# Estimating Canopy Resistance Using Machine Learning and Analytical Approaches

^{*}

## Abstract

**:**

## 1. Introduction

_{a}) and surface resistance (r

_{s}). When the surface is covered by water, r

_{s}is equal to zero. When the surface is densely covered with vegetation, r

_{s}represents the canopy resistance (r

_{c}). The aerodynamic resistance can be calculated using the MOST. The major challenge with the P–M equation is the requirement for accurate surface (or canopy) resistances [2,3,4].

_{c}(=70 s/m) in the P–M equation to estimate the reference evapotranspiration. This reference evapotranspiration is further multiplied by a crop coefficient to calculate the actual evapotranspiration. While this method is straightforward, it is considered less accurate [5,6]. In addition to using a fixed (constant) canopy resistance in the P–M equation, using variable canopy resistance estimated from models is also adopted.

_{c}as a function of meteorological variables (such as net radiation, wind speed, relative humidity, and air temperature). The Jarvis multiplicative model [7] stands as a representative example of such models. However, calibrations of regression coefficients for different vegetation types and meteorological conditions are required before using such models. Founded upon certain assumptions, Todorovic [12] proposed a mechanistic approach (analytical solution) to calculate canopy resistance; his model is also a function of climatic variables and aerodynamic resistance, but no calibration coefficients are needed. Pauwels and Samson [13] found that r

_{c}calculated from Todorovic’s equation has some problems above a wet, sloping grassland. On the other hand, Lecina et al. [14] and Perez et al. [15] recommended Todorovic’s model for semiarid conditions. Perez et al. [15] also found that through a constant canopy resistance, there could be an underestimate of evapotranspiration in the summer and an overestimate in the winter; it was shown that using a monthly averaged surface resistance instead of a constant value would lead to a better estimation of evapotranspiration at seasonal time scale. Li et al. [16] studied surface resistances under arid conditions with a dense canopy (maize field) and a partial canopy (vineyard). Their results showed that Todorovic’s method performed poorly in both cases, while the Jarvis multiplicative model performed better on dense canopy than partial canopy.

_{c}estimated from Todorovic’s or parameterization methods, machine learning (ML) models have also gained widespread use for the estimations of surface fluxes in recent times [17,18]. However, as far as our current knowledge goes, there are no published studies using ML approaches to estimate canopy resistance. Huang and Hsieh [19] employed five ML models, including support vector machine (SVM), random forest, multi-layer perceptron, deep neural network, and long short-term memory, for the gap-filling of surface fluxes. They concluded that the five ML models performed well and similarly for estimating sensible heat, latent heat, and CO

_{2}fluxes, and SVM is a little bit better than the other four ML models in LE estimations.

## 2. Sites and Data

#### 2.1. Dripsey Grassland Experiment

^{2}/m

^{2}and 0.5–1.0 m

^{2}/m

^{2}, respectively.

#### 2.2. Chi-Lan Forest Experiment

^{3}/m

^{3}, and this value remains relatively stable. The soil moisture conditions have a minimal impact on daily evapotranspiration and do not play a dominant role in the water resource cycle in this Cypress forest. The canopy in this study site was closed and uniform, with an average canopy height of 10.3 m and a leaf area index of 6.3 m

^{2}/m

^{2}. As a result, the soil heat flux at the surface was quite low, ranging from −15 to 25 W m

^{−2}throughout the year. The soil heat flux at 0.1 m and soil temperature at 0.05 m below the ground were measured using a heat flux plate (HFP01) and a soil temperature sensor.

_{2}concentrations. The sampling frequency of the eddy-covariance system was 10 Hz, and the data were averaged and recorded every 30 min. More detailed information can be found in Chu et al. [21]

#### 2.3. Sitou Forest Experiment

_{2}and H

_{2}O concentrations. Additionally, a data logger was used to collect data at 10 Hz, with the averages taken every 30 min. For meteorological data, a net radiometer (NR-LITE) and a rain gauge (TE525MM) were installed to measure net radiation and precipitation. At 27.5 m above the ground, temperature and humidity measurements were collected using a temperature and humidity sensor (HMP45C); at the same height, a precision infrared temperature sensor (IRTS-P) was used to measure canopy surface temperature. Soil heat flux at 0.05 m below the ground was measured using a heat flux plate (HFP01); a soil temperature sensor was also installed at 5 cm depth to measure the soil temperature for calculating the heat storage in this soil layer. All meteorological data were connected to a data logger (CR23X) with a sampling frequency of 30 s and an averaging period of 30 min. The data collection period for this study was from 22 May 2009 to 31 July 2010.

#### 2.4. Data Processing

- (1)
- In the evening, due to small evapotranspiration, the measured canopy resistance calculated from Equation (3) can be unreasonably large; hence, only daytime data (Rn > 0) were used for this study.
- (2)
- If a measured r
_{c}is less than 0 or larger than 1050 s/m, this data point is not reliable and excluded from further analysis. - (3)
- Obvious outliers and anomalies values in the data were removed.

_{c}are (1) time factor (i.e., Julian day) and (2) meteorological factors (i.e., available energy, air temperature, wind speed, and vapor pressure deficit). These input factors for training the ML model are also summarized in Table 2. To build the ML model, the ratio of training and testing datasets is 6:4. Each dataset is sorted chronologically, with the first 60% used for training and the remaining 40% used for validation.

## 3. Methods

#### 3.1. Penman–Monteith Equation

^{2}), Δ (kP

_{a}K

^{−1}) is the slope of the saturation vapor pressure-temperature curve calculated at the air temperature T

_{a}, γ (=ρC

_{p}/0.622L

_{v}) (kP

_{a}K

^{−1}) denotes psychrometric constant, ρ is the air density, C

_{p}(=1005 J kg

^{−1}K

^{−1}) is the specific heat of air, L

_{v}(=2.46 × 10

^{6}J kg

^{−1}) represents the latent heat of vaporization, Q

_{n}(=R

_{n}–G) is the available energy, where R

_{n}(W m

^{−2}) and G (W m

^{−2}) stand for the net radiation and soil heat flux at the ground surface, respectively, D (kP

_{a}) represents vapor pressure deficit, r

_{a}(s m

^{−1}) is the aerodynamic resistance, and r

_{c}(s m

^{−1}) is the canopy resistance. Note that for a vegetated surface and the canopy is closed, the surface resistance equals the canopy resistance, and the soil evaporation can be ignored. The term r

_{a}in Equation (1) can be calculated using MOST [26]:

_{0m}(m) is the roughness length for momentum (≈0.1 h), z

_{0v}(m) is the roughness length for water vapor (≈0.01 h), k (=0.4) is the von Kármán constant.

