should use:
•Green-Ampt ?
•SCS ?
•Topmodel ?
I use all of them when I find convenient.
However, all of them are even more “wrong” than Richards. So for the first
part of this talk I stick with Richards’ assumptions.
Take it as my null hypothesis
Better wrong than “not even wrong”
R. Rigon

10
Richards’s eq. was said to be too much computational
expensive
This statement is also true but, the real truth is that up to 1990 (M.
Celia et al.) we did not have an appropriate numerical method to
solve them, and the right numerics penetrated slowly in the community.
Now we have many (including the recents Casulli and Zanolli, 2010, 2012),
and we can, at least, explore with some confidence their behavior.
R. Rigon

11
To exaggerate
•energy budget: turbulent flows, heat equation, soil
freezing, snow budget
we added
still Freeze and Harlan, 1969 ?
EndrizziandMarsh,2010;Dall’Amicoetal.,2011,Endrizzietal.,2013
Bertoldietal.,2010a,b
R. Rigon

12
Is it feasible ?
Is it usable ?
Does it works ?
We did it !
It is useful ?
e.g Beven, 2000, 2001 (for instance) criticized this approach but we
needed anyway a reference model to start with
Yes, it is!
We “forecasted” decently well: water flows, soil moisture,
landslides, terrain temperatures, evapotranspiration, snow
cover ...
R. Rigon

13
For instance a service for forecasting
snow heights
based on GEOtop is currently operational
seehttp://www.mountain-eering.com
R. Rigon

14
So that’s the end of the story ?
certainly not !
The criticism to this type of modelling have foundations.
GEOtop NewAge Boussinesq PeakFlow
SHALSTAB GEOtop-FS The Horton Machine
and we have several models that we use at different scales and for
different purposes
We did not marry process based models
R. Rigon

15
.. In practice, when in the hands of hydrologists both the approaches
contaminate each other, and represent some compromise among
experimental evidence, scientific knowledge, mathematical
convenience, and computational tractability ... and the natural laziness
that everybody has.
GEOtop NewAge Boussinesq PeakFlow
SHALSTAB GEOtop-FS The Horton Machine
R. Rigon

16
In some models we
treat just one process,
i n o t h e r , l i k e i n
NewAGE*, we treat
them all again.
The goal here was to simplify the
equations as much as possible but
maintaining a spatially variable
description of the models
Formetta et al., 2011, 2013a,b,c
But I am not going to talk about it!
Please look at the Poster session.
R. Rigon

17
Using
models
Simula'on:

imitate
one
process
by
another
process
Process:

temporal
sequence
of
states
of
a
system
“In
computer
simula'ons
of
physical
systems,
the
construc'on
of
models
is
guided,
but
not
determined,
by
theory.
At

the
same
'me
simula'ons
models
are
o>en
constructed
precisely
because
data
are
sparse.
They
are
meant
to
replace

experiments
and
observa'ons
as
sources
of
data
about
the
world;
hence
they
cannot
be
evaluated
simply
by
being

compared
to
the
world.
So
what
can
be
the
source
of
credibility
for
simula'on
models?
I
argue
that
the
credibility
of
a

simula'on
model
comes
not
only
from
the
creden'als
supplied
to
it
by
the
governing
theory,
but
also
from
the

antecedently
established
creden'als
of
the
model
building
techniques
employed
by
the
simula'onists.
In
other
words,

there
are
certain
sorts
of
model
building
techniques
which
are
taken,
in
and
of
themselves,
to
be
reliable.
Some
of

these
model
building
techniques,
moreover,
incorporate
what
are
some'mes
called
‘‘falsifica'ons’’.
These
are

contrary-‐to-‐fact
principles
that
are
included
in
a
simula'on
model
and
whose
inclusion
is
taken
to
increase
the

reliability
of
the
results.
The
example
of
a
falsifica'on
that
I
consider,
called
ar'ficial
viscosity,
is
in
widespread
use
in

computa'onal
fluid
dynamics.
Ar'ficial
viscosity,
I
argue,
is
a
principle
that
is
successfully
and
reliably
used
across
a

