SEBS is a one-source energy balance algorithm that is arguably one of the most widely used energy balance approaches to derive turbulent fluxes. The SEBS model calculates the sensible heat flux (H) based on the Monin and Obukhov theory : θ0-θa=Hku*ρcpln⁡z-d0z0h-Γhz-d0L+Γhz0hL, u=u*kln⁡z-d0z0m-Γmz-d0L+Γmz0hL,L=-ρcpu*3θvkgH, where u is the wind speed, u* is the friction velocity, k is the von Kármán constant, z is the height above the surface, d0 is the zero plane displacement height, z0m and z0h are the roughness heights for momentum and heat transfer, and Γm and Γh are the stability correction functions for momentum and sensible heat transfer, respectively. L refers to the Obukhov length, ρ is the air density, θ0 is potential land surface temperature, θa is the potential air temperature at height z, g is the gravity acceleration and θv is the potential virtual air temperature at level z. When the suitable data are available, the only unknowns are H, u* and L. This allows the calculation of H and the further estimation of ET based on closing the energy balance at the surface, i.e., ET is estimated as the difference between net radiation (Rn) and the sum of the calculated H and ground flux (G).

Additionally, in order to constrain H estimates, two limiting cases are considered that set upper and lower bounds for the evaporative fraction. Under very dry conditions, ET becomes 0 and H is at its maximum, set by Rn - G. Under wet conditions, ET occurs at potential rates and therefore H is at its minimum. In this wet case, H is calculated via reverse application of the Penman–Monteith equation (see Sect. ) assuming that the surface resistance is zero.

SEBS has been extensively validated against tower measurements and has proved to estimate realistic evaporation rates on a variety of scales, ranging from local to regional . As an example, recently reported an average correlation of ∼ 0.8 and root mean square difference (RMSD) of 0.7 mm day-1 against eddy-covariance measurements in a validation of monthly SEBS ET aggregates over China.

As a one-source energy model, SEBS does not separate the contributing components of ET (i.e., transpiration, interception loss, bare-soil evaporation), unlike the other models studied in WACMOS-ET, which provide independent estimates of these vapor sources. Although not examined here, we note that two-source energy balance models can also treat the soil and vegetation components separately (e.g., ) but have had limited application on the global scale.

PM-MOD is based on the Penman–Monteith equation. It estimates ET as the sum of interception loss (I), transpiration (ETt) and evaporation from the soil (ETs). The interception loss is modeled as I=fwetfcΔRn-G+ρcpVPD/rawcλΔ+γrswcrawc, where Δ is the slope of the curve relating saturated water vapor pressure to temperature, VPD is the vapor pressure deficit, γ is the psychrometric constant, fc is the canopy fraction, fwet is the wet cover fraction based on the derivation by (see Eq. () in Sect. ), and rawc and rswc are aerodynamic and surface resistances against evaporation of intercepted water (calculated as a function of air temperature and leaf area index, LAI).

Canopy transpiration is estimated as ETt=1-fwetfcΔRn-G+ρcpVPD/ratλΔ+γrstrat, where rat and rst are the aerodynamic and surface resistances against transpiration. rat is determined in a similar way to rawc, and rst is a function of stomatal conductance, biome-constant values of cuticular conductance and canopy boundary-layer conductance. The values of stomatal conductance are a function of air temperature, VPD and LAI.

Evaporation from the soil surface is the sum of evaporation from wet soil and evaporation from saturated soil, which are both calculated separately based on Eq. () with specific values of aerodynamic and surface resistances for bare soils and a soil moisture constraint (fsm) depending on relative humidity (taken from , see Sect. ).

The daily ET estimates have been previously validated against EC measurements from 46 FLUXNET towers in North America, reporting for the daily estimates an average RMSD of ∼ 0.9 mm day-1 and a ∼ 0.6 average correlation coefficient.

