Using Median Point in Keeling Plot to Reduce the Uncertainty of the Isotopic Composition of Evapotranspiration

Yusen Yuan; Lixin Wang; Zhongwang Wei; Hoori Ajami; Honglang Wang; Taisheng Du

doi:10.1175/JHM-D-23-0133.1

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Fig. 1.

The comparisons of the isotopic composition of ET based on the mean method [δ_ET (mean)] and the median method [δ_ET (median)] using all six sites for both (a) ¹⁸O and (b) ²H. Daily and hourly δ_ET results were not separated in this figure.

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Fig. 2.

The degree of variance reduction (Δσ²/σ²) among different Keeling plot r² filters. The input data resolutions are 1 Hz (blank bar), 1 min (line bar), 5 min (cross bar), and 1 h (cross bar), respectively. The output resolutions are 1 day for (a) ¹⁸O and (b) ²H and 1 h for (c) ¹⁸O and (d) ²H.

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Fig. 3.

The rate of the mean method variance greater than the median method variance [ $R (σ_{ET}^{2})$ ] had a proportional relationship with the rate of the individual measurement of vapor concentration distribution that passed the normality tests [R(C_υi)].

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Fig. 4.

The relationship between the degree of variance reduction (Δσ²/σ²) and the median value of inverse vapor concentration over the mean value of inverse vapor concentration [C_υ (mean)/C_υ (median)].

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Fig. 5.

(a) The three illustrated normal distribution (N) density formulas and (b) their corresponding IG distribution density formulas. The dash dot, solid, and short dot lines in (a) represent N (μ = 1, σ² = 4), N (μ = 1, σ² = 1), and N (μ = 1, σ² = 0.25), respectively. The dash dot, solid, and short dot lines in (b) represent IG (μ = 1, λ = 0.25), IG (μ = 1, λ = 1), and IG (μ = 1, λ = 4), respectively, where λ = μ²/σ².

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Fig. 6.

The relationship between the degree of variance reduction (Δσ²/σ²) and the average vapor concentration C_υ.

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Fig. 7.

The Keeling plot regression of (a) ¹⁸O and (b) ²H on 17 May 2012 at the HH site, when the degree of variance reduction (Δσ²/σ²) reached the maximum value 33.83%.

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Editorial Type:: Article

Article Type:: Research Article

Using Median Point in Keeling Plot to Reduce the Uncertainty of the Isotopic Composition of Evapotranspiration

Yusen Yuan

Yusen Yuan ^aCenter for Agricultural Water Research in China, China Agricultural University, Beijing, China
^bDepartment of Earth and Environmental Sciences, Indiana University Indianapolis, Indianapolis, Indiana
^cDepartment of Plant and Environmental Sciences, New Mexico State University, Las Cruces, New Mexico
^dDepartment of Environmental Sciences, University of California Riverside, California

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https://orcid.org/0000-0001-5086-0023

Lixin Wang

Lixin Wang ^bDepartment of Earth and Environmental Sciences, Indiana University Indianapolis, Indianapolis, Indiana

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Zhongwang Wei

Zhongwang Wei ^eGuangdong Province Key Laboratory for Climate Change and Natural Disaster Studies, School of Atmospheric Sciences, Sun Yat-sen University, Guangzhou, Guangdong, China

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Hoori Ajami

Hoori Ajami ^dDepartment of Environmental Sciences, University of California Riverside, California

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Honglang Wang

Honglang Wang ^fDepartment of Mathematical Sciences, Indiana University Indianapolis, Indianapolis, Indiana

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Taisheng Du

Taisheng Du ^aCenter for Agricultural Water Research in China, China Agricultural University, Beijing, China

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Online Publication:: 19 Apr 2024

Print Publication:: 01 Apr 2024

DOI:: https://doi.org/10.1175/JHM-D-23-0133.1

Page(s):: 637–649

Received:: 23 Aug 2023

Final Form:: 21 Nov 2023

Accepted:: 06 Mar 2024

Published Online:: 19 Apr 2024

Displayed acceptance dates for articles published prior to 2023 are approximate to within a week. If needed, exact acceptance dates can be obtained by emailing amsjol@ametsoc.org.

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Abstract

The isotopic composition of evapotranspiration δ_ET is a crucial parameter in isotope-based evapotranspiration (ET) partitioning and moisture recycling studies. The Keeling plot method is the most prevalent method to calculate δ_ET, though it contains large extrapolated uncertainties from the least squares regression. Traditional Keeling regression uses the mean point of individual measurements. Here, a modified Keeling plot framework was proposed using the median point of individual measurements. We tested the δ_ET uncertainty using the mean point [σ_ET (mean)] and median point [σ_ET (median)]. Multiple resolutions of input and output data from six independent sites were used to test the performance of the two methods. The σ_ET (mean) would be greater than σ_ET (median) when the mean value of inverse vapor concentration ( $1 / \bar{C_{υ}}$ ) is greater than the median value of inverse vapor concentration [ $1 / C_{υ} (median)$ ]. When applying the filter of r² > 0.8, around 70% of σ_ET (mean) was greater than σ_ET (median). This phenomenon might be due to the normality of the vapor concentration C_υ producing the asymmetric distribution of 1/C_υ. The median method could perform significantly better than the mean method when inputting high-resolution measurements (e.g., 1 Hz) and when the water vapor concentration C_υ is relatively low. Compared to the mean method, applying the median method could on average reduce 6.88% of ET partitioning uncertainties and could on average reduce 9.00% of moisture recycling uncertainties. This study provided a new insight of the Keeling plot method and emphasized handling model output uncertainty from multiple perspectives instead of only from input parameters.

