the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
| 20 Mar 2025
New derivation and interpretation of the complementary relationship for evapotranspiration
Abstract. The complementary relationship (CR) between actual evapotranspiration (ET) and apparent potential evapotranspiration (PETa) is widely used as a simple yet effective method for ET estimation. However, most existing CR formulations are empirical, lacking rigorous derivation based on physics. In this study, the complementary relationship was derived analytically with a physically meaningful parameter: the wet Bowen ratio, defined as the Bowen ratio when the surface becomes saturated. This parameter can be computed from observations without calibration. Fundamentally, the CR is shown to originate from partitioning of the net radiation, with ET directly linked to the latent heat and PETa proportional to the sensible heat. Additionally, ET is linearly related to and constrained by the energy-based potential evapotranspiration (PETe). The physically-based relationship among ET, PETa, and PETe has important implications for our understanding of the spatial and temporal variations in ET and would promote practical application of the complementary relationship for ET estimation across different environments.
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Status: open (until 26 May 2025)
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RC1: 'Comment on egusphere-2025-1124', Anonymous Referee #1, 21 Mar 2025
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The authors claim to have derived a new physically-based CR via ‘rigorous derivation based on physics’, unlike previous versions, which they deem only empirical.
In fact, what the authors achieve is making use of several hypothetical and highly speculative assumptions (lines 210-215):
i) The surface temperature of a small, freely evaporating water body is always the same as that of the surrounding drying land (this would require a heat conduction as effective as evaporative cooling, which is highly unlikely under realistic conditions, thus the corresponding potential evaporation rate remains speculative only);
ii) The Bowen ratio (βw) written for such a small water body does not change during drying of the environment (contradicting the constant surface net radiation assumption stated).
None of the above assumptions are valid in general and none been ever confirmed rigorously by any study.
They proceed further and claim that neither the Penman nor the Priestley-Taylor equation is appropriate for estimating the corresponding apparent potential evaporation rate or the evaporation rate of the wet environment, even though that these equations are the backbone of practically all existing CR methods. Yet, when they decide to discuss the practical applications of their version of the CR they turn to a modified version of the Penman equation with an empirical coefficient (k’) to be determined from measurements (eqs. 25 & 26). Note that the original Penman equation does not have this additional coefficient. Also, as the land surface temperature is typically unknown in practical applications, they introduce another empirical coefficient (α) to convert the Bowen ratio of equilibrium evaporation into βw in eq. (24).
One would expect that when a new method is introduced then its practical predictive superiority is showcased over existing similar methods it is supposed to replace. Such a validation is completely missing here.
The authors’ main equation (eq. 17), when combined with eqs. 12 and 15 yields simply: ET = Rn – H, which is a rather trivial formulation of the energy balance equation. All the authors do is combine this energy balance equation with the definition of the Bowen ratio and express them in a way that looks like a CR equation, i.e., their eq. 17. For βw they use the actual land surface and air temperature plus vapor pressure values (i.e., eq. 11) by capitalizing on assumption ii). An additional problem is that they still need to know H unless they employ the above mentioned modified Penman equation.
So what is the new insight from the authors’ ‘theoretically sound’ CR? I am not sure.
