Dynamic Scaling of the Generalized Complementary ...

Dynamic Scaling of the Generalized Complementary RelationshipImproves Long-term Tendency Estimates in Land Evaporation

Jozsef SZILAGYI12 Richard CRAGO3 and Ning MA4

1Department of Hydraulic and Water Resources Engineering Budapest University of

Technology H-1111 Budapest Hungary2Conservation and Survey Division School of Natural Resources University of

Nebraska-Lincoln Lincoln Nebraska 68583 USA3Department of Civil and Environmental Engineering Bucknell University Lewisburg Pennsylvania 17837 USA

4Key Laboratory of Water Cycle and Related Land Surface Processes Institute of Geographic Sciences and

Natural Resources Research Chinese Academy of Sciences Beijing 100101 China

(Received 23 March 2020 revised 18 June 2020 accepted 28 June 2020)

ABSTRACT

Most large-scale evapotranspiration (ET) estimation methods require detailed information of land use land coverandor soil type on top of various atmospheric measurements The complementary relationship of evaporation (CR) takesadvantage of the inherent dynamic feedback mechanisms found in the soilminusvegetationminusatmosphere interface for itsestimation of ET rates without the need of such biogeophysical data ET estimates over the conterminous United States by anew globally calibrated static scaling (GCR-stat) of the generalized complementary relationship (GCR) of evaporationwere compared to similar estimates of an existing calibration-free version (GCR-dyn) of the GCR that employs atemporally varying dynamic scaling Simplified annual water balances of 327 medium and 18 large watersheds served asground-truth ET values With long-term monthly mean forcing GCR-stat (also utilizing precipitation measurements)outperforms GCR-dyn as the latter cannot fully take advantage of its dynamic scaling with such data of reduced temporalvariability However in a continuous monthly simulation GCR-dyn is on a par with GCR-stat and especially excels inreproducing long-term tendencies in annual catchment ET rates even though it does not require precipitation informationThe same GCR-dyn estimates were also compared to similar estimates of eight other popular ET products and theygenerally outperform all of them For this reason a dynamic scaling of the GCR is recommended over a static one formodeling long-term behavior of terrestrial ET

Key words landminusatmosphere interactions land evaporation evapotranspiration complementary relationship of evapo-ration

Citation Szilagyi J R Crago and N Ma 2020 Dynamic scaling of the generalized complementary relationshipimproves long-term tendency estimates in land evaporation Adv Atmos Sci 37(9) 975minus986 httpsdoiorg101007s00376-020-0079-6

Article Highlights

bull A temporally variable dynamic scaling of the GCR yields better long-term behavior than a static onebull The dynamic scaling accounts for the aridity of the environment in each time step and thus improves land evaporation

estimatesbull The dynamic scaling does not require precipitation information

1 Introduction

Land surface evapotranspiration (ET) is a central compon-

ent in the Earthrsquos energy water and carbon cycles (Wang

and Dickinson 2012 Fisher et al 2017) Accurate ETinformation is therefore essential for a better understandingof landminusatmosphere interactions (Seneviratne et al 2006)and the biospherersquos waterminuscarbon coupling (Biederman etal 2016 Feng et al 2016) It also improves drought predic-tions (Pendergrass et al 2020) and helps to find answers towater resources sustainability issues (Condon et al 2020)

Corresponding author Jozsef SZILAGYI

Email jszilagyi1unledu

ADVANCES IN ATMOSPHERIC SCIENCES VOL 37 SEPTEMBER 2020 975ndash986 bull Original Paper bull

copy The Author(s) 2020 This article is published with open access at linkspringercom

