Theoretical and operational equations

H. L. Penman, in 1948, was the first to obtain an equation that combines the energy required to sustain evaporation and an empirical description of the diffusion mechanism by which energy is removed from the evaporation surface as water vapor (^{Shuttleworth, 1993}). Penman's formula led to a new evaporation estimation criterion called the Combination Method. Several researchers modified the Penman formula to take into account the effects of the evolution of aerodynamic conditions on the growth of the crop, the former through resistance factors. The resistance of the surface (r _s ) on the water vapor flow in the stomata of the leaves and on the soil surface is distinguished from the aerodynamic resistance (r _a ) that occurs due to the friction of the air flow over the vegetable surface. Although the exchange processes in the vegetation layer are much more complex, the measurements and calculations of latent heat flow, λET, have shown a high correlation, at least for a uniform grass surface. With such modifications the theoretical Penman-Monteith formula was obtained (^{Allen et al., 1998}):

where λET is the speed of evapotranspiration in megajoule per m² per day (MJ/m²/d), Δ is the slope at one point of the saturation vapor curve versus the temperature in kilopascal per °C (kPa/°C), Rn is the net solar radiation in MJ/m²/d, G is the flow of heat from the ground in MJ/m²/d, ρa is the average density of air at constant pressure in kg/m³, c _p is specific heat of the air at constant pressure in MJ/kg/°C, (e _s - e) is the vapor pressure deficit of air in kPa, r _a is the aerodynamic resistance in s/m, γ is the psychrometric constant in kPa/°C, and r _s the surface resistance in s/m.

When considering a hypothetical vegetation surface that is 12 cm high, with a fixed surface resistance of 70 s/m, and an albedo of 0.23 in active growth that completely shades the ground and does not lack water, the following operational Penman-Monteith formula (^{Allen et al., 1998}) is obtained:

where ETo is the reference evapotranspiration in millimeters per day (mm/d) and the two new terms are Tt, which is the average air temperature at 2 meters high in °C, and u ₂, which is the average wind speed at 2 m high in m/s. The value r _s = 70 s/m corresponds to a moderately dry soil surface resulting from frequent irrigation, approximately weekly.

To get Equation A.2 from A.1, the depth of water in mm/d can be expressed in terms of energy received per unit area. This energy refers to the heat needed to evaporate the specified water depth, and is known as latent heat of evaporation (λ), which is a function of the water temperature (Ta) and is calculated with the following equation in MJ/kg (^{Allen et al., 1998}):

Since the value of λ does not change much with Ta, Ta = 20 °C is used, and then λ is approximately 2.45 MJ/kg, that is, 2.45 MJ are required to evaporate one kilogram of water or one liter. Then, a 1 mm depth of water is equivalent to 2.45 MJ/m², since 1 mm per m² is a cubic decimeter, that is, one liter. The first numerical coefficient in Equation A.2 converts the radiation, expressed in MJ/m²/d, to evaporation, in mm/d, and is equivalent to the inverse value of λ (1/λ = 0.408).

Equation (A.2) can be applied at intervals of one day, ten days, one month or even the total duration of crop growth or one year. To obtain ETo in mm/h, the numerator in the rectangular parenthesis is changed to 37 and all the variables are per hour rather than per day. For verification of the results, in humid tropical regions with a moderate average temperature (Tt≈20 °C), ETo varies from 3 to 5 mm/d; and with a hot climate, (Tt >30°C) it ranges from 5 to 7 mm/d, these intervals increase by one unit in the arid zones (^{Allen et al., 1998}). In Mexico, applications of Equation A.2 have already been done by ^{González-Camacho, Cervantes-Osornio, Ojeda-Bustamante and López-Cruz (2008)}, and ^{Chávez-Ramírez et al. (2013)}.

Estimation of parameters Δ and γ

All the expressions presented below are from ^{Allen et al. (1998)} and are used to estimate the potential evapotranspiration ( PETji) in month j of each year i, which is required for the application of Equation 1. The slope (Δ in kPa/°C) in the vapor pressure curve of saturation at a point relative to the average air temperature (Tt) in °C, is calculated with the expression:

The psychrometric constant (γ in kPa/°C) is determined with the following expression:

where c _p = 1.013·10^-3 MJ/kg/°C and ε = 0.622 are the quotients of the molecular weight of water vapor to that of air, λ is estimated with Equation A.3 for the value of Ttji in °C, and P is the atmospheric pressure at the site in kPa. This is estimated with the equation:

in which, z is the altitude in meters above sea level.

