# Feasibility of Estimating Turbulent Heat Fluxes via Variational Assimilation of Reference-Level Air Temperature and Specific Humidity Observations

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

_{HN}) and evaporative fraction (EF). EF estimates were found to agree well with observations in terms of magnitude and day-to-day fluctuations in wet/densely vegetated sites but degraded in dry/sparsely vegetated sites. Similarly, in wet/densely vegetated sites, the variations in the C

_{HN}estimates were found to be consistent with those of the leaf area index (LAI) while this consistency deteriorated in dry/sparely vegetated sites. The root mean square errors (RMSEs) of daily H and LE estimates at the Arou site (wet) were 25.43 (Wm

^{−2}) and 55.81 (Wm

^{−2}), which are respectively 57.6% and 45.4% smaller than those of 60.00 (Wm

^{−2}) and 102.21 (Wm

^{−2}) at the Desert site (dry). Overall, the results show that the VDA system performs well at wet/densely vegetated sites (e.g., Arou and Willow Creek), but its performance degrades at dry/slightly vegetated sites (e.g., Desert and Audubon). These outcomes show that the sequences of reference-level air temperature and specific humidity have more information on the partitioning of available energy between the sensible and latent heat fluxes in wet/densely vegetated sites than dry/slightly vegetated sites.

## 1. Introduction

_{HN}) (that scales the sum of H and LE) and evaporative fraction (EF) (that scales the partitioning of available energy between H and LE). Tajfar et al. [81] tested their VDA approach only at a grass-dominated sub-humid site in Kansas, and showed that sequences of the reference-level air temperature and specific humidity have implicit information for constraining C

_{HN}and EF, and retrieving turbulent heat fluxes.

_{HN}and EF, and consequently the turbulent heat fluxes estimates from the VDA approach, will be close to their true values. If these measurements do not have enough information, the C

_{HN}, EF, H, and LE estimates will be poor.

## 2. Materials and Methods

#### 2.1. Sensible and Latent Heat Fluxes

^{10Ri}). C

_{HN}is the first unknown of the VDA approach, which depends on the characteristics of the landscape, and is assumed to be constant during each month [49,50,53,57]. C

_{HN}is mainly a function of LAI and to a lesser extent wind speed, friction velocity, solar elevation, and the structure and shape of vegetation (i.e., crown density and vertical distribution of foliage elements) [50,84,85,86,87,88].

#### 2.2. Atmospheric Boundary Layer (ABL) Model

#### 2.2.1. Energy and Moisture Budget Equations

#### 2.2.2. Radiative Fluxes

#### 2.2.3. Mixed-Layer Height

_{v}is the virtual heat flux at the surface; E is the rate of evaporation from ground; and ${\delta}_{\theta}$ is the potential temperature inversion strength at the top of the mixed layer.

#### 2.2.4. Inversion Strengths of $\theta $ and $q$

#### 2.2.5. Entrainment Fluxes

#### 2.3. Variational Data Assimilation (VDA) Approach

_{HN}and EF) are obtained by minimizing the objective function J. Two different integral time scales are used in the objective function J. The first one covers the entire assimilation period in which C

_{HN}is assumed to be constant (N = 30 days). The second time scale constitutes the assimilation window in which EF is presumed to be constant [t

_{0},t

_{1}] = [09:00–16:00 LT]. The objective function J is expressed as:

_{HN}strictly positive, it is transformed to R via C

_{HN}= e

^{R}. The third and fourth terms are the quadratic errors of the unknown parameters (i.e., R and EF) with respect to their prior values (${R}^{\prime}$ and $E{F}^{\prime}$). The last two terms are the physical constraints adjoined to the model through the Lagrange multipliers ${\lambda}_{1}$ and ${\lambda}_{2}$. ${R}_{\theta}^{-1}$ and ${R}_{q}^{-1}$ are the inverse error covariance matrices of $\theta $ and $q$, respectively. ${B}_{R}^{-1}$ and ${B}_{EF}^{-1}$ are the inverse background error covariance matrices of $R$ and $EF$, respectively. Following Tajfar et al. [81,83], the diagonal elements of ${R}_{\theta}^{-1}$, ${R}_{q}^{-1}$, ${B}_{R}^{-1}$, and ${B}_{EF}^{-1}$ are set to 10

^{−1}K

^{−2}, 10

^{5}(kg/kg)

^{−2}, 10

^{8}, and 10

^{9}, respectively.

## 3. Study Sites

**,**and ${\delta}_{q}$, and the magnitudes of ${\gamma}_{\theta}$ and ${\gamma}_{q}$ were varied from 100 to 500 m, 2 to 6 K, −4.8 $\times $ 10

^{−3}to −0.5 $\times $ 10

^{−3}kg kg

^{−1}, 2 to 8 K km

^{−1}, and −7 $\times $ 10

^{−3}to −0.5 $\times $ 10

^{−3}kg kg

^{−1}km

^{−1}with the increment of 100 m, 0.4 K, 0.4 $\times $ 10

^{−3}kg kg

^{−1}, 0.5 K km

^{−1}, and 0.5 $\times $ 10

^{−3}kg kg

^{−1}km

^{−1}, respectively. For each study site, the magnitudes of $h\left(t={t}_{o}\right)$, ${\delta}_{\theta}\left(t={t}_{o}\right)$, and ${\delta}_{q}\left(t={t}_{o}\right)$, ${\gamma}_{\theta}$, and ${\gamma}_{q}$ that lead to a minimum cost function (J) are reported in Table 2.

