Water Budget by residence time

Hillslope hydrological budget including
ET by residence times
Marialaura Bancheri & Riccardo Rigon
Trento, January 2015
YvesKlein,1960

!2
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)
Bancheri and Rigon
The problem

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Mass balance
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The problem
Deep percolation excluded.

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Definitions
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Notation
We can consider various distributions. But we are interested mainly in:

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More definitions
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Notation

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Meaning of the probability
implies that the exit time is conditional to te injection time. However, this
does not explain which is the way this dependence is obtained.
In Botter et al., 2010, for instance this figure is plotted (e.g. Figure 2)
Bancheri and Rigon
Interpretation

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This means (and is explicated ‘;-)’ by Authors’ words) that first any water in
input is mixed in the control volume, and then “sampled”. Sampling is
proportional to the quantity of water present of any age, and happens any
time an event triggers it. Eventually sampled water is the one that exit the
control volume. The full understanding of the process derives, obviously,
from its mathematical formalisation, which will follow. So, even if the
notation in the previous slide is more general, using
would be more appropriate. Here S(t,ti) is the water amount inside the
control volume at time t that can be dated to injection time ti .
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Interpretation

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Above S(t,ti) is the water amount inside the control volume at time t that can
be dated to injection time ti . The total water inside the volume is:
or
if discrete injection times are considered as in Figure above.
Bancheri and Rigon
Interpretation

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Comparing:
And therefore, the fraction of water of a certain age over the total, at a
certain time is:
a relation that will be useful later
with:
we obtain
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Definition

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Volumes and Probabilities
Where:
is the probability that a water particle injected at time ti is still within the
control volume at time t
Differentiating with respect to time, using the Leibeniz rule, and comparing
with the mass balance :
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Setting up the “algebra”

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If we consider two output fluxes, i.e. Q(t) and E(t), we must consider that the
probability of exit time must be split in two components. The formal way to
do it is to introduce a partition function:
from which:
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We obtain then:
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Now consider this ansatz: " The fraction of the water mobilised at time t
injected at time ti to the total water mobilised (for each flux partition), is equal
to the volume fraction of water inside the control volume injected at time ti."
So, for instance:
is the flux (discharge) due to the water
injected at time ti
is the discharge
so the first fraction is
Bancheri and Rigon
Some physics hidden

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while, the second fraction is, as from previous slides
giving:
from which:
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The same, changed what has to be, happens for ET, obtaining:
The latter two equations allows for the determination of the conditional
probabilities for each type of flow, given:
• the exit time distribution (an unknown)
• ET (and/or Q). Which means the outputs, which, in principle could be
quantities either measured or produced by another model.
• The partition coefficient (for which we will be able to produce another
equation)
Please notice that the partition coefficient can vary from 0 to 1 making the
denominator of the fraction very small. However the fluxes are not
independent from it, since:
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Exit time Pdf’s and Unit Hydrographs
20
In general, it is:
However if the travel time distribution is time invariant, then
Then:
after having defined
The latter is the traditional form of the instantaneous unit hydrograph.
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Good old boys

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Back the water budget
To close the budget without being circular, we MUST do hypothesis about ET
and Q(t) as function of the storage. Simple possible hypothesis:
A and b are coefficients. And the latter is the Priestley Taylor equation
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Putting all together

