The Budyko framework posits that a catchment's long-term
mean evapotranspiration (ET) is primarily governed by
the availabilities of water and energy, represented by long-term mean
precipitation (

The Budyko framework represents a catchment's long-term mean
evapotranspiration (ET) as a function of the aridity index (

Despite this widespread application, several doubts have been raised about the robustness of the assumptions and interpretations that underpin this vast and growing literature, particularly with respect to the parametric Budyko equations. For example, Gentine et al. (2012) suggested that the aggregate Budyko curve behavior already reflects the interdependence among vegetation, soil, and climate; therefore, they proposed that the inclusion of the catchment-specific parameter into the Budyko framework is unnecessary. However, this interpretation was partially based on catchment data with limited scatter in Budyko space. Additionally, Greve et al. (2015) highlighted that the catchment-specific parameter has no a priori physical meaning, cannot be estimated for ungauged catchments, and its specific dependence on biophysical features can vary substantially between catchments. Furthermore, Padrón et al. (2017) undertook a comprehensive overview of the wide variety of biophysical features proposed to control the catchment-specific parameter, finding that most proposed features did not actually correlate with the parameter and the types of features that were correlated varied substantially between climatic regions. Finally, Sposito (2017a, b) suggested that the interpretation of the catchment-specific parameter as representing biophysical features does not arise from physical reasoning; thus, identified statistical relationships between the parameter and biophysical features may be spurious and premature.

Given the recent resurgence of the Budyko framework and its importance to catchment hydrology, we build upon these previous critical observations, presenting a retrospective review of the framework's assumptions and development, with the overarching goals of harmonizing historical and current interpretations as well as understanding their implications. Specifically, we critically reinterpret two key and interrelated assumptions of the current framework: (1) the concept that explicit curves represent trajectories of individual catchments through climate space, and (2) the parametric forms of the Budyko equation themselves. We contend that many current interpretations of these assumptions are unsupported by the underlying framework, potentially leading researchers to spurious conclusions about catchment hydrology. However, we stress that the aim of this reinterpretation is not to discard the voluminous efforts put forth using current interpretations of the Budyko framework but rather to recontextualize the conclusions obtained from them. Additionally, we emphasize that the Budyko framework based on the curve-like clustering pattern observed across multiple catchments is a powerful and useful concept when used appropriately and within the proper context.

We first reexamine interpretations of Budyko curves that ascribe physical
meaning to the functional form of the curve, thus implying that explicit
curves govern catchment evapotranspiration (e.g., C. Wang et al., 2016; Wang
and Hejazi, 2011; Jiang et al., 2015; Liang et al., 2015; Jaramillo et al.,
2018; Zhang et al., 2004, 2018). This concept is typically
articulated through the suggestion that an individual catchment undergoing
only changes in the aridity index will follow an explicit Budyko curve
trajectory (“the catchment trajectory conjecture”). However, we note that
it is mathematically impossible for the aridity index to vary independently
of other climate variables that impact PET or

Second, we revisit the parametric Budyko equations that are currently
interpreted by most authors to represent more generalized forms of the
nonparametric Budyko equations (Budyko, 1974), and which can thus be
used to separate the effects of changes in the average climate (i.e.,
changes in the aridity index

Additionally, while the catchment-specific parameters in the parametric
Budyko equations are typically regarded as empirical, “effective”
parameters analogous to, for example, Manning's roughness coefficient in
open channel flow or hydraulic conductivity in groundwater flow, we
demonstrate that this is not the case, as their values are not transferable
between catchments or across time for individual catchments. For an
empirical parameter to be transferable, the specific functional form of the
mathematical relationship in which it is contained must be empirically
valid. In such cases (e.g., Manning's formula and Darcy's Law), the
validated functional form contains information about the physics of its
respective system, allowing for the empirical parameter to be consistently
and independently related to physical properties of the system (e.g.,
channel surface roughness for Manning's roughness coefficient and soil pore
size for hydraulic conductivity). In these cases, the effective empirical
parameters can be estimated a priori, allowing their respective empirical
relationships to be used for making quantitative predictions under future
conditions (e.g., different hydraulic gradients). We test the empirical
validity of the parametric Budyko equations, with results suggesting that
the catchment-specific parameter is nontransferable. Thus, the value of
the catchment-specific parameter cannot be determined without first obtaining
estimates of