PT-JPL is based upon the Priestley and Taylor equation. As in PM-MOD, ET is estimated as the sum of interception loss I, transpiration ETt and evaporation from the soil ETs. The driving equations in the model are ETt=1-fwetfgfTfMαΔΔ+γRnc,ETs=fwet+fsm1-fwetαΔΔ+γRns-G,I=fwetαΔΔ+γRnc, where α is known as the Priestley and Taylor (PT) coefficient and is considered here as a constant value (1.26) that aims to summarize the atmospheric term in the Penman–Monteith equation (Eq. ), λ is the latent heat of vaporization and fwet, fg, fM, fsm and fT are ecophysiological constraint functions with values between 0 and 1 referred to as f functions. The f functions are given by fwet=RH4,fg=fAPAR/fIPAR,fM=fAPAR/fAPARmax,fsm=RHVPD,fT=e-Ts-ToptTopt2, where fwet is the relative surface wetness, fg is green canopy fraction, fAPAR (fIPAR) is the fraction of absorbed (intercepted) photosynthetically active radiation, fM is a plant moisture constraint, fAPARmax is the maximum of fAPAR, fsm is a soil moisture constraint, fT is a plant temperature constraint and Topt is the optimum plant growth temperature, estimated as the air temperature at the time of peak canopy activity when the highest fAPAR and minimum VPD occur. Note that as the input data set does not include fIPAR; fIPAR is derived from the rescaled project LAI by inverting the model original relationships between LAI and fIPAR.

Using this methodology, monthly estimates of ET were tested against EC measurements from 16 FLUXNET towers worldwide with a reported average RMSD of ∼ 0.4 mm day-1 and a ∼ 0.9 average correlation coefficient . Note that unlike the above statistics reported for SEBS and PM-MOD, these numbers come from the model run with the tower meteorology, instead of global forcings.

GLEAM calculates ET via the PT equation, a soil moisture stress computation and a Gash analytical model of rainfall interception loss . In the absence of snow, evaporation from land is calculated as ET=ETtc+ETsc+ETs+βI, where ETtc is transpiration from tall canopy, ETsc is transpiration from short vegetation, ETs is soil evaporation and I is tall canopy interception loss. β is a constant used to account for the times at which vegetation is wet; thus, transpiration occurs at lower rates (β = 0.93) .

The first three terms in Eq. () are derived using the Priestley and Taylor equation, so ET becomes ET=ΔftcStcαtcRntc-Gtc+fscSscαscRnsc-Gsc+fsSsαsRns-Gsλ(Δ+γ)+βI,, where the subscripts “tc”, “sc” and “s” correspond to tall vegetation, short vegetation and bare soil (respectively) and the fraction of each of these three cover types per pixel is represented by f. Different cover types have different values of α and parameterizations of G; additionally, Rn is distributed within the cover fractions using average albedo ratios from the literature. S represents the evaporative stress due to soil moisture deficit and vegetation phenology. Soil moisture deficit is estimated using a multilayer running water balance to describe the infiltration of observed precipitation through the vertical soil profile. Microwave soil moisture observations are assimilated into the soil profile . In vegetated land covers, phenology effects on ET are based on microwave observations of vegetation optical depth, used as a proxy of vegetation water content.

I is independently derived using a Gash analytical model , in which a running water balance for canopies and trunks is driven by observations of precipitation. The derivation of the parameters and the global implementation and validation of this I model are described in . For regions covered by ice and snow, sublimation is calculated based on a PT equation parameterized for ice and supercooled waters .

The ET estimates from GLEAM have been validated against eddy covariance towers worldwide; reported average correlations are 0.83 and 0.90 for daily and monthly estimates, respectively, with an average RMSD of ∼ 0.3 mm day-1, based on a sample of 43 towers , and correlations of 0.71–0.75 and 0.81–0.86 for daily and monthly estimates, respectively, based on a sample of 163 towers and different satellite products as forcing .

