Uncertainties Caused by Resistances in Evapotranspiration Estimation Using High-Density Eddy Covariance Measurements

Wen Li Zhao School of Environment and Energy, Peking University Shenzhen Graduate School, Peking University, Shenzhen, China
Department of Earth and Environmental Engineering, Columbia University, New York, New York

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Guo Yu Qiu School of Environment and Energy, Peking University Shenzhen Graduate School, Peking University, Shenzhen, China

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Yu Jiu Xiong School of Civil Engineering, Sun Yat-Sen University, Guangzhou, China
Department of Land, Air and Water Resources, University of California, Davis, Davis, California

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Kyaw Tha Paw U Department of Land, Air and Water Resources, University of California, Davis, Davis, California

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Pierre Gentine Department of Earth and Environmental Engineering, Columbia University, New York, New York

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Bao Yu Chen Research Institute of Agricultural Resources and Environment, Jilin Academy of Agricultural Sciences, Key Laboratory of Plant Nutrition and Agro-Environment in Northeast Region, Ministry of Agriculture, Changchun, Jilin, China

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Abstract

Quantifying the uncertainties caused by resistance parameterizations is fundamental for understanding, improving, and developing terrestrial evapotranspiration (ET) models. Using high-density eddy covariance (EC) tower observations in a heterogeneous oasis in northwest China, this study evaluates the impacts of resistances on the estimation of latent heat flux (LE), the energy equivalent of ET, by comparing resistance parameterizations with different complexities under one- and two-source Penman–Monteith (PM) equations. The results showed that the mean absolute percent error (MAPE) for the LE estimates from the one- and two-source PM equations varied from 32% to 53%, and the uncertainties were caused mainly by the resistance parameterizations. Calibrating the parameters required in the resistance estimations could improve the performance of the PM equations; specifically, the MAPEs for the one-source PM equations were approximately 16%, whereas they were 38% for the two-source PM equations, emphasizing that multiple resistances result in increased uncertainties. The following conclusions were reached: 1) the empirical and biophysical parameters required in resistance estimations were responsible for the uncertainty; 2) increasingly complex resistance parameterizations resulted in greater uncertainties in LE estimates; and 3) models without resistance parameterizations exhibited reduced uncertainties in LE estimates.

Supplemental information related to this paper is available at the Journals Online website: https://doi.org/10.1175/JHM-D-19-0191.s1.

© 2020 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Yu Jiu Xiong, xiongyuj@mail.sysu.edu.cn

Abstract

Quantifying the uncertainties caused by resistance parameterizations is fundamental for understanding, improving, and developing terrestrial evapotranspiration (ET) models. Using high-density eddy covariance (EC) tower observations in a heterogeneous oasis in northwest China, this study evaluates the impacts of resistances on the estimation of latent heat flux (LE), the energy equivalent of ET, by comparing resistance parameterizations with different complexities under one- and two-source Penman–Monteith (PM) equations. The results showed that the mean absolute percent error (MAPE) for the LE estimates from the one- and two-source PM equations varied from 32% to 53%, and the uncertainties were caused mainly by the resistance parameterizations. Calibrating the parameters required in the resistance estimations could improve the performance of the PM equations; specifically, the MAPEs for the one-source PM equations were approximately 16%, whereas they were 38% for the two-source PM equations, emphasizing that multiple resistances result in increased uncertainties. The following conclusions were reached: 1) the empirical and biophysical parameters required in resistance estimations were responsible for the uncertainty; 2) increasingly complex resistance parameterizations resulted in greater uncertainties in LE estimates; and 3) models without resistance parameterizations exhibited reduced uncertainties in LE estimates.

Supplemental information related to this paper is available at the Journals Online website: https://doi.org/10.1175/JHM-D-19-0191.s1.

© 2020 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Yu Jiu Xiong, xiongyuj@mail.sysu.edu.cn

1. Introduction

The Penman–Monteith (PM) equation (Monteith 1965), which is considered to adequately represent the evapotranspiration (ET) process, is one of the preeminent and most extensively used techniques for estimating ET (Allen et al. 1998; Mallick et al. 2014; Ershadi et al. 2015). However, the accurate estimation of terrestrial ET with the PM equation is challenged substantially by inaccuracies in the parameterizations associated with the resistance terms (e.g., Liu et al. 2012; Wang and Dickinson 2012; Zhang et al. 2016; Mallick et al. 2018a,b; Sullivan et al. 2019) that are used to describe the combined effects of stressors, such as water stress and stomatal closure. The resistance terms can refer to both canopy resistance and aerodynamic resistance in a one-source model (Wang and Dickinson 2012); in addition, these terms can refer to a more complicated resistance network that includes surface resistance, canopy bulk stomatal resistance, and other sources of resistance in a so-called multilayer/multisource model, such as those described in Shuttleworth and Gurney (1990), Norman et al. (1995), Zhang et al. (2016), and Deng et al. (2017).