Our reinterpretation is demonstrated theoretically using a stochastic soil moisture model (Porporato et al., 2004) as well as empirically using data from 728 reference catchments in the United Kingdom (UK) and United States (US). To provide context for these analyses, we first provide a brief background of the Budyko framework, describe its current dominant interpretations in the literature, and recall Budyko's own interpretation of explicit curves.

In its foundation, the Budyko framework is an expression of the water
balance for a catchment. Over long time periods, it is reasonable to assume
that positive and negative short-term changes in catchment storage average
to negligibly small values (

Given this limitation, the original Budyko hypothesis has been modified in
an attempt to explain deviations of individual catchments from the explicit
Budyko curves by invoking a function that is implicit in ET
(Yang et al., 2008):

Equation (4) has been interpreted as indirectly capturing unknown
catchment-specific factors impacting ET, other than

Most current interpretations of the functional forms of Budyko curves explicitly acknowledge their semi-empirical nature; however, many studies simultaneously ascribe specific physical meaning to the mathematical expressions. This interpretation suggests that the curves represent trajectories within Budyko space that a catchment will follow if its aridity index changes, which supposedly allows one to make predictions about ET under different climates (e.g., Roderick and Farquhar, 2011; Wang and Hejazi, 2011; Yang and Yang, 2011; W. Wang et al., 2016; Zhou et al., 2016; Shen et al., 2017; Zhang et al., 2016; Milly et al., 2018). Critically, this interpretation extends the concept of an explicit curve from its representation of an emergent global behavior of multiple catchments to the behavior of individual catchments, implying that the mathematical expressions describing Budyko curves represent fundamental catchment hydrological processes associated with the aridity index. The specific details of these catchment processes are considered to be unknown, but their integrated effects are represented in the functional form of the explicit curves.

Current interpretations of the catchment-specific parameter follow from the application of explicit curves to individual catchment behavior. Generally, these interpretations can be grouped into four distinct viewpoints. (1) The catchment-specific parameter is an effective empirical parameter related to biophysical features, and it is possible to discern and understand that relationship (e.g., C. Wang et al., 2016). (2) The parameter is related to biophysical features, but it may not be possible to determine an explicit relationship; therefore, it should be treated probabilistically (Gudmundsson et al., 2016; Greve et al., 2015; Singh and Kumar, 2015). (3) The catchment-specific parameter and parametric forms of the Budyko equation contradict the Budyko hypothesis (Sposito, 2017a, b; Gentine et al., 2012). (4) The parameter is an arbitrary empirical constant that is generated as a part of the solution to Eq. (4), but it has no a priori physical meaning (Greve et al., 2015; Sposito, 2017b; Daly et al., 2019a). In particular, the idea that the catchment-specific parameter is an effective empirical parameter related to biophysical features (i.e., interpretation 1) has been widely embraced by the catchment hydrology community, which has identified and grouped relevant biophysical features into three categories (Donohue et al., 2012; Harman and Troch, 2014): (1) climate variability, (2) catchment physical processes, and (3) vegetation structure and function. While it is generally well acknowledged that certain climatic variables (e.g., precipitation variability or the fraction of precipitation falling as snow) can influence the catchment-specific parameter (e.g., Roderick and Farquhar, 2011; Berghuijs and Woods, 2016), in practice, many studies effectively neglect this, instead focusing primarily on the role of landscape features or vegetation functioning (C. Wang et al., 2016; Zhang et al., 2004, 2018; Yang et al., 2008, 2016; Greve et al., 2015; Xu et al., 2013; Donohue et al., 2012; Knighton et al., 2020; Gao et al., 2020; Chen et al., 2020; Wu et al., 2019; Qiu et al., 2019; J. Liu et al., 2019b; Liu et al., 2020; Guo et al., 2019).

