## Sunday, September 8, 2013

### One of two topics about Evapotranspiration (and the Penman Monteith equation)

Evapotranspiration (e.g. Brutsaert, Evaporation into the atmosphere,1980, or my lectures - in Italian - in English) is the phenomenon that describes collectively evaporation from water surfaces, and plants. Hydrologists used to used to neglect its estimation when they were concentrated on rainfall-runoff, and many, even nowadays, tends to forget its existence.

Once considered a stationary energy budget, as Penman (1948) showed, its estimation is reduced to a formula, the Penman formula, subsequently modified by Monteith (1965) to account for resistances due to soil suction and plants reaction to water stress (e.g. Rodriguez-Iturbe and Porporato, 2004).  The formula is universally knows as Penman-Monteith formula. Nobody exactly knows what the resistances  means, or better, how to estimate them, since they depends on many factors.

However, being a formula, it does not require the solution of differential equations, and its evaluation depends only on the measure or the spatial extrapolation of the terms it contains.

Radiation has actually its own difficulties to be estimated, but a reasonable way to do it, was found and improved during the years and is available (e.g. Formetta et al., 2013) .  Wind velocity can be either simulate or interpolate from measurements. On resistance terms either we can do educated guesses, or, after having estimated the potential evapotranspiration (PET), at least a first approximation can be found in appropriate factor of reduction linear proportional to the water content both for the soil cases (where the water content can vary between the residual water content and water content at saturation)  and vegetation case (where the extremes are, for each species, the wilting point and a critical water content, something below the air entry level). I would not have idea , I admit,  how to estimate  air humidity, if not interpolating measurements, but the air moist at saturation, thanks to the Clausius-Clapeyron relation depends only on temperature.

For each of these quantities actually, in our ignorance we could give an estimate. If we further think to be able to assess the estimate on the measurements, we could think to be able to propagate di error to obtain (under the hypothesis of parameters distributed according to the Normal distribution) the "error" of our estimate.

Starting from the formula in the figure, an from the hypothesis of normality, it is quite easy to find the distribution of evapotranspiration, which is actually itself a normal distribution whose mean is equal to the deterministic expression of ET, given, precisely, by formula in Figure above.

So why don't try it ?

A more complex situation derives if we take into account space. In this case, any of the parameters in the game is a function of location, and the error of estimate too. However, using an appropriate linear model (for instance Kriging) is not overwhelming difficult to derive an estimate of both, and therefore coming out with a spatial estimate of ET and its error.

Space clearly bring into the game the spatial variability of ET, evaporation rather than evaporation from soil or transpiration from plants (grass, shrubs and trees, or crops at least).  So, if the error of the parameters can be estimated, for estimating evapotranspiration itself further guesses need to be done about the spatial variability of soil cover and use. This, obviously,  brings inside other sources of uncertainty. However, at least some playing around with the appropriate tools (i.e. Jgrass-NewAGE) could be made.

Obviously if stationarity of the energetic fluxes is not assumed, all it is another game (but you can use GEOtop, then).