For the case study, we employed the calibration-free CR formulated by Szilagyi et al. (2017). It describes the atmospheric self-adjustment to surface moisture conditions using three evaporation rates, namely, ETa, ETw, and the potential evaporation (ETp). ETa is the actual moisture flux from a land surface to the atmosphere, and ETw is the hypothetical ETa rate that should occur with ample water availability. ETp is the atmospheric capacity to receive water vapor that responds actively to soil moisture conditions. By defining the two dimensionless variables as x≡ETw/ETp and y≡ETa/ETp, Szilagyi et al. (2017) derived a polynomial function from four definitive boundary conditions.

Under ample water conditions, ETp does not deviate from ETw and ETa (i.e., ETp = ETw = ETa); hence, the corresponding zero-order boundary condition is (i) y=1 for x=1. In contrast, ETa must be nil over a desiccated surface (i.e., y=0), and by energy balance, the surface net radiation should be fully transformed to the sensible heat flux. Then, the atmospheric VPD would be amplified at the maximum level with the same net radiation and wind speed. Defining the maximum ETp rate as Epmax, another zero-order boundary condition is given as (ii) y=0 for x=xmin≡ETw/Epmax. When x=1 (i.e., ample water), changes in ETa would be controlled by changes in ETw, yielding a first-order boundary condition as (iii) dy/dx=1 for x=1. Over a desiccated surface, ETa stays at zero, even when ETw changes; thus, another first-order boundary condition becomes (iv) dy/dx=0 for x=xmin. The simplest polynomial equation satisfying the four boundary conditions is 1y=2X2-X3, where X rescales the variable x into [0, 1] as 2X=x-xmin1-xmin=Epmax-ETpEpmax-ETwETwETp. Equation (1) allows users to estimate ETa with no land surface information because ETp, ETw, and Epmax are all obtainable from a set of net radiation, air temperature, dew-point temperature, and wind speed data. ETp and Epmax can be estimated by the Penman (1948) equation: 3ETp=ΔTaΔTa+γRnλv+γΔTa+γfuVPD,4Epmax=ΔTdryΔTdry+γRnλv+γΔTdry+γfuesTdry, where Δ(⋅) is the slope of the saturation vapor pressure curve (kPa ∘C-1); Ta is the mean air temperature (∘C); γ is the psychrometric constant (kPa ∘C-1); Rn is the surface net radiation less the soil heat flux (MJ m-2 d-1); λv is the latent heat of vaporization (MJ kg-1); fu=2.6 (1+0.54u2) is the Rome wind function (mm d-1 kPa-1), where u2 is the 2 m wind speed (m s-1); and VPD is calculated by es (Ta) minus es (Tdew), where es(⋅) is the saturation vapor pressure (kPa), and Tdew is the dew-point temperature (∘C).

Tdry in Eq. (4) is the air temperature (∘C) at which the lower atmosphere is devoid of humidity presumably by the adiabatic drying process: 5Tdry=Twb+esTwbγ=Ta+esTdewγ, where Twb is the wet-bulb temperature (∘C) at which the saturation vapor pressure curve intersects with the adiabatic wetting line. Thus, it is obtained by 6γTwb-TavgesTwb-es(Tdew)=-1. To estimate ETw in Eq. (2), the Priestley–Taylor (1972) equation has been a typical choice (e.g., Brutsaert, 2015; Crago et al., 2016; Han and Tian, 2018; Szilagyi et al., 2017): 7ETw=αeΔTwΔTw+γRnλv, where αe is the Priestley–Taylor coefficient ranging usually within [1.10, 1.32] (Szilagyi et al., 2017), and Tw is the wet-environment air temperature (∘C). Tw can be approximated with the wet-surface temperature (Tws) because the vertical air temperature gradient is negligible under a wet environment. Given its independence on areal extent (Szilagyi and Schepers, 2014), Tws can be approximated by the implicit Bowen ratio (β) of a small wet patch: 8β=Rn-ETpETp≈γTws-TaesTws-esTdew. Equation (8) assumes that the available radiation for the wet patch is close to that of the drying surface (Szilagyi et al., 2017). Tws might be higher than Ta when the air is close to saturation. In such a case, Tws should be capped by Ta when calculating ETw.