While the globally distributed eddy-covariance flux towershave contributed significantly to our knowledge of ETacross a wide range of ecosystems [see a recent review by Bal-docchi (2020)] the spatiotemporal variation of global ETand its response to the changing climate remains highly uncer-tain (Mueller et al 2011 Liu et al 2016) because the estima-tion of long-term spatially resolved ET is yet laden by diffi-culties in parameterizing the biophysical processes (egroot water uptake stomatal resistance and its response toCO2 concentration changes) that control ET in the currentland surface models (LSMs) (Ukkola et al 2016 Ma et al2017) and remote sensing algorithms (Vinukollu et al2011 Velpuri et al 2013) In addition to possible modelstructural errors the uncertainties in the estimated ET canalso arise from errors in their gridded vegetation (Fang etal 2019) and soil (Zheng and Yang 2016) parameters dueto the large degree of complexityheterogeneity found in ter-restrial ecosystems For example most LSMs withinNLDAS-2 (the North American Land Data Assimilation Sys-tem phase 2) still utilize the NOAA normalized differencevegetation index data developed by Gutman and Ignatov(1998) on a five-year-mean monthly basis without any interan-nual variation as input (Xia et al 2012) failing to reason-ably capture the impact of vegetation changes on ETBesides a recent sensitivity study by Li et al (2018) demon-strated that the Noah-MP LSM cannot always capture theeffect of spatial changes in forest andor soil types on the sim-ulated ET because of the inherent uncertainties in multipleland cover and soil texture data

As an alternative the complementary relationship (CR)(Bouchet 1963) of evaporation inherently accounts for thedynamic feedback mechanisms found in the soilminusvegeta-tionminusatmosphere interface and thus provides a viablerobust alternative for land ET estimation relying solely onstandard atmospheric forcing without the need for any soilor vegetation data The description in the next two para-graphs of the applied CR method parallels that of Ma and Szil-agyi (2019)

The generalized nonlinear version of the complement-ary relationship (GCR) by Brutsaert (2015) relates twoscaled variables x = Ew Ep

minus1 and y = E Epminus1 as

y = (2minus x) x2 (1)

Here E (mm dminus1) is the actual ET rate while Ep (mm dminus1)is the potential ET rate ie the ET rate of a plot-sized wetarea in a drying (ie not completely wet) environment typic-ally specified by the Penman (1948) equation as

Ep =∆ (RnminusG)∆+γ

+γ fu (elowastminus e)∆+γ

(2)

∆where (hPa degCminus1) is the slope of the saturation vapor pres-sure curve at air temperature T (degC) and γ is the psychromet-ric constant (hPa degCminus1) Rn and G are net radiation at theland surface and soil heat flux into the ground respectively

(the latter is typically negligible on a daily or longer timescale) in water equivalent of mm dminus1 The e term denotesthe saturation while e [= e(Td)] is the actual vapor pres-sure of the air [hPa their difference is called the vapor pres-sure deficit (VPD)] Td is the dewpoint temperature and fuis a wind function often formulated (eg Brutsaert 1982) as

fu = 026(1+054u2) (3)

where u2 (m sminus1) is the 2-m horizontal wind speedThe so-called wet-environment ET rate Ew (mm dminus1)

of a well-watered land surface having a regionally signific-ant areal extent is specified by the Priestley and Taylor(1972) equation

Ew = α∆(T w)∆(T w)+γ

(RnminusG) (4)

The dimensionless PriestleyminusTaylor (PT) coefficient αin Eq (4) normally attains values in the range of [11minus132](Morton 1983) For large-scale model applications of grid-ded data Szilagyi et al (2017) proposed a method of find-ing a value for α thus avoiding the need for any calibration

Very soon after the publication of the GCR Crago etal (2016) and Szilagyi et al (2017) introduced a necessaryscaling into Eq (1) by means of a time-varying wetnessindex w = (Epmax minus Ep)(Epmax minus Ew)minus1 to define the dimen-sionless variable X as X = wx by which Eq (1) keeps its ori-ginal nonlinear form ie

y = (2minusX)X2 (5)

Note that Eq (4) in Priestley and Taylor (1972) wasdesigned with measurements under wet environmental condi-tions therefore Δ should be evaluated at the wet-environ-ment air temperature Tw (degC) instead of the typical drying-environment air temperature T (Szilagyi and Jozsa 2008Szilagyi 2014) By making use of a mild vertical air temperat-ure gradient (de Vries 1959 Szilagyi and Jozsa 2009 Szil-agyi 2014) observable in wet environments (as Rn is con-sumed predominantly by the latent heat flux at the expenseof the sensible one and water representing an unusuallyhigh latent heat of the vaporization value found in nature)Tw can be approximated by the wet surface temperature Tws

