# Estimation of Latent Heat Flux Using a Non-Parametric Method

^{1}

^{2}

^{*}

## Abstract

**:**

_{s}). This method is relatively new and attractive for estimating evapotranspiration, especially for T

_{s}measurements from remote sensing. The purpose of this study is to investigate the limitations of this method and compare its performance with those of the Penman–Monteith (P–M) and Priestley–Taylor (P–T) equations. Field experiments were carried out to study the evapotranspiration rates and sensible heat fluxes above three different ecosystems: grassland, peat bog, and forest. The results show that above the grassland and peat bog, the evapotranspiration rates were close to the equilibrium evaporation. Though the forest is humid (average humidity ≈ 89%; annual precipitation ≈ 2600 mm), the evapotranspiration was only 69% of the equilibrium evaporation. In terms of model predictions, it was found that the P–M and P–T equations were able to predict the water vapor and sensible heat fluxes well (R

^{2}≈ 0.60–0.92; RMSE ≈ 30 W m

^{−2}) for all the three sites if the canopy resistance and the P–T constant of the ecosystem were given a priori. However, the N-P method only succeeded for the grassland and peat bog; it failed above the forest site (RMSE ≈ 60 W m

^{−2}). Our analyses and field measurements demonstrated that for the N-P method to be applicable, the actual evapotranspiration of the site should be within 0.89–1.05 times the equilibrium evaporation.

## 1. Introduction

## 2. Experiments

#### 2.1. Grassland

_{2}/H

_{2}O gas analyzer (Li-7500) and an ultrasonic anemometer (CSAT), was installed at 5 m above the ground. The sampling frequency and averaging period for this system were 10 Hz and 30 min, respectively.

_{n}), ground heat flux (G), air temperature (T

_{a}), surface temperature (T

_{s}), and relative humidity (RH). Soil thermometers were set up at 1.5, 5, and 7.5 cm below the ground, respectively. The measured frequency was one datum per minute, and the data was then averaged every 30 min. The site description and instrumentation are also summarized in Table 1. Additional experimental details can be found in Peichl et al. [20].

_{n}= H+LE+G). For the purpose of this study, only data that satisfy the energy closure condition within 90% were selected. In other words, the difference between R

_{n}and H+LE+G should be within ±10%. This data selection process was also applied to the other two sites (peat bog and forest). Figure 1 shows the scatter plots between R

_{n}and H, LE, G, and H+LE+G. From Figure 1, we notice that in this grassland site, about 40% of the energy was used for warming the air (Figure 1a); 48% was distributed to evapotranspiration (Figure 1b); less than 10% of the energy was transported to the ground (Figure 1d); and the overall energy closure rate was about 97% after data selection using the energy closure condition. These energy partition coefficients are also summarized in Table 2.

#### 2.2. Peat Bog

^{−3}), the porosity is 95%, and the depth of the peat layer varies from about 2 to 5 m [21]. In the southern part of the bog, there is a drainage river with a catchment area of 74 ha, of which about 85% of the surface is blanket peat, and the other 15% is grazing grassland and drained peat soils.

_{2}/H

_{2}O infrared gas analyzer (Li-7500) to measure water vapor and carbon dioxide concentrations. The above variables were collected at a frequency of 10 Hz and averaged every 30 min. Additionally, air temperature and relative humidity were measured at 3 m above the ground; net radiation was measured by a net radiometer (CNR1) installed at 2 m. In addition, soil heat flux was measured 10 cm below the ground, and a soil thermometer was installed at the same depth to measure soil temperature. These measurements were collected every 1 min and averaged every 30 min [24,25]. This study uses data collected throughout year 2013. The site and instrumentation features are also listed in Table 1. Further experimental details can be found in McVeigh et al. [25].

_{n}and H, LE, G, and H+LE+G above this peat bog site. As seen in Figure 2, about 43% of the net radiation was used for warming the air and about 34% of the energy was distributed to evapotranspiration; 21% of the energy was transported to the ground; and the overall energy closure rate was about 98% after data selection using the energy closure condition. Compared with the grassland site, much more energy (21% vs. 8%) was directed into the ground due to a shorter vegetation canopy height. The regression analyses of Figure 2 are also listed in Table 2.