The single parameter of the polynomial CR, i.e., αe, is analytically obtainable by inserting the Priestley–Taylor equation into the Bowen ratio of a wet environment (Szilagyi et al., 2017) as 9αe=ΔTa+γesTws-esTdewΔTaesTws-esTdew+γTws-Ta, where αe must fall within the theoretical limit of [1, 1+γ/Δ(Ta)] (Priestley and Taylor, 1972).

Since Eq. (9) is only applicable in a wet environment, Szilagyi et al. (2017) identified wet locations in a continental area based on the fact that the air close to saturation should have high relative humidity (RH) with Tws>Ta. Thus, they calculated αe values at locations with RH > 90 % and Tws>Ta+2 ∘C, and the average value was used to predict ETa for a continental area. However, the spatially constant αe is unlikely suitable in such a large area under diverse climates because the equilibrium between the atmosphere and the underlying surface is intertwined with the partitioning of P to ETa and Q over the surface.

Kim and Chun (2021) analytically related Eq. (1) with the traditional Turc–Mezentsev equation and found that the self-adjustment of ETp (i.e., x) is tightly linked with climatological aridity and land properties. For the independence between P and “the possible maximum ETa” of the Budyko framework, Kim and Chun (2021) reformulated the traditional equation with Φ0≡ETw/P instead of the commonly used aridity index (Φ≡ETp/P) as 10ETaP=ETwP11+ETwPn1n=xETpP11+xETpPn1n, where the parameter n implicitly represents the factors affecting the P partitioning other than the climatic drivers. When dividing Eq. (10) by Φ, it is found that the Budyko Eq. (10) is intertwined with the Eq. (1) of the CR: 11y=ETaETp=2X2-X3=xn1+xnΦn1n. Equation (11) implies that the self-adjustment of ETp (i.e., x) is tightly related with the climatic condition (i.e., Φ) and the implicit land property (i.e., n).

While the x and n can be achievable from a set of ETa, ETp, Epmax, and P values by inverting Eq. (11), such an approach is not applicable in locations with no ETa data. To quantify x values only using ETp, Epmax, and P, Kim and Chun (2021) developed a regression equation between x and Φ, xmin, and n values from 513 gauged river basins around the world. We used the same regression-based regionalization. Considering xmin=xETp/Epmax, the nonlinear expression in Eq. (11) can be approximated by a multiple regression as 12x̃=b0+b1ln⁡Φ+b2ln⁡ETp/Epmax+b2ln⁡n, where x̃ is the approximate ratio of ETw to ETp, and b0, b1, and b2 are the intercept and the regression coefficients. Since the implicit parameter n is unavailable in ungauged locations, Eq. (12) needs to be further simplified by neglecting the last term: 13x̃≈c0+c1ln⁡Φ+c2ln⁡ETp/Epmax, where c0, c1, and c2 are the intercept and the coefficients of the approximated regression.

If x̃ is known by the regression by Eq. (13), the parameter αe can be estimated using the Priestley–Taylor equation as 14α̃e=x̃ETpETeq15ETeq=ΔTwΔTw+γRnλv, where α̃e is the Priestley–Taylor coefficient that approximately satisfies the CR and the Budyko equations together, and ETeq is the equilibrium ETa (mm d-1) at which VPD is nil under a wet environment. It should be noted that P, ETp, Epmax, and ETeq within Eqs. (10)–(13) must be on a timescale where the Turc–Mezentsev equation is valid (typically longer than a year), and α̃e is still bounded within [1, 1+γ/Δ(Ta)].

We examined the CR–Budyko combined framework in the Australian continent lying within (10–45∘ S, 113–155∘ E). The required atmospheric forcing data (Rn, Ta, Tdew, and u2) were collected from the advanced ERA5-Land reanalysis archive (Muñoz-Sabater et al., 2021) of the European Centre for Medium-Range Weather Forecasts (https://cds.climate.copernicus.eu; last access: 10 December 2021). The monthly averages of surface latent and sensible heat fluxes, 2 m air temperature, 2 m dew-point temperature, and 10 m U and V wind speed components at 0.1∘×0.1∘ were downloaded for 1981–2020. Rn was calculated by summing the two heat fluxes, and the 10 m wind speed components were converted to u2 using the logarithmic wind profile (Allen et al., 1998).