The resistances cannot be measured directly at or beyond the canopy scale and thus are often indirectly estimated. The aerodynamic resistance ra is often estimated using the Monin–Obukhov similarity theory (MOST), where the accuracy of the estimate depends on parameters such as the roughness length for heat (or momentum) transfer and atmospheric stratification (Huang et al. 2016; Rigden et al. 2018; Gerhards et al. 2019). For canopy resistance, its inverse, namely, canopy conductance, is often estimated based on leaf stomatal conductance gs (Baldocchi et al. 1991; Leuning et al. 1995; Zhang et al. 2011; Rodrigues et al. 2016) using empirical or physical models with varying degrees of complexity, such as the Jarvis–Stewart framework (Jarvis 1976; Stewart 1988; Mu et al. 2007; Zhang et al. 2010; Tan et al. 2019) or the Ball–Berry framework (Ball et al. 1987; Medlyn et al. 2011; Morillas et al. 2013; Jefferson 2017). Compared to calculating ra, estimating canopy resistance is more problematic because the leaf-level gs is commonly measured at small scales in situ (e.g., Irmak et al. 2008; Zhang et al. 2011) or in the laboratory under controlled conditions (e.g., Ball et al. 1987; Tyree et al. 2005), and these environments do not capture the spatiotemporal variations in the land surface and atmospheric conditions (Irmak et al. 2008; Franks et al. 2017). Additionally, gs values for plant species under different water and climatic combinations are scarce. Moreover, current leaf-level models fail to capture the complex biological controls on water and carbon fluxes throughout the canopy and landscape (Damour et al. 2010; Matheny et al. 2014; Mallick et al. 2015, 2016; van Dijk et al. 2015; Buckley 2017; Bhattarai et al. 2018; Franks et al. 2018) and may contain empirical functions that need to be calibrated prior to application. However, the calibration process is sensitive to the different variables and requires field observations, which are unavailable in most cases.

In addition to the uncertainties generated by the resistance parameterizations, one-source models are suitable for short and dense canopies or bare soil surfaces (e.g., Shuttleworth and Wallace 1985; Rana and Katerji 1998), and uncertainties in ET estimates can be caused by the big-leaf assumption for sparse or tall canopies (Ershadi et al. 2015). Therefore, a two-source model structure has been proposed to better represent the energy and water flux exchanges among the soil, vegetation, and atmosphere via a network of resistances configured architecturally either in series or in parallel (e.g., Shuttleworth and Wallace 1985; Norman et al. 1995; Boulet et al. 2015; Yao et al. 2017a; Y. Li et al. 2019; X. Li et al. 2019). Although the soil and canopy resistances are separately estimated using methods similar to those used in the one-source models, additional hypotheses are introduced to represent two-source conditions, introducing more model complexity. However, a more complex model is likely more difficult to parameterize and apply than a simpler model. As a consequence, the accurate estimation of ET requires an in-depth identification of the sources of uncertainty (Long et al. 2014; Ershadi et al. 2015; Zhang et al. 2016; Yao et al. 2017b; Mallick et al. 2018a,b; Gerhards et al. 2019). Therefore, the objectives of this study are threefold: 1) to evaluate the uncertainties caused by resistances in ET estimates by employing one- and two-source PM models with different parameterizations and complexities, 2) to identify the sources of uncertainty in resistance parameterizations, and 3) to discuss possible solutions for reducing such uncertainties.

The remainder of this paper is organized as follows. We first describe the study methods, including the resistance models, study sites, and datasets (section 2). Then, in section 3, we present the results regarding resistance-related uncertainties in ET estimates with different degrees of complexity under one- and two-source conditions. We then discuss possible solutions in section 4. The abbreviations and variables used in this study are listed in Table A1 in the appendix.

2. Methods and datasets

a. Study framework

To assess the impacts of resistances on the estimation of ET [hereby represented by the latent heat flux (LE), ET’s energy equivalent], we evaluated these influences under one- and two-source networks for the PM equation (Fig. 1). We simplified the evaluation by focusing on rs and neglecting the uncertainty caused by ra because considerable uncertainty will be generated if ra is improperly parameterized (Liu et al. 2007; Mallick et al. 2018b).

Fig. 1.
Fig. 1.

A schematic showing the workflow and scientific questions addressed in this study. Note that this study focuses on canopy resistance (black boxes), and complexity is defined as the number of empirical (or biophysical) variables required in the canopy resistance parameterization (see section 2a for details).