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Priestley Taylor parameters
*Botter et al. 2010 use a simpler relation which depends, however, from storage S(t)
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Budget equation
This is a first order ordinary non linear differential equation that can be
solved iteratively once parameters are known (for instance with the Newton
method). Once S(t) is known, also Q(t) is know. However, to know the travel
time distributions, we have to observe that the very own definition of PE
implies a further relation between the probabilities.
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Let consider
with, by definition
We have then
or
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If we consider the mixing hypothesis, from which:
and
Then the Master equation (?) reduces to:
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Is a linear partial differential equation which is integrable.
[If we make the assumptions explicated before, Q(t), ET(t) and S(t) can be
assumed to be known]
The logical initial condition is:
And the solution is:
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Is
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Consequently
using again the mixing hypotheses, also the other probabilities can be derived.
From
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From
Is
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A quantity is still to be determined, which is the coefficient of partition
It can be actually be determined, by imposing the normalisation of the
probability
This finally implies:
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To sum up
To close the budget, we used:
• The mass budget
• The mixing hypothesis
• The relation between the probabilities
• A normalisation condition
• An assumption on Q(t) form
• An assumption on ET form
The last two implies an assumption for any output (if we would have a third
we would have required a third one). Besides we have as many
normalisation conditions as the outputs mins one. With this specification,
the procedure is easily generalised.
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Summary

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Here it is the whole set of equations 1/2
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Summary

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Here it is the whole set of equations 2/2
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Summary

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Some appointment and some disappointment
Bancheri and Rigon
Notes
I cannot avoid some disappointment after having understood the main lines of
the arguments of the papers we read.
The papers introduce a formal approach that extracts residence time
information (whatever we call it) from models, i.e. from the water budget. As
presented. So in principle any model if the hydrological budget can be fitted in
it. In principle, even a complicate, process-based, distributed model like
GEOtop.
So there are opportunities in it. The disappointment come from the fact the
easily, the complication of the formalism, prevented to use, for instance, the
good stuff derived from the theory of GIUH where the information about the
spatial distribution of pathways was used ! This can be actually re-inserted, with
some little work.

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Claims
Bancheri and Rigon
Notes
The claim that the exit time distribution function is not anymore linear and
time invariant, is just a consequence of the hypotheses made, when the water
budget is written, and of the mixing hypothesis used. It is not the unexpected
outcome of the theory formalism itself. Different hypothesis bring to
different results.

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There is no other choice to determine them than having some measure to
compare the model’s outcomes with. Usually while Q(t) is measured, ET is
not. But we can try the several sets of parameters that give a better
forecasting of Q(t), according to some goodness of fit algorithm (GoF).
Assuming we have implemented the above equations in some OMS
components, JGrass-NewAGE can do it.
To really solve it, we have to determine its parameters, a, b, , if we neglect
other parameters that can be hidden in the determination of net radiation or
other energy fluxes.
Bancheri and Rigon
“Back to see the stars”

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We can make the Linear Reservoir approximation
Linear reservoir approximation
Which means to assume:
Then, the water budget becomes
This is a linear ordinary differential equation that can be integrated
by variable separation as follows. First we introduce the
multiplicative exponential below
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Then we integrated over the whole domain
The first member is integrable by definition
So we obtain the final result:
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!34
The set of equation then become:
and the following do not vary from the general case
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Bancheri and Rigon
Summary

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Please notice
that when radiation oscillates, even if dumped, also S(t) oscillates
There is at least a reason for which the previous model is not so realistic:
measures shows that ET oscillates with radiation, Q(t) not so much regularly.
So oscillations in ET should not cause oscillations in Q in a realistic model
We try to explanations:
• phases mixing
• decoupling of vadose zone and water table
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Phase mixing
The model above represent the hillslope like the
Figure aside. As there was a single plant
evapotranspirating, and a single control volume.
No space is present.
Multiple reservoirs
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Phase mixing
Instead space exists, an
the new figure is much
more representative of a
hillslope where many
trees and reservoirs are
present.
Multiple reservoirs
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Here it is the whole set of equations for two “reservoirs” -
1/2
Multiple reservoirs
For the first reservoir, equations remain the same as in the single reservoir
case. (We use here the linear reservoir approximation for simplivcity)
..…
For the second reservoir, the water budget is:
This equation is not fundamentally different from the one of
the single reservoir, except for the added input A S(t)
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Multiple reservoirs
Therefore, the solution for the budget is
Without solving it, we can observe than when S(t) decreases radiation
increases, so the overall S2(t) is less oscillating than S(t). Any subsequent
Si(t) should be less oscillating, and an approximation with just a few of
“reservoirs” should be fine.
Here it is the whole set of equations for two “reservoirs” -
2/2
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The other equations remain the same. You must
include the subscript “2”, because they are specific
for the second reservoir, and add AS(t) where
needed, i.e.
Multiple reservoirs
or
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The set of equation then become:
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Multiple reservoirs with the linear hypothesis

