### Abstract

- Top of page
- Abstract
- 1. Introduction
- 2. Methods
- 3. Experiment Description
- 4. Results and Discussion
- 5. Conclusions
- Acknowledgments
- References

[1] The residual of the surface energy budget is represented as the linearized sum of energy losses due to storage, advection and flux underestimation. Individual contributions to the residual can be quantified through constrained multiple linear regression which identifies the site specific processes that are responsible for the lack of energy budget closure. This residual decomposition approach is applied to energy balance data from the Surface Layer Turbulence and Environmental Science Test (SLTEST) site at the Dugway Proving Grounds in the Utah Salt Flats. In this case, energy storage in the soil and underestimation of the soil heat flux accounted for 89% of the residual variance. Underestimation of the sensible and latent heat fluxes had no apparent contribution to the residual, and the contribution of advection to the residual was not statistically significant.

### 1. Introduction

- Top of page
- Abstract
- 1. Introduction
- 2. Methods
- 3. Experiment Description
- 4. Results and Discussion
- 5. Conclusions
- Acknowledgments
- References

[2] Measurements of the Earth's surface energy budget do not close at timescales less than several hours [*Wilson et al.*, 2002; *Oncley et al.*, 2007; *Foken*, 2008; *Foken et al.*, 2010; *Kidston et al.*, 2010; *Foken et al.*, 2011; *Leuning et al.*, 2012]. Several experiments have been carried out to determine the root cause of this imbalance by targeting specific processes: storage [*Oliphant et al.*, 2004; *Jacobs et al.*, 2008; *Moderow et al.*, 2009; *Lindroth et al.*, 2010], advection [*Aubinet et al.*, 2010; *Kochendorfer and Paw*, 2011]. Spatial variability [*Steinfeld et al.*, 2007; *Mauder et al.*, 2010], footprint issues [*Schmid*, 1997], flux measurement corrections [*Mauder and Foken*, 2006], and meteorological conditions [*Franssen et al.*, 2010]. In some cases, the authors do close the energy budget within reasonable limits [e.g., *Jacobs et al.*, 2008], however, these successes are rare.

[3] The apparent lack of closure impacts many techniques that estimate fluxes at the Earth's surface based on an assumed energy balance. In irrigation scheduling, FAO-56 Penman-Monteith, the recommended way to estimate evaporation by the Food and Agriculture Organization of the United Nations [*Trajkovic and Kolakovic*, 2009] and the American Society of Civil Engineers [*Allen*, 2000] relies on an assumption of energy balance [*Brutsaert*, 2005]. Satellite estimates of evapotranspiration routinely rely on energy budget assumptions [*Allen et al.*, 2007; *Compaore et al.*, 2008; *Long and Singh*, 2010]. Carbon flux measurements, important in determining the net ecosystem exchange of CO_{2}, are sometimes corrected by assuming energy budget closure [*Massman and Lee*, 2002; *Wilson et al.*, 2002].

[4] Despite the lack of closure, and the myriad of studies devoted to the investigation of the individual factors that lead to missed energy, a unified approach to diagnose the cause of insufficient energy budget closure *a-posteriori* does not exist. Each field site is different, and factors contributing to incomplete closure can be caused by site specific or measurement specific effects. Direct measurement of some energy pathways such as advection or energy storage can be expensive and data intensive. The salient question is: can we evaluate the energy budget closure mismatch in a diagnostic way such that the source(s) of the mismatch is identified for a particular field site? In this way experimentalists can diagnose the site specific closure problem and invest in the appropriate instrumentation needed to capture the missed energy pathway(s).

