Version 3 of the Global Aridity Index and Potential Evapotranspiration Database

Calculating Potential Evapotranspiration using Penman-Monteith

Among several equations used to estimate PET, an implementation of the Penman-Monteith equation originally presented by the Food and Agriculture Organization FAO-56¹, is considered a standard method^3,12,13,49. FAO-56¹ defined PET as the ET of a reference crop (ET₀) under optimal conditions, in this case with the specific characteristics of well-watered grass with an assumed height of 12 centimeters, a fixed surface resistance of 70 seconds per meter and an albedo of 0.23¹. Less specifically, “reference evapotranspiration”, generally referred to as “ET₀”, measures the rate at which readily available soil water is evaporated from specified vegetated surfaces^2,13, i.e., from a uniform surface of dense, actively growing vegetation having specified height and surface resistance, not short of soil water, and representing an expanse of at least 100 m of the same or similar vegetations^1,13. ET₀ is one of the essential hydrological variables used in many research efforts, such as study of the hydrologic water balance, crop yield simulation, irrigation system management and in water resources management, allowing researchers and practitioners to study the evaporative demand of the atmosphere independent of crop type, crop development and management practices^2,4,13,49. ET₀ values measured or calculated at different locations or in different seasons are comparable as they refer to the ET from the same reference surface. The factors affecting ET₀ are climatic parameters, and crop specific resistances coefficients solved for reference vegetation. Other crop specific coefficients (K_c) may then be used to determine the ET of specific crops (ET_c), and which can in turn be determined from ET₀¹.

As the Penman-Monteith methodology is predominately a climatic approach, it can be applied globally as it does not require estimations of additional site-specific parameters. However, a major drawback of the Penman-Monteith method is its relatively high need for specific data for a variety of parameters (i.e., windspeed, relative humidity, solar radiation). Zomer et al.¹⁸ compared five methods of calculating PET with parameters from data available at the time and settled upon using a Modified Hargreaves-Thornton equation⁵⁰ which required less parametrization to produce the Global-AI_PET_v1^16,17,18. Several other attempts to produce global PET datasets with concurrently available global datasets came to similar conclusions^51,52,53. The Modified Hargreaves-Thornton method required less parameterization with relatively good results, relying on datasets which were available at the time for a globally applicable modeling effort. The Global-AI_PET_v1 used the WorldClim_v1.4²⁰ downscaled climate dataset (30 arcseconds; averaged over the period 1960–1990) for input into the global geospatial implementation of the Modified Hargreaves-Thornton equation, applied on a per grid cell basis at approximately 1 km resolution (30 arcseconds). More recently, the UK Climate Research Unit released the “CRU_TS Version 4.04”, which now includes a Penman-Monteith calculated PET (ET₀) global coverage, however at a relatively coarse resolution of 0.5 × 0.5 degrees. A number of satellite-based remote sensing datasets^{22,54,55,56,57} are now available and in use to provide the parameters for ET₀ estimates, in some cases providing high spatial and/or temporal resolution and are likely to become increasingly utilized as the historical data record lengthens and sensors improve.

The latest 2.0 versions of WorldClim⁵⁸ (currently version 2.1; released January 2020), in addition to being updated with improved data and analysis, and a revised baseline (1970–2000), includes several additional primary climatic variables, beyond temperature and precipitation, namely: solar radiation, wind speed and water vapor pressure. The addition of these variables allowed that the global data now available was sufficient to effectively parameterize the FAO-56 equation to estimate ET₀ globally at the 30 arc seconds scale (~1 km at equator).