wide
domain
of
fluid
dynamical
applica'ons,
but
it
does
not
offer
even
an
approximately
‘‘realis'c’’
or
true
account
of

fluids.
Ar'ficial
viscosity,
therefore,
is
a
counter-‐example
to
the
principle
that
success
implies
truth
–
a
principle
at
the

founda'on
of
scien'fic
realism.
It
is
an
example
of
reliability
without
truth.”
(Winsberg,
2006)
Hartmann,
S.
(1996),
The
World
as
a
Process:
Simula=ons
in
Natural
and
Social
Sciences,
in:
Hegselmann,
R.,
U.
Mueller,
K.
Troitzsch
(eds.),

Simula'on
and
Modelling
in
the
Social
Sciences
from
the
Philosophy
of
Science
Point
of
View,
Kluwert,
77-‐100.
Winsberg,
E.
(2006),
Models
of
success
versus
the
success
of
models:
Reliability
without
truth,
Synthese,
152,
1–19.
Computa=onal
era
I like philosophers
robbed from M.Toffolon

18
A warning to myself
When hydrologists play to do philosophers, even
the best, they do not do their job
I like philosophers but
R. Rigon

19
... The modeling of some processes, i.e. rainfall-runoff, soil storm
flow, snowpack evolution, are presented here according to different
degree of simplifications, and the simplifications briefly
discussed. ...
As in Ezio work
We were attracted however by determining the structure of models
simplification by theory, more than “inventing” by analogy
“processes lines” to simplify
Less is more
R. Rigon

20
2D - de Saint Venant equations
with some smart subgrid parameterization
(e.g. Casulli, 2009)
1D - Kinematic equation
So many to cite here but ... Liu and
Todini, 2002
Various aggregation strategies
for runoff, including residence
time theories (a.k.a GIUH)
Rodriguez-Iturbe and Valdes, 1979;
Rinaldo et al., 1991,
D’Odorico and Rigon, 2003
R. Rigon
Less is more

21
3D-Richards’ equation
(Richards, 1931; Celia et al. 1990)
1D-Richards + Boussinesq
Topkapi
HsB
Topog/Topmodel
CordanoandRigon,2008
(Citations from Cordano and Rigon, 2013)
Liu and Todini, 2002
Troch et al., 2003
O’Loughlin, 1986; Beven and Kirkby, 1979
Less is more
R. Rigon

22
Dalton’s Equation
e.g. Brutsaert 1982
Penman
Penman, 1948
Monteith
Monteith, 1965
Priestley-Taylor
Priestley and Taylor, 1972
Less is more
R. Rigon

23
Energy Budget
Jordan, 1991
Radiation + Temperature
Brubaker et al., 1996
Degree-day (Just
temperature)
Martinec and Rango, 1975
Less is more
R. Rigon

24* But not anymore necessarily
R. Rigon
Less is more

24
Models “complexity” and computational time increase
going from bottom up.
More complexity, more processes physics.
Scales of application usually* decrease from top to
bottom
* But not anymore necessarily
R. Rigon
Less is more

25
Less is more
R. Rigon

25
Parameters pretend to be estimated ex-ante
(measured) in more complex models (with a lot of
disclaimers ... obviously)
Are certainly calibrated (ex-post) in the simplest
models (but in some models preserve a physical
significance)
From top to bottom heuristic and statistics
substitute processes analysis
Less is more
R. Rigon

Just an example of top down derivation
The case of Richards’ equation
ChimpanzeeCongopainting

Iverson,2000;CordanoeRigon,2008
27
The Richards equation on a plane hillslope
Richardsoniana
R. Rigon

28
The Richards equation made dimensionless
Richardsoniana
R. Rigon

29
Richards eq. solution expressed in terms of
the asymptotic hydrostatic solution and a transient
term:
See also. D’Odorico et al., 2003
Richardsoniana
R. Rigon

and one equation for
30
So Richards equation is
divided into one equation for
Richardsoniana
R. Rigon

31
In turn
“Short term
solution” Taylor’s
expansion
Water table
equation Taylor’s
expansion
Slope normal flow
time scale Lateral flow
time scaleSee also. D’Odorico et al., 2003
Richardsoniana
R. Rigon