### 2. Methods

- Top of page
- Abstract
- 1. Introduction
- 2. Methods
- 3. Experiment Description
- 4. Results and Discussion
- 5. Conclusions
- Acknowledgments
- References

[5] The surface energy balance is written as:

Where *R*_{n} is the net radiation, *H* is the sensible heat flux, *LE*, is the latent heat exchange due to evaporation, *S* is the energy storage in the air, soil and plant canopy, *A* is the advection, *W* is the total measurement error, and *O*_{T} are other terms not considered in this study (soil water transport, freeze/thaw in a snowpack, energy used for photosynthesis, entropy production, mismatched measurement footprints etc.)*.* In the simplest case *S* + *A* + *W* + *O*_{T} is assumed to be small and the energy balance is considered in the following way:

where *η* is the residual. The residual can also be written as:

The time series of the residual has a functional form that is the linear combination of the time series behavior of the storage, advection, errors, and other terms. If the behavior of these terms could be mapped to specific, independent, measured quantities, it would be possible to attribute a fraction of the residual to each physical process. Proceeding term by term in equation (3):

The storage is equal to the total storage in the air, soil, and the plant canopy. These storages are proportional to the time derivative of the air temperature *T*_{a}, the skin temperature *T*_{s}, the leaf temperature *T*_{l}, the bark temperature *T*_{b}, and the core trunk temperature *T*_{c}. *C*_{S,air}, *C*_{S,soil}, *C*_{S,leaf}, *C*_{S,bark}, and *C*_{S,core} are related to the density and heat capacity each component respectively, and are assumed to be constant over the time interval of analysis.

[6] Following *Leuning et al.* [2012] the advection of a scalar, *χ*, can be estimated by

To precede, an approximation of the stream-wise scalar gradient is required. The scalar transport equation for neutral atmospheric stability conditions,

was solved for idealized surface conditions [*Sutton*, 1934], and for general surface conditions [*Polyanin*, 2002], thus the horizontal scalar gradient can be obtained under neutral conditions given a surface boundary condition. Coupled with the assumption of stationarity already invoked, it follows that each possible wind angle is associated with a unique, unknown surface condition which is in turn associated with a scalar gradient. In addition to wind direction, the advection likely has a strong dependence on atmospheric stability [*Aubinet et al.*, 2000] that was not considered in the above analysis. Modeling the stream wise scalar gradient as an unknown function of both wind direction and stability, and combining with equation (5) yields:

Where *f*(*θ*, *z*/*L*) is an unknown function of wind direction, *θ*, and atmospheric stability, *z*/*L.* Here, z is the measurement height and L is the Obukhov length. Since *f* is periodic in *θ* (the upwind topography associated with *θ* is the same upwind topography associated with *θ* + 360°) the natural course of action is to approximate *f* with a truncated Fourier series.

Where *a*_{0}(*z*/*L*), *a*_{1}(*z*/*L*), and *a*_{2}(*z*/*L*) are unknown Fourier coefficients that are functions of stability. Here only the first order terms of the series are used. Analysis of measurements taken above highly variable surfaces may require additional terms.

[7] Errors can be organized as systematic errors [*Moncrieff et al.*, 1996] caused by imperfect sensor alignment, flux underestimates caused by sensor separation [*Kristensen et al.*, 1997], sampling issues [*Lenschow et al.*, 1994; *Lee et al.*, 2004; *Kidston et al.*, 2010], and random error [*Salesky et al.*, 2012]. Sampling issues and sensor separation issues can be analyzed with a transfer function approach [*Lee et al.*, 2004] which allows the residual to be expressed as a fraction of the measured flux. Sensor alignment issues are expressed geometrically.

Where *ϕ*, *λ*, *ς*, and *ψ* are the angles of imperfect alignment between the fluxes and the instrumentation. *C*_{H,sample}, *C*_{LE,sample}, and *C*_{separation} are the positive constants that link sampling and sensor separation issues to underestimation of fluxes, and *W*_{random} is the expected random error in the flux measurements, characterized by *Salesky et al.* [2012] and is expected to be ∼10%. Combining equations (3), (4), (8), and (9), allowing for a constant offset, aggregating constants and neglecting the contribution of random noise and canopy storage yields:

If a constant Bowen ratio is observed during the course of the experiment, *H*(*t*) and *LE*(*t*) are no longer linearly independent and *C*_{H}*H*(*t*) + *C*_{LE}*LE*(*t*) should be replaced by (*C*_{H} + *C*_{LE}/*β*_{0})*H*(*t*) = *C*_{β}*H*(*t*). The data are conditionally sampled based on stability regime, and the unknown coefficients in equation (10) are determined with constrained multiple linear regression (‘lsqlin’ function in Matlab™). The resulting contribution of each physical process to the residual is estimated for stable, unstable and near neutral atmospheric stability. Any variance in the residual not explained by equation (10) that is greater than the expected random error of measurement is attributed to *O*_{T}. Note that the fundamental assumption in the above analysis is that the coefficients in equation (10) do not change over the analysis timescale. For this reason, analysis of long time series is discouraged. The shortest time series that yields converged statistics in the regression should be used.

[8] To facilitate the linear regression, realistic limits are set on the values of the unknown coefficients in equation (10). *C*_{S,air} and *C*_{S,soil} are positive as they are related to the physical properties: *ρc*_{p} where *ρ* is the density and *c*_{p} is the specific heat. Tilt errors, are constrained by setting a reasonable maximum sensor misalignment. Sampling errors are constrained by the methodology in *Lee et al.* [2004]. For advection, the methodology proposed in *Kochendorfer and Paw* [2011] can be used. Care must be taken to ensure that spurious values of flux are not included in the regression as spurious values have a disproportionate effect on the results.

### 4. Results and Discussion

- Top of page
- Abstract
- 1. Introduction
- 2. Methods
- 3. Experiment Description
- 4. Results and Discussion
- 5. Conclusions
- Acknowledgments
- References

[10] The residual (shown in Figure 1) was decomposed into components corresponding to storage, advection, and flux underestimation using the *a-posteriori* analysis outlined above (results shown in Figure 2). After the analysis, each coefficient is analyzed for statistical significance at a 99% confidence. Fifty nine percent of the residual's variance can be attributed to energy storage in the soil layer, thirty percent is attributed to underestimates of the soil heat flux, and one percent is attributed to advection (not statistically significant).

[11] None of the residual is attributed to air column storage or underestimation of the sensible or latent heat flux measured by eddy covariance *C*_{S,air} = *C*_{LE} = *C*_{H} = 0. From this analysis we can conclude 1) the energy storage in the soil and underestimation of the soil heat flux are responsible for the lack of energy budget closure at the SLTEST experiment site, and 2) fluxes measured with the eddy covariance technique were not underestimated. Advection did have a strong dependence on atmospheric stability. The coefficients in equation (10)differed by more than a factor of 2 across atmospheric stability classifications. Finally, a constant offset of 50 W-m^{−2} was observed across the entire data set. The mechanism responsible for a constant offset is unclear, and could potentially be attributed to the outdated radiation measurements, but the contribution to the residual and ultimate closure of the surface energy budget is significant. A direct comparison between the measured residual and the sum of ground storage, ground heat flux underestimate and the offset is shown in Figure 3. The RMS error between the linear form and the measured residual is 19 W-m^{−2}, well within the combined error limits of the sum of the measurements.

[12] The purpose of an *a posteriori* energy budget analysis is to identify weaknesses in experimental design that can be corrected in future experiments. For the example presented, the experimental design should be adapted to resolve the soil heat flux and the energy storage in the soil layer above the soil heat flux plate with greater accuracy. The logical course of action is to implement a soil temperature profiling strategy to explicitly measure the heat storage term and to resolve the thermal gradients that give rise to the soil heat flux. Furthermore, the observed constant offset is indicative of a biased energy measurement, likely due to the outdated and inaccurate Q7.1 radiation sensor. Future experiments should include a more precise instrument for net radiation.

[13] The purpose of this technique is not to force a closure of the energy budget. To do so would be reckless. Rather, the analysis presented provides clues into the physical processes that should be monitored in more detail at a specific experimental location. Only 5–7 days of data are needed for the analysis; therefore the analysis can be performed during an experiment, and the setup modified in an iterative fashion until a satisfactory energy budget closure is attained.