The FAO-56 Penman-Monteith equation, described in detail below, has been implemented on a per grid cell basis at 30 arc seconds resolution, using the Python programming language (version 3.2). The data to parametrize the various components equations required to arrive at the ET₀ estimate were obtained from the Worlclim 2.1⁵⁸ climatological dataset, which provides values averaged over the time period 1970–2000 for minimum, maximum and average temperature; solar radiation; wind speed, and water vapor pressure. Subroutines in the program include calculation of the psychrometric constant (aerodynamic resistance), saturation vapor pressure, vapor pressure deficit, slope of vapour pressure curve, air density at constant pressure, net shortwave radiation at crop surface, clear-sky solar radiation, net longwave radiation at crop surface, net radiation at the crop surface, and the calculation of daily and monthly ET₀. This process is described below. Geospatial processing and analysis were done using ArcGIS Pro v 2.9 (ESRI, 2020), Python (ArcPy) programming language (version 3.2), and Microsoft Excel for further data analysis, graphics and presentation.

Global Reference Evapotranspiration (Global-ET₀)

Penman⁵⁹, in 1948, first combined the radiative energy balance with the aerodynamic mass transfer method and derived an equation to compute evaporation from an open water surface from standard climatological records of sunshine, temperature, humidity and wind speed. This combined approach eliminated the need for the parameter “most difficult” to measure, surface temperature, and allowed for the first time an opportunity to make theoretical estimates of ET from standard meteorological data. Consequently, these estimates could also now be made retrospectively. This so-called combination method was further developed by many researchers and extended to cropped surfaces by introducing resistance factors. Among the various derivations of the Penman equation is the inclusion of a bulk surface resistance term⁶⁰, with the resulting equation now called the Penman-Monteith equation³, as standardized in FAO-56¹ and subsequently by the American Society of Civil Engineers – Technical Committee on Standardization of Reference Evapotranspiration^12,13,49,61. The FAO-56 Penman-Monteith form of the combination equation to estimate ET₀ is calculated as:

Where

ET₀ is the evapotranspiration for reference crop, as mm day⁻¹

R_n is the net radiation at the crop surface, as MJ m⁻² day⁻¹

G is the soil heat flux density, as MJ m⁻² day⁻¹

c_p is the specific heat of dry air

p_a is the air density at constant pressure

e_s is the saturation vapour pressure, as kPa

e_a is the actual vapour pressure, as kPa

e_s – e_a is the saturation vapour pressure deficit, as kPa

(Delta ) is the slope vapour pressure curve, as kPa °C⁻¹

(gamma ) is the psychrometric constant, as kPa °C⁻¹

r_s is the bulk surface resistance, as m s⁻¹

r_a is the aerodynamic resistance, as m s⁻¹

Psychrometric Constant (γ)

The Atmospheric Pressure (Pr, [KPa]) is the pressure exerted by the weight of the atmosphere and is thus dependent on elevation (elev, [m]). To a certain (and limited) extent evaporation is promoted at higher elevations:

Instead, the psychrometric constant, [γ, kPa C⁻¹] is expressed as:

Where c_p is the specific heat at constant pressure [MJ kg⁻¹ °C⁻¹] and is equal to 1.013 10⁻³, λ is the latent heat of vaporization [MJ kg⁻¹] and is equal to 2.45, while ε is the molecular weight ratio between water vapour and dry air and is equal to 0.622.

Elevation data has been obtained from the Shuttle Radar Topography Mission (SRTM) aggregated to 30 arc-second spatial resolution⁶² and combined with the USGS GTOPO30⁶³ database for the areas north of 60°N and south of 60°S where no SRTM data was available (available at https://worldclim.org).

Air Density at Constant Pressure [ρ_a]

The mean Air Density at Constant Pressure [ρ_a, Kg m⁻³] can be represented as:

While R is the specific heat constant (0.287, KJ Kg⁻¹ K⁻¹), the virtual temperature T_Kv can be represented as well as:

With T_avg as the mean daily air temperature at 2 m height [C°].