_{c}is computed as

_{n}and LE.

#### 3.2. Todorovic’s Analytical Equation

_{c}> 0), part of the available energy is required to heat the canopy to extract water. Therefore, this additional energy (H

^{’}) would raise the canopy temperature by an amount t, which is the temperature difference between the mean level (d + z

_{0m}) of source or sink for H and LE and the level in canopy (i.e., saturated level). Todorovic [11] then expressed LE as the difference between the potential evapotranspiration (PET) and H

^{’}, which is

_{0m}. A linear relationship between saturation vapor pressure and temperature and neutral atmospheric conditions were assumed by Todorovic [11]. As a result, the temperature difference, t, in Equation (5) could be calculated as

_{c}and then r

_{c}could be solved analytically by a quadratic equation as

_{c}is

_{i}is the isothermal resistance first introduced by Monteith [24] and defined as

_{i}) is simply the sum of r

_{c}+ r

_{a}under the isothermal condition $\left(\partial {\mathrm{T}}_{\mathrm{a}}/\partial \mathrm{z}=0\right)$, in which H = 0 and LE = R

_{n}−G.

_{c}and no training data are needed a priori.

_{c}= 0; in other words, Todorovic’s analytical solution is valid for the sites where r

_{c}is small. Equation (14) provides the limitation where this analytical solution for r

_{c}can be applied.

#### 3.3. Constant Canopy Resistance Method

_{c}in the P–M equation for estimating LE was also used in this study. The determination of the constant r

_{c}followed an average calculation approach. That is, the preprocessed data underwent Equation (3) to calculate the measured canopy resistance for each time step. Subsequently, the average of all calculated canopy resistances was taken as the constant r

_{c}.

#### 3.4. Support Vector Machine

- (1)
- Balancing model accuracy and complexity: SVM utilized the principle of structural risk minimization to estimate the classification or regression hyperplane. By finding a decision boundary that maximizes the separation between classes or regression targets, SVM effectively delineates data points of different categories within a high-dimensional plane. This approach sets SVM apart from other ML methods as it simultaneously balances accuracy and model complexity. While enhancing model accuracy, it also helps mitigate issues like overfitting and excessive computational time.
- (2)
- Simplified weight estimation process: The process of weight determination in SVM is simplified into a quadratic programming problem, making it solvable using a standardization procedure.

^{–5}and 2

^{5}. For a more comprehensive understanding of SVM, the details can be found in Cork and Vapnik [27].

## 4. Results and Discussion

#### 4.1. Diurnal Variations in Observed r_{c} and LE

_{c}and LE during the daytime (Grassland site: 06:30–21:00; forest sites: 06:30–17:30), we plotted the mean (averaged over the entire observation period) r

_{c}and LE as a function of local time (Figure 1). Notice that for the grassland, r

_{c}was larger in the early morning and then decreased to its lowest value (around 150 s/m) at 12:00 and maintained this value till 18:00 (Figure 1a).

_{c}was the lowest during the sunrise period (around 06:30) and then increased to its largest value around noon (12:00). This phenomenon is attributed to the presence of abundant dew on forest vegetation and the prevalence of fog, creating an environment akin to a wet surface in the early morning. As a result, the r

_{c}was lower, then subsequent to the evaporation of dew and the dispersal of fog, the r

_{c}gradually increased as the canopy became drier.

_{n}–G). From Figure 2a,b, it can be inferred that in the Dripsey grassland, approximately 31% of the available energy is allocated to sensible heat flux, while about 41% is distributed to LE. The overall energy closure rate is 72%. On the other hand, for the Chi-Lan forest, the available energy is distributed with approximately 53% and 19% to sensible heat and latent heat fluxes, respectively. As for Sitou forest, the distribution of available energy is approximately 34% to H and 18% to LE. In the grassland, the available energy contributed in similar proportions to heating air and evapotranspiration. However, in the forests, a majority of the energy was utilized for sensible heat flux, with only a small portion being allocated to evapotranspiration. The energy closure ratios for Dripsey grassland and Chi-Lan forest are around 72% and are normal compared with the literature. The discrepancy in energy closure rate (only 52%) in the Sitou forest might predominantly arise from the storage of some R

_{n}energy within the canopy layer. The lack of energy closure indicates that in these study areas, when using the P–M equation with available energy (R

_{n}–G) for predicting LE, there might be an overestimation of LE.

#### 4.2. Model Performance in Estimating r_{c}

#### 4.2.1. Dripsey Grassland

_{c}at the grassland. Figure 3 shows the comparison between the measured and model-estimated r

_{c}above the grassland in a 1:1 plot. At this grassland, the average of measured r

_{c}was 163 s/m, and this value was used to estimate LE in the constant canopy resistance method. From Table 3, it is evident that none of the two methods (SVM and Todorovic’s equation) were able to accurately estimate r

_{c}; R

^{2}are both less than 0.22, and RMSEs are larger than 165 (s/m) for an average r

_{c}of 163 s/m.

_{c}within specific ranges (the constant canopy resistance method uses fixed value). The SVM method has a wider prediction range, primarily falling within the range of 100 to 200 s/m. On the other hand, Todorovic’s analytical equation tends to predict very low canopy resistance values (mostly between 0–100 s/m), as explained in Equations (13) and (14). In other words, it can be inferred that Todorovic’s method tends to provide more accurate predictions of LE when dealing with smaller measured r

_{c}(r

_{c}< 100). However, when encountering medium to high measured r

_{c}, larger errors might occur.

