# Chapter 6 Estimation of potential evapo-transpiration

Package meteoland allows calculating daily potential evapo-transpiration (PET) using Penman’s formulation or Penman-Monteith formulation. PET is automatically calculated after meteorological data have been interpolated (i.e. within functions interpolationpoints(), interpolationpixels() and interpolationgrid()) or bias-corrected (i.e. within function correctionpoint() or correctionpoints()), but PET values can also be calculated for a single point using functions penman() or penmanmonteith(). For other formulations of PET, the reader is referred to the R package Evapotranspiration.

## 6.1 Penman formulation

Penman (1948) proposed an equation to calculate daily potential evaporation that combined an energy equation based on net incoming radiation with an aerodynamic approach. The Penman or Penman combination equation is: $\begin{equation} E_{pot} = \frac{\Delta}{\Delta+\gamma}\cdot \frac{R_n}{\lambda}+\frac{\lambda}{\Delta + \lambda}\cdot E_a \end{equation}$ where $$PET$$ is the daily potential evaporation (in $$mm \cdot day^{-1}$$) from a saturated surface, $$R_{n}$$ is the daily radiation to the evaporating surface (in $$MJ\cdot m^{-2}\cdot day^{-1}$$), $$\Delta$$ is the slope of the vapour pressure curve ($$kPa\cdot ^\circ C^{-1}$$) at air temperature, $$\gamma$$ is the psychrometric constant ($$kPa\cdot ^\circ C^{-1}$$), and $$\lambda$$ is the latent heat of vaporization (in $$MJ\cdot kg^{-1}$$). $$E_a$$ (in $$mm \cdot day^{-1}$$) is a function of the average daily windspeed ($$u$$, in $$m\cdot s^{-1}$$), and vapour pressure deficit ($$D$$, in $$kPa$$): $\begin{equation} E_a = f(u) \cdot D = f(u) \cdot (v_a^*-v_a) \end{equation}$ where $$v_a^*$$ is the saturation vapour pressure ($$kPa$$) and $$v_a$$ the actual vapour pressure ($$kPa$$) and $$f(u)$$ is a function of wind speed, for which there are two alternatives : $\begin{eqnarray} f(u) &=& 1.313 + 1.381 \cdot u\\ f(u) &=& 2.626 + 1.381 \cdot u \end{eqnarray}$ If wind speed is not available, an alternative formulation for $$E_{pot}$$ is used as an approximation : $\begin{equation} PET \simeq 0.047\cdot R_s \cdot (T_a+9.5)^{0.5}-2.4\cdot (\frac{R_s}{R_{pot}})^2+0.09\cdot(T_a-20)\cdot(1-\frac{RH_{mean}}{100}) \end{equation}$ where $$R_s$$ is the incoming solar radiation (in $$MJ\cdot m^{-2}\cdot day^{-1}$$), $$T_a$$ is the mean daily temperature (in $$^\circ C$$), $$R_{pot}$$ is the potential (i.e. extraterrestrial) solar radiation (in $$MJ\cdot m^{-2}\cdot day^{-1}$$) and $$RH_{mean}$$ is the mean relative humidity (in percent).

## 6.2 Penman-Monteith formulation

The Penman-Monteith combination equation: $\begin{equation} E_{pot} = \frac{1}{\lambda} \cdot \frac{\Delta \cdot R_{n} + D \cdot (\rho \cdot C_p/r_a)}{\Delta + \gamma \cdot (1 + r_c/r_a)} \end{equation}$ where $$D$$ is the vapour pressure deficit (in kPa), $$\Delta$$ is the slope of the saturated vapor pressure (in $$Pa \cdot K^{-1}$$), $$\gamma$$ is the psychrometer constant (in $$kPa\cdot K^{-1}$$), $$\lambda$$ is the latent heat vaporization of water (in $$MJ\cdot kg^{-1}$$) and $$C_p$$ is the specific heat of air (in $$MJ\cdot kg^{-1}\cdot K^{-1}$$). $$r_c$$ is the canopy resistance (in $$s\cdot m^{-1}$$). For simplicity, aerodynamic resistance ($$r_a$$) is currently set to $$r_a = 208.0/u$$ where $$u$$ is the input wind speed.

### Bibliography

Penman, H. L. 1948. Natural evaporation from open water, bare soil and grass.” Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 193: 129–45.
———. 1956. Evaporation: An introductory survey.” Netherlands Journal of Agricultural Science 4: 9–29.
Valiantzas, John D. 2006. Simplified versions for the Penman evaporation equation using routine weather data.” Journal of Hydrology 331 (3-4): 690–702. https://doi.org/10.1016/j.jhydrol.2006.06.012.