The Budyko formulation adopted for our analysis was originally formulated by as traced back by but used afterwards by, amongst others, and : 1Ea‾=Ep‾P‾P‾n+Ep‾n1/n, with E‾p the mean annual potential evaporation, E‾a the mean annual evaporation, P‾ the mean annual precipitation and n a shape factor, assumed to represent catchment characteristics (e.g., vegetation, soils). This equation can be reformulated by dividing the left-hand side and right-hand side by P‾, followed by dividing the nominator and denominator on the right-hand side by P‾ as well, leading to 2E‾aP‾=E‾pP‾E‾pP‾n+1-1/n.

In a similar way, Eq. () can be expressed by the ratio of P‾/E‾p as the dependent variable. First, both sides of Eq. () are divided by Ep‾ again, followed by dividing the nominator and denominator on the right-hand side by Ep‾ (see also Supplement S1): 3E‾aE‾p=P‾E‾pP‾E‾pn+1-1/n.

These two formulations are often used interchangeably, and data can be plotted in figures based on Eqs. () or (). We will adopt here dryness index projection and wetness index projection throughout the paper for projections based on Eqs. () and (), respectively, to refer to these different ways of applying the Budyko framework.

The exponent n in Eqs. () and () was fit to data of multiple catchments with a least-squares fit based on the Levenberg–Marquardt algorithm python scipy.optimize.curve_fit, https://docs.scipy.org/doc/scipy/reference/generated/scipy.optimize.curve_fit.html, last access: 10 February 2022, . Normally, this algorithm minimizes the sum of the squared residuals; i.e., it uses a linear least-squares loss function. Afterwards, instead of using a linear least-squares loss function, other loss functions to minimize the residuals were used, in order to obtain a robust fit. These loss functions ρ(z) are summarized in Table , and the final, resulting loss function is defined as 4ρ′=C2⋅ρxr2/C2, with xr the residual of data point x, C a scale parameter, ρ′ the resulting loss and ρ() the loss function (see Table ). The scale parameter C generally separates outliers from the data and was given different values between 0.1 and 1 in order to vary the data points that are considered as outliers, where low values of C classify the most data points as outliers. Note that C=1 with a linear loss function results in an ordinary least-squares fit again.

In order to test the different hypotheses, the CAMELS data were used, as they provide a large dataset of 671 catchments across the contiguous United States. For each catchment in this dataset, daily discharge, rainfall, potential evaporation and air temperature are available. Eventually, 357 catchments were selected based on several conditions similar to .

positive long-term mean discharge: Q‾ ≥ 0 mmyr-1

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: 5E‾a=P‾-Q‾, with P‾ the mean annual precipitation, Q‾ the mean annual discharge and E‾a the mean annual actual evaporation. In this way, all water balance components are known to plot the data in the Budyko space.

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. , in the two different projections. This was done for the selected 357 catchments all together, as well as for catchments grouped by a high aridity (Ep/P>1, 247 catchments) and a low aridity (Ep/P≤1, 110 catchments), the latter to assess whether differences start to occur when catchments are dominantly in either the contracted side of the framework (i.e., Ep/P≤1 or P/Ep≥1) or the non-contracted side of the framework. The vertical distances to the curve as well as the distances to the envelope of the physical limits were calculated for the different projections.

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.

Fitting the selected catchments of the CAMELS dataset to the two different projections led to different values for the n exponent in Eqs. () and () (Fig. a and b, n = 2.254 and n = 2.037, respectively). These n values differed even stronger when the catchments were separated into two groups based on their aridity (Ep/P>1 and Ep/P≤1, respectively, Fig. c and d). In particular, for the energy-limited catchments (Ep/P≤1, shown in red), the values changed strongly from an n value of 2.181 in the projection with a dryness index (Fig. c) to a value of 1.967 in the projection with a wetness index (Fig. d). The differences that occurred when the curves with the two different n values from the two different projections were used in the same projection and subtracted from each other (Fig. e and f) also show that especially the curves based on energy-limited catchments strongly deviated (Ep/P≤1, shown in red). In contrast, the curves obtained for water-limited catchments (Ep/P>1, blue) remained more similar, with negligible differences.

The results presented in Fig.  also strongly depended on the choice of the method, which was here a linear least-squares fit. Repeating the analysis with more robust methods (see Table ) led to smaller differences between n values in the two projections, even though differences were still present (Fig. a). In particular, the scale parameter C (Eq. ) that identifies data points as outliers had a strong effect on the resulting n values when set to a larger value. Nevertheless, differences still occurred for small values of this scale parameter, i.e the most stringent values that classify the most data points as outliers, even though these differences became relatively minor. In contrast to what was found with the linear least-squares method, the robust methods resulted in differences in n values for the water-limited catchments (Fig. c, differences between blue and red points) that are generally bigger than the differences in n values for the energy-limited catchments (Fig. b, differences between blue and red points).