#### 4.2.2. Chi-Lan Forest

_{c}at the Chi-Lan forest. Figure 4 illustrates the comparison of measured and model-estimated r

_{c}in a 1:1 plot. From Table 4, both methods failed to reproduce r

_{c}well in the Chi-Lan forest. In terms of correlation, the R

^{2}(=0.13) of the estimated r

_{c}using the SVM is even lower than that at the grassland. Similarly, the R

^{2}for Todorovic’s analytical solution estimations is only 0.01. The RMSEs for the SVM and Todorovic’s methods are 248 and 425 s/m, respectively, with an average r

_{c}of 346 s/m at this Cypress forest.

_{c}values. On the other hand, Todorovic’s method tends to predict extremely low r

_{c}, with most of the predictions less than 75 s/m. The majority of predicted r

_{c}values from Todorovic’s method are below 50 s/m.

#### 4.2.3. Sitou Forest

_{c}in the Sitou forest, while Figure 5 illustrates the comparison of predictions against the measured r

_{c}in a 1:1 plot. From Table 5, the performances for r

_{c}prediction in the Sitou forest are quite similar to those in the Chi-Lan forest. Both methods failed to reproduce r

_{c}. At this Cryptomeria forest, the R

^{2}value for r

_{c}prediction using the SVM method is 0.14, while Todorovic’s analytical solution predictions showed no correlation with the measurements (R

^{2}= 0). The RMSEs of the SVM and Todorovic’s methods are 150 and 373 s/m, respectively, while the average r

_{c}for this site is 321 s/m.

_{c}, the SVM estimated r

_{c}ranged from 0 to 400 s/m. On the other hand, same as the grassland and Cypress forest sites, Todorovic’s method tends to predict low r

_{c}values (<100 s/m).

_{c}estimations, though Todorovic’s method works better at the grassland than the two forests. The failure of Todorovic’s analytical solution comes from the assumptions of this method, which were not satisfied above these three sites. The uncertainty of SVM estimations comes from the measured r

_{c}. Detail discussions are provided in Section 4.5 and Section 4.6.

#### 4.3. Model Performance in Estimating LE

_{c}, the estimated LE was in better agreement with the measurements. In the following sections, we will provide detailed explanations for the Dripsey grassland, Chi-Lan forest, and Sitou forest, respectively.

#### 4.3.1. Dripsey Grassland

_{c}values from the SVM, Todorovic’s analytical solution, and the constant (i.e., the average) canopy resistance at the grassland. The regression analyses between LE measurements and predictions are also summarized in Table 3. All LE predictions from the three methods are strongly correlated with the measurements (R

^{2}= 0.83 to 0.88); however, Todorovic’s method tends to overestimate LE by 58.7% (slope = 0.63, see Table 3) and results in a high RMSE (=64 W/m

^{2}) while the other two methods’ RMSEs are less than 30 W/m

^{2}.

- (1)
- Surface energy imbalance. In this grassland, the energy closure rate is approximately 72% (H+LE is 72% of Q
_{n}). Hence, if we take this into account and force the energy to be closed, the ratio of measured LE to the estimated LE by Todorovic’s method combined with the P–M equation should be 87.5% (slope = 0.63/0.72 = 0.875); in other words, the overestimation rate is only 14%. In both the SVM and constant canopy resistance methods, since the training target values used for both methods are all based on the r_{c}calculated from Equation (3), the energy imbalance in Equation (1) is compensated automatically. - (2)
- The study area does not meet the assumption of Todorovic’s method. According to the assumption in Todorovic’s study [11], the pseudo resistance (r′) should be the same as the r
_{c}, that is, r′ = r_{c}. However, this is not the case in Dripsey grassland. A detailed discussion is provided in Section 4.5.

#### 4.3.2. Chi-Lan Forest

^{2}between measured and estimated LE from the three methods are low: 0.46 (SVM), 0.21 (Todorovic), and 0.41 (constant r

_{c}); nevertheless, these low values are comparable to the R

^{2}(=0.35) between measured LE and Q

_{n}(Figure 2d). Similar to the results at the grassland, Todorovic’s analytical solution overestimated LE (regression slope = 0.20) and resulted in a high RMSE (=233 W/m

^{2}), while the other two methods’ RMSEs are less than 60 W/m

^{2}at this Cypress forest site. As indicated by Equation (14), Todorovic’s method is more suitable for environments with smaller measured r

_{c}, and in the Chi-Lan forest, the average r

_{c}(=346 s/m) is quite large, indicating that this Cypress forest is not suitable for the application of Todorovic’s method and thus exhibits larger errors. Even after taking into account the effect of energy imbalance (closure rate = 72%), the regression slope is still very low (0.20/0.72 = 0.278), showing a strong overestimation by Todorovic’s method.

_{c}in the training dataset is slightly different from the average r

_{c}in the testing dataset. In other words, the average r

_{c}in this forest changed with time (seasons).

#### 4.3.3. Sitou Forest

_{c}from SVM, Todorovic, and constant canopy resistance methods at the Sitou forest. The performances of these three methods in this study area are quite similar to the results observed at the Chi-Lan forest (see regression summary in Table 5). From Table 5, the RMSE values obtained by the three methods range from 33.74 to 218.53 (W/m

^{2}), which are slightly lower than those in the Chi-Lan forest. The R

^{2}values ranged from 0.50 (Todorovic) to 0.64 (SVM) and are comparable to the R

^{2}(0.55) between measured LE and Q

_{n}(Figure 2f) at Sitou forest. Hence, the performance of the P–M equation in this Cryptomeria forest is better than that in the Chi-Lan forest (R

^{2}= 0.20–0.46).

_{c}in this Cryptomeria forest varied with time (seasons).