Citation: Journal of Hydrometeorology 21, 6; 10.1175/JHM-D-19-0191.1

For the one-source PM equation [Eq. (1)], rs is estimated using the empirical Jarvis–Stewart framework, which links canopy resistance to the leaf-level gs and environmental stress functions (Jarvis 1976; Stewart 1988), because this framework is one of the most widely used methods (Damour et al. 2010). Furthermore, this approach will continue to be used in estimating resistance (Buckley 2017), as no single method is currently available to estimate resistances accurately (Matheny et al. 2014; van Dijk et al. 2015; Mallick et al. 2016; Bhattarai et al. 2018). Although different versions of stress functions exist for the Jarvis–Stewart framework, the functions in Eq. (3) (PM_JA) were adopted here because the functions have been tested and have performed well in arid northwest China, including our study area and an adjacent experimental site consisting of maize (e.g., S. Li et al. 2013; Li et al. 2015; Hu and Jia 2015). Moreover, the empirical method of Katerji and Perrier (1983) (abbreviated PM_KP), which requires fewer empirical variables than the PM_JA model and has been calibrated at the abovementioned experimental maize site (Li et al. 2015), was included.

For the two-source PM model, the algorithms proposed by Mu et al. (2007) (abbreviated PM_Mu) and Zhang et al. (2009, 2010) (abbreviated PM_PLSH) were used. In PM_Mu, canopy resistance is estimated using the leaf area index (LAI)-based Jarvis–Stewart framework with stress functions associated with the air temperature and vapor-pressure deficit (VPD) [Eq. (13)] (Mu et al. 2007), whereas canopy resistance is calculated based on the normalized vegetation difference index (NDVI) in PM_PLSH [Eq. (15)] (Zhang et al. 2009, 2010). This difference in the canopy resistance parameterization is beneficial for investigating the impact of the parameterization on the ET estimation. Additionally, the uncertainties caused by resistance parameterizations from the empirical PM_KP method to the phenomenological Jarvis–Stewart framework with increasing complexity (defined as the number of empirical variables required for parameterizing the canopy resistance) are also discussed.

b. Study area and evaluation data

The study area is the Zhangye Oasis in the middle of the Heihe River basin (Fig. 2), and it is located at 100°6′–100°52′E and 38°32′–39°24′N, Gansu Province, northwest China. The mean annual air temperature, precipitation, and pan evaporation are 7.3°C, 100–250 mm, and 1200–1800 mm, respectively. The elevations of the oasis and the adjacent Gobi Desert range from 1400 to 1600 m, and the area is generally flat. Maize, spring wheat, vegetables, orchards, and residential areas are the main land-use types in the oasis (Fig. 2).

Fig. 2.
Fig. 2.

Study area: (a) locations of the Zhangye Oasis and the Heihe River basin in China, (b) 21 EC flux towers in the HiWATER project, and (c) detailed locations of the 17 EC systems within the key experimental area in the Zhangye Oasis. The figure is modified from Y. Q. Wang et al. (2016). AWS and LAS are abbreviations for automatic weather station and large-aperture scintillometer, respectively.

Citation: Journal of Hydrometeorology 21, 6; 10.1175/JHM-D-19-0191.1

The datasets consisted of daytime observations from 17 flux towers deployed inside the oasis over an area covering 5.5 km × 5.5 km from May to September 2012; the data were provided by the Heihe Watershed Allied Telemetry Experimental Research (HiWATER) project (X. Li et al. 2013; Liu et al. 2018). At each flux tower site, both an eddy covariance (EC) system (LI-COR Co., Ltd.) and an automatic weather station (AWS) (Campbell Co., Ltd.) with sampling frequencies of 10 Hz and 10 min, respectively, were installed [see Table S1 in the online supplemental material and Liu et al. (2016) for details]. Of the 17 EC systems, 14 were placed in maize fields while the other 3 were installed in a vegetable field, a residential area, and an orchard. The HiWATER project provided quality-controlled LE values every 30 min, and these data were processed by a research group (Liu et al. 2016). If the energy balance closure rate [i.e., (LE + H)/(RnG)] was less than 0.8, the LE from the HiWATER project was forcibly changed to achieve closure using the Bowen ratio method (Twine et al. 2000), which has been used extensively (e.g., Barr et al. 2006; Liu et al. 2011; Xu et al. 2013).

