The Penman-Monteith equation (P-M)

The CR10x team of ^{Campbell (1995)} has programmed the algorithm to calculate the ETo with the Penman Monteith (P-M) model presented by ^{Allen et al. (2006)}.

For manual calculation of ETo with the Penman-Monteith method for conditions similar to greenhouses, in FAO Bulletin 56 (^{Allen et al., 2006}) it is recommended to use 0.5 m s^-1 as the minimum value for speed of wind, which improves ETo estimates.

Models selected to calculate ETo

Several models based on temperature and radiation are included (Table 1), which were proposed for conditions other than those of a greenhouse, but that do not include wind speed, to estimate ETo on a daily basis.

ETo is the reference evapotranspiration in (mm d^-1), Tmed the average temperature (°C), Tmax is the maximum temperature (°C), Tmin the minimum temperature (°C), TD is the daily thermal oscillation (°C), RH the relative humidity in (%), is the saturation vapor pressure (kPa) e_a the real vapor pressure (kPa), ∆ is the slope of the saturation pressure curve γ is the psychometric constant, Ra is the solar radiation extraterrestrial in MJ m^-2 d^-1, Rs is the solar radiation in MJ m^-2 d^-1, Rn is the net radiation in MJ m^-2 d^-1, G is heat flow in the ground ENT#91;MJ m^-2 d^-1ENT#93; and T1 is an empirical coefficient.

Two series of daily meteorological data of (T, RH, Rs) and calculated (ETo P-M) were determined: one from September 2 to December 12, 2017 and another from March 4 to April 23, 2018. With the data of the first one, the parameters (Table 1) of the models were modified and with those of the second the models were evaluated with the parameters defined in this work. The data used for parameter estimation and evaluation are presented in Figure 3.

Figure 3 Data used for the estimation of parameters from (a) to (d) with n= 102 and those used for the evaluation of (e) to (h) with n= 51. 

With the values obtained from ETo with the models of Table 1 and calculated with the first series of data and those of P-M that were calculated by the automatic station, dispersion diagrams were constructed between the ETo data of the models and P-M, and the statistics were calculated: root of the mean square of the error (RSE) and coefficient of determination R².

Parameter estimation

To estimate the parameters of the ETo models (Table 1), the meteorological data of the first series and those calculated by the Campbell CR10x station for this task were used. An R language program was carried out for this purpose, in which one model was worked at a time, with the following procedure.

The parameters of one of the proposed models (ETox) were identified for modification. For each parameter a range was defined in which the lower limit was the nominal value of the parameter minus 10 and the upper limit the nominal value plus 10. Once defined the ranges were divided into 80 000 parts with which a vector was constructed, which defined parameter proposals with increments of 0.00025.

20 000 cycles were programmed for each model, where the program assigned a value to each parameter randomly taken from the corresponding vector. The values of the parameters selected in the model were replaced and evaluated with the 2017 meteorological data, obtaining 102 ETox data. With the ETox data and ETo data calculated with the Penman-Monteith method (P-M), the correlation coefficient (R) between them was determined.

The numerical value of R is compared with the value of Ro (which at the beginning of the program was assigned the numerical value of zero), if R> Ro, then Ro= R and the parameter values are stored in defined programming variables For this purpose, these values are updated only if in the cycle an R greater than the one stored in Ro is obtained. The cycle is repeated 20 000 times and at the end of this computational process the correlation coefficient value and the corresponding randomly proposed parameter values are obtained.

In the model, the parameters stored at the end of the 20 000 cycles are replaced, which correspond to those that generated the highest value of R and with the meteorological data the ETox were generated. With the data of ETo P-M (dependent variable) and those calculated with the model and the stored parameters (ETox independent variable) a linear regression was performed. Finally, the numerical values of the parameters were obtained by being affected by the regression coefficient and with the sum of the independent term.

The models obtained to be used in the estimation of the ETo in greenhouse, were validated with data estimated with Penman-Monteith with the series (second) of 2018 (51 data) the coefficient of determination (R²) and the root of the square medium of error (RSE) were determined (^{Draper and Smith, 2014}).

ETo estimation with the models with their modified parameters

The root values of the mean square of the error (RSE) of the ETo models with their modified parameters based on Baier-Robertson, Romanenko and Hargreaves temperature were: 0.21, 0.23, 0.20 mm d^-1 and the R² of 0.501, 0.599 and 0.626 respectively (Table 2).

Of the models for estimating ETo based on radiation, the RSEs were: 0.19 and 0.09 mm d^-1 for Priestley-Tylor and Abtew respectively and 0.08 mm d^-1 for the others. The Irmak method presented the largest adjustment to ETo values calculated with Penman-Monteith. In general, radiation-based methods were better than with respect to temperature (Table 2).

