# Reference and potential evaporation

Despite the good physical underlying theory of the Penman–Monteith equation (Allen et al. 1998) for calculating the reference evapotranspiration (ET), its major limitation is the high data demand for energy-based methods. This brought Hargreaves and Samani (1985) to derive the modified Hargreaves equation that is based on temperature only. For this reason, this equation has also been implemented in the SPHY model, according to:

*Equation 1*

$$
ET\_r = 0.0023 \* 0.408 \* Ra (T\_{avg}+17.8) \* TD^{0.5}
$$

with $$Ra \space(MJm^{-2}d^{-1})$$ the extraterrestrial radiation, $$T\_{avg} \space(\degree C)$$ the average daily air temperature, and  $$TD (\degree C)$$the daily temperature range, defined as the difference between the daily maximum and minimum air temperature. The constant 0.408 is required to convert the units to mm, and Ra can be obtained from tables (Allen et al. 1998) or equations using the day of the year and the latitude of the area of interest.

According to Allen et al. (1998), is the evapotranspiration rate from a reference surface with access to sufficient water to allow evapotranspiration at the potential rate. The reference surface is a hypothetical grass reference crop with specific characteristics. The potential evapotranspiration has no limitations on crop growth or evapotranspiration from soil water and salinity stress, crop density, pests and diseases, weed infestation or low fertility. Allen et al. (1998) determined by the crop coefficient approach, where the effects of various weather conditions are incorporated into and the crop characteristics in the crop coefficient (Kc), using:

*Equation 2*

$$
ET\_{p,t}=ET\_{r,t}\*K\_c
$$

with $$ET\_p (mm)$$ the potential evapotranspiration on day $$t$$, $$ET\_{r,t} (mm)$$  the reference evapotranspiration on day $$t$$, and Kc (–) the crop coefficient. The effects of both crop transpiration and soil evaporation are integrated into the Kc.

If the dynamic vegetation module in SPHY is not used, then the user can opt (i) to use a single constant Kc throughout the entire simulation period or (ii) to use a pre-defined time series of crop coefficients as model input. Plausible values for Kc can be obtained from the literature (Allen et al. 1998; FAO 2013). However, vegetation is generally very dynamic throughout the year. It is therefore more realistic to use a pre-defined time series of crop coefficients or to use the dynamic vegetation module, instead of a single constant Kc. This can be adjusted according to the user’s preferences.

Kc can be estimated using remotely sensed data (Rafn et al., 2008; Contreras et al., 2014). In the dynamic vegetation module, Kc is scaled throughout the year using NDVI and the maximum and minimum values for Kc, which are crop specific. These values for Kc can easily be obtained from Allen et al. (1998). Then Kc is calculated using

*Equation 3*

$$
Kc = Kc\_{min} +(Kc\_{max}-Kc\_{min})\*\frac{(NDVI-NDVI\_{min})}{(NDVI\_{max}-NDVI{min})}
$$

with $$NDVI\_{max}$$(-) and $$NDVI\_{min}$$(-) the maximum and minimum values for NDVI (vegetation type dependent). This approach shows the flexibility of SPHY in using remote sensing data (e.g., NDVI) as input to improve model accuracy.