The LAI was measured by an LAI-2000 instrument (LI-COR Co., Ltd.) in a 5 m × 5 m vegetated area near each EC system (Qu et al. 2014). Twenty measurements were collected during the HiWATER project, but LAI values were unavailable for most sites on several days. Ultimately, 16 days of LAI measurements, representing the seedling, shooting, heading, filling, and maturity stages of maize, were used in this study. The LAI values for the maize sites are shown in Fig. 3a, and the weather conditions corresponding to the LAI measurement are shown in Fig. 3b.

Fig. 3.
Fig. 3.

(a) Variations in the average leaf area index (LAI) from 14 maize fields in the 2012 growing season and (b) box-and-whisker plots for solar radiation (Rs), air temperature (Ta), relative humidity (RH), and wind speed (u) for the days corresponding to panel a using daytime (0700–1900 UTC + 8) data from the 15th EC tower. The error bars in (a) indicate the standard deviations of the observations. In the box-and-whisker plot, the boxes show the interquartile range (25th–75th percentiles), the dashed line through a box represents the median, and the white and gray dots represent the mean and outlier values, respectively.

Citation: Journal of Hydrometeorology 21, 6; 10.1175/JHM-D-19-0191.1

c. Model and resistance parameterization descriptions

1) One-source PM equation

The one-source PM equation is as follows (Monteith 1965):
L(ET)=ΔA+ρaCPVPD/raΔ+γ(1+rs/ra).
Parameter ra was calculated, neglecting the effect of atmospheric stability, using Eq. (2) (Brutsaert and Stricker 1979; Irmak et al. 2008):
ra=ln[(zrd)/zom]ln[(zrd)/zoh]k2uzr,
where k is von Kármán’s constant (k = 0.4); d is the zero-plane displacement height and commonly taken as 2/3 of the vegetation height; z0m is the local surface roughness length for momentum transport (in meters) and assumed to be 0.13 times the vegetation height; and z0h is the local surface roughness for heat (in meters) and assumed to be equal to 0.1z0m (Brutsaert and Stricker 1979).
(i) rs parameterization in PM_JA
Parameter rs in Eq. (1) was estimated using the Jarvis–Stewart framework:
rs=rsminLAI1[f(Rs)f(VPD)f(Ta)f(θ)],
where rsmin is the minimum stomatal resistance under optimal conditions (i.e., none of the controlling variables are limiting), and the following stress functions that have performed well in the study area (e.g., S. Li et al. 2013; Li et al. 2015; Hu and Jia 2015; Bai et al. 2017, 2019) were used:
f(Rs)=Rs(1000+k1)1000(Rs+k1),
f(VPD)=1k2VPD,
f(Ta)=(TaTL)(THTa)t(TopTL)(THTop)t,wheret=THTopTopTL,and
f(θ)=θθwθfθw.
(ii) rs parameterization in PM_KP
In the PM_KP model, rs is expressed as follows:
rs=c1r*+c2ra,
where c1 and c2 are empirical coefficients, and r* is related to climatic variables:
r*=Δ+γΔγ×ρaCPVPD(RnG).

In Eq. (1) the available energy, equaling the difference between net radiation and soil heat flux, was obtained directly from the flux tower observations, whereas the VPD, psychrometric constant, and slope of the saturation vapor-pressure curve were indirectly estimated from meteorological observations according to FAO paper 56 (Allen et al. 1998). The determination of the empirical coefficients (e.g., c1 and c2) and biophysical values (e.g., rsmin and Top) required in the rs parameterization [Eqs. (3)(5)] is listed in Table 1 in section 2c(3).

2) Two-source PM equation

In the PM_Mu (Mu et al. 2007) and PM_PLSH (Zhang et al. 2009, 2010) algorithms, total ET is regarded as the sum of canopy transpiration Ec and soil evaporation Es as follows:
L(ET)=LEs+LEc,
where LEs is estimated by combining a modified PM equation with the complementary relationship theory according to Mu et al. (2007) and Zhang et al. (2009, 2010):
LEs=Δ×As+ρa×CP×VPD/ra,sΔ+γ(1+rs,s/ra,s)×(Rh100)VPD/β,
where the surface resistance of the soil surface rs,s and aerodynamic resistance ra,s are estimated as follows:
rs,s=rtot×rcorr,
rcorr=1101300P×(Ta+273.15293.15),
where rtot is the uncorrected total aerodynamic resistance, which is assumed to be constant based on the biome type (Table 1):
ra,s=rsc×rssrsc+rss,
rsc=1gl_sh,and
rss=ρCp4σTa3.
Table 1.

Biophysical parameters used for the resistance parameterizations.