The European Centre for Medium-range Weather Forecasts (ECMWF) Re-Analysis-Interim (ERA-Interim) was selected to provide the near-surface meteorology every 3 h at a spatial resolution of ∼ 75 km. The use of reanalysis data is necessary as satellite observations are generally unable to retrieve the surface variables needed, such as temperature, humidity and wind speed, with sufficient accuracy or at a suitable sub-daily temporal resolution. Although products of near-surface air temperature and humidity derived from satellite sounders exist , atmospheric reanalyses have the advantage of providing regular sub-daily estimates for all weather conditions. ERA-Interim is also used in the derivation of the land surface temperature product (see Sect. ), to ensure inter-product consistency between air and surface temperatures. In terms of accuracy, ERA-Interim data have been evaluated through comparison with other reanalyses and weather stations over specific areas, showing a good general performance e.g.,.

Land surface temperature (LST) estimates have been internally generated by the project from level 1 radiances from the Advanced Along-Track Scanning Radiometer (AATSR) onboard ESA's Environmental Satellite (EnviSat) polar-orbiting satellite, from Multi-functional Transport Satellites (MT-SAT) 2 (over Australia), Meteosat Second Generation (MSG) 2 and Geostationary Operational Environmental Satellite (GOES) 12. The data sets are provided over a sinusoidal grid with 1 km resolution for AATSR at the two satellite overpasses per day (∼ 10:00 LT descending node) and 5 km for the remaining sensors (1-hourly estimates for MSG and MT-SAT, 3-hourly for GOES). Ancillary atmospheric information for the inversion of the L1 radiances comes from ERA-Interim. Estimates of surface emissivity are taken from the Global Infrared Land Surface Emissivity UW-Madison Baseline Fit Emissivity Database developed by .

The National Aeronautics and Space Administration (NASA)–GEWEX Surface Radiation Budget (SRB) satellite product version 3.1 is used to provide the surface net radiation input to the ET models. The SRB product is used by a large number of global ET algorithms to characterize the radiation at the surface, given its relatively long data record and sub-daily temporal resolution. SRB data sets include global 3-hourly averages of surface and top-of-atmosphere longwave and shortwave radiative parameters on a ∼ 100 km grid.

To characterize the vegetation state using visible and near-infrared wavebands, estimates of LAI and fAPAR have been derived by applying the Joint Research Centre (JRC) two-stream inversion package (hereafter TIP) to the ESA GlobAlbedo bihemispherical reflectances. Here, LAI is defined as the one-sided leaf area per unit ground area and fAPAR as the fraction of absorbed photosynthetically active radiation in the 400–700 nm region.

The application of the TIP LAI and fAPAR with our ET models required some LAI and fAPAR calibration. The TIP LAI is a one-dimensional (1-D) equivalent LAI for solving the radiative transfer in a three-dimensional (3-D) medium, and it is consistent with the fluxes from which it is derived. It is not consistent with LAI derived using a 3-D radiative scheme that allows some form of horizontal canopy clumping (e.g., the MODIS MOD15A2 LAI product). In practical terms, this means that if an ET model was constructed to use a MODIS-like LAI and fAPAR, a straight use of the project LAI and fAPAR will result in the ET model producing lower values than expected for those biomes where horizontal clumping is significant (e.g., for forests). While the ET dynamics may not be highly affected (there is a high degree of correlation between different LAI and fAPAR estimates), the absolute values would be. As the SEBS, PM-MOD and PT-JPL models have typically been used with the MODIS vegetation product, a rescaling of our TIP-derived LAI and fAPAR products against the MODIS product has been undertaken. For running the models on the tower scale, a local rescaling is conducted by a linear regression between the MOD15A2 and the TIP values co-registered at each tower. For global model simulations, individual rescaling per biome or climate classification is conducted. For PT-JPL, given the model internal relationships between these variables and the vegetation indexes used as model inputs (see Table  in ), it can be discussed whether the original TIP LAI and fAPAR or the rescaled LAI and fAPAR are the most appropriate to be used as model inputs. For simplicity we will apply also the rescaled LAI and fAPAR, but this choice will be further evaluated in future applications of the model with the TIP LAI and fAPAR inputs.

