Non-closure of the surface energy balance, i.e., the sum of latent (LE) and sensible (H) heat exchange falling short of available energy (A), is a common issue in eddy covariance flux (EC) measurements. Available energy equals net radiation (RN) minus the soil heat flux (G) and any other energy storage (Wohlfahrt and Widmoser, 2013). At the majority of eddy covariance flux sites, it is the rule rather than the exception to find that the sum of the turbulent fluxes LE+H underestimates A by 20 %–30 % (Leuning et al., 2012; Wilson et al., 2002). This apparently systematic bias has been extensively discussed in the literature (see reviews by Foken, 2008; Foken et al., 2011; Leuning et al., 2012; Mauder et al., 2020). In the most recent review, the following classification of reasons for the energy gap problem is listed: (1) instrument error, (2) data processing error, (3) additional sources of energy and (4) secondary circulation of energy. Our own hourly observations show that the bulk of LE+H underestimates is detected around noon, whereas during sunrise and sunset overestimates are also observed.

There are two practical approaches to deal with the energy imbalance problem: (1) comparing EC measurements with concurrent lysimeter measurements and (2) using models.

Lysimeters (LY) have a long tradition in hydrology and micrometeorology, and their limitations and sources of uncertainty are well known. There usually is a very strong correlation between concurrent LY- and EC-based evaporation data, with the LY values generally being higher. An overview of efforts to compare EC evaporation to lysimeter measurements can be found in Gebler et al. (2015). A few of the studies related to this article are described below.

Chavez and Howell (2009) hint at various error sources for the LY and EC measurements. EC observations on cotton fields in Texas with quarter-hourly measurements resulted in an energy balance gap of 22.0 % to 26.8 %. Those gaps were closed assuming Bowen ratio preservation and correct measurements of the available energy. After forced closure of the energy balance, the difference between daytime LY and EC data on two fields could be reduced from -28.8 % to 6.2 % and from -26.0 % to -12.3 %, respectively, with an accuracy of 0.03±0.5 mmd-1 (≈0.9±14Wm-2) and -0.1±0.4 mmd-1 (≈-2.8±11Wm-2), respectively. Negative values indicate that the lysimeter values were higher on average than the EC values.

Evett et al. (2012), using data from the same site as Chavez and Howell (2009), report errors of daytime EC measurements for latent heat flux of 1.9 to 2.7 mmd-1 (≈55 to 78 Wm-2) and for sensible heat flux of 1.4 to 1.9 mmd-1 (≈40 to 55 Wm-2). They reported substantially larger LY evaporation rates compared to the EC measurements due to differences in plant growth in the LY and the EC footprint. After forced closure of the energy gap, as done by Chavez and Howell (2009), mean differences from -17.4 % to -18.7 % were found between the two measurement methods after correcting for plant growth.

In the same way, Ding et al. (2010) closed the energy gaps using half-hourly daytime data on irrigated maize in an arid area in northwest China. Differences in daily measurements were also reduced there by forced Bowen ratio closure of the EC gap. Differences could be reduced from -22.4 % to -6.2 %, with 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 only caused by an underestimation of the turbulent fluxes, which are corrected according to the evaporative fraction LE/(LE+H) averaged over 7 d. After correction, they find an agreement between the LY and EC values with a total difference 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 % relative errors. The remaining differences are suspected to be due to different plant height within the 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) and (2) the method by Charuchittipan et al. (2014), which attributes a larger portion of the residual to the sensible heat flux. They also compare the EC values with the results of the hydrological model GEOtop 2.0 (Endrizzi et al., 2014). They found that a daily adjustment factor leads to less scatter than a complete partitioning of the residual for every 0.5 h time interval.

In the compilation of the literature above, the LY–EC comparisons relied on the assumptions that the available energy observations are correct and that the Bowen ratio can be preserved. In contrast to the closure method used by the abovementioned authors, Widmoser and Wohlfahrt (2018) achieved a partial latent heat closure of the energy balance by combining both the model and lysimeter approach, which is afterwards fully closed under the assumption of preservation of the Bowen ratio.

The objective of this article is to extend the abovementioned method, which was applied to only one station, to more stations in order to test its applicability and compare its results.

The method applied offers two results: (1) corrected LEEC_c values as given by LEEC_c=LEEC_o+wLEε and (2) adjusted LEEC_a values as given by LEEC_a=LEEC_c+d. One may consider LEEC_c as weakly linked to the LY measurements via the wL regression and LEEC_a as strongly linked to LELY via both wLE as well as d. Differences between the two mostly range within the measurement accuracies (Table 3a and b).

