Utility of the two-source energy balance (TSEB) model in vine and interrow flux partitioning over the growing season

Abstract

For monitoring water use in vineyards, it becomes important to evaluate the evapotranspiration (ET) contributions from the two distinct management zones: the vines and the interrow. Often the interrow is not completely bare soil but contains a cover crop that is senescent during the main growing season (nominally May–August), which in Central California is also the dry season. Drip irrigation systems running during the growing season supply water to the vine plant and re-wet some of the surrounding bare soil. However, most of the interrow cover crop is dry stubble by the end of May. This paper analyzes the utility of the thermal-based two-source energy balance (TSEB) model for estimating daytime ET using tower-based land surface temperature (LST) estimates over two Pinot Noir (Vitis vinifera) vineyards at different levels of maturity in the Central Valley of California near Lodi, CA. The data were collected as part of the Grape Remote sensing Atmospheric Profile and Evapotranspiration eXperiment (GRAPEX). Local eddy covariance (EC) flux tower measurements are used to evaluate the performance of the TSEB model output of the fluxes and the capability of partitioning the vine and cover crop transpiration (T) from the total ET or T/ET ratio. The results for the 2014–2016 growing seasons indicate that TSEB output of the energy balance components and ET, particularly, over the daytime period yield relative differences with flux tower measurements of less than 15%. However, the TSEB model in comparison with the correlation-based flux partitioning method overestimates T/ET during the winter and spring through bud break, but then underestimates during the growing season. A major factor that appears to affect this temporal behavior in T/ET is the daily LAI used as input to TSEB derived from a remote sensing product. An additional source of uncertainty is the use of local tower-based LST measurements, which are not representative of the flux tower measurement source area footprint.

Appendix 1: TSEB model

The basic equation of the energy balance at the surface can be expressed following Eq. (3).

$$\begin{aligned} & {R_{\text{n}}} \approx H+{\text{LE}}+G \\ & {R_{{\text{n,S}}}} \approx {H_{\text{S}}}+{\text{L}}{{\text{E}}_{\text{S}}}+G \\ & {R_{{\text{n,C}}}} \approx {H_{\text{C}}}+{\text{L}}{{\text{E}}_{\text{C}}} \\ \end{aligned}$$

(3)

with ${R_{\text{n}}}$ being the net radiation, $H$ the sensible heat flux, LE the latent heat flux or evapotranspiration, and $G$ 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 CO₂ (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 (${T_{\text{S}}}$ and ${T_{\text{C}}}$, 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 is considered in series. In that case $H$ can be estimated as in Eq. (4) [Norman et al. 1995, Eqs. (A.1)–(A.4)]

$$\begin{aligned} H & ={H_{\text{C}}}+{H_{\text{S}}}={\rho _{{\text{air}}}}{C_{\text{p}}}\frac{{{T_{{\text{AC}}}} - {T_{\text{A}}}}}{{{R_{\text{a}}}}} \\ & ={\rho _{{\text{air}}}}{C_{\text{p}}}\left[ {\frac{{{T_{\text{C}}} - {T_{{\text{AC}}}}}}{{{R_{\text{x}}}}}+\frac{{{T_{\text{S}}} - {T_{{\text{AC}}}}}}{{{R_{\text{s}}}}}} \right], \\ \end{aligned}$$

(4)

where ${\rho _{{\text{air}}}}$ is the density of air (kg m⁻³), ${C_{\text{p}}}$ is the heat capacity of air at constant pressure (J kg⁻¹ K⁻¹), ${T_{{\text{AC}}}}$ is the air temperature at the canopy interface (K), equivalent to the aerodynamic temperature ${T_0}$, computed as follows (Norman et al. 1995, Eq. A.4):

$${T_{{\text{AC}}}}=\frac{{\frac{{{T_{\text{A}}}}}{{{R_{\text{a}}}}}+\frac{{{T_{\text{C}}}}}{{{R_{\text{x}}}}}+\frac{{{T_{\text{S}}}}}{{{R_{\text{s}}}}}}}{{\frac{1}{{{R_{\text{a}}}}}+\frac{1}{{{R_{\text{x}}}}}+\frac{1}{{{R_{\text{s}}}}}}}.$$

(5)

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

$$\begin{aligned} {R_{\text{a}}} & =\frac{{\ln \left( {\frac{{{z_T} - {d_0}}}{{{z_{0{\text{H}}}}}}} \right) - {\Psi _{\text{h}}}\left( {\frac{{{z_T} - {d_0}}}{L}} \right)+{\Psi _{\text{h}}}\left( {\frac{{{z_{0{\text{H}}}}}}{L}} \right)}}{{\kappa ^{\prime}~{u_*}~}} \\ {R_{\text{s}}} & =\frac{1}{{c{{\left( {{T_{\text{S}}} - {T_{\text{A}}}} \right)}^{1/3}}+b{u_{\text{s}}}}} \\ {R_{\text{x}}} & =\frac{{C^{\prime}}}{{{\text{LAI}}}}{\left( {\frac{{{l_{\text{w}}}}}{{{U_{{d_0}+{z_{0{\text{M}}}}}}}}} \right)^{1/2}}, \\ \end{aligned}$$

