water budget by residence time

Hillslope hydrological budget including

ET by residence times

Marialaura Bancheri & Riccardo RigonTrento, January 2015

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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)

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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)

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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.

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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 theprobability 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

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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

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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

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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

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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.

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“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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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

hil lslope 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.

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There were made very many simplifications. There are at least two that needs

to be further discussed:

Too simplistic ?

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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 ?

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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 ?

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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 ?

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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 ?

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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

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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

Ulr

ici, 2

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Other material at

Thank you audience !

Bancheri and Rigon

http://www.slideshare.net/posterVienna

water budget by residence time

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