#### 4.4. Model Performance in Different Ranges of Canopy Resistance

_{c}values. Particularly, it shows high sensitivity in the range of r

_{c}= 0–100 (s/m), medium sensitivity in the range of 100–200 (s/m), and low sensitivity for r

_{c}larger than 200 (s/m). Since Todorovic’s analytical solution tends to produce a small r

_{c}, it can be inferred that with smaller measured r

_{c}values (r

_{c}< 100), the error between LE estimations and measurements would be smaller. However, when encountering situations with larger measured r

_{c}, significant errors would occur and lead to a decrease in the overall model performance. In this section, we compiled the performance metrics of SVM and Todorovic’s methods for the above three sensitivity ranges at the three study sites.

_{c}: 0–100 s/m), medium (r

_{c}: 100–200 s/m), and low (r

_{c}> 200 s/m) sensitivity ranges above the Dripsey grassland. From the analysis of r

_{c}estimations in Section 4.2, it was noticed that both SVM and Todorovic’s methods tend to produce r

_{c}within a certain interval. Therefore, at this grassland, as the SVM tends to reproduce r

_{c}in the medium range (where the RMSE for estimated r

_{c}is the smallest among the three ranges), the RMSE (=21.59 W/m

^{2}) of LE estimation in this interval is then the lowest compared with the high sensitivity interval (RMSE = 27.25 W/m

^{2}) and the low sensitivity interval (RMSE = 27.66 W/m

^{2}).

_{c}within the high sensitivity interval; therefore, the RMSE of predicted LE in the high sensitivity interval is also the lowest (26.32 W/m

^{2}) compared with the other two intervals. As the measured r

_{c}increases to the medium and low sensitivity intervals, the RMSE of LE predictions also increases to 70 and 90 (W/m

^{2}), respectively. Detailed scatter plots of LE v.s. R

_{n}–G and comparisons between measurements and predictions of r

_{c}and LE by SVM and Todorovic’s methods for these three intervals at the Dripsey grassland are presented in Appendix A.1.

_{c}more accurately in the high sensitivity interval, resulting in the lowest error of LE estimations with an RMSE of 55.48 (W/m

^{2}). However, as r

_{c_m}increases to the medium and high sensitivity intervals, the RMSE significantly rises to 132 and 279 (W/m

^{2}), respectively. On the contrary, due to being trained by historical data, the SVM tends to predict r

_{c}mainly ranging from 150 to 600 (s/m), with maximum values reaching around 750 (s/m). It is observed that the SVM method achieves the lowest RMSE in the low sensitivity interval (=32.87 W/m

^{2}). As r

_{c_m}decreases to medium or high sensitivity intervals, the errors in predicting LE also increase to 60 and 91 (W/m

^{2}), respectively. Noticed that, for the SVM model, the best r

_{c}estimation is in the medium range; however, the best LE estimation is in the low sensitivity range. This is because, in the low range, the LE estimation is not sensitive to the r

_{c}value [6].

^{2}) between the two sites. From Table 7 and Table 8, in all three intervals, the R

^{2}values for LE estimations by the SVM model are higher (0.77–0.89) at the Sitou forest and smaller (0.64–0.76) at the Chi-Lan forest. The same results are found in the LE estimations by Todorovic’s analytical solution. This phenomenon can be explained by noting that the correlation between LE and R

_{n}–G is higher at the Sitou forest site. From Figure 2, Figure A1, Figure A4 and Figure A7, it is evident that the correlation between LE and R

_{n}–G determines the R

^{2}between LE measurements and estimates by the P–M equation. This is because the P–M equation uses R

_{n}–G to estimate LE.

_{c}estimation is in the medium range; however, the best LE estimation is in the low sensitivity range. Detailed scatter plots of LE v.s. R

_{n}–G and comparisons between measured and predicted r

_{c}and LE by SVM and Todorovic’s methods for these three intervals at the Chi-Lan forest and Sitou forest are provided in Appendix A.2 and Appendix A.3, respectively.

#### 4.5. Uncertainty of Todorovic’s Analytical Solution

^{’}can be represented by r

_{c}? (2) if the approximation for t, i.e., Equation (6), is valid?

_{c}, Equation (5) becomes

_{n}, and r

_{c}, Equation (15) provides an analytical solution for t (canopy temperature increase). Now by substituting Equation (6) into Equation (5), we have

_{n}, Equation (17) provides an analytical solution for r′.

_{n}, and r

_{c}, Figure 9a shows the t comparison between Todorovic’s expression (t

_{TOD}, i.e., Equation (6)) and the analytical solution (t

_{ana}, i.e., Equation (15)) at the Dripsey grassland. It is clear from Figure 9a that the t value obtained from Todorovic’s expression is systematically lower than the t value required from the analytical solution. This indicates that if r′ = r

_{c}, then Todorovic’s expression (i.e., Equation (6)) for t is not valid at this grassland.

_{n}, and t from Todorovic’s expression (i.e., Equation (6)), Figure 9b shows the comparison between r′ (calculated from Equation (17)) and r

_{c}(calculated from Equation (3)). It is clear that r′ is much smaller than the measured r

_{c}in Figure 9b. The measured r

_{c}during the early morning period was around 100 (s/m) and maintained around 90 (s/m) for the noon and afternoon periods. In contrast, the r

^{’}was only about 20 (s/m) for most of the daytime. This indicates that if Todorovic’s expression (i.e., Equation (6)) is valid, then the assumption of r′ = r

_{c}is not satisfied at this grassland. Notice that the r

_{c_m}in Figure 9b is different from the r

_{c_m}in Figure 1a, where r

_{c_m}is first calculated by Equation (3) for each data and then averaged.

_{ana}) calculated from Equation (16) was very high and not reasonable (maximum = 74.52 °C at 11:30); this high value of t resulted from the high value of r

_{c}at this site and assuming r’ = r

_{c}. However, the t values calculated using the approximation proposed by Todorovic are small (maximum around 1 °C) and similar to those in the grassland.

_{c}and is much smaller. The disagreement between r′ and r

_{c_m}for this Cypress forest is much larger than that observed in the Dripsey grassland, which further indicates that Todorovic’s analytical solution for r

_{c}is not suitable for this forest site.