Table 1.
A brief introduction to LEc is provided below:
LEc=Δ×Ac+ρa×CP×VPD/ra,cΔ+γ(1+rs,c/ra,c),
where ra,c is assumed to be equal to ra,s and estimated using Eq. (10) (Zhang et al. 2009, 2010), whereas rs,c is the inverse of canopy conductance gc and described separately in PM_Mu and PM_PLSH:
rs,c=1/gc.
(i) rs,c parameterization in PM_Mu
In PM_Mu, gc is estimated as follows:
gc=LAI×cL×m(Tmin)×m(VPD),
m(Tmin)={1.0,TminTmin_openTminTmin_closeTmin_openTmin_close,Tmin_close<Tmin<Tmin_open0.1,TminTmin_close,
m(VPD)={1.0,VPDVPD_openVPD_closeVPDVPD_closeVPD_open,VPD_close<VPD<VPD_open0.1,VPDVPD_close.
(ii) rs,c parameterization in PM_PLSH
gc=g0(NDVI)×m(Tday)×m(VPD),
g0=1/[b1+b2×exp(b3×NDVI)]+b4,
m(Tday)={0.01,TdayTclose_maxexp[(TdayToptk3)2],Tclose_min<Tday<Tclose_max0.01,TdayTclose_min.

VPD, Δ, and γ in Eqs. (8) and (11) were estimated with the same methods used in Eq. (1), whereas the terms describing the available energy of the soil and vegetation components, As and Ac, were estimated using the fractional vegetation cover fc (Mu et al. 2007; Zhang et al. 2009). The fractional vegetation cover fc was deduced from the measured LAI according to Walthall et al. (2004). The determination of the empirical coefficients (e.g., b) and biophysical values (e.g., cL and Tmin_open/Tmin_close) required in the rs,c parameterization [Eqs. (13)(16)] is described in section 2c(3).

3) Calibration method used in resistance parameterizations

The empirical coefficients and biophysical values required in resistance (i.e., rs and rs,c) estimation were determined using two methods. First, we used prior knowledge (precalibrated values) from the literature as measurements of biophysical values are scarce, especially in data-poor arid northwest China, due to a lack of observations. For the one-source PM equations, values from an experimental site consisting of maize adjacent to our study area in Li et al. (2015) were used (Table 1). For the two-source PM_Mu and PM_PLSH, values were cited from Mu et al. (2007) and Zhang et al. (2009), respectively (Table 1). Second, we calibrated the empirical and biophysical values through a nonlinear least squares method [Eq. (17)], which is a nonlinear regression approach widely applied for constrained problems:
minθi=1n(MiOi)2=minθf(θ,x)O22=minθi=1n[f(θ,xi)Oi]2,
where Oi and Mi are paired LE observations and model estimates, respectively; f(θ, x) is the nonlinear objective function and defined as the PM equations in the above sections, where θ are parameters to be calibrated in the PM equation; x represents the remaining variables; and n is the number of datasets.

Calibration was performed for the maize sites because data were limited for the orchard and vegetable areas, with fewer than 200 observations each, as only one EC system was installed in each. During the calibration, the entire dataset was disrupted randomly and divided into calibration and validation groups with fixed proportions. For example, if half of the available datasets were classified into the calibration group, the remaining data belonging to the validation group were used for independent validation. To obtain the best optimization, we performed nine tests for each calibration by selecting a calibration group with a ratio ranging from 10% to 90% (with an increment of 10% for each test). The trust-region-reflective algorithm, based on the interior-reflective Newton method, was used to solve the nonlinear least squares Eq. (17) in MATLAB (MathWorks Inc., United States).

3. Results

a. Differences among the LE estimates

The LE values estimated from the one- and two-source PM equations with the default resistance methods listed in section 2b exhibited significant differences for a given land type in most cases (Fig. 4). Generally, the mean absolute percent error (MAPE) was greater than 32% for the PM equations, and the one-source PM equation performed slightly better than the two-source PM equation (Fig. 5). The one-source PM_JA method (with an MAPE of 31.7%) displayed the smallest errors in the LE estimates, whereas the empirical PM_KP approach (with an MAPE of 35.5%) led to the largest uncertainty. For the two-source method, the MAPE values were 41.5% and 52.7% for PM_Mu and PM_PLSH, respectively.

Fig. 4.
Fig. 4.

Box-and-whisker plots showing the latent heat flux (LE) values of different land types in the 2012 growing season. Note that EC represents observational data, and the other four labels are estimates based on the default PM methods using uncalibrated parameters; the legends of the figures are similar to those for Fig. 3b, and an asterisk indicates that the estimates are significantly different from the corresponding observations (ANOVA, p < 0.05).

Citation: Journal of Hydrometeorology 21, 6; 10.1175/JHM-D-19-0191.1

Fig. 5.
Fig. 5.