32
Neglecting some details
that can be found in Cordano and Rigon, 2008
Zeroth perturbation order
First perturbation order
+ analogous for d*
Richardsoniana
R. Rigon

33
Integrating zeroth order solution in the column
Making a long story short
Topkapi model
Liu and Todini, 2002
Richardsoniana
R. Rigon

34
Integrating first order solution slope-parallel
Making a long story short - II
Boussinesq equation
(e.g. Cordano and Rigon, 2013)
Richardsoniana
R. Rigon

35
Integrating Boussinesq
Making a long story short - III
HsB
Troch et al. 2003
Richardsoniana
R. Rigon

36
Simplifying HsB assuming stationarity of fluxes
and neglecting diffusive terms
Making a long story short - IV and V
Topog
O’Loughlin, 1986
assuming an exponential decay of vertical hydraulic
conductivity
Topmodel
Beven and Kirkby, 1979
Richardsoniana
R. Rigon

37
That is how we obtained:
Richardsoniana
R. Rigon

Just kidding!

39
Did you care about hypotheses ?
Is it for any occasion realistic ? Look at the following sandy-loam:
Hypotheses counts
R. Rigon

constant diffusivity
40
The Decomposition of the Richards equation
is possible under the assumption that:
Time scale of infiltration
soil depth
time scale of lateral flow
hillslope length
reference conductivity
reference hydraulic capacity
Iverson,2000;CordanoandRigon,2008
Hypotheses counts
R. Rigon

Assuming hydrostatic conditions
41
Initial condition is then:
Consequently, at surface
Hypotheses counts
R. Rigon

42
For the sandy-loam soil
assuming the water table at one meter depth
we have a vertical variation of hydraulic conductivity of one order of magnitude !
Hypotheses counts
R. Rigon

43
D0 which characterizes the time scales of flow is varying
with depth
Hypotheses counts
R. Rigon

44
Therefore
at surface
so, lateral flow at the water table level
has the same time scale vertical flow at
the surface (at least if we believe to
Richards’ equation)
Hypotheses counts
R. Rigon

45
igure 2: Experimental set-up. (a) The inﬁnite hillslope schematization. (b) The initial suction head pr
il-pixel hillslope numeration system (the case of parallel shape is shown here). Moving from 0 to 900
sponds to moving from the crest to the toe of the hillslope
The OpenBook hillslope in a 3D
simulation
Comparing with 3D
R. Rigon

46
- 54 LANNI ET AL.: HYDROLOGICAL ASPECTS IN THE TRIGGERING OF SHALLOW LANDSLIDES
(a) DRY-Low (b) DRY-Med
Simulations result
Comparing with 3D
R. Rigon

47
At the beginning the pressure is constant
along the whole transect (except for
phenomena at the divide’s edge
Comparing with 3D
R. Rigon

48
After a certain amount of time (25h in this
simulation) pressures along the slope
differentiate. With a little of analysis we
c a n d i s t i n g u i s h t w o r e g i o n s o f
differentiation. One controlled by the
boundary conditions at the bottom.
The second generated by lateral water
flow accumulation.
Comparing with 3D
R. Rigon

49
(a) (b)
Figure 6: Temporal evolution of the vertical proﬁle of hydraulic conductivity (a) and hydraulic conductivity at the soil-bedrock interface
Hidraulic conductivity is varying by three order of magnitude
at the bedrock interface.
The key to understand this phenomenology
Lannietal.,2012
Comparing with 3D
R. Rigon

50
When simulating is understanding
courtesyofE.Cordano
T’L can be very small indeed .....
Interpretations
R. Rigon

51
Understanding from simulations
At the beginning of the infiltration process the situation in surface is
marked by the blue line, the situation at the bedrock is marked by the
red line
courtesyofE.Cordano
R. Rigon
Interpretations

52
When lateral flow start we are in the following situation
courtesyofE.Cordano
Understanding from simulations
R. Rigon
Interpretations