© 2024 American Meteorological Society. This published article is licensed under the terms of the default AMS reuse license. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding authors: Yusen Yuan, yuseny@nmsu.edu; Taisheng Du, dutaisheng@cau.edu

Abstract

Corresponding authors: Yusen Yuan, yuseny@nmsu.edu; Taisheng Du, dutaisheng@cau.edu

Keywords: Databases; In situ atmospheric observations; Isotopic analysis; Uncertainty

1. Introduction

As a crucial transport pathway between the land surface and atmosphere (Jung et al. 2010), terrestrial evapotranspiration (ET) is the sum of physically controlled evaporation E and biochemical promoted transpiration T (Scott et al. 2006), as well as an integral input to the free atmosphere (Trenberth 1999). Although ET has become widely accessible through cooperated networks using the eddy covariance approach (Novick et al. 2018), researchers are unable to partition ET only using water vapor fluxes (Scott et al. 2021). The individual measurements of E and T could be a solution for ET partitioning studies, but the disagreements between E + T and ET brought about large uncertainties (Herbst et al. 1996; Li et al. 2010; Singer et al. 2010; Wei et al. 2017). The hydrogen and oxygen isotopes (²H and ¹⁸O) have been applied in hydrological studies since the 1960s. With the advent of the high-frequency isotope analyzer (Wang et al. 2009) allowing an output of 1-Hz atmospheric vapor concentration C_υ and its isotopes δ_υ, each source water could be either measured or simulated with relatively high precision. As such, the isotope method is becoming a powerful method for ET partitioning (Wang et al. 2010; Wei et al. 2018; Xiao et al. 2018), which bypassed the disagreements between E + T and ET.

The isotopic composition of ET (δ_ET) is an important parameter when calculating T of total ET (F_T) and moisture recycling using the stable isotope-based method. Although some novel δ_ET algorithms have been developed in recent two decades (Griffis et al. 2010; Lee et al. 2007), the Keeling plot method is the most prevalent one (Keeling 1958; Moreira et al. 1997; Yakir and Wang 1996; Yakir and Sternberg 2000), which occupied 43% of studies reviewed by Rothfuss et al. (2021), as its uncertainty is the same as the gradient method but significantly smaller than the eddy covariance isotopic flux method (Good et al. 2012). However, the δ_ET uncertainty contributed to F_T uncertainty was 49%–74% depending on sites (Chen et al. 2022; Cui et al. 2020; Yuan et al. 2022), which is far more than that of the other two parameters, the isotopic composition of evaporation δ_E and the isotopic composition of evaporation δ_T. To make it worse, the δ_ET uncertainty contribution would increase with the increase of F_T (Chen et al. 2022). As T dominates ET at a global scale (Jasechko et al. 2013; Wang et al. 2014), δ_ET would significantly transfer uncertainty to F_T among most terrestrial ecosystems.

Previously, researchers made more efforts to reduce uncertainties on δ_E and δ_T reviewed by Xiao et al. (2018), while rarely on δ_ET. After Phillips and Gregg (2001) proposed the first-order Taylor series expansion formula of isotope-based F_T uncertainty, researchers attempted to reduce δ_ET uncertainty by using the correlation coefficient r² filter (Wei et al. 2015), using data from various heights (Griffis et al. 2010), and avoiding averaging initial data (Good et al. 2012). The main challenge to reduce δ_ET uncertainty nowadays is less likely to further optimize the input parameters: the individual measurement of the vapor concentration C_υi and the isotopic composition of water vapor δ_υi, respectively. The optimizations of the δ_ET derivation algorithms could be a more realistic choice.

In this study, a modified Keeling plot framework was proposed using the median point of individual measurements, whose δ_ET uncertainty would decrease, resulting in a decrease of F_T uncertainty. Theoretical derivations were presented and verified by vapor concentration and isotope datasets from six sites with multiple input and output resolutions.

2. Materials and methods

a. Traditional Keeling plot method for δ_ET

From the principle of isotopic mass balance, we have

δ_{υ} = \frac{S}{C_{υ}} + δ_{ET} .

(1)

With multiple measurements of C_υi and δ_υi collected at various heights, the traditional Keeling plot framework considered that the intercept I of the ordinary least squares (OLS) regression between 1/C_υi and δ_υi is δ_ET, and assuming the slope S = C_A(δ_A − δ_ET) of this OLS regression is constant. More recently, Yuan et al. (2022) proved that if I = δ_ET and S = C_A(δ_A − δ_ET), where C_A is the concentration of advection water vapor and δ_υ and 1/C_υ in Eq. (1) has to be the mean values of 1/C_υi (

1 / \bar{C_{υ}}

) and the mean values of δ_υi (

{\bar{δ}}_{υ}

), respectively. This is an equivalent condition when applying traditional intercept-based Keeling plot:

{\bar{δ}}_{υ} = \frac{S}{\bar{C_{υ}}} + I .

(2)

The traditional Keeling plot uses a linear regression superficially, which is derived by the mean values of vapor concentration and its isotopic composition:

δ_{ET} (mean) = I = {\bar{δ}}_{υ} - \frac{S}{\bar{C_{υ}}} .

(3)

As a result, the traditional intercept-based δ_ET is the δ_ET generated from the mean values of vapor concentration and its isotopic composition [δ_ET (mean)], replacing δ_υ in Eq. (1) with

{\bar{δ}}_{υ}

in Eq. (3) and replacing 1/C_υ in Eq. (1) with

1 / \bar{C_{υ}}

in Eq. (3). Although the mean values of observation data are considered to be representative, it would be better to use medians to strengthen the robustness against outliers (Hatfield and Prueger 2015).

b. Median method for δ_ET

In this study, a new insight into the δ_ET algorithm was established using the median value of observation data [δ_ET (median)]:

δ_{ET} (median) = δ_{υ} (median) - \frac{S}{C_{υ} (median)},

(4)

where 1/C_υ (median) is equal to the median values of 1/C_υi and δ_υ (median) is equal to the median values of δ_υi.

c. The uncertainty of δ_ET (mean) and δ_ET (median)

Following Good et al. (2012) to assume the variance of the individual measurement points (1/C_υi, δ_υi) is caused by the independent variable δ_υi, the variance of δ_ET (mean) [

σ_{ET}^{2} (mean)

] and the variance of

S (σ_{S}^{2})

can be shown as the following form (details in the appendix):

σ_{ET}^{2} (mean) = ψ^{2} [\frac{1}{n} + \frac{{(\frac{1}{\bar{C_{υ}}})}^{2}}{\sum_{i = 1}^{N} {(\frac{1}{C_{υ i}} - \frac{1}{\bar{C_{υ}}})}^{2}}],

(5)

σ_{S}^{2} = ψ^{2} \frac{1}{\sum_{i = 1}^{N} {(\frac{1}{C_{υ i}} - \frac{1}{\bar{C_{υ}}})}^{2}},

(6)

where

ψ^{2} = [1 / (n - 2)] {\sum_{i = 1}^{n} [δ_{υ i} - δ_{ET} - (k / C_{υ i})]}^{2} = [1 / (n - 2)] \sum_{i = 1}^{n} R_{δ_{υ i}}^{2}

, and n is the number of the observation points (1/C_υi, δ_υi). The

\sum_{i = 1}^{n} R_{δ_{υ i}}^{2}

is the residual sum of squares of the Keeling plot regression.

