Comment on “ Technical Note : On the Matt – Shuttleworth approach to estimate crop water requirements ” by Lhomme et al . ( 2014 )

It is clear from Lhomme et al. (2014) that aspects of the explanation of the Matt–Shuttleworth approach can generate confusion. Presumably this is because the description in Shuttleworth (2006) was not sufficiently explicit and simple. This paper explains the logic behind the Matt–Shuttleworth approach clearly, simply and concisely. It shows how the Matt–Shuttleworth can be implemented using a few simple equations and provides access to ancillary calculation resources that can be used for such implementation. If the crop water requirement community decided that it is preferable to use the Penman–Monteith equation to estimate crop water requirements directly for all crops, the United Nations Food and Agriculture Organization could now update Irrigation and Drainage Paper 56 using the Matt– Shuttleworth approach by deriving tabulated values of surface resistance from Table 12 of Allen et al. (1998), with the estimation of crop evaporation then being directly made in a one-step calculation using an equation similar to that already recommended by the United Nations Food and Agriculture Organization for calculating reference crop evaporation.


Introduction
It is clear from Lhomme et al. (2014) that aspects of the explanation of the Matt-Shuttleworth approach can generate confusion.Presumably this is because the description in Shuttleworth (2006) was not sufficiently explicit and simple.I welcome the opportunity to redress this and I am grateful to Lhomme et al. (2014) for bringing this need to my attention.
The fundamental premise of the Matt-Shuttleworth approach is that describing evapotranspiration from a crop canopy using the Penman-Monteith equation (Monteith, 1964) (hereafter referred to as PM) is theoretically superior to describing evapotranspiration using the formula given (for example) as Eq. ( 1) of Lhomme et al. (2014) (hereafter referred to as FAO).Justifications for this premise are as follows.
1.It is now widely accepted in advanced models of surface-atmosphere energy exchanges that the control exerted by vegetation can be represented by a plantdependent surface resistance, r sc , which is broadly equivalent to the canopy average effect of stomatal resistance, and that the control exerted by turbulent transfer can be represented by an aerodynamic resistance, r ac , dependent on wind speed and aerodynamic properties of the canopy.PM merges these two resistances with the surface energy balance.In such advanced models r sc is either assumed fixed or dependent on incanopy environmental variables and soil moisture if these affect stomatal resistance.
2. The capability of PM to describe crop evapotranspiration is now explicitly accepted as being appropriate by the crop water requirement community because the United Nations Food and Agriculture Organization recommends that reference crop evaporation is calculated from PM (Allen et al., 1998).
via a field experiment (in much the same way that the fixed value 70 s m −1 was defined for the reference crop).Shuttleworth (2006) recommends this, but then seeks to make interim estimates of r sc from the seasonal models of K c in the existing literature.Presumably there must be environmental conditions when there is a definable pairing between the effective values of K c and r sc , specifically the prevailing meteorological conditions when the field experiment to determine K c was carried out.If these meteorological conditions were known, then the same data used to specify the particular value of K c relevant in these conditions could alternatively be used to specify the equivalent value of r sc used in the Matt-Shuttleworth approach.But unfortunately the meteorological conditions when tabulated values of K c were defined are not available -hence assumptions are required.Allen et al. (1998) deem the tabulated values of K c most reliable in "sub-humid conditions" for wind speed 2 m s −1 , and provide an empirical formula to correct K c for crops with different heights exposed to different wind speeds in different atmospheric aridity conditions.(Note that such an empirical correction is not required when using the Matt-Shuttleworth approach because crop-specific aerodynamics rather than reference crop-specific aerodynamics are used.)In the Matt-Shuttleworth approach the preferred wind speed of 2 m s −1 is adopted, but specification of "sub-humid" is also required to establish pairing between the tabulated value of K c on a particular day and required value of r sc .Allen et al. (1998) specify sub-humid conditions as being when average minimum daytime relative humidity is 45 %.However, this specification does not recognize the long established fact that both available energy and vapour pressure deficit control evapotranspiration (Penman, 1948), which suggests that characterizing the influence of