Based on these observations I can only recommend rejection of the manuscript. A thoroughly revised version of the manuscript that is not based on highly questionable assumptions [i.e., i) and ii)] could only be publishable if the authors demonstrate its practical predictive superiority (i.e., that it indeed leads to better ET estimates when differences in the number of parameters to calibrate and input requirements are properly accounted for) over existing CR models and drops any claim that it is a ‘theoretically sound’ and ‘rigorously derived’ CR version (in opposition to other existing CR versions) as all versions of the CR today are empirical to varying degrees, if not else then for the Penman equation (with its empirically derived wind-function) they employ.Citation: https://doi.org/10.5194/egusphere-2025-1124-RC1 -
AC1: 'Reply on RC1', Sha Zhou, 24 Mar 2025
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Please see the supplement for our response letter
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RC2: 'Reply on AC1', Anonymous Referee #1, 25 Mar 2025
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Reply to Zhou and Yu’s reply
I would like to reiterate (after reading the reply to a certain point, see below) that I maintain rejection of this MS. All assertive and unsubstantiated claims about what the authors propose as being the only physically meaningful CR and any claims that discredit established and thoroughly validated equations (i.e., Penman and Priestley-Taylor) should be dropped in a version that I could potentially recommend for publication, but that only after it has been shown that it indeed outperforms all existing (and not only those the authors cherry-picked) CR versions. Compared to other existing CR versions that are based on thermodynamics (see e.g., Crago, Qualls, Szilagyi and coworkers) this work does not add or reveal anything about the physics behind the CR so its novelty is minimal (if any). Its only novelty could be if it was shown to outperform other existing CR versions under the same number of calibrated parameters.
PET represents the rate of ET that would occur when water supply is unlimited at the
evaporative surface (i.e., a fully saturated surface). Since PET is constrained by available
energy (the net radiation), it is estimated using the energy-based approach and termed
as PETe in our study. PET cannot be directly observed unless the entire surface is
saturated, such as over a lake or the ocean. Our comparison of PET equations, including
Priestley-Taylor and Penman equations, demonstrates that PETe, estimated using net
radiation and the wet Bowen ratio (𝛽!), is the most reliable PET estimator (Zhou and Yu,
2024; 2025).
I checked out Zhou and Yu (2024) but I do not see a single direct comparison of ocean ET (from ERA5 or any other source) against estimates of PET whether it is by Priestley-Taylor or the wet Bowen ratio. So the above claim is not substantiated.
PETa represents the ET rate from a small, saturated surface within a larger, unsaturated
area (e.g., an evaporation pan), with energy supplied by both net radiation and the
surrounding environment. However, the energy transfer from the surroundings varies with
the pan size, leading to variations in the surface temperature of the wet area and the
corresponding evaporation rate. Due to the inherent ambiguity in the definition of PETa,
it is essentially indeterminate in practice. The lack of a definitive estimator further
complicates PETa estimation, making it challenging to develop a physically-based CR
formulation. To address this issue, we estimate the upper limit for PETa by assuming that
the surface temperature of the small wet area is maximized and equal to that of the
surrounding dry area (Ts).
This is not true either. The wet surface temperature is independent of its size, as it has been shown by Szilagyi and Schepers (2014) for large center-pivot irrigated circles. The Penman equation is the basis for all potential evaporation estimation based methods used for e.g., finding water demand of crops (see Allen et al., 1998, ISBN 92-5-104219-5), therefore PETa is far from being ‘indeterminate in practice’. It has well defined very practical formulations.
Complementary Relationship Framework:
The complementary relationship in essence describes the interplay between ET, PET,
and PETa. While their exact quantitative relationships remain uncertain across different
formulations (Table S2), all valid CR formulations must satisfy two boundary conditions:
• ET = PET = PETa under wet conditions.
• ET < PET < PETa as the surface dries up.
Comparison with Previous CR formulations:
We derived the CR formulation using two well-defined estimators of PET and PETa, along
with a physically meaningful parameter (𝛽!):
2
• Assumption ii) is used to estimate PET, i.e., PETe.
Maybe so, but it does not come from ‘rigorous derivation based on physics’ as the authors claim. There is no physical law that would stipulate that the wet Bowen ratio would remain valid during drying of the environment. In fact, the Bowen ratio for a realistic small wet surface in a hot dry environment is negative. This is so because the advected energy by the warmer wind over the cooler evaporating surface boosts the evaporation rate by providing energy toward the wet surface. This is the reason that over real wet surfaces ET rate decreases along the wet surface. In fact the entire CR is based upon this decrease. An evaporating surface having the exact same surface temperature as the surrounding drying surface is just an abstraction with little connection to the real world. But again, I am open to any abstractions as long as it results in improved ET estimates. But such validation is missing in this manuscript. Until a very credible, rigorous validation, all these assumptions hang in the air as their usefulness is unknown. After all, the proof of the pudding is in the eating.
