The widely used Budyko framework defines the water and energy limits of catchments. Generally, catchments plot close to these physical limits, and

In this way, we found that differences in

We argue here that the non-contracted side of the framework in the two projections should always be assessed, especially for data points that appear as outliers. At least, one should consider the additional uncertainty of the projection and assess the robustness of the results in both projections.

Even though often referred to as the Budyko framework, the base of the framework was formed by the work of

More recently, the Budyko framework has gained popularity with several studies that use the framework for water balance assessment

A large number of studies consider the parameter in the Budyko framework to be catchment-specific and a function of local catchment characteristics. It
has been argued that this parameter explains local climatic and environmental conditions combined

Budyko formulated his curve with an aridity index as the independent variable, and most other publications followed that definition. From the older
and traditionally cited publications, only

The choice of the projection may depend on the purpose of a given study. Often, the projection with an aridity index is used as it allows for a
straightforward estimation of the runoff ratio (

Generally, the projections should not make a large difference, as the equations can be rewritten in the different formats

In order to address the research question, the Budyko framework was applied to a selection of catchments across the contiguous United States. An open
science approach was followed using the platform Renku (

The Budyko formulation adopted for our analysis was originally formulated by

In a similar way, Eq. (

These two formulations are often used interchangeably, and data can be plotted in figures based on Eqs. (

The exponent

Loss functions used for fitting the Budyko curves, from

In order to test the different hypotheses, the CAMELS data

positive long-term mean discharge:

positive long-term mean precipitation:

runoff ratio smaller than unity:

long-term actual evaporation not exceeding potential evaporation:

no lakes: water fraction

no snow-dominated catchments: mean elevation

relatively large catchments: area

Afterwards, the actual evaporation was determined based on the long-term water balance, assuming that storage change is negligible over a longer period
of time:

The research question was addressed by a simple approach. First, the Budyko curves were fit to the CAMELS data with the different loss functions as
defined in Sect.

CAMELS dataset plotted in different projections of the Budyko space, with

In the next step, the uncertainty in the estimated mean annual actual evaporation due to the different projections was assessed. This was done by selecting one catchment for the prediction of mean annual actual evaporation, whereas the remaining 356 catchments were used to fit the Budyko curve. This was again carried out in a projection based on a wetness index and a dryness index. As both estimates can be considered equally likely, the uncertainty was defined as the relative difference from the mean of the two estimates (i.e., the difference between the estimates equals 2 times the uncertainty). In addition, the predictions were evaluated by the relative error compared with the water balance based observed evaporation. The procedure was repeated for each catchment, leading to uncertainty estimates and relative errors for each catchment. Eventually, predictions were also made by just using the non-contracted side of the framework.

Fitted

Fitting the selected catchments of the CAMELS dataset to the two different projections led to different values for the

The results presented in Fig.

The above results clearly show that the projection used to fit the Budyko curve leads to different

Once a Budyko curve is fit to the data, the distance to this curve is often used as a metric for catchment analysis

Vertical distances to

Uncertainty bounds of predicted values of

These findings imply as well that exchanging projections of the Budyko curve is not as straightforward as it seems and may result in different
outcomes. Moreover, different interpretations can be given to these distances, as an increased distance to

Therefore, also here a consistent use of the framework is needed. As an aridity of 1.0 introduces a clear distinction between under- and
overestimating the distances to the curve and envelope in Fig.

The Budyko framework is often used to predict values of

Vertical distances to the envelope for a projection with a dryness index

The uncertainty in predicted values of

An important cause of the different

The outliers also influenced the relative errors when the curve was used to predict

The Budyko framework was applied to a selection of catchments across the contiguous United States, with two different ways to plot the framework. The first projection used a wetness index, whereas the second projection used a dryness index. First, curves were fit with a standard linear least-squares algorithm, followed by more robust methods afterwards. Distances of individual catchments to the curves and envelopes were determined, in order to assess to effects of the different projections. In the next step, we assessed the uncertainty in predicted values of actual evaporation due to the different projections.

In this way, we gained the following insights:

The differences in

Robust fitting algorithms reduced the differences in

The distances to the curve had a systematic dependence on the projection, with larger differences for the non-contracted side of the framework,
i.e.,

The resulting uncertainty in predicted values of

Data points can appear as outliers in one projection but not in the other, causing differences in the fitting of the curves
(Fig.

These findings show that the projection used needs to be considered carefully. Here, we would like to argue to assess always the non-contracted side
of the framework in the two projections. Catchments that seem close to the curve and the limits on the contracted side can easily appear as strong
outliers on the non-contracted side of the framework, as the absolute value of the relative errors changes on the

The full analysis, including all scripts and data, is available on Renku (

The supplement related to this article is available online at:

Analyses, preprocessing and post-processing of data were carried out by RCN. SJS and RCN contributed to the final text.

At least one of the (co-)authors is a member of the editorial board of

Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This study is part of the WAVE project, funded by the Luxembourg National Research Fund (FNR) ATTRACT programme (A16/SR/11254288). We thank NCAR for making the CAMELS data available (

This research has been supported by the Fonds National de la Recherche Luxembourg (FNR) ATTRACT programme (grant no. A16/SR/11254288).

This paper was edited by Adriaan J. (Ryan) Teuling and reviewed by two anonymous referees.