In terms of the principle of the univariate linear regression (Montgomery et al. 2021), we have

δ_{υ} \sim N (I + \frac{S}{C_{υ}}, [\frac{1}{n} + \frac{{(\frac{1}{C_{υ}} - \frac{1}{\bar{C_{υ}}})}^{2}}{\sum_{i = 1}^{N} {(\frac{1}{C_{υ i}} - \frac{1}{\bar{C_{υ}}})}^{2}}] ψ^{2}) .

When δ_υ is

{\bar{δ}}_{υ}

, we have

{\bar{δ}}_{υ} \sim N [I + (S / {\bar{C}}_{υ}), ψ^{2} / n]

whose variance

{\bar{σ}}_{υ}^{2}

ψ^{2} / n

. When δ_υ is δ_υ (median), we have

δ_{υ} (median) \sim N (I + \frac{S}{C_{υ} (median)}, {\frac{1}{n} + \frac{{[\frac{1}{C_{υ} (median)} - \frac{1}{\bar{C_{υ}}}]}^{2}}{\sum_{i = 1}^{N} {(\frac{1}{C_{υ i}} - \frac{1}{\bar{C_{v}}})}^{2}}} ψ^{2}),

whose variance

[σ_{υ}^{2} (median)]

is shown as

σ_{υ}^{2} (median) = ψ^{2} {\frac{1}{n} + \frac{{[\frac{1}{C_{υ} (median)} - \frac{1}{\bar{C_{υ}}}]}^{2}}{\sum_{i = 1}^{N} {(\frac{1}{C_{υ i}} - \frac{1}{\bar{C_{υ}}})}^{2}}} .

(7)

According to the error transfer function, the variance of δ_ET (median) [

σ_{ET}^{2} (median)

] can be formed as

\begin{array}{l} σ_{ET}^{2} (median) = σ_{υ}^{2} (median) + {[\frac{1}{C_{υ} (median)}]}^{2} σ_{S}^{2} = ψ^{2} {\frac{1}{n} + \frac{{[\frac{1}{C_{υ} (median)} - \frac{1}{\bar{C_{υ}}}]}^{2} + {[\frac{1}{C_{υ} (median)}]}^{2}}{\sum_{i = 1}^{N} {(\frac{1}{C_{υ i}} - \frac{1}{\bar{C_{υ}}})}^{2}}} \\ = ψ^{2} {\frac{1}{n} + \frac{{(\frac{1}{\bar{C_{υ}}})}^{2} - 2 \frac{1}{C_{υ} (median)} [\frac{1}{\bar{C_{υ}}} - \frac{1}{C_{υ} (median)}]}{\sum_{i = 1}^{N} {(\frac{1}{C_{υ i}} - \frac{1}{\bar{C_{υ}}})}^{2}}} . \end{array}

\sum_{i = 1}^{N} {[(1 / C_{υ i}) - (1 / \bar{C_{υ}})]}^{2} = {CoV}^{2} \times (n - 1) \times (1 / {\bar{C_{υ}}}^{2})

, where CoV is the coefficient of variation of 1/C_υi, the variance of δ_ET could be standardized as

σ_{ET}^{2} (mean) = ψ^{2} [\frac{1}{n} + \frac{1}{{CoV}^{2} (n - 1)}],

(9)

σ_{ET}^{2} (median) = ψ^{2} {\frac{1}{n} + \frac{1 - 2 \frac{\bar{C_{υ}}}{C_{υ} (median)} [1 - \frac{\bar{C_{υ}}}{C_{υ} (median)}]}{{CoV}^{2} (n - 1)}},

(10)

so that the uncertainty of δ_ET (mean) and δ_ET (median) could be

σ_{ET} (mean) = ψ \sqrt{\frac{1}{n} + \frac{1}{{CoV}^{2} (n - 1)}} .

(11)

σ_{ET} (median) = ψ \sqrt{\frac{1}{n} + \frac{1 - 2 \frac{\bar{C_{υ}}}{C_{υ} (median)} [1 - \frac{\bar{C_{υ}}}{C_{υ} (median)}]}{{CoV}^{2} (n - 1)}} .

(12)

Then, the uncertainty reduction Δσ in terms of Eqs. (11) and (12) and the degree of variance reduction (

Δ σ^{2} / σ^{2}

) according to Eqs. (9) and (10) are able to be defined:

Δ σ = σ_{ET} (mean) - σ_{ET} (median) = ψ \sqrt{\frac{2 \frac{\bar{C_{υ}}}{C_{υ} (median)} [1 - \frac{\bar{C_{υ}}}{C_{υ} (median)}]}{{CoV}^{2} (n - 1)}},

(13)

\frac{Δ σ^{2}}{σ^{2}} = \frac{σ_{ET}^{2} (mean) - σ_{ET}^{2} (median)}{σ_{ET}^{2} (mean)} = \frac{2 \frac{\bar{C_{υ}}}{C_{υ} (median)} [1 - \frac{\bar{C_{υ}}}{C_{υ} (median)}] n}{{CoV}^{2} (n - 1) + n} .