(degC) Note that Tws may still be larger than the drying-environ-ment air temperature T when the air is close to saturationbut the same is not true for Tw due to the cooling effect ofevaporation In such cases the estimated value of Tw shouldbe capped by the actual air temperature T (Szilagyi 2014Szilagyi and Jozsa 2018) Szilagyi and Schepers (2014)demonstrated that Tws is independent of the size of the wetarea Thus Tws can be obtained through iterations from theBowen ratio (β) of the sensible and latent heat fluxes(Bowen 1926) when applied over a small plot-sized wetpatch (by the necessary assumption that the available energyfor the wet patch is close to that for the drying surface) thePenman equation is valid for ie

976 DYNAMIC SCALING IMPROVES EVAPORATION ESTIMATES VOLUME 37

β =RnminusGminusEp

Epasymp γ TwsminusT

elowast (Tws)minus elowast (Td) (6)

Epmax in the definition of X within Eq (5) is the maximumvalue that Ep can achieve (under unchanging availableenergy for the surface) during a complete dry-out (iewhen e becomes close to zero) of the environment ie

Epmax asymp∆(Tdry

)(RnminusG)

∆(Tdry

)+γ

+γ fuelowast

(Tdry

)∆(Tdry

)+γ (7)

in which Tdry (degC) is the so-achieved dry-environment air tem-perature The latter can be estimated from the (isoenthalp)adiabat of an air layer in contact with the drying surface (Szil-agyi 2018a) ie

Tdry = Twb+elowast (Twb)γ

(8)

where Twb (degC) is the wet-bulb temperature Twb can beobtained with the help of another iteration of writing out theBowen ratio for adiabatic changes (eg Szilagyi 2014)such as

γ (TwbminusT )elowast (Twb)minus elowast (Td)

= minus1 (9)

For a graphical illustration of the saturation vapor pres-sure curve the different temperatures and the related ETrates defined please refer to Ma and Szilagyi (2019) Thesame source also includes a brief description of how the CRevolved into Eq (5) over the past 40 years Additionally itplots selected historical CR functions over sample data andexplains how assigning a value of α is performed withoutresorting to any calibration A sensitivity analysis of the ETrates in Eq (5) to their atmospheric forcing is found in Maet al (2019)

While Brutsaert et al (2020) realized the necessity ofscaling x with the help of a static aridity index Crago et al(2016) Szilagyi et al (2017) Szilagyi (2018a b) Szilagyiand Jozsa (2018) Ma and Szilagyi (2019) and Ma et al(2019) performed one (and the same one) via a dynamic wet-ness index Whereas the wetness index assigns increasing val-ues to wetter environmental conditions the aridity indexdoes the same to drier ones Brutsaert et al (2020) did notinclude this dynamic wetness index method in their studyand therefore the present work was initiated to fill this gap

2 Model applications

The time-varying (and thus dynamic) scaling of x(Crago et al 2016 Szilagyi et al 2017) by the wetnessindex w [= (Epmax minus Ep)(Epmax minus Ew)minus1] in Eq (5) is neces-sary because the GCR of Brutsaert (2015) unrealistically pre-dicts near-zero land evaporation only when Ew in x itselfapproaches zero This is because the potential evaporationrate Ep in the denominator of x always assumes well-

bounded values due to physical limits on the range of its con-stituents ie net radiation soil heat flux air temperaturewind speed and VPD

An alternative static scaling of x by Brutsaert et al(2020) takes place via an adjustable parameter αc that actsas the PT-α value for wet environments Since Eq (4) canalso be written as Ew = αEe where Ee is the equilibrium evap-oration rate of Slatyer and Mcllroy (1961) thus the scaledvariable X becomes X = αc Ee Ep

minus1 = αcxαminus1 The spatiallyvariable (but constant through time at a given location)value of αc was then related to a long-term aridity index byBrutsaert et al (2020) with the latter defined as the ratio ofthe mean annual Ep and rain depth and globally calibratedwith the help of additional water-balance data requiring alto-gether seven parameters in highly nonlinear equations