We also collected the Australian edition of the Catchment Attributes and Meteorology for Large sample Studies (CAMELS; Fowler et al., 2021) series of datasets (10.1594/PANGAEA.921850). The CAMELS datasets comprise daily time series of 19 hydrometeorological variables at 222 unregulated river basins in Australia up to 2014, and we selected the 71 basins larger than 500 km2 to contain at least five CR ETa estimates within the boundaries. The water-balance ETa (ETwb) (i.e., ETwb≈∑P-∑Q) of each basin was calculated for the two periods of 1981–1997 and 1998–2014. The mean annual ETwb for the former period was used for the regressions with Eqs. (12) and (13), and the predicted ETa was evaluated against the latter.

As a point-scale evaluation dataset, the annual flux observations were taken from the 15 eddy-covariance stations (Table 1) included in the FLUXNET2015 archive (https://fluxnet.org/; last access: 1 July 2021). We chose the flux towers with two or more annual means and adopted the energy-balance-corrected latent heat flux observations with the quality measures “LE_F_MDSQC” higher than 0.70. Given the fine resolution of the ERA5-Land forcing data, we believed that the ETa estimates by CR could be directly compared with the point-scale observations.

In addition, as a grid-scale evaluation reference, the SILO P data at 0.01∘×0.01∘ were collected from the Queensland government (https://www.longpaddock.qld.gov.au/silo/gridded-data; last access: 1 June 2021) together with the Global RUNoff (GRUN) ENSEMBLE (Ghiggi et al., 2021) (https://doi.org/10.6084/m9.figshare.12794075; last access: 1 October 2021). The global Q data were produced at 0.5∘×0.5∘ using a machine-learning algorithm trained by in situ streamflow observations, and potential biases were reduced by simulations with 21 sets of atmospheric forcing (Ghiggi et al., 2021). The SILO P was used to calculate Φ at each grid of the forcing data. After bilinearly unifying the resolutions of SILO P and GRUN Q data, we also calculated the mean annual ETwb for 1998–2014 at 0.5∘×0.5∘ over the entire Australian continent.

Against the grid-scale ETwb estimates, performance of the polynomial CR was also compared with three ETa products from a physical, a machine-learning, and a land surface model. The physical model was the Global Land Evaporation Amsterdam Model (GLEAM) v3.2 (Martens et al., 2017; https://www.gleam.eu; last access: 3 June 2020) based on the Priestley–Taylor equation constrained by microwave-derived soil moisture, surface temperature, and vegetation optical depth. The machine-learning ETa product was the FluxCom (http://www.fluxcom.org/; last access: 18 March 2019) that upscaled in situ observations at 224 eddy-covariance towers using 11 algorithms (Jung et al., 2019). We used the version forced by the CRUNCEPv8 that has the longest data length from 1950 to 2016. The land surface model product was the ERA5-Land monthly ETa (https://cds.climate.copernicus.eu; last access: 7 July 2021) simulated by the advanced Hydrology Tiled ECMWF Scheme for Surface Exchanges over Land (Balsamo et al., 2015). All the modeled ETa datasets were bilinearly regridded to 0.5∘×0.5∘ for 1998–2014 to be compared with the grid-scale ETwb data.

Figure 1a depicts the spatial distribution of the inverted aridity index (Φ-1=P/ETp) that can traditionally categorize climate conditions. The mean ratios between SILO P and ETp for 1998–2014 indicated that 83 % of the Australian land surfaces were under arid (0.05<Φ-1<0.2) and semi-arid climates (0.2<Φ-1<0.5). Semi-humid (0.5<Φ-1<0.65) and humid climates (Φ-1>0.65) were only found in the northern and southeastern coastal areas and the southwestern edge where major cities and agricultural lands have developed. Despite the high aridity, hyper-arid climates (Φ-1>0.05) were not found in Australia.