_{TOD}and t

_{ana}in Figure 11a are quite similar to those in Figure 10a, but the maximum t

_{ana}(around 140 °C) is even higher than that in the Chi-Lan forest; part of this is caused by the larger energy imbalance at the Sitou forest. As to the comparison between r′ and r

_{c}, similar results to those in Chi-Lan Forest were also found in Sitou Forest (Figure 11b). Figure 11a,b demonstrates that Todorovic’s method is not suitable for this Cryptomeria forest.

#### 4.6. Uncertainty of the P–M Equation, Support Vector Machine, and Constant Canopy Resistance

_{c}mainly comes from the measured r

_{c}for training the SVM model. From Equation (3), the canopy resistance is a function of available radiation energy, LE, air temperature, wind speed, and vapor pressure deficit. In Appendix B we plotted measured r

_{c}as a function of these meteorological variables for Dripsey grassland, Chi-Lan forest, and Sitou forest, respectively. From Appendix B, it is clear that r

_{c}spreads out a lot with the meteorological variables; this explains why the SVM model was not able to predict r

_{c}well.

_{c}in this study was calculated from the reverse of Equation (1), i.e., Equation (3), the energy imbalance portion has been taken into account in the SVM and constant canopy resistance methods (recall: the SVM model were trained by the historical observed r

_{c}; the constant canopy resistance method takes the average of the historical observed r

_{c}as the constant r

_{c}). However, the LE calculated from the r

_{c}estimated by Todorovic’s analytical solution would suffer from this energy imbalance problem. From Figure 2, Table 3, Table 4 and Table 5, and Appendix A, it is evident that the R

^{2}between measured and predicted LE depends on the correlation of measured LE v.s. R

_{n}–G, and not sensitive to the accuracy of r

_{c}estimation; also, from Table 3, Table 4, Table 5, Table 6, Table 7 and Table 8, the RMSE of LE estimation depends on the accuracy of r

_{c}.

_{n}, in Equation (3) is replaced by H+LE; then the percentage errors are 6.2%, 111.8%, 79.3%, respectively.

## 5. Conclusions

- (1)
- The estimated r
_{c}from Todorovic’s analytical solution exhibits no correlation with the observed r_{c}. On the other hand, the support vector machine’s r_{c}estimation is slightly better, with R^{2}values ranging between 0.13 and 0.22 across the three research areas. Both methods tend to reproduce the canopy resistances within a certain range of intervals. - (2)
- Contrary to r
_{c}estimations, the LE estimations are in better agreement with the observations when the estimated r_{c}are adopted in the Penman–Monteith equation. In general, the SVM model performs better than the analytical solution. - (3)
- The failure of the analytical solution in estimating r
_{c}is attributed to the assumption of r′ = r_{c}. Using this method will lead to a significant underestimation of canopy resistance and subsequently an overestimation of latent heat flux. This discrepancy is more obvious in forests where r_{c}is in a bigger value (around 320 s/m). - (4)
- The coefficient of determination (R
^{2}) of regression between the measured and P–M equation estimated LE is strongly dependent on the correlation between measured LE and available energy (R_{n}–G) and not sensitive to the accuracy of r_{c}estimation; however, the RMSE of LE estimation depends on the accuracy of r_{c}.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Energy Closure and Latent Heat Flux Estimations in Different Canopy Resistance Intervals

#### Appendix A.1. Dripsey Grassland Experiment

**Figure A1.**Comparisons of H+LE v.s. R

_{n}–G and LE v.s. R

_{n}–G under different ranges of measured canopy resistance (r

_{c}): (

**a**,

**b**) r

_{c}< 100 (s/m), (

**c**,

**d**) r

_{c}=100–200 (s/m), and (

**e**,

**f**) r

_{c}> 200 (s/m) above Dripsey grassland.

**Figure A2.**Comparisons of measured canopy resistance (r

_{c_m}) and latent heat flux (LE

_{_m}) with estimations by SVM model (r

_{c_svm}, LE

_{_SVM_rc}) under different ranges of measured canopy resistance (r

_{c}): (

**a**,

**b**) r

_{c}< 100 (s/m), (

**c**,

**d**) r

_{c}=100–200 (s/m), and (

**e**,

**f**) r

_{c}> 200 (s/m) above Dripsey grassland.

**Figure A3.**Comparisons of measured canopy resistance (r

_{c_m}) and latent heat flux (LE

_{_m}) with estimations by Todorovic’s method (r

_{c_TOD}, LE

_{_TOD_rc}) under different ranges of measured canopy resistance (r

_{c}): (

**a**,

**b**) r

_{c}< 100 (s/m), (

**c**,

**d**) r

_{c}= 100–200 (s/m), and (

**e**,

**f**) r

_{c}> 200 (s/m) above Dripsey grassland.

#### Appendix A.2. Chi-Lan Forest Experiment

**Figure A4.**Comparisons of H+LE v.s. R

_{n}–G and LE v.s. R

_{n}–G under different ranges of measured canopy resistance (r

_{c}): (

**a**,

**b**) r

_{c}< 100 (s/m), (

**c**,

**d**) r

_{c}= 100–200 (s/m), and (

**e**,

**f**) r

_{c}> 200 (s/m) above Chi-Lan forest.

**Figure A5.**Comparisons of measured canopy resistance (r

_{c_m}) and latent heat flux (LE

_{_m}) with estimations by SVM model (r

_{c_svm}, LE

_{_SVM_rc}) under different ranges of measured canopy resistance (r

_{c}): (

**a**,

**b**) r

_{c}< 100 (s/m), (

**c**,

**d**) r

_{c}= 100–200 (s/m), and (

**e**,

**f**) r

_{c}> 200 (s/m) above Chi-Lan mountain.

**Figure A6.**Comparisons of measured canopy resistance (r

_{c_m}) and latent heat flux (LE

_{_m}) with estimations by Todorovic’s method (r

_{c_TOD}, LE

_{_TOD_rc}) under different ranges of measured canopy resistance (r

_{c}): (

**a**,

**b**) r

_{c}< 100 (s/m), (

**c**,

**d**) r

_{c}= 100–200 (s/m), and (

**e**,

**f**) r

_{c}> 200 (s/m) above Chi-Lan forest.