Note that the X = wx scaling by Crago et al (2016) andSzilagyi et al (2017) requires only the forcing variables (RnG T u2 and VPD) without the need for external precipita-tionrain data which is significant as precipitation is pos-sibly the most uncertain meteorological variable to predictin climate models It is important to mention that w changeswith each value of x unlike αc As Szilagyi et al (2017)demonstrated a (temporally and spatially) constant value ofthe PT α necessary for x can be set by the sole use of the for-cing variables without resorting to additional water-bal-ance data of precipitation and stream discharge thus mak-ing the approach calibration-free on a large scale (Szilagyi2018b Ma et al 2019 Ma and Szilagyi 2019) where wet-environmental conditions necessary for setting the value ofα can likely be found Note that setting a constant value ofα is also necessary for Brutsaert et al (2020) in order toforce their spatially variable but temporally constant αc val-ues to reach a predetermined value of about 13 under wetconditions Despite almost half a century of research follow-ing publication of the Priestley and Taylor (1972) equationthere is still no consensus about what environmental vari-ables (atmospheric radiative andor surficial properties)and exactly how their spatial and temporal averaging influ-ence the value of the PT α Until compelling information isavailable on these variables a spatially and temporally con-stant α value may suffice for modeling purposes

As was found by Szilagyi (2018b) the value of the PTα depends slightly on the temporal averaging of the forcingdata ie whether or not the monthly values are long-termaverages [yielding α = 113 (Szilagyi et al 2017) and 115(Szilagyi 2018b) respectively] Therefore here it is testedif such is the case for the globally calibrated model of Brut-saert et al (2020) Namely if its performance is affected bysimilar changes (from long-term mean monthly values tomonthly values) in the inputforcing variables then some cau-tion must be exercised during its routine future applicationand recalibration of its seven parameters may be necessaryNote that besides the different scaling of x everything is thesame (including data requirements) in the two GCR modelversions applied here except that Δ in Ee is evaluated at themeasured air temperature in Brutsaert et al (2020) while the

SEPTEMBER 2020 SZILAGYI ET AL 977

same in Ew (= αEe) is evaluated at an estimated wet-environ-ment air temperature (Szilagyi et al 2017) explainedabove

Both model versions (denoted for brevity by GCR-statand GCR-dyn respectively) were tested over the contermin-ous United States first with long-term averages(1981minus2010) of monthly 32-km resolution North AmericanRegional Reanalysis (NARR) (Mesinger et al 2006) radi-ation and 10-m wind (u10) data [reduced to 2-m values viau2 = u10 (210)17 (Brutsaert 1982)] as well as with 42-kmPRISM air and dewpoint temperature values (Daly et al1994) followed by a continuous 37-year simulation ofmonthly values over the 1979minus2015 period The NARRdata were resampled to the PRISM grid by the nearest neigh-bor method Monthly soil heat fluxes were considered negli-gible Evaluation of the model estimates were performed bywater-balance estimates of basin-representative evaporationrates (Ewb) with the help of United States Geological Sur-vey two- and six-digit Hydrologic Unit Code (HUC2 andHUC6) basin (Fig 1) discharge data (Q) together withbasin-averaged PRISM precipitation (P) values as Ewb = P minusQ either on an annual (for trend analysis) or long-termmean annual basis The application of a simplified water bal-ance is justifiable as soil-moisture and groundwater-storagechanges are typically negligible over an annual (or longer)scale (Senay et al 2011) for catchments with no signific-ant trend in the groundwater-table elevation values

3 Results and discussion

With the long-term mean monthly data GCR-stat per-formed slightly but consistently better than GCR-dyn (Fig 2)reflected best in the NashminusSutcliffe model efficiency (NSE)and root-mean-square error (RMSE) values both modelsproviding unbiased basin-averaged mean annual ET estim-

ates This outcome is unsurprising as GCR-stat takes advant-age of measured precipitation while GCR-dyn does not