We first examined the calibration-free approach by Szilagyi et al. (2017) that only uses the meteorological forcing inputs. The blue-colored points in Fig. 1a are the locations with RH > 90 % and Tws > Ta+2 ∘C, at which the αe values from Eq. (9) were within 1.15 ± 0.047 (mean ± standard deviation). Though the two conditions were met in some mountainous areas in the southeastern part, we excluded them because unexpectedly high αe values were obtained. The mean αe=1.15 fell within the theoretical limits and was equal to the value used in the prior studies in China (Ma et al., 2019) and the conterminous United States (Ma and Szilagyi, 2019).

Using the CR with αe=1.15, we predicted ETa over the entire Australian continent (Fig. 1b). The distribution of the resulting mean ETa for 1998–2014 was coherent with that of Φ-1. The mean CR ETa ranged in 262 ± 85.3 and 547 ± 173 mm a-1 under arid and semi-arid climates, respectively. On the other hand, CR ETa in semi-humid and humid locations was much higher, at 886 ± 187 mm a-1 and 1010 ± 213 mm a-1, respectively. The calibration-free CR predicted the continental mean ETa to be as high as 489 mm a-1 for 1981–2012, and it was about 11.3 % higher than the estimate for the same period (439 mm a-1) by Zhang et al. (2016). The mean fraction of ETa to P for 1998–2014 (97 %) was larger than the typical ETa value in Australia (∼90 %; Glenn et al., 2011), implicating that the constant αe=1.15 seemed to make the CR overrate ETa.

The overestimation of the calibration-free CR was confirmed by the flux observations and the basin-scale ETwb (Fig. 2). The percent bias (p-bias) of CR ETa to the point-scale annual ETa was +10.4 %, while it became more than doubled when compared to the basin-scale ETwb. Though the Pearson correlation coefficients (Pearson r) were significantly high between the CR ETa and the two evaluation references, the low Nash–Sutcliffe efficiency (NSE) to ETwb implies that the CR method could perform poorly in wet river basins. The regression slopes in Fig. 2 also indicate that the calibration-free CR tends to increasingly overestimate as climate becomes wetter. The root mean square error (RMSE) of CR ETa to ETwb was higher than to the point observations. Although it appeared to perform acceptably at the 15 flux towers, the CR method produced considerable biases in the 71 CAMELS basins. The performance measures were not as excellent as the same CR method had shown in the United States (Ma et al., 2021; Ma and Szilagyi, 2019; Kim et al., 2019) and in China (Ma et al., 2019).

One may argue that the mean αe derived from fractional wet areas is unlikely representative of the large Australian continent, and this might introduce the biases to CR ETa estimates. Hence, we re-simulated CR ETa with the estimate by Ma et al. (2021) (αe=1.10) from a global-scale analysis. Figure 3a shows that the predicted ETa became nearly unbiased at the 15 flux tower locations and seemingly suggests that the decreased αe could become a solution to improving the CR method. Nevertheless, the fixed αe still made the CR overestimate ETa in the CAMELS basins under (semi-)humid climates, albeit slightly ameliorated (Fig. 3b).

Figures 2 and 3 imply that the calibration-free CR with a fixed αe was unlikely good at closing the local water balance in (semi-)humid river basins. To resolve this problem with the CR–Budyko framework, first we estimated the climatological x and the parameter n of the CAMELS basins using Eq. (11) with the mean annual ETwb, P, ETp, and Epmax for 1981–1997. Figure 4a–c illustrate the scatter plots between the resultant x and corresponding Φ, ETp/Epmax, and n values. Pearson r values between the x and the other three variables were -0.88, -0.59, and 0.44, respectively (significant at the 1 % level), suggesting that the self-adjustment of ETp is not only correlated with climate conditions, but with land surface properties at least in part. By regressing between the x values and the log-transformed Φ, ETp/Epmax and n, we obtained an empirical relationship that enables us to spatially predict the mean annual ratio of ETw to ETp as 16x̃=0.949-0.204ln⁡Φ+0.231ln⁡ETp/Epmax+0.0712ln⁡n. The regression coefficients were all significant at the 1 % level, and the coefficient of determination (R2) was 0.98. The regression equation was further approximated by discarding n from the explanatory variables: 17x̃=1.023-0.220ln⁡Φ+0.210ln⁡ETp/Epmax. The R2 value of Eq. (17) declined to 0.93. We found that the simple regression between x and Φ further reduced R2 to 0.90. While the heterogeneous land properties exert non-negligible influences, the regression analyses indicate that the climatic condition dominantly explains the spatial variation of the atmospheric self-adjustment.

