*2.3. Surface Renewal (SR) Method*

The SR method is based on the turbulent exchange of the sensible heat between the plant canopy and the atmosphere, caused by the instantaneous replacement of an air parcel interacting with the surface (Figure 2). The air parcel exchanges energy between air and canopy elements; then, the parcel is detached from the surface, and a new air parcel swings in to renew the removed air. Thus, understanding the features of this turbulence mechanism is vital for correct operation and analysis of this method. The signals of air temperature display well-managed coherent structures which resemble ramp events [30–32]. The SR method is constructed on the investigative energy budget of the coherent structures that exist within the crop canopy [31,32]. The exchange of air parcels between the surface and atmosphere is established as ramp-like shapes in the turbulence temperature; *H*SR can be estimated as Equation (4):

$$H\_{SR} = \ a H\_{SR}' = \alpha z \left(\rho c\_p\right) \frac{a}{\tau} \tag{4}$$

where *α* is the calibration coefficient obtained through the slope of regression between *HSR* (calibrated sensible heat flux) and the *H SR* (uncalibrated sensible heat flux); *z* is the measurement height (m), *<sup>ρ</sup>* is the specific air density (kg·m−3), *cp* is the specific heat capacity (J·kg−1·K−1), *<sup>a</sup>* is the ramp amplitude (*K*) and *<sup>τ</sup>* is the ramp period (*s*). The SR estimations require calibration against any independent method (i.e., EC and BR).

**Figure 2.** The concept of the surface renewal theory.

The calibration factor (*α*) is obtained as the slope of the linear regression forced through the origin, between the estimations of *HSR* and the *HEC*; the value of the calibration coefficient depends on the measurement height, canopy height and architecture, atmospheric stability, turbulence characteristics, and sensor dynamic response characteristics [9]. The structure–function analysis is calculated by Equation (5) [8]:

$$\overline{S^n(r)} = \frac{1}{m-j} \sum\_{k=1}^{m-j} \left[ \left( T\_k - T\_{k-j} \right)^n \right] \tag{5}$$

where *S* denotes the structure function, *n* is the order of the structure function (2nd, 3rd, or 5th in this case), *r* is the order of function, *m* is the total number of data points, *j* is the sample lag, and *Tk* is the *k*th element in the calculated temperature data. The structure– function values are used to determine the coefficient in the following cubic polynomial expressed as Equation (6) [8]:

$$a^3 + pa + q = 0\tag{6}$$

where *p* is obtained as Equation (7) [8]:

$$p = \left[10S^2\_{(r)} - \frac{S^5\_{(r)}}{S^3\_{(r)}}\right] \tag{7}$$

Here, *q* is obtained as Equation (8) [8]:

$$q = 10 \text{S}^3\_{(r)} \tag{8}$$

These equations can be solved analytically to obtain the ramp amplitude. The ramp period is calculated from the ramp amplitude, time lag, and third-order structure function using Equation (9) [8]:

$$\pi = -\frac{a^3(r)}{S^3(r)}\tag{9}$$

The shortened energy closure was used for the estimation of *LE* above the plant canopy using *HSR* and the remaining parameters, including *Rn* and *G*, using Equation (10) [8]:

$$LE\_{SR} = \begin{array}{c} \text{Rn} \ - \ G \ - \ H\_{SR} \end{array} \tag{10}$$

The performance of the SR method was analyzed by linear regression analysis against the EC system with statistical errors including root mean square error (RMSE), the slope of regression, intercept, coefficient of determination, and relative error (RE), which indicate the performance of the SR method for estimating *H* and *LE*. The SR method overcomes the issues related to the fetch, leveling, orientation, and instrument placement, which can overcome the potential uncertainties in the EC system [33–37].