The above results clearly show that the projection used to fit the Budyko curve leads to different n values. Hence, n values that are found by fitting Budyko-type curves include a rather high uncertainty, and the interpretation should be carried out with care. This does not necessarily lead to large issues when n values are considered a characteristic for one single catchment e.g.,, as the equations can be solved analytically when just one data point is considered. However, the two formulations of the curve (Eqs.  and ) stem from the same original equation (Eq. ), meaning that the definition and value of the parameter should, in principle, not change when projections are changed. For this reason, the different values of the n parameter found here for the different projections express an additional uncertainty due to the choice of projection. When a Budyko curve is fit to multiple catchments and the resulting n values are used for interpretation, this additional uncertainty should be considered.

Once a Budyko curve is fit to the data, the distance to this curve is often used as a metric for catchment analysis e.g., and supposed to tell something about the state of the catchment, catchment characteristics or the local climate. However, the distance to the curve strongly changed depending on the projection, and the differences in distances depended on the aridity of the catchments (Fig. a). For energy-limited catchments (Ep/P≤1), the distances to the curve were lower for the projection with a wetness index in comparison with the projection with a dryness index (i.e., catchments plot left of the 1:1 line in Fig. a), whereas the opposite was true for the water-limited catchments (right of the 1:1 line in Fig. a). This was also more generally confirmed when random samples in the Budyko space were used; see Supplement S2. The distances to the physical boundaries are less often used as a metric for catchment analysis, but these changed similarly (Fig. b).

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 E/Ep=1 indicates a decreased energy use efficiency, whereas an increased distance to E/P=1 indicates a decreased rain use efficiency by evaporation. In the literature, several studies focus on explaining these distances to the curve e.g., , rather than the n values, but usually only consider one specific projection. Thus, one needs to be aware that these explanations are only valid for that specific projection because the meaning, as well as the value of these distances, changes for a different projection.

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. a and b, one may consider using only the side of the curve with Ep/P>1.0 in the dryness index projection or P/Ep<1.0 in the wetness index projection. In this way, the contracted side of the curve is not used, which could lead to errors due to seemingly low absolute deviations that are in relative terms clearly present.

The Budyko framework is often used to predict values of Ea for ungauged catchments, but the uncertainty in predictions of Ea due to the projection used exceeded 1.5 % for catchments with an aridity around 1.0 (Fig. a). In addition, the relative error compared with the observed Ea was especially large for energy-limited catchments of the CAMELS dataset (Ep/P≤1, Fig. b). However, the differences in the relative errors between the dryness and wetness index based estimates remained rather small (Fig. b).

The uncertainty in predicted values of Ea due to the choice of projection in the Budyko framework has not received much attention to date. Uncertainty evaluations do exist for the Budyko framework or derivatives thereof e.g.,, but these studies did not consider the influence of different projections. Only noted that the chosen projection may lead to ambiguities, especially related to leaky or gaining catchments. Implicitly, others may include the projection-related uncertainty indirectly by defining the curves in a more statistical way , but we would still argue that the influence of the projection used needs more consideration.

An important cause of the different n values in the different projections are data points that appear as outliers in one projection but not in the other projection. For example, several data points have short vertical distances to the envelope in a dryness index projection but have large distances to the envelope in a wetness index projection and could be considered as outliers (red points in Fig. a and b). Vice versa, one data point appears as an outlier in a dryness-index-based projection (blue point in Fig. a), but this is not apparent in the other projection (Fig. b).

The outliers also influenced the relative errors when the curve was used to predict Ea. The group of catchments identified as outliers in a wetness index projection (i.e., red points in Fig. ) led to lower n values with a lower curve (see also Fig. ) and a predicted Ea that is more often underestimated (blue boxes in Fig.  shifted downwards). Once only the non-contracted side of the framework was used for predictions, the relative errors became either more negative (for Ep/P≤1) or improved and approached 0 (for Ep/P>1). However, this was merely a result of the absence of the group of outliers (with Ep/P≤1) for the predictions of the catchments with Ep/P>1. Thus, using only the contracted sides of the framework does not necessarily improve predictions of Ea. Nevertheless, we would still argue that plotting the framework in the two projections and, at least, inspecting the non-contracted sides for outliers is a valuable and necessary step in Budyko applications.