Evapotranspiration (ET): Calculation Method

This article is the technical reference for how ZENTRA Cloud calculates Reference Evapotranspiration. If you just want to create an ET model, see Evapotranspiration (ET). This article covers the underlying equation, inputs, constants, and assumptions for users who need the full methodology.

Overview

ZENTRA Cloud computes Reference Evapotranspiration using the ASCE Standardized Reference Evapotranspiration Equation (ASCE-EWRI, 2005), which is the Penman-Monteith combination equation applied to a standardized reference surface. This is the same method described in FAO Irrigation and Drainage Paper No. 56 (Allen et al., 1998) and standardized by the American Society of Civil Engineers Environmental and Water Resources Institute (ASCE-EWRI, 2005).

Reference evapotranspiration (ET_ref) represents the evapotranspiration from a hypothetical, well-watered reference crop of uniform height, actively growing and completely shading the ground. It is a purely climatic quantity: it expresses the evaporative demand of the atmosphere independent of crop type, soil, or management. Actual crop water use is obtained downstream by multiplying ET_ref by a crop coefficient (K_c), which is outside the scope of this document.

ZENTRA Cloud supports two standardized reference surfaces:

The equation is evaluated on an hourly (or shorter) time step and then summed to the reporting interval. All Reference ET is reported in millimetres (mm) of water depth (convertible to inches).

The governing equation

The ASCE Standardized Reference ET Equation is:

              0.408 · Δ · (Rₙ − G)  +  γ · [Cₙ / (T + 273)] · u₂ · (eₛ − eₐ)
          ET_ref  =  ────────────────────────────────────────────────────────────────────
                                    Δ  +  γ · (1 + C_d · u₂)

The result ET_ref has units of mm h⁻¹ when the equation is applied on an hourly basis with the hourly constants (see Standardization constants).

Variable definitions

The two constants Cₙ and C_d are what make the equation “standardized”: rather than modeling the aerodynamic and surface (canopy) resistances explicitly, ASCE fixes them to values representative of each reference surface and time step. Their values are given in the Standardization constants section.

Input measurements

The equation is driven by four measured meteorological variables plus site metadata. In ZENTRA Cloud these are supplied by a co-located weather station (e.g. the ATMOS family of all-in-one sensors).

Measured variables

Site metadata

Latitude, longitude, and elevation are taken from the configured device location. Wind speed is adjusted from the anemometer measurement height to the standard 2 m reference height (see Wind speed adjustment).

Component calculations

Each term in the governing equation is derived from the measured inputs as follows. These sub-equations follow ASCE-EWRI (2005).

Atmospheric pressure and the psychrometric constant

Atmospheric pressure is estimated from site elevation z (m) using the simplified universal-atmosphere relationship:

P  =  101.3 · [ (293 − 0.0065 · z) / 293 ]^5.26          [kPa]

The psychrometric constant is then:

γ  =  0.000665 · P                                        [kPa °C⁻¹]

Saturation vapor pressure and its slope

Saturation vapor pressure at air temperature T (°C) uses the Tetens formulation:

eₛ  =  0.6108 · exp[ 17.27 · T / (T + 237.3) ]            [kPa]

The slope of the saturation vapor pressure curve at T is:

        2503 · exp[ 17.27 · T / (T + 237.3) ]
          Δ  =  ─────────────────────────────────────────           [kPa °C⁻¹]
                            (T + 237.3)²

(Equivalently, Δ = 4098 · eₛ / (T + 237.3)².)

Actual vapor pressure

If vapor pressure (eₐ) is measured directly, it is used as-is. If relative humidity (RH, as a fraction between 0 and 1) is supplied instead, actual vapor pressure is computed from the saturation vapor pressure at air temperature:

eₐ  =  RH · eₛ                                            [kPa]

Wind speed adjustment to 2 m

Wind speed measured at anemometer height z_w (m) is adjusted to the 2 m standard reference height using a logarithmic wind profile:

              4.87
          u₂  =  u_z · ─────────────────────                        [m s⁻¹]
                       ln(67.8 · z_w − 5.42)

If the anemometer height is unrealistically low (below ≈ 0.08 m, where the logarithm is undefined), the wind adjustment is treated as invalid.