Comparison between the estimated and measured daytime (0700–1900 UTC + 8) latent heat flux (LE) values at vegetated sites. Note that EC represents eddy covariance observations, while the other four labels are estimates based on the default PM methods using uncalibrated parameters; MAE and MAPE represent the mean absolute error and mean absolute percent error, respectively. The temporal scale of the data is 30 min.

Citation: Journal of Hydrometeorology 21, 6; 10.1175/JHM-D-19-0191.1

b. Sources of uncertainty in the LE estimates

Notably, the differences in the LE estimates and model performances between the one-source and two-source PM equations were caused by the different resistance parameterizations. Specifically, the difference for the one-source model was between the resistance parameterizations [Eqs. (3) and (5)], whereas the difference was caused by Eqs. (13) and (15) for the two-source model. Nonetheless, it should be noted that the LE values measured from the EC towers may have uncertainties varying from 1% to 16% (Xu et al. 2013; Wang et al. 2015); as is the usual practice in the research community (e.g., Zhang et al. 2010; Ershadi et al. 2015; Mallick et al. 2015), we treated the measurements as true and did not include the observational uncertainty in the following analyses.

1) Uncertainty in determining the biophysical parameters

When parameterizing resistances, different biophysical parameters (shown in Table 1) are required both in the Jarvis–Stewart framework and in other physical methods (Franks et al. 2017). However, these biophysical parameters likely depend on the species and life cycles of different plants. Although leaf-level values can be measured, the representativeness of those measurements is questionable, and measurements for plant species under different water and climatic combinations are scarce. As such, uncertainties in ET estimation are likely to be generated when biophysical boundaries are required for parameterizing resistances, especially in areas without (or with limited) observations and a priori knowledge.

Taking the one-source PM_JA equation as an example, measurements of the minimum stomatal resistance rsmin require optimal conditions, which are difficult to find (or determine). For example, when estimating the canopy resistance of maize at an experimental station in northwest China, the optimal rsmin was 20 s m−1 based on observations and simulations between 2007 and 2008 (S. Li et al. 2013), but the optimal value was found to be 150 s m−1 in another study by the same research group (Li et al. 2016). In this study, the default rsmin was set to 20 s m−1; if the value had been set to 150 s m−1, obvious differences would have been generated in the LE estimation (Fig. 6a). These results and others (e.g., Riveros-Burgos 2019) indicate that if some key biophysical variables (e.g., rsmin) are not properly determined (i.e., deduced from site experiments rather than from an understanding of the mechanistic theory), significant errors will be generated in the ET estimation, even when applied to other sites with similar environmental conditions, as shown here.

Fig. 6.
Fig. 6.

Differences in the latent heat flux (LE) estimates from modifications to the rs calculation in PM_JA: (a) modifying rsmin, (b) modifying the optimal temperature, and (c) modifying the parameterization of f(VPD). Note that the LE values were estimated using PM_JA; MAE represents the mean absolute error; LE_JA_20 and LE_JA_150 represent the estimated LE values when setting the minimum stomatal resistance rsmin to 20 and 150 s m−1, respectively; LE_JA_10 and LE_JA_35 represent the estimated LE values when parameterizing f(Ta) using optimal temperatures of 10° and 35°C, respectively; and LE_JA_L and LE_JA_Exp represent the estimated LE values when parameterizing f(VPD) using linear and exponential algorithms, respectively [the values were derived from the literature; see section 3b(1) for details].

Citation: Journal of Hydrometeorology 21, 6; 10.1175/JHM-D-19-0191.1

Similar problems are encountered when attempting to obtain the lower, optimal, and upper air temperature limits (or VPD limits) of stomatal activity shown in Table 1. Certain values are typically assumed for each plant functional type because these parameters for each species can vary temporally because of plant acclimation (Kumarathunge et al. 2019). For example, Hatfield et al. (2011) summarized the optimal temperatures for several crops with values ranging from 26°C (wheat) to 37°C (cotton). In an ET model called ETmonitor, the lower, optimal, and upper air temperatures for C4 plants, such as maize, were set to 5°, 35°, and 55°C, respectively (Hu and Jia 2015). S. Li et al. (2013) suggested that the optimal temperature is 10°C for maize. Hence, if the optimal temperature is changed from the default value of 35° to 10°C, rs increases, producing lower LE values with a mean difference of 208 W m−2 (Fig. 6b).