#### Appendix A.3. Sitou Forest Experiment

**Figure A7.**Comparisons of H+LE v.s. R

_{n}–G and LE v.s. R

_{n}–G under different ranges of measured canopy resistance (r

_{c}): (

**a**,

**b**) r

_{c}< 100 (s/m), (

**c**,

**d**) r

_{c}= 100–200 (s/m), and (

**e**,

**f**) r

_{c}> 200 (s/m) above Sitou forest.

**Figure A8.**Comparisons of measured canopy resistance (r

_{c_m}) and latent heat flux (LE

_{_m}) with estimations by SVM model (r

_{c_svm}, LE

_{_SVM_rc}) under different ranges of measured canopy resistance (r

_{c}): (

**a**,

**b**) r

_{c}< 100 (s/m), (

**c**,

**d**) r

_{c}= 100–200 (s/m), and (

**e**,

**f**) r

_{c}> 200 (s/m) above the Sitou forest.

**Figure A9.**Comparisons of measured canopy resistance (r

_{c_m}) and latent heat flux (LE

_{_m}) with estimations by Todorovic’s method (r

_{c_TOD}, LE

_{_TOD_rc}) under different ranges of measured canopy resistance (r

_{c}): (

**a**,

**b**) r

_{c}< 100 (s/m), (

**c**,

**d**) r

_{c}= 100–200 (s/m), and (

**e**,

**f**) r

_{c}> 200 (s/m) above the Sitou forest.

## Appendix B. Scatter Plots of Canopy Resistance and Meteorological Variables

**Figure A10.**Scatter plots of measured canopy resistance (r

_{c_m}) as a function of (

**a**) net radiation (R

_{n}), (

**b**) available energy (R

_{n}–G), (

**c**) latent heat flux (LE), (

**d**) wind speed (U), (

**e**) vapor pressure deficit (D), and (

**f**) air temperature (T

_{a}) above Dripsey grassland.

**Figure A11.**Scatter plots of measured canopy resistance (r

_{c_m}) as a function of (

**a**) net radiation (R

_{n}), (

**b**) available energy (R

_{n}–G), (

**c**) latent heat flux (LE), (

**d**) wind speed (U), (

**e**) vapor pressure deficit (D), and (

**f**) air temperature (T

_{a}) above Chi-Lan forest.

**Figure A12.**Scatter plots of measured canopy resistance (r

_{c_m}) as a function of (

**a**) net radiation (R

_{n}), (

**b**) available energy (R

_{n}–G), (

**c**) latent heat flux (LE), (

**d**) wind speed (U), (

**e**) vapor pressure deficit (D), and (

**f**) air temperature (T

_{a}) above Sitou forest.

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**Figure 1.**Diurnal variations in canopy resistance (r

_{c}) and latent heat flux (LE) above (

**a**) Dripsey grassland, (

**b**) Chi-Lan forest, and (

**c**) Sitou forest.

**Figure 2.**Scatter plots of sensible heat (H) and latent heat (LE) fluxes as a function of R

_{n}−G above the Dripsey grassland (

**a**,

**b**), Chi-Lan forest (

**c**,

**d**), and Sitou forest (

**e**,

**f**).

**Figure 3.**Comparisons between measured canopy resistance (r

_{c_m}) and estimations by (

**a**) SVM (r

_{c_SVM}) model and (

**b**) Todorovic’s analytical solution (r

_{c_TOD}) above Dripsey grassland.

**Figure 4.**Comparisons between measured canopy resistance (r

_{c_m}) and estimations by (

**a**) SVM (r

_{c_SVM}) model and (

**b**) Todorovic’s analytical solution (r

_{c_TOD}) above Chi-Lan forest.

**Figure 5.**Comparisons between measured canopy resistance (r

_{c_m}) and estimations by (

**a**) SVM (r

_{c_SVM}) model and (

**b**) Todorovic’s analytical solution (r

_{c_TOD}) above Sitou forest.

**Figure 6.**Comparisons between measured latent heat flux (LE

_{_m}) and estimations by (

**a**) SVM (LE

_{_SVM_rc}) model, (

**b**) Todorovic’s analytical solution (LE

_{_TOD_rc}), and (

**c**) constant r

_{c}(LE

_{_avg_rc}) above the Dripsey grassland.

**Figure 7.**Comparisons between measured latent heat flux (LE

_{_m}) and estimations by (

**a**) SVM (LE

_{_SVM_rc}) model, (

**b**) Todorovic’s analytical solution (LE

_{_TOD_rc}), and (

**c**) constant r

_{c}(LE

_{_avg_rc}) above Chi-Lan forest.

**Figure 8.**Comparisons between measured latent heat flux (LE

_{_m}) and estimations by (

**a**) SVM (LE

_{_SVM_rc}) model, (

**b**) Todorovic’s analytical solution (LE

_{_TOD_rc}), and (

**c**) constant r

_{c}(LE

_{_avg_rc}) above Sitou forest.

**Figure 9.**Comparisons between (

**a**) canopy temperature increase calculated by Todorovic’s expression (t

_{TOD}, Equation (6)) and the analytical solution (t

_{ana}, Equation (16)); (

**b**) pseudo resistance (r’

_{TOD}, calculated by Equation (17)) and measured canopy resistance (r

_{c_m}, calculated by Equation (3)) above Dripsey grassland.

**Figure 10.**Comparisons between (

**a**) canopy temperature increase calculated by Todorovic’s expression (t

_{TOD}, Equation (6)) and the analytical solution (t

_{ana}, Equation (16)); (

**b**) pseudo resistance (r’

_{TOD}, calculated by Equation (17)) and measured canopy resistance (r

_{c_m}, calculated by Equation (3)) above Chi-Lan forest.