However the picture changes when switching fromlong-term mean monthly forcing values to monthly valuesin a continuous simulation (Fig 3) GCR-dyn with aslightly changed PT-α value [from 113 to 115 using the pro-cedure of Szilagyi et al (2017)] continues to produceunbiased estimates of basin-averaged mean annual evapora-tion values However the globally calibrated GCR-statmodel underestimates the water-balance-derived values byabout 10 [ie relative bias (RB) of minus009 for both basinscales] and produces reduced interannual variability (see thehorizontally elongated ldquocrosses rdquo for the HUC2 basins inFig 3) in comparison with GCR-dyn Reduced model per-formance of GCR-stat is also apparent in the long-term lin-ear tendencies (obtained as least-squares fitted linear trends)of the basin-averaged annual evaporation values (Fig 4) bybeing less effective than GCR-dyn in reproducing theobserved linear trends in the water-balance data

As on a mean-annual basis GCR-stat performs betterthan (with mean monthly values) or about equal to (in a con-tinuous simulation) GCR-dyn by exploiting precipitationdata (which on the watershed scale naturally serves as anupper bound for land ET) its weakened performance intrends can only be explained by the same reliance on thelong-term means of the precipitation (and Ep) rates in the(therefore) static αc values that will hinder its response toslow (decadal) changes in aridity The same problem can-not occur in GCR-dyn since its wetness index (w) isupdated in each step of calculations

The current GCR-dyn model has already been shown to(a) yield correlation coefficient values in excess of 09 withlocal measurements of latent heat fluxes across diverse cli-mates in China (Ma et al 2019) in spite of large differ-ences in spatial representativeness (ie grid resolution vsflux measurement footprint) and (b) outperform several popu-

Fig 1 Distribution of the 18 HUC2 (outlined in red) and 334 HUC6 basinsacross the conterminous United States Seven HUC6 basins marked bygreen yielded outlying water-balance-derived evaporation estimates and wereleft out of model comparisons


lar large-scale ET products over the conterminous UnitedStates (Ma and Szilagyi 2019) These products includethree LSMsmdashnamely Noah (Chen and Dudhia 2001) VIC(Liang et al 1994) and Mosaic (Koster and Suarez 1996)two reanalysis productsmdashnamely NCEP-II (Kanamitsu etal 2002) and ERA-Interim (Dee et al 2011) another tworemote-sensing based approachesmdashnamely GLEAM (Mir-alles et al 2011 Martens et al 2017) and PML (Zhang etal 2017 Leuning et al 2008) and a spatially upscalededdy-covariance measurement product FLUXNET-MTE(Jung et al 2011) In a comparison with water-balancedata GCR-dyn turns out to produce even better statistics onthe HUC2 scale than the spatially interpolated eddy-covari-

ance measured ET fluxes (Fig 5) which is remarkable froma calibration-free approach GCR-dyn especially excels inthe long-term linear tendency estimates of the HUC2 ETrates demonstrated by Figs 6 and 7 As FLUXNET-MTEemploys several temporally static variables for its spatial inter-polation method its ability to capture long-term trends issomewhat limited (Jung et al 2011) On the contrary thedynamic scaling inherent in GCR-dyn automatically adaptsto such trends and identifies them rather accurately

Among the different popular large-scale ET modelsGCR-dyn produces multi-year mean annual ET rates clos-est in its spatial distribution to those of FLUXNET-MTE(Fig 8) with a spatially averaged ET value almost identical

Fig 2 Regression plots of model estimates [ES (a c) from GCR-dyn EB (b d) from GCR-stat) against water-balance (Ewb)evaporation rates Long-term mean (1981minus2010) monthly values served as model forcing α = 113 in GCR-dyn (a c) NSENashminusSutcliffe model efficiency R linear correlation coefficient RB relative bias RMSE root-mean-square error (mm) SR ratioof standard deviations of the mean annual model and water-balance values


(both in its spatial average and standard deviation) to that ofGLEAM (Fig 8) which is a remote-sensing based approachNote that all models of the comparison (except GCR-dyn)rely on precipitation data as input which greatly aids ETestimations as on a regional scale and long-term basis precip-itation forms an upper bound for terrestrial ET rates plusit may drive any required soil-moisture water-balance calcula-tions