Estimation of radiations Rn and G

The net radiation (Rn) is equivalent to the difference between the net short-wave incident solar radiation (Rns) and the net long-wave solar radiation that is emitted or released (Rnl), that is:

The Rns is the difference between the incident radiation solar (Rs) and the reflected one, so it is estimated with the expression:

where α is the albedo or coefficient of reflection of the vegetation cover, which is dimensionless. A value of 0.23 is adopted for the hypothetical reference grass. Rs _j must be expressed in MJ/m²/d, therefore, the average monthly values, in cal/cm²/d, from the maps proposed by ^{Almanza and López (1975)}, or by ^{Hernández et al. (1991)}, must be multiplied by 0.041868 to obtain MJ/m²/d.

The long-wave energy emission rate is proportional to the absolute temperature of the surface raised to the fourth power. This relationship is known as the Stefan-Boltzmann's Law. Since water vapor, clouds, carbon dioxide, and dust absorb and emit long-wave radiation, their balance or net flow that leaves the earth's surface is estimated by correcting the Stefan-Boltzmann law for relative humidity and cloudiness, according to the following equation:

where, σ = 4.903·10^-9 MJ/K⁴/m²/d is Stefan-Boltzmann's constant, Ttji is the average temperature of the month, in degrees Kelvin, equal to the degrees centigrade (°C) plus 273.16, eji is the current partial vapor pressure in kPa and Rso is the solar radiation on clear days or without cloudiness, in MJ/m²/d. This is estimated with the expression:

where, z is the altitude of the site and Re _j is the what is known as extraterrestrial radiation, in MJ/m²/d. The quotient Rs/Rso must be less than one. The estimates of eji, Re _j , and of the two missing terms in Equation A.2 (e _s and u ₂) are detailed in the following Appendix.

By considering that the heat flow from the ground (G) is less than Rn, a very simple expression is used for its estimation, which considers the ground temperature to be similar to that of air; this is:

where, c _s = 2.10 MJ/m³/°C is the caloric capacity of the ground, Δd is the interval in days, and Δz is the soil depth affected, which for lapses of one month or more is considered equal to 2 meters. Based on these numerical values, Equation A.11 for the first month, subsequent, and last month are:

In equation A.14, NA is the number of years in the climatic record processed.

Extraterrestrial radiation

Re _j is the solar radiation at the top of the atmosphere in cal/cm²/d. It is tabulated monthly and is a function of the latitude of the site (φ), in degrees. To avoid interpolation of Re _j , 12 third-degree Newton polynomials were developed. Their formula is applicable at latitudes from 10 to 40 degrees north (^{Campos-Aranda, 2005a}):

Los b _i coefficients are as follows:

The values of Re _j estimated with Equation A.15 must be multiplied by 0.041868 to obtain them in MJ/m²/d.

Partial vapor pressure

To estimate eji with Equation A.9, we remember that hotter air may contain more water vapor, whose maximum is the partial pressure of saturation vapor (e _s), in kilopascal (kPa), which is a function of temperature and is estimated with the following expression:

When there is less amount of water vapor than the maximum, the partial pressure of water vapor is designated by e, and then the relative humidity (HR) in percentage is:

If the air that contains any amount of water vapor equal to e is cooling, it reaches a point where e becomes e _s, and that temperature is called the dew point (t*), which is estimated with Equation A.16. Solving for this gives us:

Having found that t* is very close to the average monthly minimum temperature (tji), a simple way of estimating the value of eji is established. This was verified almost three decades ago by ^{Arteaga-Ramírez (1989)}, and more recently by ^{Cervantes-Osornio, Arteaga-Ramírez, Vázquez-Peña, Ojeda-Bustamante and Quevedo-Nolasco (2013)}.

Table A.1 shows the monthly average relative humidity data (Equation A.17), partial vapor pressure (e), and minimum temperature (t) for five meteorological observatories (^{SARH, 1982}), which are surrounding the three weather stations that will be processed. Based on Equation A.18, using 6.108 instead of 0.6108, since e is in mbar, the corresponding dew point temperatures (t*) were obtained. Then, the differences between t and t* were calculated for their inspection and to determine the corrections to the value t, since theoretically these differences should be close to zero.

The corrections in °C suggested in Saltillo are combined for the Villa de Arriaga station (September to February with zero and March to August with -1.00) and for Aguascalientes (July to February zero and March to June -2.00), proposing: September to February with zero and March to August with -1.50. Data from the San Luis Potosí observatory were not used since they are not considered to be fully reliable. For the observatory of Río Verde, the recommended corrections are: March to August zero and September to February +1.20. Finally, in the Xilitla station, the correction suggested for the Tampico observatory, which is to add 0.60°C in all months, will be applied.