#### 2.3. Forest

_{n}and H, LE, G, and H+LE+G above the forest site are shown in Figure 3. For this site, the energy partitions for H and LE were about 57% and 35%, respectively; only 4% of the energy was transported into the ground due to the high forest canopy; and the energy closure rate was 97%. These analyses for energy partitions above the forest site are also summarized in Table 2.

## 3. Methodology

#### 3.1. Penman–Monteith Equation

_{p}P/0.622L

_{v}) is the psychrometric constant (kPa K

^{−1}), C

_{p}is the specific heat of air (J kg

^{−1}K

^{−1}), P is the atmospheric pressure (kPa), L

_{v}is the latent heat of evaporation (J kg

^{−1}), Δ is the slope of the saturated vapor pressure (kPa K

^{−1}) calculated at T

_{a}, D is the vapor pressure deficit (kPa), r

_{c}is the canopy resistance (s m

^{−1}), and r

_{av}is the aerodynamic resistance (s m

^{−1}).

^{−1}), z is the measurement height, z

_{o}is the surface roughness for momentum (≈ 0.1 h, h is canopy height), and z

_{ov}is the surface roughness for water vapor (≈ 0.01 h,).

_{c}. Here, the r

_{c}value, which could result in a minimum RMSE (root mean square error) between the observed LE and the predicted LE by Equation (1), was the one adopted. The remaining 75% of the data was then used for evaluating the performance of the P–M equation with this r

_{c}value. The determined r

_{c}values for the grassland, peat bog, and forest sites were 60, 63, and 134 (s m

^{−1}), respectively. These r

_{c}values for the three sites are also listed in Table 1.

#### 3.2. Priestley–Taylor Equation

_{eq}) and can be calculated as

_{eq}. The α values for the grassland, peat bog, and forest sites were found to be 0.962, 0.956, and 0.692, respectively. These values indicate that the evapotranspiration at the grassland and peat bog sites are both close to the equilibrium evaporation, but the forest’s evapotranspiration is only around 70% of LE

_{eq}. This value is close to the α values (0.74 and 0.64) of the spruce forest and Scots Pine forest in dry conditions [28]. These α values for the three sites are also summarized in Table 1.

#### 3.3. Non-Parametric Method

^{−8}) is the Stefan–Boltzmann constant (W m

^{−2}K

^{−4}), T

_{s}is the surface temperature (K), T

_{a}is the air temperature (K), and ε is the land surface emissivity. In this study, ε for grassland, peat bog, and forest are 0.95, 0.99, and 0.98, respectively. The parameters required in the non-parametric method are R

_{n}, G, T

_{s}, and T

_{a,}which can be easily measured. No additional empirical or semi-empirical parameter is required.

_{eq}); the second term (II) accounts for the energy adjustment by the long-wave radiation difference between the surface and atmosphere; and the third term (III) represents the flux change caused by the ground heat flux and the temperature difference between the surface and the air. Generally speaking, during the daytime, term I is the major component contributing to LE and term III is minor More details about the derivation and the principle can be found in Liu et al. [1].

## 4. Results and Discussion

#### 4.1. Performances of P–M, P–T, and N-P Methods

#### 4.1.1. Grassland

^{2}between 0.90–0.92; RMSE around 30 W m

^{−2}) and H (R

^{2}between 0.87–0.89; RMSE around 30 W m

^{−2}) accurately above the grassland site.

_{n}− G) at the grassland, as shown in Figure 5. For the cases of low available energy (R

_{n}− G < 300), all three models resulted in similar and slightly underestimated LE values. This indicated that the actual evapotranspiration rate was slightly higher than the equilibrium evaporation rate when the available energy was small. This extra evapotranspiration was due to the demand of unsaturated atmosphere.

#### 4.1.2. Peat Bog

^{2}between 0.82–0.89); similar results are also found for H (R

^{2}between 0.79–0.90). The RMSEs for LE (between 29–33 W m

^{−2}) and H (between 27–31 W m

^{−2}) for each of the models are approximately the same as those at the grassland site. Overall, these three models performed well above the peat bog site.