The spatial distribution of the modeled multi-year lin-ear ET trends is displayed in Fig 9 Again GCR-dyn pro-duces results closest in spatial distribution to FLUXNET-MTE in terms of the statistically significant trends and to

GLEAM in general For a more detailed discussion ofmodel comparisons (including additional model descrip-tions) please refer to Ma and Szilagyi (2019)

In conclusion it can be stated that the GCR of evapora-tion (Brutsaert 2015) is a very effective tool in land ET mod-eling as it requires only a few largely meteorological for-cing variables and avoids the need for detailed soil-mois-ture and land-cover information Although attractive as its(altogether seven) parameters have already been globally pre-calibrated the GCR model version (GCR-stat) of Brutsaertet al (2020) may however not perform optimally in estimat-ing long-term tendencies in basin-wide ET rates This is par-

Fig 3 Regression plots of model estimates [ES (a c) from GCR-dyn EB (b d) from GCR-stat] against water-balance (Ewb)evaporation rates Monthly (1979minus2015) values served as model forcing for the continuous simulation of monthly evaporationrates α = 115 in GCR-dyn (a c) The vertical and horizontal bars represent the standard deviation of the annual modeled andwater-balance HUC2 values respectively The large number of data points hinders a similar plot for the HUC6 values


ticularly the case in comparison to an earlier calibration-free version (GCR-dyn) having no precalibrated parametervalues but requiring that its sole temporallyminusandspatiallyminusconstant parameter (ie the PT coefficient) be setusing the actual forcing data through a largely automatedmethod described in Szilagyi et al (2017) Since in a continu-ous monthly simulation both models performed about thesame while the GCR-dyn produced better long-term tenden-cies a dynamic scaling of Ew Ep

minus1 is recommended over a

static one in future applications of the GCR of evaporationAs has been recommended before (Szilagyi 2018b Szil-

agyi and Jozsa 2018 Ma and Szilagyi 2019) it is reiter-ated here that GCR-dyn due to its minimal data require-ment calibration-free nature and dynamic scaling may con-tinue to serve as a diagnostic and benchmarking tool formore complex and data-intensive models of terrestrial ETrates including calibrationverification (for past values) andreality checking (for future scenario values) of the LSM-

Fig 4 Regression plots of the linear trends (1979minus2015) in annual modeled [ES (a c) from GCR-dyn EB (b d) from GCR-stat]and water-balance values The vertical and horizontal bars represent the standard error in the trend-value estimates for themodeled and water-balance HUC2 values (a b) respectively The large number of data points hinders a similar plot of theHUC6 values (c d) RMSE is now in mm yrminus1


Fig 5 Regression plots of the HUC2-averaged multiyear mean annual ET rates (Emod) of GCR-dyn (a) andeight other (bminusi) popular large-scale ET models against the simplified water-balance (Ewb) estimatesTemporal averaging follows the availability of data displayed in parentheses for each product The length ofthe whiskers denotes the standard deviation of the HUC2-basin averaged annual ET values The long blue linerepresents a 11 relationship while the least-squares fitted linear relationships are shown in maroon color(after Ma and Szilagyi 2019)

Fig 6 Regression plots of the linear trend values (mm yrminus1) in modeled HUC2-averaged annual ET sums(Emod) against those in Ewb over 1979minus2015 The length of the whiskers denotes the standard error in theestimated slope value The long blue line represents a 11 relationship while the least-squares fitted linearrelationships are shown in maroon color (after Ma and Szilagyi 2019)


Fig 7 Regression plots of the linear trend values (mm yrminus1) in GLEAM- PML- and FLUXNET-MTE-modeled (aminusc) HUC2-averaged annual ET sums (Emod) against those in Ewb over the different modelperiods (shown in parentheses) For comparison regressions for the GCR-dyn ET values over the sameperiods are also displayed (dminusf) The length of the whiskers denotes the standard error in the estimatedslope value The long blue line represents a 11 relationship while the least-squares fitted linearrelationships are shown in maroon color (after Ma and Szilagyi 2019)