Net radiation

Net radiation Rₙ is the difference between net incoming shortwave and net outgoing longwave radiation:

Rₙ  =  R_ns  −  R_nl                                      [MJ m⁻² h⁻¹]

Net shortwave radiation from the measured incoming solar radiation Rₛ (W m⁻²), corrected for surface albedo α. The factor 0.0036 converts an average power flux in W m⁻² over one hour to an energy total in MJ m⁻² h⁻¹:

R_ns  =  0.0036 · (1 − α) · Rₛ                            [MJ m⁻² h⁻¹]

Extraterrestrial (potential) radiation Rₐ — the theoretical clear-atmosphere solar radiation on a horizontal surface at the top of the atmosphere — is computed from solar geometry for the site and time step. It uses the day of year (Julian day), latitude φ, longitude λ, the solar declination, and an equation-of-time correction to locate true solar noon:

δ_s  =  0.409 · sin( 2π · J / 365 − 1.39 )                [solar declination, rad]

with a solar constant of 1360 W m⁻². Rₐ is evaluated at the midpoint of each time step and is used both to establish clear-sky radiation and to distinguish daytime from nighttime periods (daytime is defined as Rₐ > 0).

Clear-sky radiation scales the extraterrestrial radiation for atmospheric transmissivity, including an elevation correction:

R_so  =  (0.75 + 2×10⁻⁵ · z) · Rₐ                         [same units as Rₐ]

Cloudiness function compares measured solar radiation to clear-sky radiation to quantify cloud cover. The ratio Rₛ/R_so is bounded to [0.3, 1.0]; during nighttime (when the ratio is undefined) the last valid daytime value is carried forward:

R_ratio  =  clip( Rₛ / R_so , 0.3 , 1.0 )
          
          f_cd     =  clip( 1.35 · R_ratio − 0.35 , 0.05 , 1.0 )

Net longwave radiation uses the Stefan-Boltzmann law with humidity and cloudiness corrections. The coefficient 2.04×10⁻¹⁰ MJ K⁻⁴ m⁻² h⁻¹ is the Stefan-Boltzmann constant expressed for an hourly time step; air temperature is converted to Kelvin (T_K = T + 273.16):

R_nl  =  2.04×10⁻¹⁰ · T_K⁴ · (0.34 − 0.14 · √eₐ) · f_cd   [MJ m⁻² h⁻¹]

Soil (ground) heat flux

For hourly and shorter time steps, soil heat flux is taken as a fixed fraction of net radiation, with different fractions for daytime and nighttime and for each reference surface:

G  =  C_g · Rₙ                                            [MJ m⁻² h⁻¹]

where C_g is the soil-heat-flux coefficient from the Standardization constants table.

Assembling the result

The component terms (Δ, γ, Rₙ, G, u₂, eₛ, eₐ) are substituted into the governing equation together with the surface- and time-step-specific constants Cₙ and C_d. The result is clamped to be non-negative:

ET_ref  =  max( 0 , ET_ref )

This prevents small negative values (physically meaningless “condensation” artifacts) that can arise at night from the linearized radiation and vapor-deficit terms.

Standardization constants

The constants Cₙ, C_d, the surface albedo α, and the soil-heat-flux coefficient C_g depend on the reference surface and on whether the period is daytime or nighttime. ZENTRA Cloud uses the following values for the hourly time step:

The numerator constant Cₙ reflects the time step and the aerodynamic characteristics of the reference surface; the denominator constant C_d reflects the bulk surface resistance and changes between day and night because canopy resistance rises sharply at night. The grass and alfalfa Cₙ/C_d values match the standard ASCE-EWRI (2005) hourly coefficients for the short and tall reference surfaces respectively.