In general, LY measured data are higher than data based on the EC method. This is in accordance with the literature (e.g., Chavez and Howell, 2009). They differ substantially less in humid climate with around 10 to 30 Wm-2 (0.35 to 1.0 mmd-1) than at the Majadas station with around 30 to 60 Wm-2 (1.0 to 2.1 mmd-1).

The adjustment of the EC to the LY data expressed by the differences Da hint at a nearly perfect match for the means (Table 3a and b). They are all in the range of the measurement accuracies. All SDs given by the difference SD(LELY)-SD(LEEC_a), respectively SD(LEEC_c), increase with adjustments, but remain less than SD(LELY) (see the SDs for Do and Da values in Table 3a and b). The difference between SD(LELY) and SD(LEEC_o) gets bigger, since SD(LEEC_o) gets smaller after correction, whereas SD(LELY) remains the same.

The effectiveness of our method is demonstrated by comparing our results given in Table 5a and b with the following previously published results:

Chavez and Howell (2009) with reductions of LY–EC differences from -28.8 % to 6.2 % and from -26.0 % to -12.3 %, respectively, with an accuracy of ≈0.9±14Wm-2 and ≈-2.8±11Wm-2, respectively

It is surprising that the systematic deviations d between the LY and EC measurements (Table 4a and b) are on average within the measurement accuracy with the exception of F1 and (marginally) M2. For the humid regions d is positive (four cases) as well as negative (one case). For Majadas d is positive only for M3, measured during the rainy season. For M4 the d values are distinctly below measurement accuracy (Table 4b; Fig. 6b). One could expect a more pronounced difference of d for the two different measurement devices (RHB and lower boundary-controlled 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 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 % occur with M4 during high and low soil moisture, respectively (Table 4a and b, lines εred).

The calculated wLE values appear nearly independent of daytime hours (Fig. 7a and b). Data from the humid climate gave hourly averaged wLE values within a surprisingly 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 (wLE around 0.78) and low soil moisture (wLE around 0.25). This discrepancy of wLE is mitigated by extending the daily time window of the Majadas data (Sect. 2.4).

The SDs of wLE for daytime hour averages change little, with a tendency of smaller values in the morning and evening. This relates to small absolute values of evaporation during stable or weakly unstable conditions.

The value of wLE seems partly positively correlated to the magnitude of evaporation. This correlation is indicated in Fig. 8b, where a drop in wLE follows LEEC_c.

We could not find any explanation for the unexpected drop of d values for G2 (Fig. 6a).

The applied partial closure gives, according to our knowledge, the first fully rational method to partially close the energy gap and a more detailed description of the correlations between the LY and EC observations. The method gives two results for improved LEEC estimates, one weakly linked and one strongly linked to the LELY readings. Their differences appear negligible in view of the inaccuracies of the input data. The method also allows a distinction between systematic and random deviations, probably for the first time. The wLE weight averages are rather stable during daytime. The systematic deviations and random deviations (Table 4a and b) are mostly below or very close to measurement accuracies.

In the future, one should try to increase the temporal resolution of the LY-EC comparison. As a first step we recommend performing the comparison of the LY and EC measurements based on 5 to 10 min lysimeter intervals and center the averaging window accordingly on the EC high-frequency data. We thereby expect an improvement of the accuracy of wLE, d and drand estimates. The benefit of using more highly resolved lysimeter data is described in Ruth et al. (2018).

In the long term, one may think of improving measurement 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 observations on a high temporal scale to identify a mass increase in the absence of rain, i.e., dew formation (Ruth et al., 2018). If a high-precision lysimeter capable of resolving evaporation as well as condensation is available complementary to an EC setup, LE can directly be obtained from the lysimeter. As long as no improvements are realized, as a pragmatic solution for full energy balance closure we recommend closing by 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 wLE values between 0.2 and 0.4 during the daytime.

We also recommend testing high-quality flag 0 datasets (Mauder et al., 2013) for plausibility by the out-of-bound method, which may be derived from Wohlfahrt and Widmoser (2013).

The method proposed here may also be applied if reliable sap flow measurements are available instead of lysimeter observations. We guess that an adoption of our method may apply to partial energy closure by heat fluxes if surface temperatures estimates are known from telemetry.