• Assumption i) is used to estimate the upper limit of PETa by maintaining the
surface temperature of the small wet area equal to that of the surrounding
environment.
See my comment above: it is just an abstraction that never happens in reality by common natural processes.
Quite dissimilar to our approach, many previous CR studies in fact involve other two
assumptions, i.e., Assumptions a) and b) below. These assumptions are so commonly
made implicitly that they have been taken for granted. In fact, they have rarely been
clearly stated and validated.
• Assumption a): PETa can be estimated using Penman equation (PETpm) or pan
evaporation.
See the widely used irrigation water demand applications by FAO based on Allen et al. (1998). The Penman equation (or its variants) have been widely used by thousands of authors and practices. How can one say that it was not validated? Penman (1948) validated it first by employing evaporation pans, and since then there have been literally hundreds if not thousands of validations. How can one make such a bold claim without any basis to that claim?
• Assumption b): PET can be estimated using the Priestley-Taylor equation (PETpt).
The same goes for the Priestley-Taylor equation: there have been a large number of papers validating it.
We adopt Assumptions (i) and (ii) instead of Assumptions (a) and (b) for three main
reasons:
1) Reliability of PETe: Assumption (ii) provides a robust PET estimator, i.e., PETe. Both
Assumption ii) and the reliability of PETe have been validated in our previous studies
(Zhou and Yu, 2024; 2025). Importantly, these studies have demonstrated that PETe is
much better than PETpt in terms of estimating ET under wet conditions (e.g., over the
ocean) and that PETe can be used for estimating the PET over land based on the Budyko
framework.
See my comment above about the missing validation of this claim.
2) Indeterminacy of PETa: PETa is indeterminate, as its value varies with the size of the
small wet area. Additionally, the Penman equation should not be used to estimate PETa
as it neglects energy transfer from the surrounding environment (Zhou and Yu, 2024).
Not true and not true again. The ET rate along a wet surface is an exponentially decreasing function which means that after a certain extent (in the range of few tens or hundreds of meters) the average ET rate of the wet surface changes little by its size. The Penman equation does account for energy transfer as it was validated by actual small-pan measurements by Penman himself and it is the exact reason that in general the Penman ET rate is larger than the Priestley-Taylor ET rate.
We define an upper limit of PETa using Assumption i) to ensure a physically meaningful
CR formulation.
Again, I do not see why an evaporating small wet surface having the same surface temperature as the surrounding drying surface (which is never observed in reality under normal circumstances) would result in more ‘physically meaningful’ (not to mention exclusivity of the claim) CR than other existing CR approaches.
As discussed in Section 4.1, only when PETa is maximized can we derive
the CR with a physical meaningful parameter (𝛽! ). This is because the empirical
parameter, 𝑘, estimated through calibration with observations would diverge from the
Bowen ratio of an evaporation pan when its temperature is lower than the surface
temperature of the surrounding environment (Ts), and they converge to and give physical
meaning to the parameter 𝑘, i.e., 𝑘 = 𝛽!, only when the surface temperature of the pan
is the same as its surrounding environment. This resolves the empirical nature of many
previous CR formulations (Table 2).
3) Consistency with the boundary conditions: Our estimates of PETe and PETa allow
CR derivation from the fundamental energy balance equation (Rn=ET+H). They also
ensure consistency with two key boundary relationships, i.e., ET=PETe=PETa under wet
conditions and ET<PETe<PETa when the surface dries up (see Table 2 and Fig. 2). In
contrast, previous CR formulations often violate these conditions, particularly when PETpt
and PETpm are directly adopted to estimate PET and PETa, respectively. For instance,
using PETpt and PETpm directly results in inconsistencies under wet conditions, i.e.,
ET≠PETpt and PETpt≠PETpm (see Fig. S3 of Zhou and Yu, 2025 and Fig. 5 of Yang et
al., 2019). Since PETpt and PETpm fail to meet the required boundary conditions, they
should not be used to formulate the CR.