_{n}− G) at the peat bog (Figure 7). From Figure 7 it appears for the cases of high available energy (R

_{n}− G > 300), all three of the models slightly overpredicted LE. This shows that the actual evapotranspiration rate was slightly lower than the equilibrium evaporation rate when the available energy was large. This reduction of evapotranspiration might have been caused by resistance from the vegetation in order to reduce water loss during higher air temperature and radiation.

#### 4.1.3. Forest

^{−2}) above the forest site. However, the LE and H predicted by the N-P method were not in agreement with the measurements (both RMSEs were around 65 W m

^{−2}).

_{n}-G) at the forest site. In Figure 9, it appears that in the cases of high available energy (R

_{n}− G > 400), the N-P method overpredicted LE and underestimated H. This reveals that the actual evapotranspiration rate was much lower than the equilibrium evaporation rate when the available energy was large. Similar to the peat bog site (although the magnitude was larger), this reduction of evapotranspiration might have been be caused by resistance from the vegetation in order to avoid a large amount of water loss (physiological control) during high air temperature and radiation.

#### 4.2. Limitation of the Non-Parametric Method

_{n}-G for latent heat and sensible heat fluxes above the grassland, peat bog, and forest. For all three sites, term III is very small and close to zero. Hence, term III can be neglected.

_{eq}in Equation (6), and neglecting term III, we have

_{eq}with LE/α in Equation (6), we get

_{eq}along with the measured LE for the three sites. In Figure 11, the slopes for the grassland and peat bog sites are positive, but those for the forest site are negative. This is because when R

_{n}for the forest site is large, most of the energy (57%, see Table 2) is distributed to H for warming up the air; hence, the difference between T

_{s}and T

_{a}is reduced which results in a smaller term II. Figure 11 also reveals that term II is about ±10% (from −0.051 to 0.083) of the equilibrium evapotranspiration (Figure 11a,c,e).

#### 4.3. Sensitivity of the N-P Method on Surface Temperature

_{c}and α) are not needed. However, compared with the P–M and P–T equations, this method requires one additional input, T

_{s}. Hence, it is worth noting the sensitivity of the N-P method to the change of T

_{s}.

_{s}, we have

_{s}. From Equation (10), it is clear that ∂LE/∂T

_{s}is a function of T

_{s}and G. From field measurements of the three sites, T

_{s}is in the range of −5 to 30 °C, and G is between −20 and 100 (W m

^{−2}). From these conditions, with ε = 0.95, Figure 12 plots ∂LE/∂T

_{s}and ∂H/∂T

_{s}as a function of T

_{s}under different G values. Figure 12 reveals that with a one-degree change in T

_{s}, the predicted LE and H are only changed by 4 to 6 (W m

^{−2}). In other words, the LE predicted by the N-P method is not sensitive to the uncertainty of T

_{s}measurements.

## 5. Conclusions

- (1)
- The evapotranspiration rates above the grassland and peat bog were close to the equilibrium evaporation (P–T constant ≈ 0.96). The forest’s evapotranspiration rate was 69% of the equilibrium evaporation, and about 60% of the net radiation energy was distributed to sensible heat flux.
- (2)
- Both the P–M and P–T equations performed well at estimating water vapor and sensible heat fluxes for all of the three sites. However, the canopy resistance in the P–M equation and the Priestley–Taylor constant in the P–T equation must be known a priori.
- (3)
- The water vapor flux predictions by the N-P method were in agreement with the measurements above the grassland and peat bog. However, this was not the case for the forest site.
- (4)
- Our analysis shows that with one degree of change in T
_{s}, the predicted LE is only changed by 4 to 6 (W m^{−2}). Hence, the LE predictions by the N-P method are not sensitive to the uncertainty of T_{s}measurements. - (5)
- Field measurements from the three sites reveal that the second term of the N-P method is about ±10% (from −0.05 to 0.08) of the equilibrium evapotranspiration. For applying the N-P method to estimate LE, the actual evapotranspiration of the site should be around 0.89–1.05 times the equilibrium evapotranspiration.