With the values calculated with the modified models of Turc, Abtew, Jensen-Haise, Caprio, Irmak, Stephens and Makkink and those estimated by the P-M method, a good fit was obtained, in all (R²) they were greater than 0.918. According to the statistics in Table 2, the ETo data obtained with the radiation-based models have a better level of adjustment with those of ETo P-M than those based on temperature. The coefficients of determination (R²), being all smaller than the unit, indicate that the data have dispersion with respect to the 1:1 line.

From the statistics (Figure 4 and Table 2), the behavior of the models to predict the values of ETo P-M inside a greenhouse is observed, with its original parameters (Figure 4) and the modified ones (Table 2). The models based on temperature went from R² values between 0.463 and 0.523 to values between 0.501 and 0.626 and their RSE from 5.14 to 15.26 to values between 0.2 and 0.23. In all these models there was an improvement of the model with the parameters modified in the estimation of the ETo.

Those based on radiation also showed an improvement to predict the values of ETo P-M, went from RSE with values between 0.31 to 0.72 to values between 0.08 to 0.19 and R² were improved from values between 0.198 to 0.939 to values from 0.665 to 0.946.

Performance evaluation of models based on temperature

In the temporal march of the values of the ETo obtained with the models based on temperature and those obtained with the expression of Penman-Monteith (P-M), it is observed that the three models overestimated the P-M values during the 51 days used to evaluate the models (Figure 5). In Table 2 the ones that show the best fit are the estimates made with the Baier-Robertson and Hargreaves models, but the data of the two models overestimate the P-M values (Figures 6a and 6c). The Romanenko Model in most of its data overestimates P-M and only in four underestimated data (Figure 6b).

Figure 5 Temporary adjustment between the ETo measurements of the automatic station (P-M) and the modified temperature-based models to predict the daily ETo inside a greenhouse of: Baier-Robertson, Romanenko and Hargreaves, n= 51. 

Figure 6 Scatter diagrams between the values predicted by the modified models and the measurements inside the greenhouse with the automatic station (P-M): a) between the Baier-Robertson method and P-M; b) between Romanenko and P-M; and c) between Hargreaves and P-M, n= 51. 

The dispersion diagrams between the ETo estimates of the models based on temperature and the estimation of ETo P-M are shown in Figure 6a, 6b and 6c. The one that presented the best trend was the one that corresponds to the data between the method of P-M and Baier-Roberston’s and the diagram has the most attached points to the 1:1 line.

The statistics of the evaluation of the models with 51-day data, the Baier-Robertson and Hargreaves models obtained coefficients of determination (R²) of 0.461 and 0.411 and their RSE of 0.31 and 0.39 mm d^-1 respectively. The statistics of the Romanenko method are (R²) of 0.3 and RSE of 0.34 mm d^-1 was the best method based on temperature.

Performance evaluation of radiation-based models

The temporal relationship between ETo of the radiation-based models and the Penman-Monteith (P-M) method, in the evaluation period, the one that showed the least adjustment was that of Turc (green line) since in all its data it underestimated the ETo P-M value (red line). The Abtew method data adhered to those of P-M and in general terms the model has adjustment to P-M values.

The models of Jensen-Haise, Caprio, Irmak, Stephens, Makkink, showed temporary marches of their data that underestimate those of P-M. In its temporary march, the Priestley-Taylor method showed adherence to the P-M model data, in values of ETo less than 1.9 mm d^-1 overestimated and greater than 2.1 mm d^-1 underestimate (Figure 7).

Figure 7 Temporary adjustment between the ETo measurements of the automatic station (P-M) and the modified models based on radiation to predict the daily ETo inside a greenhouse, n= 51. 

In the dispersion diagrams between the estimates of the modified models based on radiation and Penman-Monteith, it can be seen that they generally showed a trend and six of the eight models underestimated the P-M of ETo values (Figure 8). The models that performed better in the evaluation stage were: Abtew, Priestley-Tylor and Makkink (Figures 8b , 8g and 8h).

Figure 8 Scatter diagrams between the values predicted by the modified models and the measurements inside the greenhouse with the automatic station (P-M): a) Turc and P-M; b) Abtew and P-M; c) Jensen-Haise and P-M; d) Caprio and P-M; e) Irmark and P-M; f) Stephens and P-M; g) Pristley-Taylor and P-M; and h) Makkink and P-M, n= 51.  

The Turc model with an R² of 0.943 and an RSE of 0.21 mm d^-1 was the one that was least adjusted, the rest of the models obtained R² greater than 0.787 and RSE less than 0.16 mm d^-1 (Table 3).

When comparing the statistics of the modified models, it is appreciated that the radiation-based models are more closely attached to the values calculated with Penman-Monteith. The Abtew and Priestley-Taylor models with R² of 0.947 and 0.909 respectively and RSE of 0.06 for both were the ones that best predicted the ETo of P-M values in the greenhouse.