Saturation Vapor Pressure [KPa]

Saturation Vapor Pressure [KPa] is strictly related to temperature values (T)

Values of saturation vapor pressures, as function of temperature, are calculated for both Minimum Temperature [T_min, C°] and Maximum temperature [T_max, C°]. Due to nonlinearity of the equation, the mean saturation vapour pressure [e_s, KPa] is calculated as the average of saturation vapour pressure at minimum [e_{s_min}] and maximum temperature [e_{s_max}]

The actual vapour pressure [e_a, KPa] is the vapour pressure exerted by the water in the air and is usually calculated as function of Relative Humidity [RH]. Water vapour pressure is already available as one of the Worldclim 2.1 variables.

The vapour pressure deficit (e_s–e_a), [KPa] is the difference between the saturation (es) and actual vapour pressure (({e}_{a})).

Slope of Saturation Vapor Pressure (Δ)

The Slope of Saturation Vapor Pressure [Δ, kPa C⁻¹] at a given temperature is given as function of average temperature:

Where T_avg [C°] is the average temperature.

Net Radiation At The Crop Surface (R
_n)

Net radiation [R_n, MJ m⁻² day⁻¹] is the difference between the net shortwave radiation [Rns, MJ m⁻² day⁻¹] and the net longwave radiation [R_nl, MJ m⁻² day⁻¹], and is calculated using solar radiation (R_s). In Worldclim 2.1 solar radiation (R_s) is given as KJ m⁻² day⁻¹. Thus, for computation of ET₀, its unit should be converted to MJ m⁻² day⁻¹ and thus its value should be divided by 1000. The net accounting of either longwave and shortwave radiation sums up the incoming and outgoing components.

The net shortwave radiation [R_ns, MJ m⁻² day⁻¹] is the fraction of the solar radiation R_s that is not reflected from the surface. The fraction of the solar radiation reflected by the surface is known as the albedo [α]. For the green grass reference crop, α is assumed to have a value of 0.23. The value of Rns is:

The difference between outgoing and incoming longwave radiation is called the net longwave radiation [R_nl]. As the outgoing longwave radiation is almost always greater than the incoming longwave radiation, R_nl represents an energy loss. Longwave energy emission is related to surface temperature following Stefan-Boltzmann law. Thus, longwave radiation emission is calculated as positive in the outward direction, while shortwave radiation is positive in the downward direction. The net energy flux leaving the earth’s surface is influenced as well by humidity and cloudiness

Where σ represent the Stefan-Boltzmann constant (4.903 10-9 MJ K⁻⁴ m⁻² day⁻¹), T_max,K and T_min,K the maximum and minimum absolute temperature (in Kelvin; K = C° + 273.16), e_a is the actual vapour pressure; R_s the measured solar radiation [MJ m⁻² day⁻¹] and R_so is the calculated clear-sky radiation [MJ m⁻² day⁻¹]. R_so is calculated as function of extraterrestrial solar radiation [R_a, MJ m⁻² day⁻¹] and elevation (elev, m):

The extraterrestrial radiation, [R_a, MJ m⁻² day⁻¹], is estimated from the solar constant, solar declination and day of the year. It requires specific information about latitude and Julian day to accomplish a trigonometric computation of the amount of solar radiation reaching the top of the atmosphere following trigonometric computations as shown in Allen et al.¹.

Although the soil heat flux is small compared to R_n, particularly when the surface is covered by vegetation, changes of soil heat flux may still be relevant at monthly scale. However, accurate assessments of soil heat flux may require computation of soil heat capacity, related to its mineral composition and water content, which in turn may be rather inaccurate at global scale at resolution of 30 arc sec. Thus, for simplicity, changes in soil heat fluxes are ignored (G = 0).

Bulk Surface Resistance (r
_s)

The resistance nomenclature distinguishes between aerodynamic resistance and surface resistance factors. The surface resistance parameters are often combined into one parameter, the ‘bulk’ surface resistance parameter which operates in series with the aerodynamic resistance. The surface resistance, r_s, describes the resistance of vapour flow through stomata openings, total leaf area and soil surface. The aerodynamic resistance, ra, describes the resistance from the vegetation upward and involves friction from air flowing over vegetative surfaces. Although the exchange process in a vegetation layer is too complex to be fully described by the two resistance factors, good correlations can be obtained between measured and calculated evapotranspiration rates, especially for a uniform grass reference surface.