_{a}= T

_{s}), the actual evaporation rate is the equilibrium evaporation, and that when the surface temperature is increased (i.e., T

_{s}> T

_{a}), the actual evaporation rate will be dropped since some energy is used for sensible heat flux. Our field measurements show that this energy adjustment is small (around 10% of LE

_{eq}). Hence, the N-P method is only suitable for a system where the actual evaporation rate is close to the equilibrium evaporation (difference within ±10%). For a system (such as the forest) where the actual LE is much less than LE

_{eq}, this method will not work.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

Symbol | Description |

C_{p} | specific heat of air (J kg^{−1} K^{−1}) |

D | vapor pressure deficit (kPa) |

G | ground heat flux (W m^{−2}) |

H | sensible heat flux (W m^{−2}) |

k | von Karman constant (= 0.4) |

LE | latent heat flux (W m^{−2}) |

L_{v} | latent heat of evaporation (J kg^{−1}) |

P | atmospheric pressure (kPa) |

R_{n} | net radiation (W m^{−2}) |

r_{c} | canopy resistance (s m^{−1}) |

r_{av} | aerodynamic resistance (s m^{−1}) |

T_{a} | air temperature (°C) |

T_{s} | surface temperature (°C) |

U | wind speed (m s^{−1}) |

z | measurement height (m) |

z_{o} | surface roughness for momentum (m) |

z_{ov} | surface roughness for water vapor (m) |

α | Priestly–Taylor constant |

γ | psychrometric constant (kPa K^{−1}) |

Δ | slope of the saturated vapor pressure (kPa K^{−1}) |

ε | land surface emissivity |

σ | Stefan–Boltzmann constant (= 5.67 × 10^{−8}) (W m^{−2} K^{−4}) |

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**Figure 4.**Comparisons of measured and predicted LE and H by the P–M equation (

**a**,

**b**), P–T equation (

**c**,

**d**), and N-P method (

**e**,

**f**) above the grassland. The 1:1 line is also shown.

**Figure 5.**Scatter plots of measured and model-predicted LE (

**a**) and H (

**b**) as a function of available energy (R

_{n}− G) above the grassland.

**Figure 6.**Comparisons of measured and predicted LE and H by the P–M equation (

**a**,

**b**), P–T equation (

**c**,

**d**), and non-parametric method (

**e**,

**f**) above the peat bog. The 1:1 line is also shown.

**Figure 7.**Scatter plots of measured and model-predicted LE (

**a**) and H (

**b**) as a function of available energy (R

_{n}− G) above the peat bog.

**Figure 8.**Comparisons of measured and predicted LE and H by the P–M equation (

**a**,

**b**), P–T equation (

**c**,

**d**), and non-parametric method (

**e**,

**f**) above the forest. The 1:1 line is also shown.

**Figure 9.**Scatter plots of measured and model-predicted LE (

**a**) and H (

**b**) as a function of available energy (R

_{n}− G) above the forest.

**Figure 10.**Scatter plots of the three terms in the non-parametric method as a function of R

_{n}− G for LE and H above the grassland (

**a**,

**b**), peat bog (

**c**,

**d**), and forest (

**e**,

**f**).

**Figure 11.**Scatter plots of term II as a function of LE

_{eq}and measured LE above the grassland (

**a**,

**b**), peat bog (

**c**,

**d**), and forest (

**e**,

**f**).

Site | Grassland | Peat Bog | Forest |
---|---|---|---|

Data period | 1 January 2013–31 December 2013 | 1 January 2013–31 December 2013 | 22 May 2009–31 July 2010 |

Altitude (m) | 195 | 150 | 1252 |

Location | 59°59′ N, 8°45′ W | 51°55′ N, 9°55′ W | 23°39′51.09″ N, 120°47′44.57″ E |

