# Considerations for the energy balance residual (EBR) correction

Non-closure of the surface energy balance is a frequently observed phenomenon of hydrometeorological measurements when using the eddy covariance (EC) method to estimate sensible heat (`H`) and latent heat (`LE`) fluxes (Stoy et al., 2013; Mauder et al., 2017).

One of the most accredited hypotheses is that, for the most part, the lack of closure is due to a systematic underestimation of `H` and `LE` related to synoptic scale transport phenomena that are not captured with traditional EC systems. Under this assumption, in the past few years several methods have been proposed to correct EC-derived `H` and `LE` measurements, so as to better or completely close the energy balance.

Among others, Matthias Mauder and his group have paid significant attention to this issue and, in time, have proposed several correction methods. The main practical difference across the various methods is the way the residual energy is partitioned between `H` and `LE`. Most of the proposed methods do rely on the energy imbalance itself to compute the correction factors and therefore achieve a perfect closure by definition. Only recently, correction methods not informed by the energy imbalance have been proposed, which are currently under active development.

Mauder et al. (2013) use the energy balance residues evaluated on a daily basis and partition that energy between `H` and `LE` in a way that preserves the Bowen ratio (`H`/`LE`), according to:

5‑2

5‑3

where:

`H`is sensible heat flux (W m^{-2})`LE`is latent heat flux (W m^{-2})- “
`meas`” indicates fluxes as “measured” with EC, while “`corr`” indicates EBR-corrected fluxes `R`is net solar radiation (W m_{g}^{-2})`G`is total soil heat flux (W m^{-2})`C`is the correction factor, equal for both sensible and heat flux (adimensional)`K`is the number of valid observations of`H`and`LE`in each day, when`R`> 20 W m_{g}^{-2}

The factor `C` represents the relative amount of energy balance closure (ranging from 0 to 1) evaluated on a daily basis. Note that the correction is only computed with, and applied to, flux data corresponding to periods with `R _{g}` > 20 W m

^{-2}. With this method, the energy balance is by definition perfectly closed when evaluated on a daily time scale (i.e., daily sums). When evaluated on individual flux data points (e.g., 30-minutes) the energy balance is not necessarily perfectly closed.

Charuchittipan et al. (2014) argued that synoptic-scale transport components not captured by EC are mostly driven by convective motions and therefore proposed a correction method that assigns most of the residual energy to `H`, according to:

5‑4

5‑5

5‑6

where:

`Res`is the energy balance residual, evaluated for each data point (e.g., on a 30-minute basis)`B`is buoyancy flux (m K s^{-1})`Bo`is Bowen ratio (adimensional)`c`is air specific heat at constant pressure (J kg_{p}^{−1 }K^{−1})`L`is water vapor latent heat of vaporization (J kg^{−1})

With this method, by definition the correction closes the energy balance exactly, when the balance is evaluated on the time scale of the flux computation (e.g., 30 minutes).

Based on the results of a Large Eddy Simulation study, De Roo, et al. (2018) proposed the first correction method that is not informed by the known energy imbalance and which can therefore use the energy balance achieved after the correction as a metric of performance:

with:

5‑9

5‑10

and with:

5‑11

5‑12

where:

`h`is the effective measurement height (m)_{m}`h`is the height of the Planetary Boundary Layer (m)_{PBL}`u*`is the friction velocity (m/s)`w*`is the convective velocity scale (m/s)

The performance of this correction improves with the effective measurement height. At lower measurement heights, however, the correction tends to improve the energy balance rather poorly. For this reason, an EBR-rescaled version of the method was also proposed by the authors. In this case, equations 5‑7 and 5‑8 are replaced by:

5‑13

5‑14

with:

5‑15

Similar to the method of Mauder et al. (2013), due to the definition of `C` as the daily-scale energy imbalance, this method is expected to perfectly close the energy budget when evaluated on daily sums, while a slight departure from perfect closure is expected when the budget is evaluated per flux averaging interval.