Not true and not true again. Yang et al. (2019) used a constant value of 1.26 for the PT-alpha in their Fig. 5. Several studies have shown (e.g., Andreas et al., 2013) how the value of the PT-alpha changes with temperature and possibly with other environmental variables. With the correct alpha value, the three evaporation terms become equal under wet conditions.
We agree that Penman and Priestley-Taylor equations have been extensively used in
hydrology, climate, agriculture, and many other relevant fields. However, recent research
has questioned their reliability in estimating PETa and PET (Milly et al., 2016; 2017; Greve
et al., 2019; Zhou and Yu, 2024; 2025).
The Penman equation (PETpm) combines PETe and PETa while eliminating the term Ts,
assuming that 1) the surface is saturated and 2) PET=PETe=PETa, which are only valid
under wet conditions (see Section 2 of Zhou and Yu, 2024). Direct application of PETpm
over unsaturated land leads to PET overestimation (due to dry atmospheric conditions,
such as higher vapor pressure deficit and warmer temperatures) and PETa
underestimation (as it neglects energy transfer from the surroundings).
Not true again. The Penman equation was parameterized (i.e., its wind function) so that the measurements came (as they should) from unsaturated conditions (see Penman’s original 1948 paper). The Penman ET rate should never be taken equal to the wet environment ET rate (i.e., PET here). Of course it overestimates PET then. For wet-environment ET we have the Priestley-Taylor equation where the slope of the saturation vapor pressure curve is evaluated under actual wet-environmental conditions. Many people do not realize it (including Zhou and Yu, it appears) that the two temperatures should be (and are) different in the two equations as long as the Penman equation is used under drying conditions. See the works of Crago and coworkers and Szilagyi and coworkers explaining this multiple times and specifying how the wet-environmental air temperature can be estimated under drying conditions.
To resolve this issue, an adjustment parameter 𝑘" = #$#/&#$#/&!(where 𝛽 is the Bowen ratio and 𝛽! is the wet Bowen ratio) can modify the Penman equation for estimating PET and PETa using routine meteorological data when Ts is unknown (Zhou and Yu, 2024).
The Priestley-Taylor equation (PETpt), a simplified version of PETpm with an empirical
coefficient (α), has commonly been used for PET estimation. However, PETpt exhibits
large biases in estimating ET over the ocean (Yang et al., 2019; Zhou and Yu, 2025),
making it unsuitable for PET estimation.
Again not true. What claimed here implicitly is that Priestley and Taylor (1972) never validated their equation successfully, when in fact they did. Since then on, with the correct PT-alpha value it is remarkably successful in estimating wet-environmental ET rates, provided the correct wet-environment air temperature is used when applied under drying conditions.
Moreover, PETpt overestimates the sensitivity of
PET to temperature, leading to an exaggerated increase in PET under warming climates
(Yang et al., 2019; Zhou and Yu, 2025). These issues can be resolved by using PETe
instead of PETpt.
Yang et al (2019) is not listed here, neither in the MS. So it must be Yang and Roderick (2019). It can overestimate PET with a fixed value of its alpha parameter (as Y-R do). As Andreas et al. (2013) and other workers since showed the value of alpha decreases with temperature, so any claimed overestimation thus likely corrects itself with the correct alpha value.
This study derives a CR formulation based on PETe, PETa, and the physically meaningful
parameter (𝛽!). Multiple approaches can be used to estimate PETe, PETa, and 𝛽!,
depending on data availability:
1) When Ts and sensible heat (H) are known (e.g., flux tower sites and reanalysis
products), 𝛽! can be calculated using equation (11), while PETe and PETa can be
derived from equations (12) and (15), respectively.
ET can be derived as Rn – H, that simple. No need for CR.