The widely held interpretations of explicit curves representing individual
catchment behavior and the catchment-specific parameter representing
biophysical/landscape features has led to the development of methods to
determine the sensitivity of precipitation partitioning to climate and/or
landscape changes for individual catchments (Roderick and Farquhar,
2011; Wang and Hejazi, 2011; Yang and Yang, 2011; W. Wang et al., 2016; Zhou et
al., 2016; Shen et al., 2017; Yeh and Tsao, 2020; Zhang et
al., 2016, 2020; Sinha et al., 2020; Ning et al., 2020b; J. Liu et al., 2019a; Liu et al., 2020; Li et al., 2018, 2019; Z. Li et al., 2020b; H. Li et al., 2020; Yang et al.,
2018; Xing et al., 2018b; Xiangyu et al., 2020) and multiple
methods for decomposing anthropogenic and climatic impacts on precipitation
partitioning (Wang and Hejazi, 2011; Xing et al., 2018b; Jaramillo et
al., 2018; Mo et al., 2018; Sun et al., 2014; Jiang et al., 2015; Liang et al.,
2015; Huang et al., 2016; Zhang et al., 2020; Yeh and Tsao, 2020; Xiangyu et
al., 2020; Song et al., 2020; Sinha et al., 2020; Z. Li et al., 2020a; H. Li et al.,
2020a; Deng et al., 2020; Zhang et al., 2019a; Young et al., 2019; Xin et al.,
2019; Wang et al., 2019; Lv et al., 2019; N. Liu et al., 2019; Lee and Yeh,
2019; Kazemi et al., 2019; Y. He et al., 2019a, b; G. He et al.,
2019; Wang et al., 2018; Xu et al., 2014). Additionally, these
interpretations have led numerous studies to pursue predictive relationships
for the catchment-specific parameter based on various biophysical features
(Table S1 in the Supplemental Information) (Yang et al., 2007; Donohue et
al., 2012; Yang et al., 2009; Shao et al., 2012; Li et al., 2013; Xu et al.,
2013; Cong et al., 2015; Yang et al., 2016; Zhang et al., 2018; Abatzoglou and
Ficklin, 2017; Xing et al., 2018a; Zhao et al., 2020; Ning et al., 2020b; Ning
et al., 2020a; T. Li et al., 2020; S. Li et al., 2020; Zhang et al., 2019b; Ning et
al., 2019; Bai et al., 2019; Ning et al., 2017). However, these relationships are
all statistical or derived from curve fitting, which makes it difficult to
develop a mechanistic understanding of causal relationships between the
catchment-specific parameter and relevant biophysical features.
Additionally, interpretations of these relationships implicitly assume that
the functional forms of either Eqs. (5) or (6) represent a physically
meaningful relationship between the aridity and evaporative indices, which has not been empirically validated as previously noted by
Berghuijs et al. (2020). An explicit derivation of

It is useful to recall that Budyko (1974) considered explicit curves to be
semi-empirical. While the physical basis for explicit curves is dictated by
the conservation of mass and energy (i.e., the curves could never cross the
water and energy limits in Budyko space) and the fact that the curves must
approach the energy and water limits for increasing humidity (i.e.,

“The choice of one or another interpolation function for the transition from the first of these conditions to the second is not very important, since, over most of the range of variation in the parameters of the relationship equation, the appropriate relation deviates little from one or the other boundary condition.”.