Average wind speed

Finally, regarding the estimation of the average wind speed (v) to be used in Equation A.2, the FAO has established four values according to the type of winds that occur in the region: (1) weak from 0.50 to 1.0 m/s, (2) moderate from 1 to 3 m/s, (3) severe from 3 to 5 m/s and (4) strong ≥ 5 m/s. It also points out that the average speed at 2 m high in more than 2 000 meteorological stations in the world is 2 m/s.

^{Campos-Aranda (2005b)} processed average monthly wind speed data (m/s) from 31 meteorological observatories that had this data, with the number of records ranging from 8 to 20 years. He found that the mode of 2.05 m/s verifies the observed global mean value and results in a sample and population median of the order of 2.3 m/s. This value is recommended for the monthly average.

Solar radiation with temperature data

When the Rs _j maps of ^{Almanza and López (1975)}, or ^{Hernández et al. (1991)} are not available, or one does not want to use monthly average values, an estimate based on the monthly difference between the maximum (T) and minimum temperature (t) of the air can be made, since such difference is related to the degree of cloudiness at the site. In general, on clear days or days without cloudiness, high temperatures are generated during the day and low at night, due to the fact that long-wave radiation is not absorbed or returned. The opposite occurs on cloudy days. The empirical operational equation is (^{Allen et al., 1998}):

in which, Rsji is solar radiation, in MJ/m²/d, ka is an adjustment factor that ranges from 0.16 to 0.19, with units 1/°C, and Re _j is extraterrestrial radiation in MJ/m²/d. The value ka = 0.16 is used for inland locations, where air masses are not influenced by the sea, and ka = 0.19 is used in coastal areas where there is such an affectation. ^{Allen et al. (1998)} also indicate how Rsji data from a nearby meteorological station can be transported to the location under study.

Thornthwaite Method

This estimates the potential evapotranspiration PETji) of month j of year i based solely on the average temperature (Ttji) in °C and the latitude of the site (LAT) in degrees. Its equation is (^{Mather, 1977}; ^{Campos-Aranda, 2005a}, ^{Xu, Singh, Chen, & Chen, 2008}):

in which, IC _i is an annual heat index, equal to the sum of the 12 monthly indices, which are:

The exponent α is a function of IC _i with the following empirical equation:

Finally, Fc _j is a monthly average corrective factor function of the latitude of the site and the number of days of the month (ndm). This formula is:

where N is the maximum sunlight or maximum number of hours with average monthly sunshine. For its estimation in the Mexican Republic, ^{Campos-Aranda (2005a)} proposed the following empirical expression:

in which, nm is the number of the month, with 1 for January and 12 for December; A and B are constants, a function of the latitude of the site (LAT) in degrees, with the following expressions:

Turc Method

In the early 1960s, L. Turc proposed the following equation to estimate the monthly and 10-day PETji, which is a function of the average monthly temperature, Ttji, of the incident solar radiation Rs _j , expressed in cal/cm²/d, and of the average monthly relative humidity (^{Turc, 1961}; ^{Campos-Aranda, 2005a}; ^{Xu et al., 2008}):

The coefficient c _j takes the following values: 0.40 for months of 30 or 31 days, 0.37 for February, and 0.13 for a period of 10 days. The corrective factor is applied when the average monthly relative humidity (HRji) is less than 50%, its expression is:

Hargreaves-Samani Method

Average daily potential evapotranspiration (ETPji),in millimeters, were proposed in the early 1980s, exclusively based on the average temperature (Ttji) expressed in degrees Fahrenheit and the daily average incident solar radiation ( Rsji) expressed in millimeters of evaporated water depth. Its equation is (^{Hargreaves and Samani, 1982}, ^{Campos-Aranda, 2005a}, ^{Xu et al., 2008}):

The incident solar radiation (Rsji) can be estimated with the Angström formula (^{Jáuregui-Ostos, 1978}, ^{Allen et al., 1998}) when there is insolation or actual sunlight (n) data, or with the monthly maps available for the Mexican Republic (^{Almanza and López, 1975}; ^{Hernández et al., 1991}), which are reported in cal/cm²/d. For the transformation of Rsj into evaporated water depth per day, the following formula is used:

where Hvji is the so-called latent heat of evaporation or energy needed in calories to evaporate 1 g or cm³ of water. It is estimated with the following expression, with the average ( Ttji) monthly temperature in °C:

The other empirical formula by Hargreaves-Samani can be consulted in ^{Campos-Aranda (2005a)}, which is a function of extraterrestrial solar radiation and monthly average and minimum temperatures. An application of this criterion is given in ^{Campos-Aranda (2014)}.