(14)

As CoV² is greater than zero, and n ≥ 2, we can conclude that when

1 / \bar{C_{υ}} > 1 / C_{υ} (median)

σ_{ET}^{2} (mean) > σ_{ET}^{2} (median)

and σ_ET (mean) σ_ET (median); when

1 / \bar{C_{υ}} < 1 / C_{υ} (median)

σ_{ET}^{2} (mean) < σ_{ET}^{2} (median)

and σ_ET (mean) < σ_ET (median). The magnitude of

1 / \bar{C_{υ}}

and

1 / C_{υ} (median)

would be the only criteria to determine which method has lower variance and uncertainty. The uncertainty form σ was applied on Δσ to test whether it was over the isotope analyzer precision. The variance form σ² was applied on

Δ σ^{2} / σ^{2}

because a more simple relationship between

Δ σ^{2} / σ^{2}

and

1 / C_{υ} (median)

over

1 / \bar{C_{υ}}

would be concluded later in section 4b. Moreover, σ² was more straightforward to calculate the uncertainty of F_T compared to σ. The σ² is also more widely used in linear regression–related studies (e.g., the Keeling plot regression) than σ (Wiesel et al. 2008).

d. Description of testing datasets

Six sites (Table 1) were selected that contained datasets of C_υ and δ_υ (both ¹⁸O and ²H) to test the performance of the median method. Of these six sites, five of them were from the Stable Water Vapor Isotope Database (SWVID) hosted by Yale University and sponsored by the U.S. National Science Foundation (Wei et al. 2019). The C_υ, δ_υ, and related meteorology data were stored at hourly resolution in SWVID. Among all 42 sites in SWVID, Heihe (HH), Mase (MS), New Haven (NH), Niwot Ridge (NR), and Rosemount (RM) were selected, which were terrestrial sites, and had been used to simulate δ_ET to make ET partitioning (Berkelhammer et al. 2016; Griffis et al. 2011; Lee et al. 2006; Wei et al. 2015; Wen et al. 2016). The RM site only provided ¹⁸O data. The one site outside SWVID was Shiyanghe Experimental Station of China Agricultural University in Wuwei in northwest China. Water isotope measurements have been conducted in the central of a maize field at the Wuwei (WW) site since 2013 (Wu et al. 2018). In situ vapor isotope measurements with the output of 1-Hz C_υ and δ_υ were installed in the summer of 2017 and 2018 to quantify moisture recycling (Yuan et al. 2020) and F_T (Yuan et al. 2022).

Table 1.

List of six sites to verify the median method.

Here, 96 days of data were used in WW, 121 days of data were used in HH, 385 days of data were used in MS, 226 days of data were used in NH, 817 days of data were used in NR, and 61 days of data were used in RM. Each day, C_υi and δ_υi data from 0700 to 1900 local time were selected to capture the daytime δ_ET. The 1-Hz C_υ and δ_υ data in WW were aggregated to 1, 5, and 10 min and hourly data by averaging. The daily resolution δ_ET (mean) and δ_ET (median) at six sites using the 1-h resolution input C_υi and δ_vi data were calculated to test the applicability of the median method among different sites. The performance of the median method with hourly and daily δ_ET output using multiple input resolutions (1 Hz, 1 min, 5 min, 10 min, and 1 h) in WW was also tested. Different criteria r² of the Keeling plots were considered to evaluate the performance of the median method. As r² > 0.8 and r² > 0.6 are the two general criteria to filter the Keeling plots in previous studies (Wei et al. 2015; Yuan et al. 2022), we tested r² > 0.8, r² > 0.6, and no filter scenario in this study.

3. Results

a. Comparison between the mean and the median method

The comparisons of δ_ET (mean) and δ_ET (median) among all six sites were performed on both ¹⁸O and ²H (Figs. 1a,b). We compared two time resolutions (1 h and daily) of δ_ET calculated by various C_υi and δ_υi time resolutions (i.e., 1 Hz, 1 min, 5 min, 10 min, and 1 h) in WW and daily δ_ET calculated by hourly C_υi and δ_υi at the other five sites. The δ_ET (mean) and δ_ET (median) agreed well with each other [δ_ET (median) = 0.9995 δ_ET (mean), R² = 0.9997*** (where *** indicates statistically significant at p = 0.001)] for both ¹⁸O and ²H, which supported the validity of δ_ET (median).

Fig. 1.

Citation: Journal of Hydrometeorology 25, 4; 10.1175/JHM-D-23-0133.1

When using 1-h resolution C_υi and δ_υi input data and 1-day resolution δ_ET output (Table 2), 27.21% of Keeling plot regression r² for ¹⁸O was greater than 0.8, 44.34% of Keeling plot regression r² for ¹⁸O was greater than 0.6, 17.34% of Keeling plot regression r² for ²H was greater than 0.8, and 36.86% of Keeling plot regression r² for ²H was greater than 0.6. The $σ_{ET}^{2} (mean)$ of ¹⁸O was greater than $σ_{ET}^{2} (median)$ of ¹⁸O in 69.63%, 68.87%, and 63.23% cases when applying the filters of r² > 0.8, r² > 0.6, and no filters, respectively. The $σ_{ET}^{2} (mean)$ of ²H was greater than $σ_{ET}^{2} (median)$ of ²H in 72.22%, 68.65%, and 63.69% cases when applying the filters of r² > 0.8, r² > 0.6, and no filters, respectively. Such results suggested that ¹⁸O-based Keeling plot regression would have higher r² than that of ²H, and the δ_ET (median) method might be more useful when r² was high for both ¹⁸O and ²H.

Table 2.

The number of the isotope composition of ET (δ_ET) among six sites in 1-h input resolution and 1-day output resolution when applying various Keeling plot regression r² filters (>0.8, >0.6, and no filter), and the proportion of the variance of δ_ET using mean point [ $σ_{ET}^{2} (mean)$ ] was larger than the variance of δ_ET using median point (median). Both ¹⁸O and ²H are listed.

The proportion of the median method better than the mean method in WW (the only site with multiple resolution datasets) with multiple input and output resolution is shown in Table 3. The $σ_{ET}^{2} (mean)$ was greater than $σ_{ET}^{2} (median)$ in 81.25% of the cases when δ_ET output was 1-day resolution for both ¹⁸O and ²H, while only 49.55% of the cases showed the advantages of the median method when δ_ET output was 1-h resolution for both ¹⁸O and ²H. The proportion of the median method better than the mean method significantly decreased with the increased resolution of C_υi and δ_υi input data, except for ²H at 1-day resolution of δ_ET output. The largest proportion of the median method better than the mean method occurred when using 1-h input and 1-day output, while the lowest proportion of the median method better than the mean method occurred when using 1-Hz input and 1-h output.