2) Differences among the resistance parameterizations

Figure 7a shows the considerable differences between rs_JA, which was estimated in accordance with Eq. (3), and rs_KP, which was estimated in accordance with Eq. (5). The average difference between the rs_JA and rs_KP estimates was 267 s m−1 if values > 1000 s m−1 were assumed to be outliers, which led to substantial differences in the LE estimates. Similarly, obvious differences were noted between rs,c_Mu, which was estimated in accordance with Eqs. (12) and (13), and rs,c_PLSH, which was estimated in accordance with Eqs. (12) and (15) (Fig. 7b). Notably, several algorithms can be used for a given stress function in the Jarvis–Stewart framework (Table 2). For example, we used a linear algorithm to parameterize f(VPD) in the PM_JA model; however, the relationship between f(VPD) and VPD can be exponential (S. Li et al. 2013; Li et al. 2015). Nevertheless, the results in Fig. 7c suggest that changing the parameterization of f(VPD) from linear to an exponential algorithm can cause overestimation of LE. Thus, the method for estimating resistances and their parameterizations can be major sources of uncertainty in the estimation of ET.

Fig. 7.
Fig. 7.

Differences in surface resistance between the (a) one- and (b) two-source PM schemes. Note that rs_JA and rs_KP are rs values from the default Jarvis–Stewart framework using uncalibrated parameters, rs_PM is the inverse of the one-source PM equation, and rs,c_Mu and rs,c_PLSH are the values from the default PM_Mu and PM_PLSH algorithms using uncalibrated parameters, respectively.

Citation: Journal of Hydrometeorology 21, 6; 10.1175/JHM-D-19-0191.1

Table 2.

Different algorithms that represent the environmental forces for leaf resistance in the PM_JA model.

Table 2.

3) Complexity of resistance algorithms

The results from the one-source PM_JA method were more similar to the observations with respect to MAPE than the LE_KP values, which were estimated using the surface resistance parameterizations from the empirical PM_KP method. However, the relatively complex two-source PM equation performed more poorly than the one-source PM equation, as shown in section 3a, which was likely because the two-source PM equation requires more complex parameterizations. Nonetheless, the performance of PM_Mu was 10% better than that of PM_PLSH in terms of the MAPE (Fig. 5) because even though the only difference between these algorithms was the canopy resistance parameterization, the former requires fewer biophysical parameters than the latter and therefore avoids generating uncertainty. We performed a sensitivity test by adding a perturbation (with an increment of 10%) to each input variable in the rs parameterization and found that the most important variables influencing rs were the VPD and air temperature, with average MAPE values at least 9 times larger than those of the other variables. We also found that in the Jarvis–Stewart framework, more uncertainty was generated in more complex algorithms than in simple ones (Fig. 8).

Fig. 8.
Fig. 8.

Sensitivity analysis for (a)–(d) surface resistance and (e)–(h) latent heat flux (LE) estimates by adding an incremental perturbation to each input variable when parameterizing rs (or rs,c). Note that perturbation was set to 10%, 20%, 30%, 40%, 50%, 60%, 70%, 80%, and 90% of standard deviation. MAPE represents the mean absolute percent error between estimates with and without perturbation; MAPE values > 100% are not shown.

Citation: Journal of Hydrometeorology 21, 6; 10.1175/JHM-D-19-0191.1

4. Discussion

a. Reducing uncertainties in resistance parameterizations

To reduce the uncertainties, we calibrated the model parameters. The results in Table 1 show the differences between the calibrated and default values. The performance of the one-source PM equation improved remarkably after calibration (Fig. 9) with MAPE values of 16.3% and 12.5% for LE_JA and LE_KP, respectively. Considerable improvement was also observed for the two-source PM_PLSH, as evidenced by a decrease in the MAPE of 14%, whereas only marginal improvement (i.e., a MAPE decrease of <4%) was observed for the two-source PM_Mu equation. The results indicate that the PM equation can only be used when the exact values of the resistances are known and that the calibrated two-source PM equation does not extrapolate well, as summarized in Blöschl et al. (2019). This can explain why an analytical solution of the canopy-substrate resistance, rather than a calibration-based approach, was used for estimating ET in the PM equation (Mallick et al. 2014, 2015, 2016, 2018a,b; Bhattarai et al. 2018).

Fig. 9.
Fig. 9.

Performances of the PM equations after calibrating the biophysical parameters prior to the resistance parameterization for the maize sites. Note that MAE and MAPE represent the mean absolute error and mean absolute percent error, respectively, and the temporal scale of the data is 30 min (the same as in Fig. 5).