Fig 8 Spatial distribution of the multiyear (1979minus2015) mean annual ET (mm) rates (Emod) by GCR-dyn andeight other popular large-scale ET models (aminusi) and their spatially averaged (j) values plusminus standarddeviations (after Ma and Szilagyi 2019) The 18 HUC2 basins are also outlined


predicted ET rates of existing regional and general circula-tion models

Acknowledgements All data used in this study can beaccessed from the following websites NARR data wwwesrlnoaagovpsddatagriddeddatanarrhtml PRISM temperaturehumidity and precipitation prismoregonstateedu USGS HUC2and HUC6 runoff waterwatchusgsgov id=wwds_runoff NoahET data discgsfcnasagovdatasetsNLDAS_NOAH0125_M_V002summarykeywords=NLDAS VIC ET data discgsfcxnasagovdatasetsNLDAS_VIC0125_M_V002summarykeywords=NLDAS Mosaic ET data discgsfcnasagovdatasetsNLDAS_MOS0125_M_V002summarykeywords=NLDAS NCEP-II ETdata wwwesrlnoaagovpsddatagriddeddatancepreanalysis2html ERA-Interim ET data wwwecmwfintenforecastsdatasetsreanalysis-datasetsera-interim GLEAM ET data httpgleameuhome PML ET data datacsiroaudaplandingpagepid=csiro17375ampv= 202ampd=true FLUXNET-MTE ET data climate-dataguideucareduclimate-datafluxnet-mte-multi-tree-ensembleThe GCR-dyn modeled ET rates and the HUC2- and HUC6-aver-aged Ewb P Q data are available from httpsdigitalcommonsunledunatrespapers986 This research was supported by a BME-Water Sciences and Disaster Prevention FIKP grant of EMMI(BME FIKP-VIZ)

Open Access This article is distributed under the terms of thecreative commons attribution 40 international license (httpcreat-ivecommonsorglicensesby40) which permits unrestricted usedistribution and reproduction in any medium provided you giveappropriate credit to the original author(s) and the source providea link to the creative commons license and indicate if changeswere made

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While the globally distributed eddy-covariance flux towershave contributed significantly to our knowledge of ETacross a wide range of ecosystems [see a recent review by Bal-docchi (2020)] the spatiotemporal variation of global ETand its response to the changing climate remains highly uncer-tain (Mueller et al 2011 Liu et al 2016) because the estima-tion of long-term spatially resolved ET is yet laden by diffi-culties in parameterizing the biophysical processes (egroot water uptake stomatal resistance and its response toCO2 concentration changes) that control ET in the currentland surface models (LSMs) (Ukkola et al 2016 Ma et al2017) and remote sensing algorithms (Vinukollu et al2011 Velpuri et al 2013) In addition to possible modelstructural errors the uncertainties in the estimated ET canalso arise from errors in their gridded vegetation (Fang etal 2019) and soil (Zheng and Yang 2016) parameters dueto the large degree of complexityheterogeneity found in ter-restrial ecosystems For example most LSMs withinNLDAS-2 (the North American Land Data Assimilation Sys-tem phase 2) still utilize the NOAA normalized differencevegetation index data developed by Gutman and Ignatov(1998) on a five-year-mean monthly basis without any interan-nual variation as input (Xia et al 2012) failing to reason-ably capture the impact of vegetation changes on ETBesides a recent sensitivity study by Li et al (2018) demon-strated that the Noah-MP LSM cannot always capture theeffect of spatial changes in forest andor soil types on the sim-ulated ET because of the inherent uncertainties in multipleland cover and soil texture data

As an alternative the complementary relationship (CR)(Bouchet 1963) of evaporation inherently accounts for thedynamic feedback mechanisms found in the soilminusvegeta-tionminusatmosphere interface and thus provides a viablerobust alternative for land ET estimation relying solely onstandard atmospheric forcing without the need for any soilor vegetation data The description in the next two para-graphs of the applied CR method parallels that of Ma and Szil-agyi (2019)

