I computed the daily reference evapotranspiration $ET_0$ using the Crop evapotranspiration - Guidelines for computing crop water requirements - FAO Irrigation and drainage paper 56. The outputs seems realistic regarding the dynamic and the annual total but the time series contains negative values (about 1% of the time series).

The computation involves many equations for the sub-terms but the general formula is the following:

$$ ET_0 = \frac{0.408 \Delta (R_n - G) + \gamma \frac{900}{T + 273}U_2(e_s - e_a)}{\Delta + \gamma(1+0.34 U_2)} $$


  • $ET_0$ reference evapotranspiration [mm day-1],
  • $R_n$ net radiation at the crop surface [MJ m-2 day-1],
  • $G$ soil heat flux density [MJ m-2 day-1] (=0 in this case),
  • $T$ mean daily air temperature at 2 m height [°C],
  • $U_2$ wind speed at 2 m height [m s-1],
  • $e_s$ saturation vapour pressure [kPa],
  • $e_a$ actual vapour pressure [kPa],
  • $e_s - e_a$ saturation vapour pressure deficit [kPa],
  • $\Delta$ slope vapour pressure curve [kPa °C-1],
  • $\gamma$ psychrometric constant [kPa °C-1].

It appears that the negative values are due to a negative values of the net radiation $R_n$. A negative value of $R_n$ makes sense but what about negative $ET_0$ ? How should I deal with these as I intend to use $ET_0$ as input for hydrological modelling ?

  • $\begingroup$ Wouldn't you expect negative values if water was condensing on the ground/vegetation? Perhaps that's why the have soil heat flux? $\endgroup$ – DougS Dec 3 '17 at 17:19

Short answer, especially if it is about 1% of the time series and these values are just slightly negative, this likely indicates that the ET needs to be recalibrated slightly, and for practical purposes you can set the ET on those days to 0 mm/d. The ET calculations typically calculate potential ET, meaning that if there is no water available no ET will occur, and a similar concept applies to negative ET values (i.e. no demand for water -> no ET).

A couple of notes
- the reference $ET_0$ calculations were meant for ten-day or monthly time steps, not daily, which may be a reason for these results (see top of Ch 2)
- the equation seems to be for a specific type of reference grass, and they suggest adjusting the $ET_0$ based on crop type using a ratio; another potential adjustment to make

Overall if the negative values are infrequent and close to zero then likely setting to zero is not an issue, if the values are much less than zero than check the calculations and try the suggestions above to recalibrate the calculation.

  • $\begingroup$ Thank you for your answer. Indeed, I think setting negative ET to 0 make sense for practical purposes. I also think I should inspect further the calibration parameters of the Angstrom formula FAO ch3 eq.35 when estimating the shortwave radiation. I am not sure the pyranometer is well calibrated. $\endgroup$ – Delforge Aug 9 '17 at 13:47

We are calculating $ET_0$ on a 10 minute basis and ran into the same problem on http://lapalma.HDmeteo.com.

I can confirm this problem with the FAO formula. We are clipping negative hourly values to 0. This occurs during nighttime when there is no wind and the wind term does not compensate for the radiation term.

It is true that during the night, with high humidity and absent wind the crops get wet due to dew. But dew should be collected by the pluviometer and not accounted for by negative $ET_0$.

The weather stations from Davis with integrated $ET_0$ module seem to use clipping of negative values, too, as they never report $ET_0 < 0$ even with hourly granularity.


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