Citation: Journal of Hydrometeorology 21, 6; 10.1175/JHM-D-19-0191.1

The calibration was performed by trial and error and requires additional observation datasets, which are difficult to obtain in most applications. In this study, we were able to calibrate the biophysical parameters only for maize due to the limited data for the orchard and vegetable fields (with <200 observations). Even if data are available to perform the calibration, such calibrated biophysical values are not transferable across different time scales (even for a biome under similar environmental conditions), as shown here and in other studies (e.g., H. Wang et al. 2016). In fact, the uncertainty caused by model structure and model calibration is an unsolved problem in ET-related hydrological studies (Blöschl et al. 2019). Moreover, dividing the data into calibration and validation groups is a subjective procedure that affects both the numerical values of the calibrated parameters and the validation results. In another test, we increased the number of calibration datasets; however, we found that the performance of the PM equation exhibited only a marginal change (Fig. 10). These results indicate that performing additional parameterizations, increasing algorithmic complexity, and calibrating more parameters enhance the uncertainty and errors.

Fig. 10.
Fig. 10.

Change trends of the one- and two-source PM equation performances with an increase in the ratio of the number of calibration datasets to the total number of datasets. Note that the error bar represents the standard deviation for the estimates from data randomly sampled to perform calibration 100 times at a given ratio.

Citation: Journal of Hydrometeorology 21, 6; 10.1175/JHM-D-19-0191.1

b. Reducing uncertainties by avoiding resistance parameterizations

Avoiding the use of resistances may be a solution to reduce the uncertainties in the estimation of ET. Several methods that eliminate resistances have been proposed and provide comparable accuracy; examples include the three-temperature (3T) model (Qiu et al. 2006; Xiong et al. 2015; Y. Q. Wang et al. 2016), the surface temperature initiated closure model (Mallick et al. 2014, 2015; Bhattarai et al. 2018, 2019), the Priestley–Taylor method (Priestley and Taylor 1972; García et al. 2013; Ai and Yang 2016), the triangle or trapezoidal method (Price 1990; Long and Singh 2012), the maximum entropy production model (Wang and Bras 2011; Hajji et al. 2018), the complementary relationship model (Bouchet 1963; Brutsaert 2015; Ma et al. 2019), the nonparametric approach (Liu et al. 2012), and the surface renewal method (Paw U et al. 1995).

Figure 11 shows that the LEs estimated at the 16 vegetated flux sites from the 3T model (see the supplemental material for a detailed model description), which does not explicitly include any resistance parameterizations, were generally close to the observed values, with an MAPE of 19%. The 3T model requires net radiation, soil heat flux, air temperature, and soil and canopy temperatures as inputs, and the use of fewer model inputs leads to a decreased dependence on observational data; hence, the number of LE estimates from the 3T model was greater than those from the one- and two-source PM equations shown in Fig. 5. Moreover, the performance of the 3T model was comparable to the performances of the calibrated PM equations in Fig. 9; this finding indicates that by eliminating resistance parameterizations, uncertainties generated by such processes can be avoided.

Fig. 11.
Fig. 11.

An example showing the performance of the three-temperature (3T) model, which does not explicitly include any resistance parameterizations. The temporal scale of the data is 30 min (the same as in Fig. 5).

Citation: Journal of Hydrometeorology 21, 6; 10.1175/JHM-D-19-0191.1

5. Conclusions

This study investigated the uncertainties in the estimation of ET generated by resistance terms by comparing different resistance parameterizations under one- and two-source PM methods. The MAPE varied from 32% to 53% for the one- and two-source PM equations with different resistance parameterizations. If the parameters were calibrated properly prior to the resistance estimation, the performance of the one-source PM equation improved considerably (with MAPE values from 12% to 16%), whereas the MAPEs for the two-source PM equation were 38%. Therefore, increasing the number of resistances increases the corresponding error, and the calibrated two-source PM equation requiring overly complicated parameterizations does not extrapolate well. We conclude the following: 1) different resistance parameterizations in the PM equation can produce substantial uncertainties; 2) the empirical and biophysical values required in resistance parameterizations are responsible for uncertainty, and care should thus be taken when using prior knowledge as an alternative because the values are not transferable across temporal scales and between areas; 3) increasingly complex resistance parameterizations lead to larger uncertainties in the estimation of ET, and the large number of ambiguous parameterizations of the resistances and related uncertainties challenge the current status quo of using such formulations in the PM equation; and 4) models without resistance parameterizations exhibit reduced uncertainties in ET estimates.

Acknowledgments

We gratefully acknowledge the financial support from the National Natural Science Foundation of China (41671416) and the China Scholarship Council (201806010242, 201606380186). We also thank the Heihe Watershed Applied Telemetry Experimental Research (HiWATER) project for their support with the EC observation data. We are also grateful to the reviewers for their insightful and constructive comments.

APPENDIX

Variables and Abbreviations

Variables and abbreviations are listed in Table A1.

Table A1.

Summary and description of the variables.

Table A1.

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