From this interpretation, it is clear that the explicit curves resulting from the original Budyko hypothesis, while constrained at their limits by fundamental physical laws, are empirical in nature and not derived from catchment hydrologic processes. It should also be noted that the explicit curve relationships were developed to describe the general behavior of multiple catchments over a wide range of aridity indices. This gives the nonparametric Budyko curves (e.g., Eq. 3) some predictive power, albeit in a probabilistic sense. Any given individual catchment would, on average, be expected to fall close to the explicit curves but, in principle, could fall anywhere in Budyko space. Therefore, predictions of ET using the original Budyko curves have a quantifiable uncertainty associated with them. Budyko and Zubenok (1961) showed that this mean error was approximately 10 %, which has been confirmed more recently (Gentine et al., 2012).

Given this background, it is important to recognize the difference between applying a semi-empirical curve to describe the general behavior of aggregated catchments and using a similar curve to represent the trajectory of an individual catchment undergoing changes in aridity. The original Budyko curve emerges from the ensemble characteristics of many catchments across a range of aridity indices. Suggesting that Budyko curve behavior applies to the trajectories of individual catchments may be a reasonable conjecture, but it requires either theoretical justification or empirical validation, both of which are currently lacking. In the following sections (Sect. 3.1.1, 3.1.2), we describe our methods for testing this assumption using both theoretical models and empirical data.

To test the catchment trajectory conjecture, we employed the biophysical
stochastic soil moisture model of Porporato et al. (2004). This model,
being physically based, has been used to lend support to Budyko curves and
in developing relationships between

We first write the model of Porporato et al. (2004) in a form that can
be plotted in Budyko space:

We tested the catchment trajectory conjecture by varying the model climatic
parameters while holding the landscape parameter constant. If the resulting
trajectories are not Budyko curves, the conjecture should be rejected.
Notably, there are five qualitatively distinct ways that

The effective climate and landscape parameters in the Porporato model appear exclusively in ratios, such that only the relative magnitude between parameters is important. Moreover, the same trajectories can be made from infinite parameter combinations. For our test cases, we chose parameter values to maintain illustrative simplicity and to produce visually informative trajectories not restricted to a small portion of Budyko space.

Our empirical test of the catchment trajectory conjecture involves tracking the actual trajectories of reference catchments in Budyko space over time and quantifying whether they follow Budyko curves. Reference catchments are defined based on long-term stability of land use. Therefore, any changes to precipitation partitioning over time in reference catchments must be attributed to climatic factors, and the catchment trajectory conjecture predicts that their expected trajectories through Budyko space must be Budyko curves (i.e., those described by Eqs. 5 or 6). This prediction can be tested by comparing actual Budyko space trajectories of reference catchments computed from empirical observations against the expectation from the catchment trajectory conjecture. If the observed reference catchment trajectories are distinct from the expected Budyko curve trajectories, the conjecture should be rejected.

For a given reference catchment, estimates of

The conjecture was tested for each reference catchment by comparing all realizations of actual trajectories to the conjectured Budyko curve trajectory using the nonparametric sign test (Holander and Wolfe, 1973). This is a distribution-free test for consistent over- or underestimation between paired observations (see also Sect. S2). If the catchment trajectory conjecture is correct, then the frequency at which actual and expected Budyko space trajectories are found to be statistically indistinguishable will be higher than what is expected due to random chance (see also Sect. S2). Moreover, we calculated the maximum deviations of the actual trajectories (using the 10-year averaging window) from the expected Budyko curve trajectory for all reference catchments. These values represent the largest magnitudes of climate-induced changes in precipitation partitioning that would be misinterpreted as land-use-induced changes when subscribing to the catchment trajectory conjecture. Finally, we estimated the magnitude of the largest errors in the evaporative index that occurred when using the well-established nonparametric Budyko curve instead of the parametric form. This was done by calculating the maximum deviations between Eq. (3) and the actual trajectories (10-year averaging window) for all reference catchments.

