ET in coarse grained modelling

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Riccardo RigonTrento, January 2015

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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].

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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

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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

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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

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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

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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

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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

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If we assume the Rodriguez-Iturbe hypothesis [4]

Rigon, R.

Introducing soil moisture dependance

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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

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Therefore

should be an acceptable approximation

Rigon, R.

Introducing soil moisture dependance

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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

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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

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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

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Find this presentation at

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Other material at

Thank you audience !

Rigon, R.

http://abouthydrology.blogspot.it/search/label/Evapotranspiration