Partial energy balance closure of eddy covariance evaporation measurements using concurrent lysimeter observations over grassland

With respect to the ongoing discussion on the causes of the energy imbalance and approaches to force energy balance closure a method had been proposed which allows the partial latent heat flux closure (Widmoser and Wohlfahrt; 2018). In the 10 present paper, this method is applied to four measurement stations over grassland under humid and semi-arid climate, where lysimeters (LY) and eddy covariance (EC) measurements were taken simultaneously. Results differ essentially from the ones quoted in literature. We distinguish between resulting EC-values weakly and strongly correlated to LY-observations as well as systematic and random deviations between LYund EC-values. At the overall average, an excellent match could be achieved between LY and EC-measurements, which were partially closed with evaporation-linked 15 weights. But there remain high differences between standard deviations of LYand adjusted EC-values. For further studies we recommend data collected at time intervals even below half an hour. No correlation could be found between correction evaporation weights and weather indices. Only for some datasets a positive correlation between evaporation and the correcting evaporation weight could be found. This effect appears pronounced for cases with high radiation and plant water stress. 20 Without further knowledge on the causes of energy imbalance one might perform full closure using equally distributed weights. Full closure, however, is not dealt with in this paper.

Bowen ratio preservation and correct measurements of the available energy. The comparison with LY-measurements on two 30 fields reduced the differences from -41.4% to -28.8%, respectively from -34.1 to -26% with an accuracy of -0.03 ± 0.5 mm d -1 (≈ -0.9 ± 14 Wm -2 ), respectively -0.1 ± 0.4 mm d -1 ( ≈ -2.8 ± 11 Wm -2 ). Negative values indicate that the lysimeter values were higher on average than EC-values. Evett et al. (2012), using data from the same site as Chavez and Howell (2009), quotes errors of EC-measurements for latent heat flux with 1.9 to 2.7 mm d -1 (≈ 55 to 78 Wm -2 ), for sensible heat flux with 1.4 to 1.9 mm d -1 (≈ 40 to 55 Wm -2 ). Since those 35 observations were made on cotton fields, an influence of the increasing plant height as against constant measurement height is suspected. After forced closure of the energy gap as done by Chavez and Howell (2009) differences between the two measurements methods were found from -17 to -19 % after correcting for plant growth, i.e. smaller than the ones mentioned by Chavez and Howell (2009).
In the same way, Ding et al. (2010) closed the energy gaps using half-hourly data on irrigated maize in an arid area in  China. There also, differences of daily measurements were reduced by forced Bowen ratio closure of the EC-gap. Differences could be reduced from -22.4% to -6.2%, the lysimeter measurements again being higher on average.
The following authors dealt with comparing measurements on grassland. Gebler et al. (2015) assumed that the energy balance deficit is caused by an underestimation of the turbulent fluxes only, which are corrected according to the evaporative fraction 45 LE/(LE+H) averaged over 7 days. After correction, they find an agreement of LY-values with EC-values of 3.8 % (19 mm) over a year. The best agreements on the basis of monthly values during summer were obtained with less than 8 % of relative errors. The remaining differences are suspected to be due to different plant height within EC-fetch and the lysimeter. Mauder et al. (2018) evaluated two adjustment methods to close the energy balance: (1) the Bowen ratio preservation adjustment, following the approach of Mauder et al. (2013); (2) the method by Charuchittipan et al. (2014), which attributes a larger portion 50 of the residual to the sensible heat flux. They also compare the EC-values with the results of the hydrological model GEO top 2.0 (Endrizzi et al.;. They found that a daily adjustment factor leads to less scatter than a complete partitioning of the residual for every half-hour time interval. In contrast to the closure method used by the above quoted authors, Widmoser and Wohlfahrt (2018) achieved a partial latent heat closure of the energy balance by a direct comparison between LY-and ECmeasurements, which is afterwards fully closed under the assumption of preservation of the Bowen ratio. 55 In this article, we concentrate on the partial evaporation closure of several datasets from four different stations by comparing concurrent LY-and EC-measurements. We close the energy gaps of the latent heat fluxes by applying the method used by Wohlfahrt and Widmoser (2013), which will be explained briefly in Sect. 2.5. The closing weights (wL) as well as systematic (d) and random deviations (dran) between LY-und EC-measurements will be presented. Results of the different datasets will 60 be compared. The results differ essentially from the ones quoted in literature. Full closure will not be dealt with in this article. https://doi.org/10.5194/hess-2020-299 Preprint. Discussion started: 14 July 2020 c Author(s) 2020. CC BY 4.0 License. Table 1 gives a list of the data used. 65 The stations RHB, G und F are within humid climate and represent typical grassland under agricultural use. For G1, F1 and F2 measurements used were between 5 am and 8 pm with a time interval of one hour. The daytimes used for G2, also with time intervals of one hour, were reduced to 9 am to 4 pm for reasons given below (Sect. 2.4).
The station Majadas represents a different situation in several aspects (Perez-Priego et al., 2015;Migliavacca et al., 2017): 80 -Climate: continental Mediterranean climate with winter rains (mean annual rainfall: ca 700 mm, mainly from November until May) and long dry periods during summer.
-Observation time was restricted from 9 am to 4 pm with half-hourly intervals. ( -Lysimeter values are the mean of four lysimeter measurements. 90