atmospheric humidity relative to the influence of available energy is arguably a more appropriate way to define the meaning of sub-humid conditions.For this reason, the climatological resistance, r clim , is adopted as a measure of atmospheric aridity in the Matt-Shuttleworth approach, this being calculated from where ρ (kg m −3 ) is the density of air, c p (J kg is the specific heat of moist air, D 2 (kPa) is the vapour pressure deficit and T C 2 ( • C) the air temperature both at 2 m, (kPa K −1 ) is the rate of change of saturated vapour pressure with temperature, γ (kPa K −1 ) is the psychrometric constant, and A (W m −2 ) is the available energy which becomes A mm (mm) when expressed as evaporated water equivalent.The pairing between the tabulated value of K c and value of r sc is then defined by specifying r clim in subhumid conditions when the tabulated value of K c in Eq. ( 1) of Lhomme et al. (2014) provides its best estimate of crop evapotranspiration.Shuttleworth (2006) does not assume that reference crop evapotranspiration rate, E o , is equal to the Priestley-Taylor estimate of evapotranspiration rate, E PT , every day.This is obviously not an acceptable assumption because, as shown in Eq. ( 23.20) of Shuttleworth (2012), the relationship between E o and E PT on a particular day can in fact be expressed as a function of r clim on that day.However, the condition E o = E PT is used to specify the default value of r clim considered characteristic of sub-humid conditions (i.e. the ratio of (ρc p D) to ( A) in sub-humid conditions).One argument for selecting this as a default condition is the history behind the calculation of reference crop evaporation.The original approach in Doorenbus and Pruit (1977) -who largely defined the tabulation of K c -allowed calculation of reference crop evaporation in several different ways, including as E PT .Later Allen et al. (1998) added the calculation based on PM, which is here referred to as E o .If either K c E o or K c E PT can be used to calculate crop evapotranspiration optimally in sub-humid conditions, then E o = E PT can presumably be used to specify the ratio of (ρc p D) to ( A) (i.e. the value of r clim ) in these sub-humid conditions.In addition, Shuttleworth (2006) describes modelling studies of feedback interactions between the atmospheric boundary layer and surface exchanges at regional scale which suggest there is a range of surface resistances and atmospheric aridity when the concept of "potential evaporation" applies, and when there is at least approximate equality between E PT and evapotranspiration calculated from PM for a range of surface resistances typical of pastureland and many agricultural crops, including 70 s m −1 .
Specifying the default value of r clim in sub-humid conditions from E o = E PT ultimately leads to the result that r sc should be calculated from K c for a preferred value of r clim given by r pref clim = 104 1.26 To calculate the required value of r sc , Shuttleworth (2006) then simply adopts the definition of K c = α a α s given (for example) as Eq. ( 7) of Lhomme et al. (2014) and inverts this to express r sc as a function of K c , but with the ratio of (ρc p D) to ( A) set equal to the (now defined) value of r pref clim in subhumid conditions.However, in order to allow this calculation for crops with height greater than 2 m the interrelationship between r sc and K c is first recast to a height of 50 m, and appropriate forms of aerodynamic resistances and vapour pressure used, these being defined using Eq. ( 24) with Z = 50 m, and Eqs. ( 27) and (29) in Shuttleworth (2006).
However, Eq. ( 2) means the default value of r clim when r sc should be calculated is not yet fully specified because it is a function of temperature.Ideally the temperature selected to calculate would be that when K c was calibrated, but this temperature is generally not known.For this reason Shuttleworth and Wallace (2010) investigated the sensitivity of the calculation of r sc to temperature and on this basis recommended using 20 • C, which implies r sc is optimally calculated from K c when r pref clim is 55 s m −1 .For this value of r pref clim and a 2 m wind speed of 2 m s −1 , the second "advective" term in the numerator of the equation used to calculate E o is, for example, around 50 % of the first "radiation" term; for this value of r  2012) then simplifying, it can be shown that where with h c being the height of the crop.Table 23.5 of Shuttleworth (2012) gives values of r sc calculated from selected values of K c and h c representative of tabulated values during Stage 3 of crop growth.For the hypothetical 1 m high crop considered in Lhomme et al. (2014), assuming the crop factor K c = 1 was calibrated in conditions when the value of preferred climatological resistance had the default value 55 s m −1 , the fixed value of surface resistance estimated using Eqs.( 3) and ( 4) is 111 s m −1 (but see below).