Table 3.

The number of the isotope composition of ET (δ_ET) at the WW site using multiple input and output resolutions when applying various Keeling plot regression r² filters (>0.8, >0.6, and no filter), and the proportion of the variance of δ_ET using mean point [ $σ_{ET}^{2} (mean)$ ] was larger than the variance of δ_ET using median point (median). Both ¹⁸O and ²H are listed.

b. Uncertainty reduction and variance reduction using the median method

The variance reduction results are shown in Tables 4 and 5, which focus on the cases of $σ_{ET}^{2} (mean) > σ_{ET}^{2} (median)$ . After applying the median method among six sites, the average uncertainty reduction [Avg(Δσ)] for ¹⁸O was 0.10‰, 0.15‰, and 0.29‰ when applying the filters of r² > 0.8 , r² > 0.6, and no filters, respectively. The Avg(Δσ) for ²H was 6.60‰, 12.03‰, and 22.90‰ when applying the filters of r² > 0.8, r² > 0.6, and no filters, respectively. The precisions of the vapor isotope analyzers (listed in Table 1) were better than 0.20‰ for ¹⁸O and 1.00‰ for ²H (Smith et al. 2021; Xu et al. 2022). The Avg(Δσ) would decrease when the stricter r² filter was applied.

Table 4.

The ¹⁸O isotope composition of ET when applying the mean method [σ_ET (mean)(¹⁸O)] and the median method [σ_ET (median)(¹⁸O)]. The corresponding ¹⁸O-based average uncertainty reduction {Avg[Δσ(¹⁸O)]}, the average degree of variance reduction {Avg[Δσ²/σ²(¹⁸O)]}, and the maximum degree of variance reduction {Max[Δσ²/σ²(¹⁸O)]} shown at different sites.

Table 5.

The ²H isotope composition of ET when applying the mean method [σ_ET (mean)(²H)] and the median method [σ_ET (median)(²H)]. The corresponding ²H-based average uncertainty reduction {Avg[Δσ(²H)]}, the average degree of variance reduction {Avg[Δσ²/σ²(²H)]}, and the maximum degree of variance reduction {Max[Δσ²/σ²(²H)]} shown on different sites.

The average variance reduction [Avg(Δσ²/σ²)] for ¹⁸O was 9.11%, 7.62%, and 6.40% when applying the filters of r² > 0.8, r² > 0.6, and no filters, respectively. Correspondingly, Avg(Δσ²/σ²) for ²H was 9.09%, 7.69%, and 6.30% when applying the filters of r² > 0.8, r² > 0.6, and no filters, respectively. The Avg(Δσ²/σ²) would increase with the stricter r² filter applied. The maximum variance reduction [Max(Δσ²/σ²)] was 33.83% among all six sites.

c. Median method performance using high-resolution input and output data

As is shown in Fig. 2, the overall Avg(Δσ²/σ²) for ¹⁸O at WW using 1-day output resolution was 17.12%, 11.54%, and 11.41% when applying the filters of r² > 0.8, r² > 0.6, and no filters, respectively, which was higher than that of 1-h output resolution. Similarly, the overall Avg(Δσ²/σ²) for ²H at WW using 1-day input resolution was 15.96%, 10.17%, and 11.41% when applying the filters of r² > 0.8, r² > 0.6, and no filters, respectively, which was also higher than that of 1-h output resolution. The overall Avg(Δσ²/σ²) for ¹⁸O at WW on 1-Hz input resolution was 16.68%, 12.69%, and 9.63% when applying the filters of r² > 0.8, r² > 0.6, and no filters, respectively, which was higher than that of 1-min or 1-h input resolution. Correspondingly, the overall Avg(Δσ²/σ²) for ²H at WW using 1-Hz input resolution was 17.83%, 10.99%, and 9.63% when applying the filters of r² > 0.8, r² > 0.6, and no filters, respectively, which was higher than that of 1-min or 1-h input resolution.

Fig. 2.

Citation: Journal of Hydrometeorology 25, 4; 10.1175/JHM-D-23-0133.1

4. Discussion

a. Why the median method has lower uncertainty and variance than the mean method in most cases

As the magnitude of $1 / \bar{C_{υ}}$ and $1 / C_{υ} (median)$ would be the only criteria to determine which method has lower uncertainty and variance, the proposition of the reason why most time the median method performs better than the mean method is equivalent to the proposition of the reason why in most time $1 / \bar{C_{υ}} > 1 / C_{υ} (median)$ .

We noticed that in either traditional Keeling plot algorithm or our median point algorithm, we do not use measured C_υi directly, but we use inversed C_υi. The C_υi normality was tested using the Kolmogorov–Smirnov (K-S) test for 1-Hz, 1-min, and 10-min resolution C_υ data and Shapiro–Wilk (S-W) for 5-min and 1-day resolution C_υ data. Generally, the S-W test is more appropriate for handling sample sizes < 50, whereas the K-S test is used for handling sample sizes > 50. (Mishra et al. 2019; Ruxton et al. 2015). Most (about 75%) C_υi distribution for 1-h input and 1-day output among all six sites were normal distribution (Table 6). As for the various input and output in WW, the rate of $σ_{ET}^{2} (mean) > σ_{ET}^{2} (median) [R (σ_{ET}^{2})]$ had a proportional relationship with the rate of C_υi distribution passing the normality tests [R(C_υi)] according to Fig. 3 [ $R (σ_{ET}^{2}) = 2.05 \times R (C_{υ i}) - 0.75 %$ , R² = 0.65]. As a result, the normality of C_υi would lead to $1 / \bar{C_{υ}} > 1 / C_{υ} (median)$ , as well as in most cases, the median method had lower uncertainty and variance than the mean method in our study.

Table 6.

The normality test passed rate of the number of the atmosphere vapor concentration C_υ among six sites.

Fig. 3.