The generalized nonlinear version of the complement-ary relationship (GCR) by Brutsaert (2015) relates twoscaled variables x = Ew Ep

minus1 and y = E Epminus1 as

y = (2minus x) x2 (1)

Here E (mm dminus1) is the actual ET rate while Ep (mm dminus1)is the potential ET rate ie the ET rate of a plot-sized wetarea in a drying (ie not completely wet) environment typic-ally specified by the Penman (1948) equation as

Ep =∆ (RnminusG)∆+γ

+γ fu (elowastminus e)∆+γ

(2)

∆where (hPa degCminus1) is the slope of the saturation vapor pres-sure curve at air temperature T (degC) and γ is the psychromet-ric constant (hPa degCminus1) Rn and G are net radiation at theland surface and soil heat flux into the ground respectively

(the latter is typically negligible on a daily or longer timescale) in water equivalent of mm dminus1 The e term denotesthe saturation while e [= e(Td)] is the actual vapor pres-sure of the air [hPa their difference is called the vapor pres-sure deficit (VPD)] Td is the dewpoint temperature and fuis a wind function often formulated (eg Brutsaert 1982) as

fu = 026(1+054u2) (3)

where u2 (m sminus1) is the 2-m horizontal wind speedThe so-called wet-environment ET rate Ew (mm dminus1)

of a well-watered land surface having a regionally signific-ant areal extent is specified by the Priestley and Taylor(1972) equation

Ew = α∆(T w)∆(T w)+γ

(RnminusG) (4)

The dimensionless PriestleyminusTaylor (PT) coefficient αin Eq (4) normally attains values in the range of [11minus132](Morton 1983) For large-scale model applications of grid-ded data Szilagyi et al (2017) proposed a method of find-ing a value for α thus avoiding the need for any calibration

Very soon after the publication of the GCR Crago etal (2016) and Szilagyi et al (2017) introduced a necessaryscaling into Eq (1) by means of a time-varying wetnessindex w = (Epmax minus Ep)(Epmax minus Ew)minus1 to define the dimen-sionless variable X as X = wx by which Eq (1) keeps its ori-ginal nonlinear form ie

y = (2minusX)X2 (5)

Note that Eq (4) in Priestley and Taylor (1972) wasdesigned with measurements under wet environmental condi-tions therefore Δ should be evaluated at the wet-environ-ment air temperature Tw (degC) instead of the typical drying-environment air temperature T (Szilagyi and Jozsa 2008Szilagyi 2014) By making use of a mild vertical air temperat-ure gradient (de Vries 1959 Szilagyi and Jozsa 2009 Szil-agyi 2014) observable in wet environments (as Rn is con-sumed predominantly by the latent heat flux at the expenseof the sensible one and water representing an unusuallyhigh latent heat of the vaporization value found in nature)Tw can be approximated by the wet surface temperature Tws

(degC) Note that Tws may still be larger than the drying-environ-ment air temperature T when the air is close to saturationbut the same is not true for Tw due to the cooling effect ofevaporation In such cases the estimated value of Tw shouldbe capped by the actual air temperature T (Szilagyi 2014Szilagyi and Jozsa 2018) Szilagyi and Schepers (2014)demonstrated that Tws is independent of the size of the wetarea Thus Tws can be obtained through iterations from theBowen ratio (β) of the sensible and latent heat fluxes(Bowen 1926) when applied over a small plot-sized wetpatch (by the necessary assumption that the available energyfor the wet patch is close to that for the drying surface) thePenman equation is valid for ie


β =RnminusGminusEp




Epmax asymp∆(Tdry

)(RnminusG)

∆(Tdry

)+γ

+γ fuelowast

(Tdry

)∆(Tdry

)+γ (7)



(8)



= minus1 (9)















































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β =RnminusGminusEp




Epmax asymp∆(Tdry

)(RnminusG)

∆(Tdry

)+γ

+γ fuelowast

(Tdry

)∆(Tdry

)+γ (7)



(8)



= minus1 (9)















































REFERENCES



























94JD00483






































































REFERENCES



























94JD00483

















































94JD00483


































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