Figure  illustrates an example of both products at two towers. The station Quebec – Eastern Boreal, Mature Black Spruce (CA-Qfo) is located in an evergreen needleleaf forest and shows that the MOD15A2 and WACMOS-ET LAI and fAPAR absolute values differ considerably. This is expected, as allowing some form of horizontal clumping (MODIS 3-D radiative transfer scheme) or not (TIP 1-D) can result in large differences in the estimated LAI and fAPAR in forests. It can be seen that the local calibration of the MODIS-like product retains the dynamics of the WACMOS-ET product, while adding absolute values close to the MODIS product. The station Brookings (US-Bkg) is situated in a cropland area, where the effects of clumping are much less severe, and the different LAI and fAPAR values are much closer.

Vegetation height on the global scale is required by SEBS. For shrubland and forest biomes the product developed by was used as static canopy height cover. For grassland and cropland biomes, where the temporal dynamics of canopy height can be more considerable, we approximated canopy height with the method by , with the minimum and maximum canopy height obtained from the static vegetation table of the North American Land Data Assimilation System (NLDAS).

A soil moisture product combining observations from active and passive microwave sensors has been developed as part of the ESA Climate Change Initiative (CCI) and is adopted here to provide the surface soil moisture data that are assimilated into GLEAM. Details on the product algorithm and evaluation can be found in . The data are provided on a regular grid with a resolution of 0.25∘ × 0.25∘. The product performs well in moderately vegetated regions but shows higher uncertainties in densely vegetated regions (as vegetation attenuates the microwave signal from the ground) and mountainous areas (due to the high surface roughness) .

The retrieval of soil moisture from passive sensors discussed in Sect.  can be accompanied by an estimation of the vegetation optical depth (VOD). VOD can be used to account for the development of vegetation over the year as it is a good proxy of vegetation water content . Although most ET models traditionally use parameters derived from visible and near-infrared wavelengths, microwave VOD is used by GLEAM. Here the long-term record by based on the application of the Land Parameter Retrieval Model by is used by GLEAM.

Observations of precipitation and snow water equivalent are also required by GLEAM only. Precipitation is used both to estimate the effects of soil water limitations on ET and to calculate interception loss. To run the model on the tower scale we use the Climate Prediction Center (CPC) Morphing Technique (CMORPH) . CMORPH transports the features of precipitation estimates derived from low orbiter satellite microwave observations using information from geostationary satellite infrared (IR) data. Precipitation estimates are available every 30 min on a grid with a spacing of 8 km at the Equator, although the resolution of the individual satellite-derived estimates is coarser at ∼ 12 km × 15 km. The spatial coverage ranges from 60∘ N–60∘ S. To run the model globally, we use the National Centers for Environmental Prediction (NCEP) Climate Forecast System Reanalysis (CFSR) land precipitation estimates . These precipitation estimates come from the hourly CFSR output but are corrected using the observation-based data sets of the CPC and the Global Precipitation Climatology Project (GPCP) . Finally, snow water equivalent estimates come from ESA GlobSnow. Since GlobSnow covers the Northern Hemisphere only, data from the National Snow and Ice Data Center (NSIDC) are used in snow-covered regions of the Southern Hemisphere . The product combines satellite passive microwave measurements with ground-based weather station data in a data assimilation scheme . The products exist at a daily resolution and a spatial resolution of 25 km.

Model simulations are evaluated by comparison with the turbulent latent fluxes measured by the eddy-covariance technique at a selection of tower sites from FLUXNET . A first sample of towers was compiled by selecting those stations from the FLUXNET La Thuile synthesis data set which contain latent flux measurements in the 2005–2007 period, as well as the meteorological and radiation inputs required to run the ET models at the towers' locations. The 24 selected stations are described in Table  and their geographical location is displayed in Fig. . While some meteorological variables such as near-surface air temperature or humidity are measured at nearly all towers, other inputs such as the surface net radiation or the ground heat flux are measured at only a few towers. Some stations that were very close to the shore or in places with regular flooding were discarded. The final selection of 24 towers represents a significant number of biomes and a reasonable sample of dry and wet climate regimes.