2) When Ts is available (from in situ or remote sensing observations) but H is not, PETa
can also be estimated using equation (14).
We have two equations, Rn = ET + H and the actual Bowen ratio, and two unknowns. No need for CR.
3) When both Ts and H are unknown, 𝛽! can be estimated from routine meteorological
observations using equation (23) (no, 24) and PETe and PETa using equations (24) (no, 25) and (25) (no, 26).
The equations are misnumbered in the MS text, starting with eq. 22 which (i.e., the numbering) appears twice in the MS. So any equation reference should be incremented by one after the first eq. 22 reference.
Based on these approaches to estimation of PETe, PETa and 𝛽! depending on data
availability, the newly derived CR formulation has significant potential for estimating ET.
We have discussed its advantages comparing with previous CR formulations (Table 2) in
Section 4.3 and discussed its practical implications in Section 4.4. However, applying the
5
CR formulation specifically for ET estimation is not the primary focus of this, for most part,
analytical work. A comprehensive and systematic evaluation of its effectiveness for ET
estimation, along with comparisons to other established ET estimation methods (such as
FLUXCOM and GLEAM), is an important next step. This should be addressed in a
dedicated future study to rigorously validate whether the CR formulation can offer
improved performance or advantages over existing approaches.
Disagree totally. There are so many bold claims here and in the MS discrediting established equations (such as the Penman and Priestley-Taylor equation) and discrediting all CR versions as unphysical as opposed to what the authors present here that the authors MUST show that their ‘superior approach’ indeed leads to better ET estimates. And that comparison must be done professionally, and in a way that unequivocally prove that what proposed in the MS indeed works in the real life.
In this study, we aim to advance our understanding of the complementary relationship by
clearly defining and estimating PETe and PETa and establishing the quantitative
relationships among ET, PETe, and PETa in a CR formulation with a physically
meaningful parameter (𝛽!). Notably, the CR formulation in equation (17) and the
relationships among ET, PETe, and PETa as shown in equations (18) and (19) have
been validated using data from 146 Fluxnet sites (see Fig. 2).
No, these are not validations. A validation involves the actual measured value of ET and its estimate and some statistical measure (e.g., regression plot, R2, MSE, MAE, NSE, slope of best fit equation, etc.).
We respectfully disagree with the characterization of the energy balance equation as a
trivial formulation. On the contrary, it plays a fundamental role in the complementary
relationship, which is governed by the partitioning of energy between latent and sensible
heat under wet and dry conditions (see Section 2). Changes in the partitioning between
the latent and sensible heat manifest themselves as a complementary relationship
between ET and PETa, as the latent heat is directly related to ET and the sensible heat
proportional to PETa with the wet Bowen ratio (𝛽!).
The key new insight is that by clearly defining and estimating PET and PETa, the
complementary relationship naturally emerges from the energy balance equation,
which serves as its foundation. This revelation and reification eliminate the need to
construct complex, non-linear relationships among ET, PET, and PETa that rely on
unknown empirical parameters, as done in many previous studies (see the review by Han
and Tian, 2020 HESS). In fact, CR formulations based on PETpt and PETpm fail to satisfy
the boundary conditions of the complementary relationship, and many existing CR
formulations are either special cases or unrealistic under certain conditions (see our
response to the first comment and Section 4.3).
Sorry, I cannot stand this assertive language any longer.Citation: https://doi.org/10.5194/egusphere-2025-1124-RC2 -
AC2: 'Reply on RC2', Sha Zhou, 26 Mar 2025
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Supplementary Response to Two Key Questions:
1. Why the Penman equation should not be directly used to estimate PET and PETa under dry (unsaturated) conditions
2. Why the Priestley-Taylor equation should not be used to estimate PET
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AC2: 'Reply on RC2', Sha Zhou, 26 Mar 2025
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RC2: 'Reply on AC1', Anonymous Referee #1, 25 Mar 2025
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AC1: 'Reply on RC1', Sha Zhou, 24 Mar 2025
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