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Being able to estimate the probabilities in each “reservoir” does not
immediately solve the problem to determine the overall probability. I.e.,
which is
knowing
and ?
Under the hypothesis of statistical independence of the variables in the
two reservoirs, it is:
Which is, is the convolution of the probabilities p1 and p2
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This mechanism has been substantially implemented in Botter et al. 2010. The
idea is that in the vadose zone water does not move laterally (see also, Lanni et
al., 2012, 2013) but only percolate down; viceversa, water move laterally just in
the water table, as in the schematic figure below.
Vadose zone/Water Table decoupling
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In its simplest interpretation the upper compartment is no different from the
one treated previously, if we exchange Je with Q.
The bottom compartment is even simpler, being without ET. Therefore
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Therefore (according to our notation), for the upper reservoir
For the lower reservoir
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If we do not make further assumption
i.e eq. (34) in Botter et al., 2010
and consequently all the other distribution functions, according to the
schemes of slides XX
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The paper by Botter et al., 2010, in reality make a more complicate
assumption, with the scope of accommodating long travel time. It introduce,
in fact the concept of waiting time, or a time expected by a water drop in a
reservoir to be mobilised. So, when injected, the water waits a random time
Tm, and the is mobilised and have a travel time TT, such that
Accordingly, the distribution function of TE is the convolution of the
distributions of the other two random variables.
But let’s forget it, at the moment.
Bancheri and Rigon

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There were made very many simplifications. There are at least two that needs
to be further discussed:
Too simplistic ?
Bancheri and Rigon
Would ot be usually accepted as a reasonable approximation. We could try
instead:
which could let the system quite undetermined. In this second case, we could,
however, try to use a width function approach as first approximation

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Too simplistic ?
Bancheri and Rigon
Not sure that it works, but we could put:
where W is a normalised width function depending on water velocities in
channels, hillslopes, and a diffusion coefficient. This would imply an iteration
over the parameters for calibration, and an iteration over the distribution
function for consistency, with
But, I confess, I have to sediment further this idea.

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Too simplistic ?
Bancheri and Rigon
There are a lot of intermediate assumptions that could be made.
One, for instance, is assuming a Nash/Gamma IUH as initial probability.
where K and n are two appropriate parameters.
I am pretty sure that it would work quite well, indeed.

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Too simplistic II ?
Bancheri and Rigon
There are other aspects that can be improved, especially for ET.
For this specific aspect, please refer to:
h t t p : / / a b o u t h y d r o l o g y . b l o g s p o t . i t / 2 0 1 4 / 1 2 /
evapotranspiration-parameters-in-coarse.html
The presentation you can find there discusses the extension of Priestley-
Taylor formula to catchments

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Too simplistic III ?
Bancheri and Rigon
Sic stantibus rebus
Another option is to use “tout court” TOPKAPI equations (plus PT) as the
basic ones, and use them as the basis for the full treatment of discharges and
schematisation of the catchment.
They would offer the advantage of a top-down derivation from mass
conservation that could be interesting to test.

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What is next
Bancheri and Rigon
Implementation
Next, before to dig further in the theory, including mixing, and
transport, and other stuff, or the latter models, like TOPKAPI, it
would necessary to devise how to implement all this theory in a way
consistent with the JGrass-NewAGE system.

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Find this presentation at
http://abouthydrology.blogspot.com
Ulrici,2000?
Other material at
Thank you audience !
Bancheri and Rigon

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