Climate type | Temperate | Temperate | Sub-tropical |

Annual rainfall (mm) | 1161 | 1834 | 2635 |

Mean temperature (°C) | 9 | 10.1 | 16.6 |

Mean humidity (%) | 92 | 82 | 89 |

Canopy height (m) | 0.3 | 0.1 | 26 |

Surface emissivity | 0.95 | 0.99 | 0.98 |

Canopy resistance (s m^{−1}) | 60 | 63 | 134 |

Priestley–Taylor constant | 0.962 | 0.956 | 0.692 |

Measurement height (m) | |||

Eddy covariance system | 5 | 3 | 28 |

Air temperature & humidity sensor | 5 | 3 | 28 |

Net radiometer | 4 | 2 | 27.5 |

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

Soil thermometer | −0.015, −0.025, −0.075 | −0.1 | −0.05, −0.1 |

Soil moisture meter | −0.05 | −0.1 | −0.05, −0.1 |

**Table 2.**Summary of linear regression between R

_{n}and H, LE, G, and H+LE+G for the grassland, peat bog, and forest sites. Y = aX + b; X is R

_{n}; Y is H, LE, and so on.

Grassland | Peat Bog | Forest | |||||||
---|---|---|---|---|---|---|---|---|---|

Flux | a | b | R^{2} | a | b | R^{2} | a | b | R^{2} |

H | 0.404 | −17.439 | 0.897 | 0.426 | −14.819 | 0.913 | 0.574 | −14.495 | 0.893 |

LE | 0.481 | 23.329 | 0.907 | 0.342 | 22.239 | 0.876 | 0.347 | 18.439 | 0.733 |

G | 0.081 | −5.916 | 0.590 | 0.210 | −6.948 | 0.808 | 0.044 | −3.459 | 0.474 |

H+LE+G | 0.966 | −0.026 | 0.997 | 0.977 | 0.472 | 0.997 | 0.965 | 0.485 | 0.997 |

**Table 3.**Summary of regression analysis between measured and model-predicted fluxes at the three sites. Y = aX + b; Y is measured flux; X is predicted flux.

Latent Heat Flux | Sensible Heat Flux | ||||||||
---|---|---|---|---|---|---|---|---|---|

Site | Model | a | b | R^{2} | RMSE | a | b | R^{2} | RMSE |

Grassland | P–M | 0.930 | 18.221 | 0.895 | 28.338 | 0.874 | −13.689 | 0.869 | 29.310 |

P–T | 0.884 | 21.336 | 0.922 | 27.362 | 1.039 | −19.791 | 0.894 | 27.251 | |

N-P | 0.906 | 23.484 | 0.913 | 29.866 | 0.974 | −21.189 | 0.882 | 30.447 | |

Peat Bog | P–M | 0.709 | 9.729 | 0.874 | 32.507 | 1.156 | 7.505 | 0.852 | 29.781 |

P–T | 0.691 | 20.482 | 0.888 | 29.047 | 1.347 | −16.744 | 0.896 | 26.967 | |

N-P | 0.675 | 18.608 | 0.826 | 33.161 | 1.133 | −1.661 | 0.797 | 31.063 | |

Forest | P–M | 0.950 | −1.269 | 0.603 | 28.799 | 0.864 | 3.026 | 0.883 | 29.575 |

P–T | 0.687 | 18.383 | 0.643 | 33.945 | 1.185 | −17.503 | 0.875 | 33.004 | |

N-P | 0.433 | 35.156 | 0.601 | 66.553 | 1.823 | −81.167 | 0.734 | 64.657 |

Site | Grassland | Peat Bog | Forest |
---|---|---|---|

α from Equation (7) | 0.929 | 0.917 | 1.051 |

α from Equation (8) | 0.928 | 0.890 | 1.037 |

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**MDPI and ACS Style**

Hsieh, C.-I.; Chiu, C.-J.; Huang, I.-H.; Kiely, G. Estimation of Latent Heat Flux Using a Non-Parametric Method. *Water* **2022**, *14*, 3474.
https://doi.org/10.3390/w14213474

**AMA Style**

Hsieh C-I, Chiu C-J, Huang I-H, Kiely G. Estimation of Latent Heat Flux Using a Non-Parametric Method. *Water*. 2022; 14(21):3474.
https://doi.org/10.3390/w14213474

**Chicago/Turabian Style**

Hsieh, Cheng-I, Cheng-Jiun Chiu, I-Hang Huang, and Gerard Kiely. 2022. "Estimation of Latent Heat Flux Using a Non-Parametric Method" *Water* 14, no. 21: 3474.
https://doi.org/10.3390/w14213474