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The computation of the reference crop evapotranspiration (ETo) using the FAO Penman-Monteith equation (PM-ETo) requires data on maximum and minimum air temperatures (T-max, T-min), vapour pressure deficit (VPD), solar radiation (R-s) and wind speed at 2 m height (u(2)). However, those data are often not available, or data sets may be incomplete or have questionable quality. Various procedures were proposed in FAO56 to overcome these limitations and an abundant literature has been and is being produced relative to alternative computational methods. Studies applied to a variety of climates, from hyper-arid to humid, have demonstrated that improved methods to compute PM-ETo from temperature only (PMT approach) have appropriate accuracy. These methods refer to estimating: (i) the dew point temperature (T-dew) from T-min or, in case of humid climates, from the mean temperature, T-mean; (ii) R-s from the temperature difference (TD = T-max-T-min); and (iii) u(2) using default global or regional values. Greater difficulties refer to the need for locally calibrating the radiation adjustment coefficient (k(Rs)) used with the R-s equation. Therefore, considering that calibrated k(Rs) values were made available by past studies for a large number of locations and diverse climates, the current study developed and tested simple computational approaches relating locally calibrated k(Rs) with various observed weather variables - TD, relative humidity (RH) and average u(2). The equations were developed using CLIMWAT monthly full-data relative to all the Mediterranean countries. The equations refer to all available data, or to data grouped as hyper-arid and arid, semi-arid, dry and moist sub-humid, and humid climates. To test those k(Rs) equations, ETo computed from temperature and using the predicted k(Rs) values were compared with ETo computed with full data sets of the same Mediterranean locations and of Iran, Inner Mongolia, Portugal and Bolivia. RMSE average values result then small, ranging from 0.34 to 0.54 mm day(-1), therefore not very far from values obtained when a trial and error procedure was used for all the same locations, from 0.27 to 0.46 mm day(-1). These indicators allow to propose the use of k(Rs) obtained from the predictive equations instead of locally calibrated k(Rs) values, which greatly eases computations and may largely favour the use of the PMT approach.