**Figure 11.**Comparisons between (

**a**) canopy temperature increase calculated by Todorovic’s expression (t

_{TOD}, Equation (6)) and the analytical solution (t

_{ana}, Equation (16)); (

**b**) pseudo resistance (r’

_{TOD}, calculated by Equation (17)) and measured canopy resistance (r

_{c_m}, calculated by Equation (3)) above Sitou forest.

Site | Dripsey Grassland | Chi-Lan Forest | Sitou Forest |
---|---|---|---|

Data period | 1 January 2013–31 December 2013 | 1 May 2005–30 April 2007 | 2 May 2009–31 July 2010 |

Altitude (m) | 200 | 1650 | 1252 |

Location | 51°59′ N, 8°46′ W | 24°35′ N, 121°30′ E | 23°39′ N, 120°47′ E |

Annual rainfall (mm) | 1222.7 | 4000 | 2635 |

Mean temperature (°C) | 9.6 | 13 | 16.6 |

Mean humidity (%) | 86 | 91 | 89 |

Canopy height (m) | 0.3 | 10.3 | 26 |

Average Canopy resistance (s m^{−1}) | 163.36 | 346.21 | 321.17 |

Measurement height (m) | |||

Eddy-covariance system | 5 | 24 | 28 |

Air temperature and humidity sensor | 2.5 | 23.6 | 27.5 |

Net radiometer | 4 | 22.5 | 27.5 |

Soil heat flux plate | −0.1 | −0.1 | −0.05 |

Soil temperature sensor | −0.015, −0.05, −0.075 | −0.05 | −0.05 |

Input Factors | Abbreviation | Definition |
---|---|---|

Time factor | JD | Julián day plus decimal time (converts 24 h in a day to a continuous value from 0 to 1) |

Meteorological factors | Q_{n} | available energy (W/m^{2}) (=R_{n}–G) |

T_{a} | air temperature (°C) | |

U | wind speed (m/s) | |

D | vapor pressure deficit (hPa) |

**Table 3.**Summary of linear regression between measured and model estimated canopy resistance (r

_{c}) and latent heat flux (LE) above Dripsey grassland. r

_{c_SVM}and r

_{c_TOD}denote the r

_{c}estimated by SVM and Todorovic’s method, respectively; LE

_{_SVM_rc}, LE

_{_TOD_rc}, and LE

_{_avg_rc}denote the estimated LE where the r

_{c}is from SVM, Todorovic’s method, and the average, respectively. RMSE: root mean square error.

r_{c} Model | Slope | Intercept | RMSE (s/m) | R^{2} | LE Model | Slope | Intercept | RMSE (W/m^{2}) | R^{2} |
---|---|---|---|---|---|---|---|---|---|

r_{c_SVM} | 1.72 | –79.19 | 165.49 | 0.22 | LE_{_SVM_rc} | 1.02 | 1.27 | 25.87 | 0.88 |

r_{c_TOD} | 0.85 | 132.93 | 210.14 | 0.03 | LE_{_TOD_rc} | 0.63 | 11.18 | 63.78 | 0.86 |

Average r_{c} = 163.36 (s/m) | LE_{_avg_rc} | 0.94 | 10.76 | 30.71 | 0.83 |

**Table 4.**Summary of linear regression between measured and model estimated canopy resistance (r

_{c}) and latent heat flux (LE) above Chi-Lan forest. r

_{c_SVM}and r

_{c_TOD}denote the r

_{c}estimated by SVM and Todorovic’s method, respectively; LE

_{_SVM_rc}, LE

_{_TOD_rc}, and LE

_{_avg_rc}denote the estimated LE where the r

_{c}is from SVM, Todorovic’s method, and the average, respectively. RMSE: root mean square error.

r_{c} Model | Slope | Intercept | RMSE (s/m) | R^{2} | LE Model | Slope | Intercept | RMSE (W/m^{2}) | R^{2} |
---|---|---|---|---|---|---|---|---|---|

r_{c_SVM} | 0.71 | 67.67 | 248.32 | 0.13 | LE_{_SVM_rc} | 1.03 | 19.90 | 57.39 | 0.46 |

r_{c_TOD} | –1.77 | 394.21 | 424.58 | 0.01 | LE_{_TOD_rc} | 0.20 | 55.96 | 233.61 | 0.21 |

Average r_{c} = 346.21 (s/m) | LE_{_avg_rc} | 0.91 | 28.10 | 59.13 | 0.41 |

**Table 5.**Summary of linear regression between measured and model estimated canopy resistance (r

_{c}) and latent heat flux (LE) above Sitou forest. r

_{c_SVM}and r

_{c_TOD}denote the r

_{c}estimated by SVM and Todorovic’s method, respectively; LE

_{_SVM_rc}, LE

_{_TOD_rc}, and LE

_{_avg_rc}denote the estimated LE where the r

_{c}is from SVM, Todorovic’s method, and the average, respectively. RMSE: root mean square error.

r_{c} Model | Slope | Intercept | RMSE (s/m) | R^{2} | LE Model | Slope | Intercept | RMSE (W/m^{2}) | R^{2} |
---|---|---|---|---|---|---|---|---|---|

r_{c_SVM} | 0.94 | 88.73 | 150.19 | 0.14 | LE_{_SVM_rc} | 1.07 | –5.51 | 33.74 | 0.64 |

r_{c_TOD} | 0.59 | 320.00 | 373.14 | 0.00 | LE_{_TOD_rc} | 0.22 | 25.92 | 218.53 | 0.50 |

Average r_{c} = 321.17 (s/m) | LE_{_avg_rc} | 0.86 | 16.54 | 37.33 | 0.58 |

**Table 6.**Summary of linear regression between measured and model estimated canopy resistance (r

_{c}) and latent heat flux (LE) above Dripsey grassland for three different ranges (0–100, 100–200, and >200) of measured canopy resistance (r

_{c_m}). r

_{c_SVM}and r

_{c_TOD}denote the r

_{c}estimated by SVM and Todorovic’s method, respectively; LE

_{_SVM_rc}and LE

_{_TOD_rc}the estimated LE where the r

_{c}is from SVM and Todorovic’s method, respectively. RMSE: root mean square error; Int.: intercept.