Possible errors of lysimeter observations
Lysimeters can achieve measurement accuracies between ca ± 15 to ± 20 Wm -2 (≈ 0.5 to 0.7 mm d -1 ), depending on their construction. Furthermore, hydraulic conditions (cylinder walls, soil conditions, ground water table) of the lysimeter do not correspond with the undisturbed surrounding. In addition to these systematic errors, random errors may occur due to 95 instabilities caused by wind gusts. One may also note that lysimeter observations generally do not include negative values (condensation). The influence of wind and dew on lysimeter observations is described in Meissner et al. (2007) and Ruth et al. (2018). The theoretical accuracy of lysimeter measurements can be calculated from the surface area and weighing accuracy.

Possible errors of EC-observations.
Systematic measuring errors of the latent heat flux (LE) may be around ± 30 Wm -2 , of sensible heat flux (H) around ± 13 Wm -105

Data selection 110
High quality data were at disposal from all the observation stations. Still we had to dismiss 2 to 5% of the EC-measurements -mostly for morning and evening hours with high instability of turbulent fluxes. We sorted them out on the basis of the Outof-Bound concept introduced by Wohlfahrt and Widmoser (2013), which excludes physically unrealistic measurements.
Furthermore, data showing big differences between LY-and EC-measurements (i.e. > 300 Wm -2 ≈> 0.44 mm hr -1 ) along with strong wind gusts ( > 2.0 ms -1 ), as well as early morning values with high air humidity and high dew formation were also 115 excluded, thus reducing the original data sets for another 5% at the average.
The overall data selection led to a reduced number of early morning and late evening data as compared to the number of data available for the rest of the day. That means that results for around sunrise and sunset are generally less reliable. In case of G2 the morning and evening data had to be reduced to such an extent that we decided to evaluate only data from 9 am to 4 pm.
For Majadas, all morning data were omitted for this reason. The numbers of data given in Table 1 correspond to the data 120 analyzed below.

Evaluation of weights wL by regression (partial closure)
Wohlfahrt and Widmoser (2013) introduced a simple framework for studying the energy imbalance (ε), i.e. 125 They proposed three dimensionless weights (wA, wH and wL) for the terms on the RHS of Eq. (1) which obey the following two constraints: (i) each weight is bound between zero and unity and (ii) the three weights sum up to unity. 130 Provided these weights are known, the terms on the RHS of Eq. (1) can be corrected for the lack of energy balance closure as: In this paper, we are concerned only with the evaluation of wL (Eq. 2c) by regressing the difference between LY and EC latent heat fluxes as a function of the energy imbalance: where wL represents the slope of the best-fit linear relationship and the y-intercept (d) can be interpreted as a systematic difference between LY and EC latent heat flux measurements. The random difference follows from For regression, data were binned according to LE-size in such a way that for each bin the same number of data pairs (LY-LE) vs ε, see Eq. (3), was available. The number of bins, i.e. 5 to 14, depended on the number of data per dataset at disposal. At least 90 data-pairs entered each regression. 150

Used parameters
The results of the partial energy closure will be represented by the following parameters:
-DLaL = LY -aLE as difference between observed LY -and adjusted aLE-values: aLE = cLE +d Furthermore we list the 160 -systematic deviations d, see intercept in Eq. (3) -εred/ε as a measure for the relative ε, remaining after adjustment; εred = (1-wL) -weights wL One may note that the DLaL-values correspond to the remaining differences after LE adjustment to the LY-data and as such 165 may be interpreted as random deviations drand or noise.