Citation: Journal of Hydrometeorology 25, 4; 10.1175/JHM-D-23-0133.1

Theoretically, for a normal distribution A ∼ N(μ, σ²), A⁻¹ is an inverse Gaussian (IG) distribution (μ > 0, λ > 0, λ = μ²/σ²) whose skewness 3 × (μ/λ)^1/2 is always greater than zero (Chhikara 1988). When the skewness is greater than zero, the mean is greater than the median. Therefore, if C_υi ∼ N(μ, σ²), 1/C_υi ∼ IG(μ, λ), then $1 / \bar{C_{υ}} > 1 / C_{υ} (median)$ . Not only for a normal distribution but also for a uniform distribution A ∼ U(a, b), A⁻¹ is a Pareto distribution (x_m > 0, α > 3) whose skewness 2(1 + α)/(α − 3) × [(α − 2)/α]^1/2 is also always greater than zero (Hosking and Wallis 1987). Therefore, when C_υi is evenly increasing or decreasing to make C_υi ∼ U(a, b), we still can conclude that $1 / \bar{C_{υ}} > 1 / C_{υ} (median)$ .

b. When shall we use the median method for δ_ET?

As is shown in Eq. (14),

Δ σ^{2} / σ^{2}

is related to CoV², n, and

\bar{C_{υ}} / C_{υ} (median)

. The CoV² value is extremely small, which had an average value of 7.56 × 10⁻¹⁰ ppm⁻² in this study. Thus, CoV²(n − 1) + n ≈ n can be applied to simplify Eq. (14) as

\frac{Δ σ^{2}}{σ^{2}} = \frac{σ_{ET} {(mean)}^{2} - σ_{ET} {(median)}^{2}}{σ_{ET} {(mean)}^{2}} \approx 2 \frac{\bar{C_{υ}}}{C_{υ} (median)} [1 - \frac{\bar{C_{υ}}}{C_{υ} (median)}],

(15)

which was also supported by our in situ data among six sites in Fig. 4. Therefore, the variance reduction is related to the ratio of

1 / C_{υ} (median)

over

1 / \bar{C_{υ}}

Fig. 4.

Citation: Journal of Hydrometeorology 25, 4; 10.1175/JHM-D-23-0133.1

Higher C_υi and δ_υi input resolution would be the one scenario to make a larger Δσ²/σ². Inputting 1-Hz data has been shown by Good et al. (2012) that could reduce 94.14% variance for ¹⁸O and reduce 95.99% variance for ²H compared with 3-h resolution input data using the traditional Keeling plot method. For a given resolution output δ_ET in this study (Figs. 2a–d), Avg(Δσ²/σ²) would increase with the increase of input C_υi and δ_υi resolution. The median method would be more useful when the variance of δ_ET had been minimized in the traditional Keeling plot algorithm. One possible reason for this phenomenon is that the kurtoses of C_υi distribution would decrease with the increase of input number (Searls and Intarapanich 1990), which would bring about an increased skewness of its inverse distribution (Figs. 5a,b). Then, $1 / C_{υ} (median)$ and $1 / \bar{C_{υ}}$ would be more separate with an increasing skewness.

Fig. 5.

Citation: Journal of Hydrometeorology 25, 4; 10.1175/JHM-D-23-0133.1

The other scenario to apply the median method would be when C_υ is relatively small. Ideally, when the expectation of C_υi distribution is closer to zero, the variance of 1/C_υi would increase, leading to an increasing distance between $1 / C_{υ} (median)$ and $1 / \bar{C_{υ}}$ . Results in our study also proved that relatively high Δσ²/σ² values occurred when average vapor concentrations were relatively small (Fig. 6). The average Δσ²/σ² values were higher in WW and HH than those of other four sites, mainly due to WW and HH being in arid areas that have lower C_υ. On 17 May 2012, Δσ²/σ² value reached its maximum (33.83%) at HH site. When inverse lower C_υi (e.g., 2000 ppm), it would be more likely to be outliers on the Keeling plot regression lines (Figs. 7a,b). ET partitioning has significant importance in quantifying biomass production and understanding the water resource allocation (Kool et al. 2014), especially in arid and semiarid ecosystems (Sun et al. 2019), which occupied 28% of all ET partitioning studies (Rothfuss et al. 2021). The higher Δσ²/σ² values would significantly reduce the uncertainty of F_T calculations in arid and semiarid ecosystems.

Fig. 6.

The relationship between the degree of variance reduction (Δσ²/σ²) and the average vapor concentration C_υ.

Citation: Journal of Hydrometeorology 25, 4; 10.1175/JHM-D-23-0133.1

Fig. 7.

The Keeling plot regression of (a) ¹⁸O and (b) ²H on 17 May 2012 at the HH site, when the degree of variance reduction (Δσ²/σ²) reached the maximum value 33.83%.

Citation: Journal of Hydrometeorology 25, 4; 10.1175/JHM-D-23-0133.1

Although Δσ²/σ² could be significantly higher when inputting higher resolution C_υi and δ_υi data, and when C_υ is relatively small, there is an upper bound limit for Δσ²/σ² when transferring Eq. (15) as follows:

\frac{Δ σ^{2}}{σ^{2}} \approx 2 \frac{\bar{C_{υ}}}{C_{υ} (median)} [1 - \frac{\bar{C_{υ}}}{C_{υ} (median)}] = - 2 {[\frac{\bar{C_{υ}}}{C_{υ} (median)} - \frac{1}{2}]}^{2} + \frac{1}{2},

(16)

where the maximum Δσ²/σ² is 0.5 when

1 / \bar{C_{υ}} = 2 \times [1 / C_{υ} (median)]

c. The benefits for ET partitioning and moisture recycling calculations

As δ_ET (mean) and δ_ET (median) were consistent, the ET partitioning results were consistent as well. However, if the median method was applied to obtain δ_ET when

1 / \bar{C_{υ}} > 1 / C_{υ} (median)

, the uncertainty of F_T would have decreased. The ET partitioning uncertainty analysis was applied based on the first-order Taylor series approximation (Phillips and Gregg 2001):

σ_{F_{T}} = \sqrt{\frac{\partial^{2} F_{T}}{\partial^{2} δ_{ET}} σ_{ET}^{2} + \frac{\partial^{2} F_{T}}{\partial^{2} δ_{E}} σ_{E}^{2} + \frac{\partial^{2} F_{T}}{\partial^{2} δ_{T}} σ_{T}^{2}},