It is important readers understand that the use of the value r pref clim = 55 s m −1 derived from E o = E PT is a default assumption whose use is recommended when the meteorological conditions prevailing when the values of K c given in Allen et al. (1998) were calibrated are not known.In fact the Matt-Shuttleworth approach is easily adapted to fine-tune estimates of r sc if additional information on or assumptions about the conditions when values of K c were calibrated are made.To do this the calculation of r sc is merely made using the value r pref clim relevant during the period of calibration.If, for example, it is known or if it can be safely assumed that the value K c = 1 on a particular day in the season for the 1 m high hypothetical crop considered by Lhomme et al. (2014) had been calibrated in the sub-humid conditions that they specify, then it is the value of climatological resistance in these specified conditions that should be used as the preferred value when calculating r sc using the Matt-Shuttleworth approach.For the purpose of illustration, assume the clear sky conditions sub-humid conditions adopted by Lhomme et al. (2014) prevailed when this calibration was made, that the crop had an albedo of 23 % and the temperature was 20 • C and wind speed 2 ms −1 .In this case, with net long-wave radiation estimated from Eq. (5.22) in Shuttleworth (2012), the preferred value of climatological resistance to be used when calculating r sc from K c would be 35.5 s m −1 (corresponding to a Priestly-Taylor α = 1.107), and the corresponding equation used to calculate r sc from K c would be Consequently, the value of r sc for this crop on this day would be 89 s m −1 .
Similarly if the values of K c and h c given by Allen et al. (1998) during Stage 3 for cassava (in year one 0.8 and 1.0 m; and in year two 1.1 and 1.5 m, respectively), banana (in year one 1.1 and 3.0 m, and in year two 1.2 and 4.0 m, respectively), and millet (1.0 and 1.5 m, respectively) were assumed to have been calibrated in these same sub-humid conditions, then the equivalent values of r sc would be for cassava 182 and 61 s m −1 in years one and two, respectively; for banana 70 and 53 s m −1 in years one and two, respectively; and for millet 92 s m −1 .These values of r sc when applied in Eq. ( 5) in the same sub-humid conditions of course give the same estimates of evapotranspiration as FAO estimates in these conditions, as they should since both the value of K c and (in effect) r sc are assumed calibrated in these conditions.In different conditions the two estimates will differ to some extent, not least because the two approaches make different assumptions regarding crop aerodynamics.A similar approach could be used to derive r sc for crops that can be safely assumed to have had K c calibrated in semi-arid conditions.
3 How is surface resistance applied in the Matt-Shuttleworth Approach?
Because some crops have a crop height greater than 2 m, it is preferable to use the value of r sc in a version of PM which applies for a reference height of 50 m.This version of PM can be written in a form similar to that recommended by the United Nations Food and Agriculture Organization for calculating reference crop evaporation (see Eq. 23.37 of Shuttleworth, 2012).Thus, where u 2 (m s −1 ) and D 2 (k Pa) are the wind speed and vapour pressure deficit at 2 m, γ m ** = γ (1+r sc u 2 /R 50 c ), and (no units).( 7) Note that in Eq. ( 6) r clim is the value calculated by Eq. ( 1) using measured values of weather variables on the day that ET c is calculated, not r pref clim .

Concluding comments
If the crop water requirement community decided that it is preferable to use PM to estimate crop water requirements directly for all crops, the United Nations Food and Agriculture Organization could now update Irrigation and Drainage Paper 56 using the Matt-Shuttleworth approach, in default conditions using Eq. ( 3) to derive tabulated values of r sc from Table 12 of Allen et al. (1998), with a one-step estimate of ET c then directly made from Eq. ( 5).However, if there is a decision to update, arguably the first step should be to define specific values of r clim corresponding to sub-humid and semi-arid conditions by also specifying the available energy and temperature in such conditions, then to attempt to define for which crops it should be assumed the calibration of K c was made in sub-humid, semi-arid, and default conditions.
To facilitate the application of the Matt-Shuttleworth approach, I provide two Excel spreadsheets (amongst other files) at http://www.hwr.arizona.edu/~shuttle/Terrestrial~Hydrometeorology/ which are ancillaries to this paper.The first spreadsheet uses Eqs. ( 3) and ( 4) to duplicate the calculations of r sc in Table 23.5 of Shuttleworth (2012): it can be modified to make calculations for other combinations of K c and h c .The second spreadsheet is edited from that used to calculate Table 23.6 of Shuttleworth (2012) and makes example calculations of ET c using the Matt-Shuttleworth approach, i.e.Eqs. ( 5) and (6), and also using the traditional FAO method for several example crops (hypothetically) growing at three example sites (Oxford, Tucson, and Manaus) on three example days.It can be modified to make (or test) such calculations with alternative data from alternative sites.
pref clim the Allen et al. (1998) criterion that relative humidity is greater than 45 % is met at 20 • C for A mm values in the range 0-7 mm day −1 .By using the preferred values of temperature (20 • C) and wind speed (2 m s −1 ) and assuming a pressure of 100 k Pa in equations (23.34), (23.35) and (23.36) of Shuttleworth (