## Abstract

The thermal-based Two-Source Energy Balance (TSEB) model partitions the evapotranspiration (ET) and energy fluxes from vegetation and soil components providing the capability for estimating soil evaporation (E) and canopy transpiration (T). However, it is crucial for ET partitioning to retrieve reliable estimates of canopy and soil temperatures and net radiation, as the latter determines the available energy for water and heat exchange from soil and canopy sources. These two factors become especially relevant in row crops with wide spacing and strongly clumped vegetation such as vineyards and orchards. To better understand these effects, very high spatial resolution remote-sensing data from an unmanned aerial vehicle were collected over vineyards in California, as part of the Grape Remote sensing and Atmospheric Profile and Evapotranspiration eXperiment and used in four different TSEB approaches to estimate the component soil and canopy temperatures, and ET partitioning between soil and canopy. Two approaches rely on the use of composite \(T_\mathrm{rad}\), and assume initially that the canopy transpires at the Priestley–Taylor potential rate. The other two algorithms are based on the contextual relationship between optical and thermal imagery partition \(T_\mathrm{rad}\) into soil and canopy component temperatures, which are then used to drive the TSEB without requiring a priori assumptions regarding initial canopy transpiration rate. The results showed that a simple contextual algorithm based on the inverse relationship of a vegetation index and \(T_\mathrm{rad}\) to derive soil and canopy temperatures yielded the closest agreement with flux tower measurements. The utility in very high-resolution remote-sensing data for estimating ET and E and T partitioning at the canopy level is also discussed.

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## Notes

- 1.
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## Acknowledgements

Partial funding provided by E.&J. Gallo Winery made possible the acquisition and processing of the high-resolution manned aircraft and UAV imagery collected during GRAPEX IOPs. In addition, thanks are given to the Utah Water Research Laboratory for the use of the AggieAir UAV platform, support personnel and partial funding. In addition, we would like to thank the staff of Viticulture, Chemistry and Enology Division of E.&J. Gallo Winery for the collection and processing of field data during GRAPEX IOPs. Finally, this project would not have been possible without the cooperation of Mr. Ernie Dosio of Pacific Agri Lands Management, along with the Borden/ McMannis vineyard staff, for logistical support of GRAPEX field and research activities. USDA is an equal opportunity provider and employer.

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## Additional information

Part of this research was conducted thanks to the MC-COFUND Talentia Program.

Communicated by N. Agam.

## Appendices

### Appendix

### A TSEB model

The basic equation of the energy balance at the surface can be expressed following Eq. 4:

with \(R_\mathrm{n}\) being the net radiation, *H* is the sensible heat flux, \(\lambda E\) is the latent heat flux or evapotranspiration, and *G* is the soil heat flux. “*C*” and “*S*” subscripts refer to canopy and soil layers, respectively. The symbol “\(\approx\)” appears, since there are additional components of the energy balance that are usually neglected, such as heat advection, storage of energy in the canopy layer, or energy for the fixation of CO\(_2\) (Hillel 1998)

The key in TSEB models is the partition of sensible heat flux into the canopy and soil layers, which depends on the soil and canopy temperatures (\(T_\mathrm{S}\) and \(T_\mathrm{C}\), respectively). If we assume that there is an interaction between the fluxes of canopy and soil, due to an expected heating of the in-canopy air by heat transport coming from the soil, the resistances network in TSEB can be considered to be in series. In that case, *H* can be estimated as in Eq. 5 (Norman et al. 1995, Eqs. A1–A3):

where \(\rho _{air}\) is the density of the air (\(\hbox {kg} hbox{m}^{-3}\)), \(C_{p}\) is the heat capacity of the air at constant pressure (\({\hbox {J} \hbox {kg}^{-1} \hbox {K}^{-1}}\)), and \(T_\mathrm{AC}\) is the air temperature at the canopy interface, equivalent to the aerodynamic temperature \(T_0\), computed with Eq. 6 (Norman et al. 1995, Eq. 4):

Here, \(R_{a}\) is the aerodynamic resistance to heat transport (\({\hbox {s} \hbox {m}^{-1}}\)), \(R_{s}\) is the resistance to heat flow in the boundary layer immediately above the soil surface (\({\hbox {s} \hbox {m}^{-1}}\)), and \(R_{x}\) is the boundary layer resistance of the canopy of leaves (\({\hbox {s} \hbox {m}^{-1}}\)). The mathematical expressions of these resistances are detailed in Eq. 7 and in Norman et al. (1995) and Kustas and Norman (2000) and discussed in Kustas et al. (2016):

where \(u_{*}\) is the friction velocity (\({\hbox {m} \hbox {s}^{-1}}\)) computed as