(6)

where ${u_{\text{*}}}$ is the friction velocity (m s⁻¹) computed as:

$${u_*}=\frac{{\kappa ^{\prime}u}}{{\left[ {\ln \left( {\frac{{{z_u} - {d_0}}}{{{z_{0{\text{M}}}}}}} \right) - {\Psi _{\text{m}}}\left( {\frac{{{z_u} - {d_0}}}{L}} \right)+{\Psi _{\text{m}}}\left( {\frac{{{z_{0{\text{M}}}}}}{L}} \right)} \right]}}.$$

(7)

In Eq. (7), ${z_u}$ and ${z_T}$ are the measurement heights for wind speed $u$ (m s⁻¹) and air temperature ${T_{\text{A}}}$ (K), respectively. ${d_0}$ is the zero-plane displacement height, ${z_{0{\text{M}}}}$ and ${z_{0{\text{H}}}}$ are the roughness length for momentum and heat transport respectively (all those magnitudes expressed in m), with ${z_{0{\text{H}}}}={z_{0{\text{M}}}}{\text{exp}}\left( { - k{B^{ - 1}}} \right)$. In the series version of TSEB ${z_{0{\text{H}}}}$ is assumed to be equal to ${z_{0{\text{M}}}}$ since the term ${R_{\text{x}}}$ already accounts for the different efficiency between heat and momentum transport (Norman et al. 1995), and therefore $k{B^{ - 1}}=0$. The value of $\kappa ^{\prime}=0.4$ is the von Karman’s constant. The ${\Psi _{\text{m}}}$ and ${\Psi _{\text{h}}}$ terms in Eqs. (6) and (7) are the adiabatic correction factors for momentum and heat, respectively, whose formulations are described in Brutsaert (1999, 2005) and are functions of the atmospheric stability. The stability is expressed using the Monin–Obukhov length $L$ (m), which has the following form:

$$L=\frac{{ - u_{*}^{3}{\rho _{{\text{air}}}}}}{{kg\left[ {H/\left( {{T_{\text{A}}}{C_{\text{p}}}} \right)+0.61E} \right]}},$$

(8)

where $H$ is the bulk sensible heat flux (W m⁻²), $E$ is the rate of surface evaporation (kg s⁻¹), and $g$ the acceleration of gravity (m s⁻²).

The coefficients $b$ and $c$ in Eq. (6) 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). $C^{\prime}$ is assumed to be 90 s^1/2 m⁻¹ and ${l_{\text{w}}}$ is the average leaf width (m).

Wind speed at the heat source sink (${z_{0{\text{M}}}}+{d_0}$) and near the soil surface was originally estimated in TSEB using the Goudriaan (1977) wind attenuation model:

$$\begin{gathered} U\left( z \right)={U_{\text{C}}}{\text{exp}}\left[ { - {a_{\text{G}}}\left( {1 - z/{h_{\text{c}}}} \right)} \right] \hfill \\ {a_{\text{G}}}=0.28{\text{LA}}{{\text{I}}^{2/3}}h_{{\text{c}}}^{{1/3}}l_{{\text{w}}}^{{ - 1/3}}. \hfill \\ \end{gathered}$$

(9)

For the vineyards, Nieto et al. (2018a) utilized a new wind profile formulation developed by Massman et al. (2017). This canopy wind profile model is a more physically based method for calculating wind speed attenuation for the canopies of vertically non-uniform foliage distribution and leaf area that often exist in orchards and vineyards.

When only a single observation of ${T_{{\text{rad}}}}$ is available (i.e., measurement at a single angle), partitioning of ${T_{{\text{rad}}}}$ into canopy and soil components (${T_{\text{C}}}$ and ${T_{\text{S}}}$) is required to estimate component sensible heat fluxes via Eq. (4). The approach developed for TSEB (Norman et al. 1995) starts with an initial estimate of plant transpiration (LE_C), as defined by the Priestley and Taylor (1972) relationship, applied to the canopy divergence of net radiation (${R_{{\text{n,C}}}}$)

$${\text{L}}{{\text{E}}_{\text{C}}}={\alpha _{{\text{PT}}}}{f_{\text{g}}}\frac{\Delta }{{\Delta +\gamma }}{R_{{\text{n,C}}}}.$$

(10)