53
At the beginning
The condition of the perturbative derivation are verified
courtesyofE.Cordano
R. Rigon
Interpretations

54
At the end
courtesyofE.Cordano
Conditions for lateral flow are dominating. Actually the same
phenomenology deducted by the perturbation theory! But obtained for a
different reason.
R. Rigon
Interpretations

55
Lateral Flow
•Can be fast, ... very fast, much faster than what happens in vadose
conditions
•In fact, to have the effects just described, we have to believe to the form
that Soil Water retention Curves have.
•Other soils behave differently
•If macropores or cracks are present, vertical infiltration can still remain
faster
R. Rigon
Interpretations

56
Inappropriate numerics (or gridding)
Can hide it!
R. Rigon
Interpretations

Further investigations
MachaelLeong-Cuttingthetimewithaknife,2012

58
CAPITOLO 5. IL BACINO DI PANOLA
Figura 5.2: Rappresentazione della profondità del suolo del pendio di Panola.
costante su un campione prelevato a 10 cm di profondità, risulta pari a 64 [cm/h]; per ciò che concerne
il valore della conducibilità idraulica a saturazione del bedrock, non esistono misure dirette e↵ettuate
su campioni prelevati in sito; tuttavia si stima che il suo valore sia 2-3 ordini di grandezza inferiore
rispetto a quella del terreno soprastante. Entrambi i valori di conducibilità idraulica satura (del bedrock
e del terreno) saranno comunque oggetto di calibrazione numerica all’atto delle simulazioni svolte con
GEOtop, utilizzando come valori di partenza quelli qui citati.
Panola’s hillslope
R. Rigon
Richards equation is still valid here ?

59
Terrain surface Bedrock surface Soil depth varies
Depression
Soil (sandy loam) Bedrock
Ksat = 10-4 m/s Ksat = 10-7 m/s
Panola’s hillslope
R. Rigon

60
Q(m3/h)
t=9h
t=18h
t=22h
With a rainfall of 6.5 mm/h and a duration of 9 hours
Lannietal.,2011
R. Rigon

61
t=6h t=9ht=7h t=14h
Lannietal.,2011
Tromp Van Meerveld et al., 2006 call it filling and spilling
R. Rigon

62
Q(m3/h)
t=9h
t=18h
t=22h
Lannietal.,2011
R. Rigon

63
1D
3D
 No role played by hillslope
gradient
First Slope Normal infiltration works
Then Lateral flow start
Infiltration front propagate
Drainage is controlled by the bedrock form
As in the open book case
Lannietal.,2011
R. Rigon

64
Now we want a model that can run 100 times faster
In which, we obviously use all the machinery of the
Richards’ equation, i.e. hydraulic conductivity and soil
water retention curves
R. Rigon

65
Ii.e. time to water table
development
Twt(x,y):= [Vwt(x,y)-V0(x,y)]/I
Initial conditions
(hydrostatic slope normal)
boundary conditions
(including rainfall, I)
t> Twt(x,y)
YES
NO
Lannietal.,2012
Slope Normal
unsaturated flow
A heuristic model
for each
time
step
Faster is better
R. Rigon

66
YES
t> Tmax
wt(x,y)
hydrologically
connected
A(x,y) >0
YES
NO
hydrologically
disconnected
A(x,y) =0
A heuristic model
Lannietal.,2012
R. Rigon
Faster is better

67
YES
update soil
pressure
start lateral flow update soil
pressure
next
time
step
A heuristic model
Lannietal.,2012
R. Rigon
Faster is better

68
* Is not completely true.
I question also of personal attitude:
I understand (fluid) mechanics through
equations and I try to interpret observations
through equations.
Someone else (i.e. many of my students)
simply did not have the training for that and
prefer to rebuilt the physics of the problem by
small pieces.
This has a certain appealing to many (especially
to natural scientists and geologists), and can
indeed be useful to see thing from different
perspectives.
Doodley,Muttley,andtheirflyingmachines
R. Rigon
Attitudes