Equation (17) performed excellently in reproducing the x values from CR with Φ and ETp/Epmax (Fig. 4d). The NSE, RMSE, Pearson r, and p-bias between the predicted x̃ and the x from CR were 0.93, 0.03, 0.96, and 0.0 %, respectively.

By multiplying x̃ by the mean annual ratio between ETp and ETeq, we determined α̃e across the Australian land surfaces. The resulting α̃e values ranged within 1.13 ± 0.114, and the median value was almost equal to the global estimate (1.10) by Ma et al. (2021). They were relatively high in the northwestern and the northern part while being below the mean in the southern and the eastern parts (Fig. 5a). On 19 % of the surfaces, α̃e values were unity, and thus they might become below the theoretical limit unless bounded.

We again generated CR ETa using the spatially varying α̃e values (Fig. 5b). The mean CR ETa for 1998–2014 ranged in 249 ± 78.8 and 530 ± 172.0 mm a-1 under arid and semi-arid climates, while it decreased to 805.2 ± 209 and 932 ± 239 mm a-1 in semi-humid and humid regions, respectively. The flux observations were still acceptably regenerated with the less biases than in the case of αe=1.15 (Fig. 6a). The α̃e based on the Budyko framework significantly reduced the biases introduced by the constant αe in (semi-)humid basins. Albeit some biases remained, the water-balance ETwb for 1998–2014 in the CAMELS basins were better reproduced by the spatially varying α̃e (Fig. 6b).

To confirm the improved performance of the combined CR–Budyko method across Australia, we resampled the new CR ETa estimates to 0.5∘×0.5∘ and compared them with the grid ETwb data. The ETa products by GLEAM, FluxCom, and ERA5-Land were evaluated with the grid evaluation reference. As shown, the CR method with a constant αe=1.15 overrated the mean annual ETa along the eastern and the northern coastlines (Fig. 7b), underperforming the physical, the machine-learning, and the land surface models (Figure 8a). Although the smaller αe=1.10 made the CR method perform better, its predictability was still poorer than the three advanced models, and the residual variance was as large as in the case of αe=1.15 (Fig. 8b).

In contrast, when employing the α̃e conditioned by local climate conditions, the same CR formulation could alleviate the overestimation along the coastlines (Fig. 7c). The Budyko-function-based α̃e led the CR ETa estimates to neatly agree with the grid ETwb, and the residual variance was much smaller than in the case of αe=1.10 (Fig. 8c). The CR method with α̃e clearly outperformed the three advanced models in reproducing the grid ETwb estimates (Fig. 8d–f). Although the referenced grid ETwb has some error sources associated with the upscaling of P and Q, our comparative evaluation suggests that conditioning αe with local climate conditions could substantially reduce the uncertainty of CR ETa estimates in ungauged areas.

The CR explains the dynamic equilibrium between the atmospheric ETp and the underlying moisture conditions, while the Budyko framework describes the steady-state water balance with climatic controls (i.e., P and ETw). The analytical link between the CR and the Budyko equations, hence, implies that the atmospheric self-adjustment needs to be conditioned by the long-term climate conditions. Constraining the Turc–Mezentsev equation by the polynomial CR, Kim and Chun (2021) found that Q changes would be more sensitive to climatic changes than when they were not linked. In the opposite direction, the CR can be constrained by the Budyko equation to determine its essential parameter.