A general equation for the bulk surface resistance (r_s, [s m⁻¹]) describes a ratio between the bulk stomatal resistance of a well illuminated leaf (r_l) and the active sunlit leaf area of the vegetation:

The stomatal resistance of a single leaf under well-watered conditions has a value of about 100 s m⁻¹. It can be assumed that about half (0.5) of the total LAI is actively contributing to vapour transfer, while it can also be roughly generalized that for short crops there is a linear relation between LAI and crop height (h):

When the evapotranspiration simulated with the Penman-Monteith method is referred to a specific reference crop, denoted as ET₀, a simplified computation of the method can occur that defines a priori specific variables into constant values. In this case, the reference surface is a hypothetical grass reference crop, well-watered grass of uniform height, actively growing and completely shading the ground, with an assumed crop height of 0.12 m, and an albedo of 0.23. The surface resistance for this hypothetical grass can be simplified to the following:

For such reference crop the surface resistance is fixed to 70 s m⁻¹ and implies a moderately dry soil surface resulting from about a weekly irrigation frequency.

Aerodynamic Resistance (r
_a)

The aerodynamic resistance [s m⁻¹] verifies the transfer of water vapour and heat from the vegetation surface into the air, and is controlled by both vegetation status but also atmospheric turbulence under theoretical aspect as:

Z_m [m] is the height [h] of wind measurements and Z_h [m] is the height of humidity measurements. These are normally set at 2 meters height, although several climate models may provide them for higher heights (e.g. 10 m). The zero plane displacement (d [m]) term can be estimated as two thirds of crop height, while Z_om is the roughness length governing momentum transfer, and can be calculated as Z_om = 0.123 * h.

The roughness length governing transfer of heat and vapour, Z_oh [m], can be approximated as one tenth of Z_om. k is the von Karman’s constant, equal to 0.41, and u_z [m s-1] is the wind speed at height z.

The reference surface, as stated, is a hypothetical grass reference crop, well-watered grass of uniform height, actively growing and completely shading the ground, with an assumed crop height of 0.12 m, and an albedo of 0.23. For such reference crop the surface resistance is fixed to 70 s m-1 and implies a moderately dry soil surface resulting from about a weekly irrigation frequency.

When crop height is equal to 0.12 and wind/humidity measurements are taken at 2 meters height, then the aerodynamic resistance can be simplified as:

Reference Evapotranspiration (ET
₀)

Given the above, and the specific properties of the standard reference crop, the FAO-56 Penman-Monteith method to estimate ET₀ then can be calculated as:

Aridity Index (AI)

Aridity is often expressed as a generalized function of precipitation and PET. The ratio of precipitation over PET (or ET₀). That is, the precipitation available in relation to atmospheric water demand⁶⁴ quantifies water availability for plant growth after ET demand has been met, comparing incoming moisture totals with potential outgoing moisture⁶⁵.

Geospatial analysis and global mapping of the AI for the averaged 1970–2000 time period has been calculated on a per grid cell basis, as:

where:

AI = Aridity Index

MA_Prec = Mean Annual Precipitation

MA_ET₀ = Mean Annual Reference Evapotranspiration

Mean annual precipitation (MA_Prec) values were obtained from the WorldClim v 2.1⁵⁸, as averaged over the period 1970–2000, while ET₀ datasets estimated on a monthly average basis by the Global-ET₀ (i.e., modeled using the method described above) were aggregated to mean annual values (MA_ET₀). Using this formulation, AI values are unitless, increasing with more humid condition and decreasing with more arid conditions.

As a general reference, a climate classification scheme for Aridity Index values provided by UNEP⁶⁴ provides an insight into the climatic significance of the range of moisture availability conditions described by the AI.

Source: Ecology - nature.com