Here, ${\alpha _{{\text{PT}}}}$ is the Priestley–Taylor coefficient, initially set to 1.26, ${f_{\text{g}}}$ is the fraction of vegetation that is green and hence capable of transpiring, $\Delta$ is the slope of the saturation vapor pressure versus temperature curve, and $\gamma$ is the psychrometric constant. This method allows the canopy sensible heat flux to be calculated using the energy balance at the canopy layer (${H_{\text{c}}}={R_{{\text{n,C}}}} - {\text{L}}{{\text{E}}_{\text{C}}}$) and hence an estimate of ${T_{\text{C}}}$ to be obtained by rewriting Eq. 10 to have the following form (Norman et al. 1995):

$${T_{{\text{Ci}}}}={T_{\text{a}}}+\frac{{{R_{{\text{n,C}}}}{R_{\text{a}}}}}{{\rho {C_{\text{p}}}}}\left[ {1 - {\alpha _{{\text{PT}}}}{f_{\text{G}}}\frac{\Delta }{{\Delta +\gamma }}} \right],$$

(11)

where T_Ci is the initial estimate of canopy temperature. Alternatively, T_Ci can be derived with the linearization approximation to the series resistance approach described in Appendix A of Norman et al. (1995), specifically Eqs. (A.10)–(A.13). Then, ${T_{\text{S}}}$ is the derived from the following using T_rad, T_C, and an estimate of ${f_{\text{c}}}\left( \theta \right)$, the fraction of vegetation observed by the sensor view zenith angle $\theta$:

$$T_{{{\text{rad}}}}^{4}\left( \theta \right)={f_{\text{c}}}\left( \theta \right)T_{{\text{C}}}^{4}+\left[ {1 - {f_{\text{c}}}\left( \theta \right)} \right]T_{{\text{S}}}^{4}.$$

(12)

The value of ${f_{\text{c}}}\left( \theta \right)$ is typically estimated as an exponential function of the leaf area index (LAI), which includes a clumping factor or index $\Omega$ for canopies where the LAI is concentrated for sparsely distributed plants or for organized canopies such as row crops (Kustas and Norman 1999; Anderson et al. 2005), and has the following form:

$${f_{\text{c}}}\left( \theta \right)=1 - \exp \left( {\frac{{ - 0.5\Omega {\text{LAI}}}}{{\cos \theta }}} \right).$$

(13)

However, due to the unique vertical canopy structure and wide row width relative to canopy height of vineyards, a new method to derive $\Omega$ had to be developed that was both based on the geometric model of Colaizzi et al. (2012a) and simple enough to be incorporated into TSEB. Parry et al. (this issue) compared radiation extinction models of different complexities and found the simplified geometric model developed by Nieto et al. (2018b) was a robust modified radiation scheme. The resulting effective LAI, ${\text{LA}}{{\text{I}}_{{\text{eff}}}}=\Omega {\text{LAI}}$, was then used as input in the Campbell and Norman (1998) canopy radiative transfer model to estimate soil and canopy net radiation, R_n,S and R_n,C, respectively (see also Kustas and Norman 2000 for details).

The final energy balance component of soil heat flux, G, is typically estimated by TSEB as a fraction of the net radiation at the soil surface, R_n,S. However, over the daytime period, the assumption of a constant ratio between $G$ and ${R_{{\text{n,S}}}}$ is unreliable (Santanello and Friedl 2003; Colaizzi et al. 2016a, b). Based on observations between the measured soil heat flux and the estimated ${R_{{\text{n,S}}}}$ in Nieto et al. (2018b), a modified formulation estimating $G$ as a function of ${R_{{\text{n,S}}}}$ was adopted that accounts for the temporal behavior of the $G/{R_{{\text{n,S}}}}$ ratio over the daytime period using a double asymmetric sigmoid function; this estimate was a significantly better fit to the observations than the sinusoidal function proposed by Santanello and Friedl (2003).

With R_n,S and G estimated and H_S computed via T_S estimated from Eqs. (11)–(13), iteratively solving for Eqs. (4)–(9) results in LE_s solved as a residual via Eq. (3) for the soil layer, namely LE_s = R_n,S − G − H_S.

In some cases, the initial ${T_{\text{C}}}$ implied by the Priestley–Taylor approximation (Eq. 11) results in deriving a relatively high value of ${T_{\text{S}}}$ for a given observed ${T_{{\text{rad}}}}$and ${f_{\text{c}}}\left( \theta \right)$ condition. This high T_S can cause a significant overestimate in H_S and therefore produce unrealistic estimates of LE_S (i.e., negative values during daytime; LE_s < 0) solved by residual. In this case, the ${\alpha _{{\text{PT}}}}$ coefficient is iteratively reduced at 0.1 intervals from its initial value ~ 1.26, effectively assuming the canopy is stressed and transpiring at sub-potential levels until LE_s ≥ 0.