Crop water requirements are commonly estimated with the
FAO-56 methodology based upon a two-step approach: first a reference
evapotranspiration (ET

The well-known FAO-56 publication on crop evapotranspiration (Allen et al.,
1998) is the outcome of a revision project concerning a previous publication
(FAO-24) on the same subject (Doorenbos and Pruitt, 1977). In FAO-56 the
current guidelines for computing crop water requirements are presented. Two
different ways of calculating crop evapotranspiration are retained and
detailed: the single crop coefficient and the dual crop coefficient. In the
single crop coefficient approach, crop evapotranspiration under standard
conditions is calculated as

The FAO-56 methodology (single or dual crop coefficients) is commonly called
the two-step approach (Shuttleworth, 2007) because ET

Consequently, many authors (e.g. Shuttleworth, 2007) have suggested that a
better approach would consist in estimating ET

Given that the familiar Penman–Monteith equation (Eq. 3) is only relevant when soil evaporation is negligible, the problem which arises from a theoretical standpoint is that the dual coefficient of the two-step approach (Eq. 2), which accounts for crop transpiration and soil evaporation, cannot be translated into the one-step approach. A physical model equivalent to the dual coefficient approach would be the one-dimensional two-source model designed for sparse crops by Shuttleworth and Wallace (1985) and revisited by Lhomme et al. (2012). Unfortunately, from an operational standpoint, the practical implementation of this two-source model can be hindered by its mathematical formalism, which is far more complex than the common Penman–Monteith equation. Following the idea of Wallace (1995), who stated that “the key to continued improvement in evaporation modelling is to attempt to simplify these complex schemes while still retaining their essential elements as far as possible”, the article aims at showing that the two-source model of evaporation can be transformed into a Penman–Monteith type equation, where foliage transpiration resistance and soil evaporation resistance are included within a bulk surface resistance. Then, it will be shown that the transpiration resistance can be inferred from the basal crop coefficient of the dual approach in a way similar to the Matt–Shuttleworth approach. Numerical simulations will be performed to illustrate the advantages of this new form of the Penman–Monteith equation to estimate crop water requirements with a one-step approach.

The so-called Penman–Monteith equation (Monteith, 1963, 1965) results from
the combination of the convective fluxes emanating from the canopy with the
energy balance. Introducing effective resistances within and above the
canopy, the convective fluxes of sensible heat (

As thoroughly explained in Lhomme et al. (2012, Sect. 4), the
within-canopy resistances (

Resistance networks and potentials for a two-layer representation of the convective fluxes (sensible heat and latent heat) within the canopy. The nomenclature used is given in the list of symbols.

The soil surface resistance (

According to FAO-56, the aerodynamic resistance above the canopy (

Similarly to the Matt–Shuttleworth method developed for a single crop
coefficient (Shuttleworth, 2006), the problem to tackle now is to infer the
values of both surface resistances (

Given that many crops have a crop height close to (or greater than) the
reference height of 2 m, the weather variables involved in the
Penman–Monteith equation should be taken at a higher level than the
reference height. This point is thoroughly developed in the
Matt–Shuttleworth method, where it is suggested that air characteristics be
taken at a blending height arbitrarily set at

Canopy evapotranspiration is the sum of foliage evaporation (ET

The two surface resistances (

The fact that surface resistances are necessarily positive imposes a
physical constraint on the values of

In the numerical simulations carried out below, the daily net radiation of
the reference crop (

The sensitivity of crop evapotranspiration ET

Simulations were undertaken to compare the proposed comprehensive
Penman–Monteith equation (Eq. 10) with the reference model represented by
the full two-layer model detailed in Appendix C. Working on a daily basis,
soil heat flux is neglected and the ratio

Relative error on crop evapotranspiration ET

Typical values at reference height of daily minimum relative
humidity (RH

As explained in Sect. 2.2, the modified roughness length

For different LAI, RE on crop evapotranspiration
ET

Foliage surface resistance

Variation of the ratio between the modified roughness length
(

For three types of climate (SA, SH,

We have shown that the FAO-56 dual crop coefficient approach, where the crop
coefficient

Variation of foliage surface resistance (

As a consequence of the above development, and following the suggestion
already made by Shuttleworth (2014) for computing crop water requirements,
we think that the United Nations FAO could
find some interest in recommending the use of the one-step approach in
replacement of the FAO-56 two-step approach. In the one-step approach, four
parameters should be adjusted to a specific crop: its albedo to estimate the
net radiation, its aerodynamic resistance and the two components of the
surface resistance (soil and vegetation). Albedo varies as a function of
green canopy cover (or LAI). The aerodynamic resistance is calculated as a
function of crop height (Eq. 14), provided the roughness length is correctly
determined (Eq. 16). The soil component of the surface resistance requires a
specific parameterization as a function of top soil layer water content.
Some empirical parameterizations already exist and should be thoroughly
examined and tested. With regard to foliage resistance, although it can be
inferred in principle from the basal crop coefficient, it is certainly more
recommendable to undertake experimental and bibliographical works in order
to determine appropriate values under standard conditions (i.e. non-stressed
and well-managed crop). Given that foliage resistance is expressed as the
simple ratio of leaf stomatal resistance to leaf area (see Eq. 12) and that
LAI is an adjustable and experimentally accessible parameter, one can
imagine that the mean leaf stomatal resistance could play the same role in
the one-step approach as (and replace) the basal crop coefficient of the
two-step approach. Tabulated values for different crops could be supplied
and organized by group type in the same way as the crop coefficients in
FAO-56. Only one value per crop could be needed, instead of the three values
generally provided for crop coefficients, given that LAI values should be
able to account for the necessary adjustment to crop cycle characteristics.
It is worthwhile stressing, nevertheless, that the leaf stomatal resistance
of a given crop under standard conditions (which represents a minimum value)
is subject to the influence of other climatic environment parameters than water
stress (i.e. temperature, humidity, radiation, CO

According to FAO-56, the daily calculation of

List of symbols.

The parameterization commonly used to simulate the component air resistances
taken and adapted from Shuttleworth and Wallace (1985), Choudhury and
Monteith (1988), Shuttleworth and Gurney (1990), Lhomme et al. (2012). The
aerodynamic resistance between the substrate (with a roughness length

Following the reformulated expression of the two-layer model proposed by
Lhomme et al. (2012), crop evaporation is given by