Our empirical tests were based on 728 UK and US reference catchments
identified from well-accepted peer-reviewed datasets. These datasets were
produced using standardized methodologies with well-documented quality
control standards. The 68 UK catchments (Fig. S1a) were from the Catchment
Attributes and MEteorology for Large-sample Studies for Great Britain
(CAMELS-GB) dataset (Coxon et al., 2020a), which also had membership
in the UK Benchmark Network (UKBN2) dataset (Harrigan et al.,
2018) and had the highest data-quality metric (a benchmark score of 6).
UKBN2 reference catchments have been identified as “near-natural” and are
intended to be used for the investigation of climate-driven changes in river
flow. The CAMELS-GB dataset contains daily time series of discharge, precipitation, and
potential evapotranspiration for each catchment with contiguous record lengths of between 12 and 45 years. The 660 US reference catchments (Fig. S1b) were from the original
CAMELS dataset (Addor et al., 2017; Newman et al., 2015). All catchments
in the CAMELS dataset are considered reference catchments, with minimal
land use changes or disturbances and minimal human water withdrawals
(Newman et al., 2015). Daily times series of discharge, precipitation,
maximum temperature, and minimum temperature with contiguous lengths between 20 and 35 years were
available for each US reference catchment. Daily potential evapotranspiration time series were
computed from the daily

The aridity indices of the 728 UK and US reference catchments span from 0.13
to 5.93 (300 are arid,

To understand the limitations of the catchment-specific parameters within
the parametric Budyko framework, it is illuminating to first review their
origin. In the derivations of both forms of the parametric Budyko equations
(Eqs. 5 and 6),

Empirical relationships with effective parameters are common and useful in hydrology (e.g., Manning's formula and Darcy's Law). The usefulness of such relationships comes from their transferability either between similar physical systems or within the same system at different times. For example, Darcy's Law states that under certain constraints (i.e., small flow velocities and laminar flow) the flux of water through a porous medium will change linearly with changes in the hydraulic gradient. As long as the flow velocities within the given medium remain small, the slope of the relationship between the hydraulic gradient and flux (i.e., the hydraulic conductivity) will remain constant, meaning its value is transferable across time for that porous medium system. The linear gradient–flux relationship holds for a wide range of different porous media, which allows the slope of the relationship to be independently related to physical properties of the various systems (e.g., pore size distributions, Wang et al., 2017). Therefore, the hydraulic conductivity can be estimated a priori from information independent of the hydraulic gradient and flux, and its value can be consistently transferred between systems with similar properties (i.e., those with similar porous media). For the parametric Budyko equations to be useful empirical relationships analogous to Darcy's Law, the functional forms of Eqs. (5) and (6) must be empirically valid. Specifically, the formulae must be shown to describe how a catchment's evaporative index changes for a given change in the aridity index (i.e., the catchment trajectory conjecture would need to be shown to be valid).

We test the empirical validity of the parametric Budyko framework and the transferability of the catchment-specific parameter with our empirical test of the catchment trajectory conjecture using the 728 UK and US reference catchments (Sect. 3.1.2). This analysis tests the hypothesis that catchments' evaporative indices follow parametric Budyko curves through Budyko space when undergoing changes in the aridity indices. Our test of the transferability of the parametric Budyko curves is directly analogous to testing the linear gradient–flux relationship for Darcy's Law.

Equations (5) and (6) are the most widely accepted and frequently used
single-parameter Budyko equations. The following properties of these
equations are used either as foundational constraints in their derivation or
to justify their validity in describing Budyko space: (1) they describe a
family of concave down, nonintersecting curves; (2) these curves satisfy
conservation of mass and energy; (3) every point within Budyko space belongs
to only one curve (i.e., the uniqueness requirement); (4) the values and
first derivatives of all curves approach 0 and 1, respectively, in the humid
limit (i.e.,

Commonly held interpretations about the parametric Budyko equations, such as the catchment trajectory conjecture, explicitly or implicitly ascribe physical meaning to the specific mathematical functions (e.g., Eqs. 5 and 6) that describe single-parameter curves (e.g., individual catchment trajectories). However, different (but valid) single-parameter Budyko curves described by nonequivalent functions will produce contradictory results when used in hydrological applications (e.g., causal attributions and sensitivity analyses). Such results suggest that the physical interpretations attributed to specific functional expressions of Budyko curves are unfounded.