69
3968 C. Lanni et al.: Modelling shallow landslide susceptibility
1
2
3
Figure 7. Patterns of Return period TR (years) of the critical rainfalls for shallow landslide4
triggering (i.e., FS≤1) and associated levels of landslide susceptibility obtained by means 5
of QDSLaM.6
7
Fig. 7. Patterns of return period TR (years) of the critical rainfalls for shallow landslide triggering (i.e. FS  1) and associated levels of
landslide susceptibility obtained by means of QDSLaM.
Table 3. Percentages of catchment area (C) and observed landslide area (L) in each range of critical rainfall frequency (i.e. return period TR)
for QDSLaM.
Susceptibility
Pizzano Fraviano Cortina
TR level Ca Lb Ca Lb Ca Lb
Years Category % % % % % %
Uncond Unstable 9.9 60.2 7.7 77.7 8.5 56.8
0–10 Very high 20.3 26.9 16.1 18.5 13.5 39.2
10–30 High 7.8 0.0 5.6 1.5 5.8 4.0
Lannietal.,2012
However, it works
R. Rigon
Faster is better if it works (Klemes fogive me!)

70
CAPITOLO 5. IL BACINO DI PANOLA
Figura 5.4: Immagine tratta da Tromp-van Meerveld e McDonnell, (2006a) [24]; (a) deflusso sub-
superficiale totale per i segmenti in cui è stata suddivisa la trincea e (b) numero di eventi meteorici che
producono deflussi misurabili.
5.2.1 Il ruolo dei macropori
TrompVanMeerveldetal.,2006
And finally macropores
R. Rigon
Macropores

71
Macropore Flow
Initiation
Water supply to the
macropores
Interaction
Water transfer between
macropores and the
surrounding soil matrix
M.Weiler,fromMochaproject
Macropores!
R. Rigon
Macropores

72
0.00
date (dd/mm) 2002
01/01 11/01 21/01 31/01 10/02 20/02 02/03 12/03 22/03 01/04 11/04 21/04 01/05 11/05 21/05
Figura 5.16: Confronto tra flussi misurati e computati attraverso la Simulazione 0 presso la trincea
alla base del pendio.
0.000.020.040.060.080.10
Simulazione 0 - evento 6 febbraio
date (dd/mm) 2002
portate[l/s]
05/02 06/02 07/02 08/02 09/02 10/02 11/02 12/02
Flussi misurati
Simulazione 0
0.000.020.040.060.080.10
Simulazione 0 - evento 30 marzo
date (dd/mm) 2002
portate[l/s]
29/03 30/03 31/03 01/04 02/04 03/04 04/04 05/04 06/04 07/04
Flussi misurati
Simulazione 0
Figura 5.17: Confronto tra flussi misurati e computati attraverso la Simulazione 0 presso la trincea
alla base del pendio: a sinistra si riporta l’evento del 6 febbraio 2002, a destra quello del 31 marzo.
può essere causata da diversi fattori, quali un’errata assegnazione delle caratteristiche del suolo o del
bedrock, oppure un errore nello stabilire la condizione iniziale circa la quota della falda.
Un aspetto decisamente importante da considerare, tanto in questi risultati quanto in quelli presentati
successivamente, è che nella creazione della geometria di calcolo 3D utilizzata da GEOtop non è
DaPrà,2013
Certainly the volumes of water cannot be
simulated with the only Richards equation
No way!
R. Rigon
Macropores

Conclusions
TowrdsthecompleteworksofShakespeare(Essayof
Monkeywriting)

74
.. It is concluded that all models, at any scale, are truly
inherently statistical, in the statisticians sense, and also in
the statistical-mechanical sense, since they derive from an
inductive-deductive process compared to some evidences,
and, at the same time, represent the emergent behavior of
some smaller physical world.
I do not think I really illustrated this: but I believe it is
true, anyway.
R. Rigon
Epilogue

75
big thanks to Ezio
Eventually
for his life-long coherent effort to work with
equations and scientific rigor in a way that was an
example for me and for many
R. Rigon
Ezio!

Thank you for your attention
G.Ulrici,2000?
76
These slides are available at http://abouthydrology.blogspot.com
Thank you
R. Rigon

Ezio Todin symposium talk

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