Global-scale calibration of three empirical PET formulas: evaluation of simple meteorological variables as proxies for solar radiation

Potential Evapotranspiration (PET) is a relevant input for hydrological modelling and assessing water balance components. PET estimation usually requires meteorological inputs to simulate key radiative and aerodynamic processes.

The reference Penman-Monteith method estimates PET as a function of net solar radiation, pressure, humidity, wind speed, maximum and minimum air temperature. Except for temperature, reliable observations of the other variables are rarely available in most regions of the world; consequently, simpler empirical formulas have been developed to reproduce PET using only a subset of variables. Net solar radiation is probably the most critical input because it plays a major role in evaporative processes, but is very challenging to measure and is often unavailable. This study focuses on three widely used empirical expressions that, in addition to temperature, apply different approaches to approximate the solar radiation at the surface:

• Oudin formula¹ uses, in addition to temperature, the extra-terrestrial radiation value, which depends only on latitude and day of the year.
• Hargreaves formula² improves the estimation of surface solar radiation using diurnal temperature variation to compute a sky clearness factor, simulating the fraction of extraterrestrial solar radiation absorbed by the atmosphere.
• a modified version of Hargreaves formula³ also includes precipitation in the sky clearness computation.

Here we test the robustness of these three formulas in estimating daily PET across diverse global regions. Our purpose is to assess whether increasing the formula complexity, estimating surface solar radiation including temperature variation and then also precipitation, can improve daily PET estimation.

Leveraging worldwide meteorological data of thousands of basins from the large-sample hydrological dataset Caravan⁴, we initially compared the daily PET estimates produced by the above cited formulas with the reference values based on the Penman-Monteith method. Significant differences were observed among climatic regions: the Hargreaves formula generally performed best, but all methods exhibit biases in particular contexts.

Secondly, we conducted a series of calibration experiments, modifying the original parameterization of the formulas by optimizing their fit to the reference Penman-Monteith in the study basins. We optimized the empirical coefficients for all the basins, both globally and within homogeneous hydro-climatic regions, and analyzed the spatial pattern of performance. Then, we divided the basins into training and validation sets using a distance-based criterion within each region. We performed new calibrations to assess whether the fitted formulas remained valid across different catchments.

 

References:

¹Oudin, L. et al. (2005). Which potential evapotranspiration input for a lumped rainfall–runoff model? Journal of Hydrology, 303(1–4), 290–306. https://doi.org/10.1016/j.jhydrol.2004.08.026

²Hargreaves, G.H. & Samani Z.A. (1985). Reference Crop Evapotranspiration from Temperature. Applied Engineering in Agriculture, 1(2), 96–99. https://doi.org/10.13031/2013.26773

³Droogers, P., & Allen, R. G. (2002). Estimating reference evapotranspiration under inaccurate data conditions. Irrigation and Drainage Systems, 16(1), 33–45. https://doi.org/10.1023/A:1015508322413

⁴Kratzert, F. et al. (2023). Caravan—A global community dataset for large-sample hydrology. Scientific Data, 10(1), 61. https://doi.org/10.1038/s41597-023-01975-w