r_{c} Range | High: r_{c_m} = 0–100 (s/m) | Medium: r_{c_m} = 100–200 (s/m) | Low: r_{c_m} > 200 (s/m) | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

r_{c} model | slope | Int. | RMSE | R^{2} | slope | Int. | RMSE | R^{2} | slope | Int. | RMSE | R^{2} |

r_{c_SVM} | –0.07 | 63.09 | 85.04 | 0.01 | 0.12 | 125.59 | 49.34 | 0.03 | 0.82 | 233.70 | 276.83 | 0.04 |

r_{c_TOD} | –0.07 | 56.80 | 42.52 | 0.01 | 0.03 | 141.92 | 111.27 | 0.00 | 1.18 | 308.98 | 369.12 | 0.06 |

LE model | slope | Int. | RMSE | R^{2} | slope | Int. | RMSE | R^{2} | slope | Int. | RMSE | R^{2} |

LE_{_SVM_rc} | 1.25 | 2.94 | 27.25 | 0.96 | 1.09 | –5.43 | 21.59 | 0.94 | 0.87 | –6.60 | 27.66 | 0.92 |

LE_{_TOD_ rc} | 0.80 | 11.51 | 26.32 | 0.94 | 0.66 | 2.19 | 70.32 | 0.96 | 0.56 | –1.82 | 89.15 | 0.90 |

**Table 7.**Summary of linear regression between measured and model estimated canopy resistance (r

_{c}) and latent heat flux (LE) above Chi-Lan forest for three different ranges (0–100, 100–200, and >200) of measured canopy resistance (r

_{c_m}). r

_{c_SVM}and r

_{c_TOD}denote the r

_{c}estimated by SVM and Todorovic’s method, respectively; LE

_{_SVM_rc}and LE

_{_TOD_rc}the estimated LE where the r

_{c}is from SVM and Todorovic’s method, respectively. RMSE: root mean square error; Int.: intercept.

r_{c} Range | High: r_{c_m} = 0–100 (s/m) | Medium: r_{c_m} = 100–200 (s/m) | Low: r_{c_m} > 200 (s/m) | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

r_{c} model | slope | Int. | RMSE | R^{2} | slope | Int. | RMSE | R^{2} | slope | Int. | RMSE | R^{2} |

r_{c_SVM} | 0.03 | 38.65 | 261.90 | 0.01 | 0.03 | 141.47 | 205.17 | 0.01 | 0.36 | 342.02 | 255.00 | 0.05 |

r_{c_TOD} | –0.13 | 52.86 | 43.00 | 0.01 | –0.01 | 149.65 | 129.81 | 0.00 | –0.68 | 503.63 | 520.41 | 0.00 |

LE model | slope | Int. | RMSE | R^{2} | slope | Int. | RMSE | R^{2} | slope | Int. | RMSE | R^{2} |

LE_{_SVM_rc} | 1.79 | 17.37 | 91.26 | 0.64 | 1.38 | 12.25 | 60.79 | 0.76 | 0.92 | 3.43 | 32.87 | 0.65 |

LE_{_TOD_ rc} | 0.75 | 21.13 | 55.48 | 0.83 | 0.46 | 29.27 | 132.06 | 0.83 | 0.21 | 22.96 | 279.33 | 0.45 |

**Table 8.**Summary of linear regression between measured and model estimated canopy resistance (r

_{c}) and latent heat flux (LE) above Sitou forest for three different ranges (0–100, 100–200, and >200) of measured canopy resistance (r

_{c_m}). r

_{c_SVM}and r

_{c_TOD}denote the r

_{c}estimated by SVM and Todorovic’s method, respectively; LE

_{_SVM_rc}and LE

_{_TOD_rc}the estimated LE where the r

_{c}is from SVM and Todorovic’s method, respectively. RMSE: root mean square error; Int. intercept.

r_{c} Range | High: r_{c_m} = 0–100 (s/m) | Medium: r_{c_m} = 100–200 (s/m) | Low: r_{c_m} > 200 (s/m) | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

r_{c} model | slope | Int. | RMSE | R^{2} | slope | Int. | RMSE | R^{2} | slope | Int. | RMSE | R^{2} |

r_{c_SVM} | 0.05 | 36.69 | 135.27 | 0.02 | 0.03 | 148.35 | 104.97 | 0.01 | 0.18 | 370.46 | 241.76 | 0.01 |

r_{c_TOD} | –0.17 | 51.10 | 41.65 | 0.02 | –0.14 | 158.75 | 128.79 | 0.01 | 1.11 | 387.56 | 431.50 | 0.02 |

LE model | slope | Int. | RMSE | R^{2} | slope | Int. | RMSE | R^{2} | slope | Int. | RMSE | R^{2} |

LE_{_SVM_rc} | 1.87 | –10.39 | 50.62 | 0.89 | 1.75 | –28.25 | 42.43 | 0.88 | 1.05 | –15.70 | 26.15 | 0.77 |

LE_{_TOD_ rc} | 0.75 | 9.71 | 35.35 | 0.93 | 0.41 | 14.40 | 133.31 | 0.88 | 0.21 | 17.80 | 247.17 | 0.61 |

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Hsieh, C.-I.; Huang, I.-H.; Lu, C.-T.
Estimating Canopy Resistance Using Machine Learning and Analytical Approaches. *Water* **2023**, *15*, 3839.
https://doi.org/10.3390/w15213839

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Hsieh C-I, Huang I-H, Lu C-T.
Estimating Canopy Resistance Using Machine Learning and Analytical Approaches. *Water*. 2023; 15(21):3839.
https://doi.org/10.3390/w15213839

**Chicago/Turabian Style**

Hsieh, Cheng-I, I-Hang Huang, and Chun-Te Lu.
2023. "Estimating Canopy Resistance Using Machine Learning and Analytical Approaches" *Water* 15, no. 21: 3839.
https://doi.org/10.3390/w15213839