To illustrate the contradictory nature of the parametric Budyko equations,
we compare behaviors of Eqs. (5) and (6) to those of two new relationships
that also conform to all of the properties of Eqs. (5) and (6) and have
analogous parameters:

Resulting trajectories of the theoretical test of the Budyko curve
conjecture plotted in Budyko space. The energy and water limits of Budyko
space are given as solid black lines.

The theoretical test of the catchment trajectory conjecture for cases 1
(variable storm size) and 2 (variable storm frequency) generally resemble
Budyko curves in that they are monotonically increasing, concave down, and
approach the energy and water limits as

The Budyko space trajectory for test case 5 (variable precipitation
flashiness) is a vertical line at

Semi-log plot of the Budyko space locations (black dots) of the 728 UK and US reference catchments and their corresponding expected Budyko curves (Eq. 5, gray dashed curves) and 10-year average actual trajectory realizations (red solid curves). The global behavior of the catchments and their actual trajectories generally agree with the nonparametric Budyko curve (Eq. 3, blue solid curve) but not with the expected parametric Budyko curves.

The empirical test of the Budyko curve catchment trajectory conjecture evaluated whether real-world reference catchments not subjected to significant land use change actually follow Budyko curve trajectories over time. The catchments investigated span a wide range of aridity indices, climate zones, latitudes, longitudes, and vegetation types, and the global behavior of their long-term mean water balances is in agreement with the nonparametric Budyko curve (Fig. 2). However, individual catchments do not generally follow parametric Budyko curve trajectories (Figs. 2, 3a), implying significant errors in the prediction of precipitation partitioning sensitivity based on the catchment trajectory conjecture (Fig. 3b). In addition to the theoretical test of the conjecture (Sects. 3.1.1, 4.1.1), the qualitative and quantitative results of this empirical test provide further evidence against the conjecture.

The data for the 728 UK and US reference catchments are shown in Budyko space with their corresponding expected and actual Budyko curve trajectories in Fig. 2. The data generally cluster in a manner reflective of the well-known nonparametric Budyko curve behavior (blue solid curve). Additionally, the aggregate behavior of the actual trajectories (red solid curves) also generally follows the nonparametric Budyko curve. However, there are significant discrepancies between the shape of the overall ensemble cloud of catchments and their actual trajectories versus the corresponding conjectured trajectories (gray dashed curves) for most individual catchments. Many of the curves that would be expected based on the catchment trajectory conjecture span regions of “unpopulated” Budyko space where actual catchments are rarely observed.

Comparison of actual catchment trajectories with their
corresponding expected Budyko curves (Eq. 5) suggested by the catchment
trajectory conjecture.

Nonparametric sign tests showed that none of the reference catchments
consistently followed the Budyko curves that would be expected based on the
catchment trajectory conjecture (i.e., for multiple realizations of actual
trajectories using different averaging window sizes). From the total of
24 501 actual trajectory realizations, 23 231 (95 %) were found to have
consistent differences (

Figure 3a gives examples of actual trajectory realizations (10-year average) that are statistically distinguishable (red curves) and indistinguishable (blue curves) from their expected trajectories (black dotted curves). The maximum deviation between the actual evaporative index (10-year average) and those determined from expected trajectories shown in Fig. 3a is 0.14, corresponding to an absolute relative error of 212 %. Figure 3b gives a histogram of the maximum absolute relative errors in the evaporative index between the 10-year average actual trajectory realizations and expected trajectories for all 728 reference catchments, truncated to a maximum value of 225 %. The locations of the errors associated with the example trajectories in Fig. 3a are given by arrows in Fig. 3b, with their colors (red or blue) corresponding to the trajectory's statistical distinguishability. The full range of evaporative index errors spanned from 0.4 % to 1991 %, with a mean of 26 %. The mean error closely agrees with the value (27.9 %) found by Berghuijs and Woods (2016) in a comparable test of the catchment trajectory conjecture using Eq. (6) and 420 catchments from the MOPEX (Model Parameter Estimation Project) dataset (Schaake et al., 2006). Importantly, the average relative error for the parametric Budyko framework (26 %) is actually larger than that for Eq. (3) (23 %), which suggests that the nonparametric Budyko curve is in better agreement with the global behavior of catchments than the ensemble of parametric curves specifically fit to the individual catchments.