In Eq. , \(z_{u}\) and \(z_{T}\) are the measurement heights for wind speed *u* (\({\hbox {m} \hbox {s}^{-1}}\)) and air temperature \(T_{A}\) (K), respectively. \(d_0\) is the zero-plane displacement height, and \(z_{0M}\) and \(z_{0H}\) are the roughness length for momentum and heat transport, respectively (all those magnitudes expressed in m), with \(z_{0H}=z_{0M}\exp \left( {-kB^{-1}}\right)\). In the series version of TSEB, \(z_{0H}\) is assumed equal to \(z_{0M}\), since the term \(R_x\) already accounts for the different efficiencies between heat and momentum transport (Norman et al. 1995), and therefore, \(kB^{-1}=0\). The value of \(\kappa '=0.4\) is the von Karman’s constant. The \(\varPsi _{m}\left( \zeta \right)\) terms in Eqs. 7a and are the adiabatic correction factors for momentum. The formulations of these two factors are described in Brutsaert (1999) and Brutsaert (2005). These corrections depend on the atmospheric stability, which is expressed using the Monin–Obukhov length *L* (m):

where *H* is the bulk sensible heat flux (\({\hbox {W} \hbox {m}^{-2}}\)), *E* is the rate of surface evaporation (\({\hbox {kg} \hbox {s}^{-1}}\)), and *g* is the acceleration of gravity (\({\hbox {m} \hbox {s}^{-2}}\))

The coefficients *b* , *c* in Eq. 7b depend on turbulent length scale in the canopy, soil-surface roughness, and turbulence intensity in the canopy, which are discussed in Sauer et al. (1995), Kondo and Ishida (1997) and Kustas et al. (2016). \(C'\) is assumed to be 90\(\hbox { s}^{^1/_2} \hbox { m}^{-1}\) and \(l_w\) is the average leaf width (m)

### B Modifications to TSEB model for row crops

### B.1 Radiation transmission in row crops

The clumping index for row crops is defined as the factor that modifies the leaf area index of a real canopy (*F*) in a fictitious homogeneous canopy with \(\mathrm{LAI}_{\mathrm{eff}}=\varOmega F\) such as its gap fraction is the same as the gap fraction of the real-world canopy (\(G\left( \theta ,\phi \right)\)):

where \(\kappa _{be}\left( \theta \right)\) is the beam extinction coefficient through a plant with an ellipsoidal inclination distribution (Campbell 1986, 1990), \(\theta\) is the zenith incidence angle, and \(\psi\) is the relative azimuth angle between the incidence beam and the row direction

Our modelled real canopy consists of a horizontally infinite long prism with a total height \(h_c\) (i.e., the canopy height) and a width \(w_c\) (i.e., canopy width) that is placed above the ground at \(h_b\) (i.e., the height of the first living branch). This canopy contains finite-sized leaves randomly placed (no clumping within the canopy) oriented according to a ellipsoidal leaf angle distribution function (Campbell 1990) with a total leaf area index *F* (Fig. 3).

Then, the real canopy gap fraction consists of the sunlit part of the bare soil that is not shaded by the canopy plus the gaps caused by the solar beam passing through the crop canopy (Eq. 11):

The solar canopy view factor \(f_{sc}\left( \theta ,\phi \right)\) is the fraction of soil that is cast by shadows (Colaizzi et al. 2012a) and in our case is estimated as

where L is the row separation (m). For a vertical projection (\(\theta =0\)), Eq. 12 reduces to \(w_{c}/L\), the fractional cover.

### B.2 Massman et al. (2017) wind attenuation profile

Compared to previously used canopy wind profiles such as Goudriaan (1977) or Massman (1987), the additional key input required in Massman et al. (2017) wind attenuation model is the relative canopy foliage distribution, computed as in Eq. 13:

where \(\tfrac{f_a\left( \xi \right) }{\int _0^1f_a\left( \xi '\right) d\xi '}\) is the relative canopy shape (i.e., \(\sum {\tfrac{f_a\left( \xi \right) }{\int _0^1f_a\left( \xi '\right) d\xi '}}=1\), and \(\xi =z/h_c\)) and *PAI* is the plant (leaves+stems) area index. Massman et al. (2017) modelled \(f_a\left( \xi \right)\) as a combination of asymmetric Gaussian curves, but \(f_a\left( \xi \right)\) can also be estimated as a continuous curve obtained from canopy structure measurements or three-dimensional cloud points, such as in Nieto et al. (2018).

The canopy wind speed profile is then the product of two terms: one logarithmic profile (\(U_b\)) that is dominant near the ground and a second a hyperbolic cosine profile (\(U_t\)) that dominates near the top of the canopy, where the canopy foliage distribution plays a major role. Ancillary input in \(U_t\) is the the drag coefficient of the individual foliage elements (\(C_d\)), which is usually considered equal to 0.2 (Massman et al. 2017; Goudriaan 1977). Massman et al. (2017) model has as well the ability to consider variations of the drag coefficient due to either wind sheltering between foliage elements, or vertical variations independently of wind blocking. This effect can usually be disregarded in most canopies (Massman et al. 2017), so was it in this study.

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Nieto, H., Kustas, W.P., Torres-Rúa, A. *et al.* Evaluation of TSEB turbulent fluxes using different methods for the retrieval of soil and canopy component temperatures from UAV thermal and multispectral imagery.
*Irrig Sci* **37, **389–406 (2019). https://doi.org/10.1007/s00271-018-0585-9

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