This document discusses the Priestley-Taylor method for estimating evapotranspiration (ET) in coarse-grained modeling. It begins by explaining the Priestley-Taylor equation and some basic notation used. It then discusses how the equation, which estimates instantaneous ET, can be adapted to provide hourly or daily ET estimates by integrating it over time. It also explains how the equation can be spatially averaged to provide estimates over larger areas. The document explores incorporating soil moisture dependence into the equation and how spatial averaging is more complex when factors are correlated in rugged terrain. It concludes by providing references for further information.

1. ET in coarse grained modelling
reforestatiom.me
Riccardo Rigon
Trento, January 2015

2. !2
Priestley-Taylor ET
Rigon, R.
The problem
It can be derived from Penman-Monteith equation by cutting the part
dependent on the so called atmospheric demand [1,2,3].

3. !3
A little of notation
Brackets indicate indicate spatial average:
the index A indicates the area of the domain of integration. In case it is
generic or obvious, it can be omitted.
Overline indicates temporal average, the time interval is usually omitted, and
must be deduced by the context
Rigon, R.
Notation

4. !4
The way it is derived implies it is, in first place, an approximated formula
for instantaneous Evapotranspiration. Therefore, to obtain it for hourly or
daily rates, we have to integrate it in time.
The nature of alpha as a parameter that is calibrated or determined “a
posteriori” make it reasonable to assume that it can be considered a
constant. The fractional term is mildly dependent on temperature, and
therefore, we can assume that it can be approximated by an appropriate
value estimated for a representative temperature (and the bias correction
included in the determination of alpha).
* I am sorry of using with two different meanings
Rigon, R.
Time averaging

5. !5
Therefore:
Or:
and the form of the formula remains approximately invariant, if we
substitute the instantaneous quantity with the hourly ones. Let’s in the
following drop the subscript and assume that the quantities are replaced
by suitably averaged quantity.
Rigon, R.
Time averaging

6. !6
If we extend our analysis to a spatial domain, we have to integrate all in space:
where HRU stands for Hydrologic Response Unit, meaning a portion of a
basin where hydrological quantities can (or must) be considered
homogeneous. For the same arguments exposed for the temporal average:
which says that the operation of spatial average (coarse graining) does not
change the form of PT formula, if we substitute in the formula, suitably
spatially averaged quantities.
Rigon, R.
Space averaging

7. !7
Algebra is trivial but indicates that
t h e f o r m u l a f o r m r e m a i n s
approximately valid if we consider
a v e r a g e s a n d n o t p o i n t
measurements !
In case, one should test how much a single point measurement is
representative of the average.
Rigon, R.
Space averaging

8. !8
Everything change
if we consider a dependance from soil moisture content. In this
case:
* However, for the concept of potential evapotranspiration, see:
http://abouthydrology.blogspot.it/2015/01/potential-evapotranspiration.html
Rigon, R.
Introducing soil moisture dependance

9. !9
If we assume the Rodriguez-Iturbe hypothesis [4]
Rigon, R.
Introducing soil moisture dependance

10. !10
Thus averaging
If we do not consider raining intervals (where we can assume to have
computed ET in the net rainfall budget, i.e. we assume that measured
rainfall is the net budget of rainfall minus ET), and if the time
interval of integration is no more than daily, we can think that f ~
const, and therefore:
Rigon, R.
Introducing soil moisture dependance

11. !11
Therefore
should be an acceptable approximation
Rigon, R.
Introducing soil moisture dependance

12. !12
Averaging over space is a little bit more tricky
From:
Now, by definition, if f and energy terms are uncorrelated, or, for instance,
either soil moisture or radiation is uniform (i.e. spatially constant):
Rigon, R.
Introducing soil moisture dependance

13. !13
In rugged terrain
Simulations with complex models [5,6] suggest that this is not usually the case.
Therefore, the average product
should be statistically modelled, after, for instance, simulations with
process-based distributed models.
Rigon, R.
Introducing soil moisture dependance

14. !14
References
[1] Priestley, C.H.B. and Taylor R. J., On the assessment of surface heat flux and evaporation using large scale
parameters, Monthly Weather Review, Vol. 100, No 2, 81-92,1972
[2] - Evapotranspiration - Slides on http://www.slideshare.net/SlidesAboutHydrology/15-evapotranspiration
[3] - Solar Radiation - Slides on http://www.slideshare.net/SlidesAboutHydrology/13-solar-radiation
[4] - Rodriguez-Iturbe, I., Porporato, A., Ridolfi, L., Isham, V., & Cox, D. (1999). Probabilistic modelling of water
balance at a point: the role of climate, soil and vegetation. Prooceedings of the Royal Society, 455, 3879–3805
[5] - Rigon R., Bertoldi G e T. M. Over, GEOtop: A distributed hydrological model with coupled water and energy
budgets, Vol. 7, No. 3, pages 371-388
[6] Bertoldi G. R. Rigon e T. M. Over, Impact of watershed geomorphic characteristics on the energy and water
budgets, Vol. 7, No. 3, pages 389-394, 2006
Rigon, R.
To deepen and fix your knowledge

15. !15
Find this presentation at
Ulrici,2000?
Other material at
Thank you audience !
Rigon, R.
http://abouthydrology.blogspot.it/search/label/Evapotranspiration