From these results, we can conclude that individual catchments do not generally or consistently follow Budyko curve trajectories as posited by the catchment trajectory conjecture. As such, the use of this conjecture in hydrological analyses (e.g., precipitation partitioning sensitivity and causal attribution to anthropogenic and climatic impacts) will likely introduce significant errors and may lead to spurious conclusions.

Illustration of the nonuniqueness of the parametric Budyko
equations using

The results of our empirical test of the Budyko curve catchment trajectory
conjecture (Sects. 3.1.2, 4.1.2) strongly suggest that the parametric
Budyko equations do not describe the long-term evaporative behavior of
individual catchments (i.e., they are not empirically valid). This further
suggests that their specific functional forms are not physically meaningful,
and the catchment-specific parameter cannot be independently related to
physical properties. Thus,

Due to this nontransferability and proxy relationship, it is not possible
to solve for

Illustration of the contradiction between different versions of the four parametric Budyko equations. Constant parameter trajectories, defined by each of the four parametric equations, cross one another. This means that if a catchment has a constant parameter trajectory in one formulation, the parameter must change for the other formulations.

The inability to estimate

In principle, with an appropriate interpretation of the catchment-specific
parameter, use of the parametric Budyko framework in landscape hydrology is
benign, if unnecessary. However, in practice, even with an appropriate
interpretation of

While the acknowledgment of the proxy nature of the catchment-specific
parameter and

If the family of curves described by parametric Budyko equations are
interpreted as trajectories for catchments undergoing changes in aridity,
then each possible parametric Budyko equation contradicts all others, as
each gives specific but nonequivalent functional forms for the trajectories.
Even Eqs. (5) and (6), which are generally regarded as essentially
interchangeable when using the approximate relationship,

The parametric Budyko equations described by Eqs. (5), (6), (9), and (10)
represent four equally valid families of curves (Fig. 4) in that they are
all monotonically increasing, concave down, and approach the energy and
water limits as

Of the previously proposed parametric Budyko equations, Eqs. (5) and (6) have been the most widely used (e.g., Donohue et al., 2012; Yang et al., 2007, 2009, 2016; Shao et al., 2012; Li et al., 2013; T. Li et al., 2020; S. Li et al., 2020; Xu et al., 2013; Cong et al., 2015; Zhang et al., 2018, 2019b; Abatzoglou and Ficklin, 2017; Xing et al., 2018a; Zhao et al., 2020; Ning et al., 2019, 2020a, b; Bai et al., 2019). Any of these studies could have justifiably used Eqs. (9) or (10) instead, as there is not an objective reason to choose any one over the others. However, each equation would lead to substantially different and potentially contradictory results. For example, methods for predicting the sensitivity of precipitation partitioning to changes in the aridity index or the catchment-specific parameter (Sect. 2.2) rely on the specified shape of the Budyko curve. The use of Eq. (5) to compute sensitivities would produce substantially different results compared with those produced from Eq. (10). Additionally, methods for attributing changes in precipitation partitioning to anthropogenic and climatic changes (Sect. 2.2) will produce contradictory conclusions when using one parametric Budyko formulation compared with using another.

It is important to note that Eqs. (5), (6), (9), and (10) are not the only potential parametric Budyko equations. In fact, the Porporato model (Eq. 8) can be manipulated into a single-parameter Budyko equation (e.g., Harman et al., 2011; Daly et al., 2019b). There are likely many more, all equally valid, versions with even starker differences in the shapes of the curves (leading to even larger discrepancies between formulations if the current interpretations of explicit Budyko curves and parametric Budyko equations are maintained). This “equifinality” and nonuniqueness of the parametric Budyko equations is incompatible with the overwhelming current interpretation of the parametric framework and lends support to our contention that the parametric Budyko formulations are better understood as arbitrary coordinate transformations between alternative representations of Budyko space.