(17)

where

σ_{E}^{2}

and

σ_{T}^{2}

represent the variance of δ_E and δ_T, respectively, and

σ_{F_{T}}

represents the uncertainty of F_T. The

σ_{F_{T}}

usually applies on the daily time scale, as

σ_{E}^{2}

and

σ_{T}^{2}

are usually the variance of subdaily δ_E and δ_T (Wei et al. 2015; Wu et al. 2018; Xiao et al. 2018). Thus, if

σ_{F_{T}}

is applied at the hourly scale, hourly

σ_{E}^{2}

and

σ_{T}^{2}

are zero, and the degree of

σ_{F_{T}}

reduction is σ_ET (median) over σ_ET (mean). Unfortunately, we do not have hourly scale δ_E and δ_T data to present hourly

σ_{F_{T}}

results. If daily scale

σ_{F_{T}}

is applied at HH (Wen et al. 2016), MS (Wei et al. 2015), and WW (Yuan et al. 2022), respectively, where δ_E and δ_T data are accessible, the daily

σ_{F_{T}}

could on average decrease 0.0061, 0.0016, and 0.0137 in HH, MS, and WW, respectively. That would be 90.56%, 96.34%, and 92.07% of the original daily

σ_{F_{T}}

applied by the mean method (Table 7). If the maximum

Δ σ^{2} / σ^{2}

is used on each site, the daily

σ_{F_{T}}

could maximally decrease 0.0170, 0.0073, and 0.0244 in HH, MS, and WW, respectively. That would be 82.34%, 85.45%, and 85.86% of the original daily

σ_{F_{T}}

applied by the mean method, respectively. Although the daily

σ_{F_{T}}

decreases, the T/ET results based on the mean method and the median method were both 0.87, 0.89, and 0.80 at HH, MS, and WW, respectively.

Table 7.

The uncertainty of transpiration over evapotranspiration based on the mean method [ $σ_{F_{T}} (mean)$ ] and median method [ $σ_{F_{T}} (median)$ ], respectively. The $σ_{F_{T}} (mean)$ and $σ_{F_{T}} (median)$ are calculated by the isotope composition of evaporation δ_E and its uncertainty σ_E, the isotope composition of transpiration δ_T and its uncertainty σ_T, the isotope composition of ET using mean method [δ_ET (mean)] and its uncertainty [σ_ET (mean)], and the isotope composition of ET using median method [δ_ET (median)] and its uncertainty [σ_ET (median)]. Note that δ_E, δ_T, σ_E, and σ_T are estimated values cited from Wen et al. (2016), Wei et al. (2015), and Yuan et al. (2022), respectively.

Similarly, for local moisture recycling rate calculation, the uncertainty of moisture recycling σ_MR at a daily time scale could decrease 0.0526 on average at WW (Yuan et al. 2020). That would be 91.00% of the original daily σ_MR applied by the mean method. Daily σ_MR could maximally decrease 0.0950 at WW, which would be 83.74% of the original daily σ_MR applied by the mean method.

d. Implementation of the median method

The traditional Keeling plot inputs $1 / \bar{C_{υ}}$ , ${\bar{δ}}_{υ}$ , and s, while the median method inputs 1/C_υ (median), δ_υ (median), and s. Although ${\bar{σ}}_{υ}^{2}$ is always smaller than $σ_{υ}^{2} (median)$ , $σ_{ET}^{2} (mean)$ is always greater than $σ_{ET}^{2} (median)$ when $1 / \bar{C_{υ}} > 1 / C_{υ} (median)$ due to the extrapolation algorithm of the Keeling plot. The $σ_{ET}^{2} (mean)$ would enlarge several times from ${\bar{σ}}_{υ}^{2}$ depending on the distance between $1 / \bar{C_{υ}}$ and the y axis (Yuan et al. 2022), bringing about a relatively large δ_ET (mean) variance. Using the median method is a compromising strategy to balance the variance of input data and the calculation process. In spite of larger uncertainty in δ_υ (median) than ${\bar{δ}}_{υ}$ , the median method finally obtains a lower variance of output δ_ET through a closer extrapolation to the y axis. We cautiously remind researchers to handle model output uncertainty from multiple perspectives rather than only from input parameters.

The traditional Keeling plot method was established by Keeling (Keeling 1958, 1961) to originally capture CO₂ emission isotope compositions on a regional basis whose theory is based on the isotope mass balance principle. Apart from water vapor isotopes and ¹³C isotope from CO₂, the Keeling plot has also been explored to intercept the isotope compositions of CH₄ fluxes (Fisher et al. 2011) and N₂O fluxes (Francis Clar and Anex 2022) to better understand the emission of greenhouse gases. Using the median method in these studies may obtain less variance of the isotope composition of greenhouse gas fluxes. The net ecosystem CO₂ exchange consists of soil respiration and gross primary productivity (Gilmanov et al. 2007). The lower variance of the isotope composition of net ecosystem CO₂ exchange may lead to less uncertainty of gross primary productivity partitioning.

5. Conclusions

In this study, a novel algorithm was established using the median point of in situ measurements to estimate the isotopic composition of evapotranspiration δ_ET. The theoretical deviation was presented and tested using observation with multiple input and output resolutions from six independent sites. The δ_ET variance $σ_{ET}^{2}$ would significantly decrease when the mean value of inverse vapor concentration $1 / \bar{C_{υ}}$ was greater than the median value of inverse vapor concentration [ $1 / C_{υ} (median)$ ]. When applying the filter r² > 0.8, around 70% of the Keeling plot regression should apply the median method rather than the traditional intercept method as well as the mean method. The criteria of when the median method is better than the mean method are also related to the normality of individual measurement of inverse vapor concentration (1/C_υi). The median method could perform even better when inputting higher time resolution measurement data and when the vapor concentration C_υ is relatively small. When applying the median method at the Wuwei, Heihe, and Mase sites, the average F_T uncertainty reduction was 6.88%, and the average moisture recycling uncertainty reduction was 9.00%. This study provided a new insight into the Keeling plot method. The median point theory in this study could be potentially applied to calculate the isotopic composition of CO₂, CH₄, and N₂O using the traditional Keeling plot method.

Acknowledgments.