In a later step, by removing the constraint of requiring local measurements of all the model inputs, the first selection of 24 towers is extended to a total of 85 stations. This second selection is used to evaluate model performance when the models are run with the satellite data used for the global runs.

While, in principle, the surface energy balance should close at the tower, this is rarely the case: a lack of closure in the surface energy balance of about 10–30 % is commonly found when comparing the EC measurements against the energy balance residual (ER) term, i.e., the difference between net radiation and the sum of the sensible, latent and ground fluxes (e.g., ). Consequently, throughout the paper the model evaluation is discussed by comparing it with both the EC measurements and the in situ ER estimates.

To run SEBS, the broadband longwave radiometer measurements need to be converted into LST estimates. This is done by inverting the equation relating the upwelling spectral radiance measured by the radiometer and the LST. Broadband emissivity is required, and it is estimated from the MODIS-based Global Infrared Land Surface Emissivity Database operated by the Cooperative Institute for Meteorological Satellite Studies (CIMSS). The estimates are calculated by following the approach suggested by using a linear combination of narrowband emissivities at 8.5, 11 and 12 µm.

SEBS also requires vegetation height to derive the surface roughness values. In most cases a mean annual value can be obtained from the tower metadata, and this value is adopted here as vegetation height at the tower. However, a clear limitation in this assumption is that it does not include dynamic changes in vegetation height over time. As discussed in Sect. , the importance of neglecting the temporal variability in height is biome-dependent; for instance, in forests the mean vegetation height is typically more constant than in, e.g., croplands, where the changes derived from agricultural practices can be large.

The model performance is investigated on sub-daily and daily timescales. The tower data are available at 0.5 h intervals and have been time-integrated to 3 hours in order to run the ET models at that sub-daily resolution. The satellite data have been time-matched to the 3-hourly or daily resolutions from their native resolution in different ways (see below), depending on the type of data and original resolution. The 3-hourly inputs were then aggregated to daily values in order to run the models with tower-based daily inputs. The tower data record is not always time-continuous, as in some instances there are gaps in the record. This is not a problem for the PM-MOD, PT-JPL and SEBS models because the ET estimates depends only on the instantaneous atmospheric or surface state. When inputs to the models and/or ET for the evaluation are missing, those three models are not run. Conversely, GLEAM requires continuous data records to update the soil moisture state variable. To facilitate running GLEAM with tower inputs, the tower measurements are gap-filled with the corresponding pixel data (see Sect. ). ET estimates from these periods are removed after the runs, so, as before, only the time steps where tower forcing data are available are used for model evaluation.

The models are validated against the tower ET only under dry (non-raining) conditions, as EC gas analyzers are not reliable during rain events due to disturbance of the infrared signal by droplets on the sensor . Therefore, any days with precipitation as indicated by the tower or satellite precipitation are removed from the validation, and the interception component from PM-MOD, PT-JPL and GLEAM is not considered in the validations.

Only PM-MOD and GLEAM specifically deal with nighttime evaporation. Nevertheless, nighttime values are required from all models to integrate the 3-hourly ET estimates to daily values. For SEBS and PT-JPL negative nighttime estimates are set to 0 to allow the daily integration for those models. To separate day and night, daylight times are identified by calculating the solar zenith angle. Time intervals, where the cosine of the zenith angle is larger than 0.2, are kept as day values. This day and night separation may be less accurate than using a solar downward radiation threshold, but it allows a day–night flag for those stations without solar radiation measurements. The impact of setting ET from SEBS and PT-JPL to 0, as these models cannot specifically simulate nighttime conditions, is addressed in Sects.  and , where sub-daily periods, including daytime, are investigated.

The following ET estimates are generated to evaluate model performance.

Tower-based ET: ET generated by the four models using the 3-hourly or daily in situ data (surface radiation, LST, air temperature, air humidity, and wind speed and precipitation), the scaled WACMOS-ET LAI and fAPAR and gridded soil moisture and VOD data. Note that the 3-hourly tower-based ET estimates are also time-integrated to daily values, so daily ET estimates exist both from the runs with daily inputs and from the integration of the 3-hourly ET estimates.