In Crago and Qualls (2018), the optimal αe for the linear CR of Crago et al. (2016) varied largely between 1.00 and 1.43. This point-scale experiment has already suggested that a constant αe is unlikely suitable for definitive CRs to predict ETa in Australia. The ratio between the aerodynamic and the radiation components of ETw is evidently affected by the heat entrainment from the top of the boundary layer (Baldocchi et al., 2016), the dissimilarity between heat and water vapor sources (Assouline et al., 2016), the large-scale synoptic changes (Guo et al., 2015), and the horizontal advection of dry-air mass (Jury and Tanner, 1975). More recently, Han et al. (2021) proved the nonlinear dependence of ETw on ETeq, and Yang and Roderick (2019) showed αe changing with Rn over ocean surfaces. Hence, the constant αe assumption underpinning the calibration-free CR is counterintuitive to the theoretical and empirical evidence. Although Ma et al. (2021) found some global applicability of the calibration-free CR, its performance remains unknown in most of the Australian surfaces and in many ungauged basins over the world.

Since ETa plays a pivotal role in the terrestrial water and energy balances, the partitioning of Rn into the latent and the sensible heat fluxes cannot be independent of the partitioning of P into ETa and Q. On a mean annual scale, P and ETw are the major determinants of the P partitioning, and thus the parameter αe might not be independent of P. Given the large variability of P, assuming a fixed αe across a continental area may introduce considerable biases to CR ETa estimates. Thus, discarding available P data may not be a good choice when predicting ETa by the CR method in ungauged areas. It is noteworthy that Φ dominantly explained the spatial variation of the mean annual x of the 71 CAMELS basins, and the α̃e values conditioned by local climates were of a large spatial variation. This suggests that the CR with a constant αe may produce unreliable ETa estimates in ungauged locations.

Nonetheless, the low performance with a constant αe does not indicate that the CR method underperforms the sophisticated ETa models. The simple polynomial CR seemed to outperform the advanced the advanced physical, machine-learning, and land surface models, when its parameter was conditioned by local climates. The proposed CR–Budyko framework enabled us to regionalize the optimal αe for the CR method from gauged basins to ungauged locations in an empirical manner. It should be highlighted that the CR with spatially varying α̃e produced much smaller residual variance than the three advanced models.

In seven Australian eddy-covariance flux towers, Crago et al. (2022) found that the optimal αe for the polynomial CR was 1.35 for predicting daily ETa in the dimensionless form (i.e., y=ETa/ETp). However, it was increased to 1.42, 1.45, 1.47, and 1.50 to simulate the dimensional latent heat fluxes at daily, weekly, monthly, and annual timescales, respectively. This implies that the timescale would largely affect the optimal αe for the definitive CRs. Though the stationary Budyko equation can become a constraint at a mean-annual scale, how to capture the scale dependence of αe is a remaining question.

Further questions can arise as to how to quantify ETp and Epmax. For example, the αe values from ETp with the Rome wind function rely upon an unrealistic assumption that the aerodynamic resistance on a vegetated surface is equivalent to that of an open-water surface. It is still unknown if this assumption is practically valid because the Penman equation with the Rome wind function may result in unrealistically high ETp, even on a large wet area (McMahon et al., 2013). Given the importance of the aerodynamic resistance in modulating surface temperature (Chen et al., 2020), ignoring its variability may become a significant error source for the CR method at both annual and subannual timescales.

In addition, there are some caveats in our case study. We employed the meteorological data different from those used in Ma et al. (2021). The ERA5-Land dataset is a downscaled version of the ERA5 data (Hersbach et al., 2020) by which Ma et al. (2021) predicted ETa globally. Ma et al. (2021) incorporated remotely sensed albedo and emissivity together with a correction factor when calculating Rn, whereas we used the sum of the ERA5-Land latent and sensible heat fluxes. Those input differences may lead to differences in CR ETa estimates.

The gridded GRUN Q, too, has some uncertainty sources, though it is the ensemble of many runoff simulations from 21 different atmospheric forcing inputs. In the machine-leaning process by Ghiggi et al. (2021), some Q observations affected by human activities (e.g., dam regulation and return flows from groundwater abstraction) might not be excluded, potentially disrupting the empirical relationship between atmospheric forcing and natural flows. In addition, the uncertainty of SILO P might be non-negligible in areas with limited weather stations and in mountainous areas (Fu et al., 2022). Though we reduced the potential biases of the gridded P and Q datasets by temporal averaging, the grid-scale ETwb estimates should be treated as plausible values rather than exact observations.