Daytime versus nighttime is determined per time step from the solar geometry: a period is classified as daytime when the computed extraterrestrial radiation Rₐ is greater than zero, and nighttime otherwise.

For reference, ASCE also defines daily coefficients (Cₙ = 900, C_d = 0.34 for grass; Cₙ = 1600, C_d = 0.38 for alfalfa). In ZENTRA Cloud the equation is always evaluated on an hourly (or shorter) basis using the hourly constants, and daily totals are obtained by summation rather than by direct daily-time-step computation.

Time step handling

Hourly base computation. The governing equation is fundamentally an hourly calculation: it uses the hourly standardization constants and the hourly Stefan-Boltzmann coefficient, and it evaluates solar geometry at the midpoint of each hour.

Sub-hourly intervals. When the underlying measurement interval is shorter than one hour (for example, 5-, 15-, or 30-minute data), the hourly ET rate (mm h⁻¹) is computed for the interval and then scaled by the interval length in hours to produce the ET depth for that interval:

ET_interval  =  ET_ref (mm h⁻¹)  ×  Δt (hours)            [mm]

For example, a 15-minute interval yields ET_ref × 0.25 mm. Interval totals are likewise clamped to be non-negative.

Daily and longer totals. Daily (and longer) Reference ET is obtained by summing the interval/hourly depths over the period. It is an aggregation of the base computation, not a separate daily-time-step calculation.

Solar time correction. Because the radiation and cloudiness terms depend on the sun’s position, the solar-geometry calculations require local standard time. Timestamps recorded during Daylight Saving Time are converted back to standard time before the solar declination, equation of time, and hour angle are evaluated, so that true solar noon is located correctly regardless of the observing period’s clock convention.

Assumptions, limitations, and references

Assumptions and scope

• Reference ET characterizes atmospheric evaporative demand over a standardized, well-watered reference surface — it is not the actual evapotranspiration of any specific crop. Actual crop water use requires multiplication by an appropriate crop coefficient (K_c) and, where relevant, a water-stress coefficient.
• The standardized formulation replaces the explicit aerodynamic and surface-resistance terms of the full Penman-Monteith equation with the fixed constants Cₙ and C_d. This is valid for the defined reference surfaces (short/grass and tall/alfalfa) and standard measurement conditions.
• The calculation assumes measurements are representative of the reference surface: air temperature, humidity, and wind measured at standard heights over a surface with adequate fetch, and unobstructed incoming solar radiation.

Limitations

• Results depend on the quality and completeness of the four input measurements; gaps or errors in temperature, humidity, wind, or solar radiation propagate directly into ET_ref.
• The wind-speed adjustment requires a valid anemometer height; unrealistically low heights invalidate the wind term.
• The non-negativity clamp removes small negative values that can otherwise occur at night.
• The nighttime cloudiness function relies on carrying forward the last valid daytime clear-sky ratio, since the solar ratio is undefined without sunlight.

Validation. The ZENTRA Cloud implementation is validated against an independent reference implementation of the hourly ASCE Standardized Penman-Monteith equation (a spreadsheet reference model), reproducing its Reference ET output to the reported precision (0.001 mm) across the daily cycle.

References

1. ASCE-EWRI (2005). The ASCE Standardized Reference Evapotranspiration Equation. Report of the ASCE Environmental and Water Resources Institute Task Committee on Standardization of Reference Evapotranspiration. R. G. Allen, I. A. Walter, R. L. Elliott, T. A. Howell, D. Itenfisu, M. E. Jensen, and R. L. Snyder (eds.). American Society of Civil Engineers, Reston, VA.
2. Allen, R. G., Pereira, L. S., Raes, D., & Smith, M. (1998). Crop Evapotranspiration — Guidelines for Computing Crop Water Requirements. FAO Irrigation and Drainage Paper No. 56. Food and Agriculture Organization of the United Nations, Rome.
3. Monteith, J. L. (1965). Evaporation and environment. Symposia of the Society for Experimental Biology, 19, 205–234.