170
Tables 2a and 2b give means and standard deviations (SD) of the observed oLE-, the corrected cLE-, the adjusted aLE-and LY-evaporations for the analyzed periods and stations along with energy balance deficit ε and correlation coefficients between LY-and LE-data.
175 https://doi.org/10.5194/hess-2020-299 Preprint. Discussion started: 14 July 2020 c Author(s) 2020. CC BY 4.0 License. One may note that F1 has the lowest evaporation rate among the humid stations. This will influence the following results throughout. 185

Differences between means and standard deviations of LY-and EC-measurements
Tables 3a and 3b show the differences between LY-and EC-parameters of Tables 2a and 2b. 200

Parameters obtained by the LY-EC-comparison
Tables 4a and 4b present the parameters d (intercept = systematic deviation), εred/ε and wL as obtained by applying Eq. (3). 220

Reduction of the LY-LE-differences by adjustment expressed in percentages.
Tables 5a and 5b give the average and standard deviation differences between LY-and EC-values as expressed in percentages of LY. The improvements made visible by comparing the differences before and after adjustments. As such, they may also be compared to the quotations in Chavez and Howell (2009)

Averaged hourly daytime values for wL
265 Figures 5a and 5b show averaged daytime-hour-values of the weights wL. Figure 5a gives the humid wL-data for bins ranging from 6 (G2) to 12 (G1, F1, F2) and 14 (RHB). The wL data for Majadas in Fig. 5b used bins varying between 5 and 12. We distinguish between the drying periods (about March to August) in red and yellow as well as the one "rainy" period M3 (end of August 2017 to beginning of January 2018) in blue. Figure 5b also splits M4 into a period with "high soil moisture" (20.04. to 23.06., red line with blue triangles) and a "low soil moisture" (01.07. to 04.09., red line with yellow triangles). Both periods 270 are under high temperatures and very sparse rainfall. For soil moisture, see Fig. 6b.
All humid averaged values of daytime-hours of wL are roughly within the range of around 0.2 und 0.4. Their standard deviation is highest in the hours around noon (not shown), and not as expected during sunrise and sunset hours. For Majadas, variations in the various datasets are higher, especially for the drying period (i.e. no rainfall, but still high soil moisture) of M4 (topmost line in Fig. 5b). 275 https://doi.org/10.5194/hess-2020-299 Preprint. Discussion started: 14 July 2020 c Author(s) 2020. CC BY 4.0 License.

285
Figures 7a and 7b illustrate the EC-deviations from the LY-values before (light green) and after (blue) EC-adjustments along the analyzed time period for F2 (7a) and M4 (7b). They demonstrate again the remaining high variation.

Correlations between wL and different evaporation terms
290 Tables 7a and 7b show correlation coefficients between wL and three estimates of evaporations. For 7 out of 11 datasets, including all three dry periods of Majadas, the correlation coefficients are rather high. We could however not find correlations between wL and other weather indicators or combinations of them (not shown).

305
The method applied offers two results: (1) corrected cLE-values as given by cLE = oLE + wL ε and (2) adjusted aLE-values as given by aLE = cLE + d. One may consider cLE as weakly linked to the LY-measurements via the wL-regression and aLE as strongly linked to LY via both wL as well as d. Differences between the two range between 1 and 15 Wm -2 (Tables 3), i.e.
within the measurement accuracies.
In general, LY-measured data are higher than data based on the EC-method. This is in accordance to literature (e.g. Chavez and Howell, 2009). They differ surprisingly little in humid climate with around 10 to 30 Wm -2 (0.35 to 1.0 mm d -1 ) in contrast to the difference at the dry station Majadas with around 30 to 60 Wm -2 (1.0 to 2.1 mm d -1 ).
The adjustment of the LE-to the LY-data expressed by the differences DLaL hint at a nearly perfect match for the means 315 (Tables 3). They are all in the range of the measurements accuracies. All standard deviations given by the difference SD(LY) -SD(aLE), respectively SD(cLE), however, increase with adjustments, but remain less than SD(LY) (see SD for DLoL-and Tables 3a and 3b).