We acknowledge major support from the National Natural Science Foundation of China (52239002). LW acknowledges partial support from the Department of Energy Grant DE-SC0024297. HA acknowledges partial support from University of California Riverside and the USDA National Institute of Food and Agriculture Hatch funds (PI Ajami CA-R-ENS-5147-H). We thank Dr. Manoj Shukla, Dr. Kenneth Carroll, and Dr. Ray Anderson for your assistance.

Data availability statement.

Data are available at http://vapor-isotope.yale.edu/.

APPENDIX

Variance of Keeling Plot Slope and Intercept

The original form of

σ_{ET}^{2} (mean)

and

σ_{S}^{2}

was proposed by Good et al. (2012):

σ_{ET}^{2} (mean) = ψ^{2} \frac{\sum_{i = 1}^{n} \frac{1}{C_{υ i}^{2}}}{n [\sum_{i = 1}^{n} \frac{1}{C_{υ i}^{2}} - \frac{{(\sum_{i = 1}^{n} \frac{1}{C_{υ i}})}^{2}}{n}]},

(A1)

σ_{S}^{2} = ψ^{2} \frac{1}{\sum_{i = 1}^{n} \frac{1}{C_{υ i}^{2}} - \frac{{(\sum_{i = 1}^{n} \frac{1}{C_{υ i}})}^{2}}{n}} .

(A2)

Note that

\sum_{i = 1}^{n} (1 / C_{υ i}^{2}) - {{[\sum_{i = 1}^{n} (1 / C_{υ i})]}^{2} / n} = \sum_{i = 1}^{n} {[(1 / C_{υ i}) - (1 / \bar{C_{υ}})]}^{2}

is based on the principle of expectation and variance. As such, Eqs. (A1) and (A2) can be reorganized as

\begin{array}{l} σ_{ET}^{2} (mean) = ψ^{2} \frac{\sum_{i = 1}^{n} \frac{1}{C_{υ i}^{2}} - \frac{{(\sum_{i = 1}^{n} \frac{1}{C_{υ i}})}^{2}}{n} + \frac{{(\sum_{i = 1}^{n} \frac{1}{C_{υ i}})}^{2}}{n}}{n [\sum_{i = 1}^{n} \frac{1}{C_{υ i}^{2}} - \frac{{(\sum_{i = 1}^{n} \frac{1}{C_{υ i}})}^{2}}{n}]} = ψ^{2} \frac{\frac{1}{n} \sum_{i = 1}^{n} {(\frac{1}{C_{υ i}} - \frac{1}{\bar{C_{υ}}})}^{2} + {(\frac{1}{n} \sum_{i = 1}^{n} \frac{1}{C_{υ i}})}^{2}}{\sum_{i = 1}^{n} {(\frac{1}{C_{υ i}} - \frac{1}{\bar{C_{υ}}})}^{2}} \\ = ψ^{2} [\frac{1}{n} + \frac{{(\frac{1}{\bar{C_{υ}}})}^{2}}{\sum_{i = 1}^{N} {(\frac{1}{C_{υ i}} - \frac{1}{\bar{C_{υ}}})}^{2}}], \end{array}

(A3)

σ_{S}^{2} = ψ^{2} \frac{1}{\sum_{i = 1}^{N} {(\frac{1}{C_{υ i}} - \frac{1}{\bar{C_{υ}}})}^{2}} .

(A4)

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Fig. 6.

The relationship between the degree of variance reduction (Δσ²/σ²) and the average vapor concentration C_υ.

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Fig. 7.

The Keeling plot regression of (a) ¹⁸O and (b) ²H on 17 May 2012 at the HH site, when the degree of variance reduction (Δσ²/σ²) reached the maximum value 33.83%.

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Which Is Better, an Ensemble of Positive–Negative Pairs or a Centered Spherical Simplex Ensemble?

Authors:

Xuguang Wang

Craig H. Bishop

, and

Simon J. Julier

METEOROLOGY IN THE COLLEGES AND UNIVERSITIES

Spectral Characteristics of Convective-Scale Precipitation Observations and Forecasts

Authors:

May Wong

and

William C. Skamarock

Chart I. Tracks of Centers of High Areas. July, 1902

Atlantic Hurricane Season of 1980

Authors:

Miles B. Lawrence

and

Joseph M. Pelissier

Using Median Point in Keeling Plot to Reduce the Uncertainty of the Isotopic Composition of Evapotranspiration

Abstract

Abstract

1. Introduction

2. Materials and methods

a. Traditional Keeling plot method for δ_ET

b. Median method for δ_ET

c. The uncertainty of δ_ET (mean) and δ_ET (median)

d. Description of testing datasets

3. Results

a. Comparison between the mean and the median method

b. Uncertainty reduction and variance reduction using the median method

c. Median method performance using high-resolution input and output data

4. Discussion

a. Why the median method has lower uncertainty and variance than the mean method in most cases

b. When shall we use the median method for δ_ET?

c. The benefits for ET partitioning and moisture recycling calculations

d. Implementation of the median method

5. Conclusions

Acknowledgments.

Data availability statement.

APPENDIX

Variance of Keeling Plot Slope and Intercept

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Using Median Point in Keeling Plot to Reduce the Uncertainty of the Isotopic Composition of Evapotranspiration

Abstract

Abstract

1. Introduction

2. Materials and methods

a. Traditional Keeling plot method for δET

b. Median method for δET

c. The uncertainty of δET (mean) and δET (median)

d. Description of testing datasets

3. Results

a. Comparison between the mean and the median method

b. Uncertainty reduction and variance reduction using the median method

c. Median method performance using high-resolution input and output data

4. Discussion

a. Why the median method has lower uncertainty and variance than the mean method in most cases

b. When shall we use the median method for δET?

c. The benefits for ET partitioning and moisture recycling calculations

d. Implementation of the median method

5. Conclusions

Acknowledgments.

Data availability statement.

APPENDIX

Variance of Keeling Plot Slope and Intercept

REFERENCES

Share Link

Headings

References

Figures

Cited By

Metrics

Related Content

AMS Publications

Get Involved with AMS

Affiliate Sites

Follow Us

Contact Us

a. Traditional Keeling plot method for δ_ET

b. Median method for δ_ET

c. The uncertainty of δ_ET (mean) and δ_ET (median)

b. When shall we use the median method for δ_ET?