Let me have a little rehearsal on

Evapotranspiration (see

my posts for "random" thinking on the topic) estimation with simplified model, like

Penman-Monteith or Priestley-Taylor. Literature reports the concept of "

Potential Evapotranspiration", ETp, but, as

Brutsaert says, the concept is a slippery one. From the conceptual point of view, from what I understand, it is the ET when no resistance is acting, except what is implied by limitations by atmospheric forcings. One can try to see it in the field or deduce from formulas. In Penman-Monteith (PM) approach:

this has a clear interpretation, which is, the ET you obtain by putting any resistance, rg (either representing soil or vegetation, indeed), to zero.

Unfortunately this is not the same in Priestley-Taylor (PT), where the resistance terms simply are not present:

where everything is lumped in the alpha coefficient. What people did in this case, I think, to assess alpha in ETp (say alphap) was to measure ET in conditions supposed to be those in which ETp realizes and estimate the alphas which are reported in literature (IMHO this is the only way it can be accomplished but I confess I did not went into the detail of that literature, see for instance the overview by

Cristea et al, 2012). Once your get alphap you can estimate ETp but still you have to introduce a further reduction to get the real ETa. The method popularized by the ecohydrology literature (e.g. read

Amilcare Porporato here) is to introduce a linear decrease of ETp with water storage in the root zone "reservoir".

Both the passage, the determinatio of ETp in the framework of PT and the linear reduction with storage have, in my view, strong drawbacks from the quantitative point of view.

One can get the alphap, but literature show a huge variability. So literature is quite useless to obtain quantitative results, ith a decent ncertainty.

The (linear) decrease of ET with soil moisture requires the determination of at least one additional coefficient. In fact, it is well known that

ET has two stage: stage one, when ET is "at the potential rate", independently from the water content up to a critical soil moisture, well below saturation, when ET is depressed, not by increasing suction (the so called

Kelvin effect, which is a second order effect) but by the fact that pores at the soil or leaf surface to which water is supplied are more and more far apart (see recent literature by

Dani Or and co-workers). This critical soil moisture, at which the second stage ET starts is a further coefficient, and its identification with saturation implies a clear underestimation of ET. It is usually given for granted by my

friends ecohydrologists and my master IRI's literature that it can be determined. But I do not have to remind to you all how much elusive it is the definition of the "root zone soil moisture" just to cite a practical aspect of it.

Even if field-fellow-scientists claim to have measured it, I know that who tried in lab of few square meters really struggled to close the water budget budget under very controlled conditions (let's say: personal communications). In nature, as my hero

Pete Eagleson teaches, interaction among plants distribution, atmosphere, and rugged terrain makes any of the above coefficient heterogeneous, and the trials to find a rational to all of it, kind of frustrating to my eyes.

Said all of this, let's go back to PT, and its use conjointly with a rainfall-runoff model. Simplifying a little the PT formula reads:

Where alpha is the PT coefficient, f(theta) a function os soil moisture content, F(T) = Delta/ (Delta + gamma) has a weak dependence on temperature T, Rn the net radiation. (The latter,

in our systems is estimated by properly accounting the topographic variability, while in most of applications is given by a rough estimate ... ). Please observe that there is no way to get rid of the alpha coefficient. So even if we would implement a way to get into account of f(theta), for instance introducing a linear decrease with soil water content, we still have to estimate alpha.

The more direct way we have for doing it, is, in my and

Wuletawu Abera opinion, using the water budget, but I will come back to this in another post.

A bare-bone formulation of f(theta), is it being either 1 if there is enough water in the soil reservoir (in whatever way we account for its content in our modelling) or cut it to the maximum amount of water available when there is not enough. This is, for instance the strategy used by the Hymod model, when working at daily time scale. Actually, to be sincere, this limitation is reasonable in the original formulation of Hymod (e.g

Formetta et al., 2011), working at the daily scale. However, I am sure is not limiting anything at hourly scale in which, for instance our Hymod implementation can work. This to say, that, in our formulation f(theta) should be essentially always one.

In principle, if, for a reason or another, we leave f(theta) =1, we can evaporate also if there is no water in soil. But what we can do if we have at most one reservoir for hillslope, or in many cases where Hymod is used, just one reservoir for catchment to account for soil misture ? This simplification itself hide the complexity of soil moisture distribution at sites scales (and we know that convex-divergent sites are potentially less wet than convergent-concave, and possibly very close to be completely dry in summer). To be realistic we should invent some further function that says which is the distribution of wetness in the catchment, given the total amount of water, and we need to invent to map a distribution into a mean reduction of ET at catchment scale .... but this is not anymore a bare-bone solution.

The often used linear dependence of f( ) on theta, is not "scale invariant". In fact, assuming it exists for any site, when coarse graining the physics at hillslope (HRUs) scale, it must coming out from integration of sites evapotranspiration over the whole space.

If we can assume that the alpha is a space-time constant (and therefore we can take it out of the integral, certainly this is not for both soil moisture content and radiation, which are known to be highly variable in space. Making the linear assumption valid again, means therefore to assume that

for some representative (space) "mean" (with bars) values of soil moisture and radiation content. This statement is certainly true, if we assume space continuity of both the quantities (given for granted for radiation, but less granted for soil moisture), but to be empirically determined case by case for any practical application (this can be explored and approximated, under general condition, with the use of suitable computational modules ... that even

Jgrass-NewAGE does not posses).

Finally and besides, establishing what is wet an what is dry implies to know how much the initial soil moisture is. So, even if we cut ET with soil moisture we cannot do it properly, if we do not do a guess about soil moisture initial condition.

So what can we do to estimate alpha of PT and having a guess of the initial soil moisture ?

The answer in one of the next posts.

P.S. - A not up-to-date series of slides on ET can be found

on Slideshare here.