The original Budyko hypothesis given in Eq. (2) and the resulting nonparametric curve (e.g., Eq. 3) provide an overarching framework for understanding catchment hydrology in terms of energy and water balances. As the development of the Budyko framework advanced over the past century, early conceptual tools, such as explicit functional curves, gained considerable influence, resulting in interpretations that are not actually supported by the framework and which may lead to spurious conclusions. In this study, we have revisited, summarized, and critically evaluated these interpretations, leading to a reinterpretation of explicit Budyko curves and the parametric Budyko equations.

It is apparent from the literature that the prevailing interpretation of
explicit Budyko curves ascribes undue physical meaning to the explicit
mathematical expression describing the curve. By returning to Budyko's own
interpretation of explicit curves, we saw that earlier conceptual frameworks
considered the specific choice of functional form to be arbitrary as long as
the curves suggested conservation of energy and mass in the humid and arid
limits and provided a good representation of the global behavior of multiple
catchments across a range of aridity conditions. We reinforce that the general global
Budyko curve behavior observed across multiple catchments is a valid, well-documented, and physically driven phenomenon. However, the attribution of
physical meaning to the specific functional forms of curves, as well as explicitly
interpreting them as trajectories for catchments undergoing changes in
aridity, is an unsupported conjecture. Our tests of this conjecture showed,
both theoretically and empirically, that conceptualizing Budyko curves as
trajectories is unjustified. Therefore, as an alternative to using explicit
Budyko curves to understand catchment trajectories, we reiterate the
long-standing suggestion (e.g., Eagleson, 1978; Milly, 1994; Daly and
Porporato, 2006; Rodriguez-Iturbe et al., 1999; Feng et al., 2015, etc.)
that process-based evapotranspiration models should be used. Additionally,
to be a valid representation of catchment evapotranspiration, process-based
models need to able to reproduce the empirically established, nonparametric
Budyko curve behavior when applied to multiple catchments across a range of
climates. Thus, the general Budyko curve behavior can serve as a global
constraint (i.e., calibration or validation) in the application of such
models (e.g., Greve et al. 2020). Furthermore, while the
parametric Budyko framework lacks predictive power, the nonparametric
framework allows for probabilistic predictions of ET and

A literature review suggests that most current interpretations view the
parametric Budyko equations as more general and versatile forms of the
nonparametric Budyko equations. We illustrated that the parametric Budyko
equations are underdetermined, lack predictive power, and are nonunique,
merely serving as a coordinate transformation between Budyko space and
Budyko curve space. Coupled to current interpretations of the parametric
equations is the idea that the catchment-specific parameter is a lumped
quantity that represents the influence of catchment biophysical features on

In closing, we recommend that improved understanding of ET should emerge from fundamental physical and biological controls, utilizing the empirically validated global Budyko curve behavior as a constraint, rather than ascribing undue meaning to arbitrary functional forms or ambiguous parameters. As with any empirical relationship, extrapolating the use of the Budyko curve beyond the regime for which is was developed is unjustified without additional evidence. By doing so we risk drawing spurious conclusions about the hydrologic functioning of landscapes. Empirical relationships, such as the Budyko curve, emerge from the underlying physics within a given context, but those relationships are susceptible to losing their physical foundations outside of that context.

The data used in this paper can be obtained from the following locations: the CAMELS-GB database
(

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NGFR conceived the study, compiled the data, performed the analyses, and drafted the paper. All authors contributed in the methodological design, interpretation of results, and paper preparation.

The contact author has declared that neither they nor their co-authors have any competing interests.

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Nathan G. F. Reaver received support from a University of Florida graduate fellowship.

This paper was edited by Luis Samaniego and reviewed by two anonymous referees.