DLaL-values in
The adjustments reached in this paper are higher (Tables 5) than the ones quoted by 320 - Chavez and Howell (2009) with reductions of LY-EC-differences from 41.4% to 28.8%, respectively from 34.1 to 26% with an accuracy of ≈ 0.9 ± 14 Wm -2 , respectively ≈ 2.8 ± 11 Wm -2 - Evett et al. (2012), mentioning LE-EC-measurements errors within ≈ 55 to 78 Wm -2 , which were reduced after forced closure of the energy gap to LY -and LE-EC differences between 17 and 19 % and - Ding et al. (2010), quoting that differences between LY-and LE-EC-measurements could be reduced from 30.2 to 10.3%. 325 It surprises that the systematic deviations d between LY-and EC measurements (Tables 4a and 4b) (Table 4b; Fig. 4b). One could expect a more pronounced difference of d for the two different 330 measurements devices (RHB and TERENO lysimeters).
The energy gaps are in the range of 25 to 100 Wm -2 for the humid stations. They are much higher for Majadas with around 120 to 180 Wm -2 . The gaps ε reduce to about 50 to 80% after partial energy closure. They appear rather constant (around 70%) for the humid regions and vary more for Majadas, for which the most striking variations, i.e. 23% and 72.6% respectively, 335 occur with M4 during high and low soil moisture (Tables 4a und 4b, lines εred).
The calculated wL-values appear nearly independent of daytime hours ( Fig. 5a and 5b). Data from humid climate gave hourly averaged wL-values within a narrow range of 0.2 to 0.4. The corresponding values for Majadas show wider variations. During the non-rainy-season, they differ more substantially for M4 with high soil moisture (wL around 0.78) and low soil soil moisture 340 (wL around 0.25). We cannot give any explanation for this.
Standard deviations of wL for daytime hours averages change little, but we were surprised to find the highest daily average standard deviations of wL at noon (Fig. 5b). We would have expected them to take place in morning and evening, when there are (1) less data available and (2) the energy fluxes are less stable. 345 Since wL-values are partly positively correlated to the height of evaporation (Tables 7a and b) and seem to depend to some extent on seasons ( Fig. 6a and 6b), one might conclude that the high standard variations are rather related to weather conditions.
No clear picture, however, can be drawn on this aspect.
We also could not find any explanation for other specific cases found, like the unexpected drop of d-values for G2 (Fig. 4a).

Summary and conclusions
The applied partial closure gives, according to our knowledge, so far the best adjustments of EC-to LY-measurements. The method gives two results for improved LE-estimates, one weakly linked and one strongly linked to the LY-readings. Their differences appear negligible in view of the inaccuracies of the input data. The method also allows a distinction between systematic (d) and random deviations (drand) for the first time, probably. The wL-weight-averages are rather stable during 355 daytime. The systematic deviations d and random deviations (Tables 3) are all below or very close to measurements accuracies.
For the future, one should try to increase the information of LY-as well as EC-measurements. In a first step we recommend to perform the comparison of LY and EC based on 5 to 10 minutes intervals of lysimeter readings instead of currently one/half hour, and center the EC averaging window accordingly. We expect an improvement of the accuracy of wL-, d-and drand 360 estimates thereby. The benefit of using higher resolved lysimeter data is described in Ruth et al. (2018).
In long terms, one may think of improving measurements accuracies of relevant input data. Lysimeter-measurements should include negative values (condensation) and consider the influence of wind. The former can be realized by including rain observation on a high temporal scale to identify a mass increase in the absence of rain, i.e., dew formation (Ruth et al.;.
As long as no improvements are realized, as a pragmatic solution for full energy balance closure we recommend closing by 365 attributing one third of the gap ε to each of the three weights. This is common practice in land surveying. This recommendation is supported by the fact that we found generally rather constant wL-values during daytime between 0.2 and 0.4.

Data availability
The data basis for the presented analyses is available at https://doi.org/10.3929/ethz-b-000420733. The dataset consists of the half-hourly or hourly, respectively, time series of lysimeter and eddy covariance evapotranspiration, as well as ancillary data