## 4. Results and Discussions

_{HN}and EF are the two key unknowns of the VDA system. Table 3 shows the estimated C

_{HN}values from the VDA approach and corresponding LAI values at the six study sites. C

_{HN}is mainly affected by vegetation phenology [44,46,47,50,52,55]. As shown in Table 3, the variations in C

_{HN}estimates are consistent with those of LAI in the wet and/or densely vegetated sites. For instance, at the Willow Creek site, C

_{HN}and LAI decrease slightly during the modeling periods. At the Arou and Brookings sites, C

_{HN}reaches its highest value in the second monthly period and decreases in the third month. A similar trend is observed in LAI. At the Bondville site, the LAI and C

_{HN}estimates remain almost constant during the modeling periods (DOYs 182–271).

_{HN}retrievals and LAI weakens in dry/sparsely vegetated sites. At Audubon, LAI values increase continuously during the modeling periods. However, C

_{HN}estimates decrease slightly in the second monthly period. At the Desert site, LAI is invariant, while C

_{HN}shows slight variations in different assimilation periods. Among the study sites, Desert has the lowest C

_{HN}estimates because it has no vegetation (LAI = 0), while Willow Creek has the highest C

_{HN}values due to its dense vegetation cover (LAI = 5.67). In general, the study sites with higher LAI values (e.g., Arou and Willow Creek) have higher C

_{HN}estimates compared to the sites with lower LAI (e.g., Desert and Audubon).

_{HN}estimates agree well with the vegetation phenology at wet and/or densely vegetated sites, although no information on vegetation density is used in the VDA approach. These findings indicate that the VDA approach can extract the implicit information contained in the air temperature and specific humidity measurements to estimate C

_{HN}at wet and/or densely vegetated sites, but its performance degrades in dry and/or slightly vegetated sites.

^{−2}and LE retrievals are far from the 45-degree line These outcomes show that the information content of the atmospheric state variables (e.g., air temperature and specific humidity) for estimating the turbulent heat fluxes significantly reduces in dry/sparsely vegetated sites (e.g., Audubon and Desert). H and LE estimates at Bondville and Brookings are comparable, and agree fairly well with the observations. Overall, the results indicate that the sequences of air temperature and specific humidity have a significant amount of information for partitioning the available energy between turbulent heat fluxes at sites with high SM and/or LAI values, but their information content significantly reduces at sites with low SM and/or LAI.

_{HN}estimates (see Equation (1)) but errors in LE retrievals stem from uncertainties in both H and EF estimates (see Equation (3)). More sources of errors increase the scattering of LE estimates [44,81,83].

_{HN}, constant daily EF, insignificant advection, and convectively well mixed boundary layer, which results in constant profiles of the potential temperature and specific humidity with height.

_{HN}and EF are not updated. In this approach, the ABL potential temperature ($\theta $) and specific humidity ($q$) are estimated by integrating Equations (4) and (5) forward in time using the initial guesses of C

_{HN}and EF. The large discrepancy between the turbulent heat fluxes estimates from VDA and the open loop at the wet and/or highly vegetated sites shows that the performance of VDA significantly improves at these sites by using the information content of atmospheric state variables.

^{−2}(113.46 Wm

^{−2}) and 45.14 Wm

^{−2}(137.52 Wm

^{−2}), respectively. The corresponding MAE and RMSE values for daily LE estimates are 59.09 Wm

^{−2}(160.88 Wm

^{−2}) and 80.37 Wm

^{−2}(182.74 Wm

^{−2}), respectively. By assimilating ${T}_{a}$ and ${q}_{a}$, on average, the MAE and RMSE of the daily H (LE) estimates from VDA are reduced by 69.4% (63.3%) and 67.2% (56%) compared to those of the open loop. By the assimilation of air temperature and specific humidity, the MAE and RMSE of the daily H (LE) estimates from the open loop is reduced by 89.5% (81%) and 87.1% (76.5%) at Arou, and 80.9% (68.3%) and 81.3% (62.8%) at Willow Creek. The corresponding values at Desert and Audubon are 25.3% (23.8%) and 25.9% (22.9%), and 38.2% (42.9%) and 36.7% (38.4%), respectively. These results show that the assimilation of ${T}_{a}$ and ${q}_{a}$ leads to a higher improvement in the H and LE estimates at wet/densely vegetated sites.

_{HN}and EF estimates by minimizing the difference between the ABL potential temperature and specific humidity estimates from Equations (4) and (5) (i.e., $\theta $ and $q$), and the corresponding values obtained from the reference-level air temperature and specific humidity via the MOST (i.e., ${\theta}_{SL}$ and ${q}_{SL}$) (Appendix B). Thus, a close agreement between the $\theta $ and ${\theta}_{SL}$, and $q$ and ${q}_{SL}$ (e.g., Arou) shows that the VDA approach can successfully update the initial guesses of C

_{HN}and EF and converges to their optimal values. On the other hand, a significant misfit between the ${\theta}_{SL}$ and $\theta $, and ${q}_{SL}$ and $q$ (e.g., Desert) implies that the VDA cannot effectively improve the initial guesses of C

_{HN}and EF and converges to the inaccurate C

_{HN}and EF values. Figure 9 shows half-hourly $\theta $ estimates from VDA versus ${\theta}_{SL}$ values. Similarly, Figure 10 indicates half-hourly $q$ estimates versus ${q}_{SL}$ values. As shown, at Arou and Willow Creek, the $\theta $ and $q$ estimates from VDA agree well with ${\theta}_{SL}$ and ${q}_{SL}$, respectively. This shows that in wet and/or densely vegetated sites, the VDA approach can take advantage of the significant amount of information in the sequences of air temperature and humidity to optimize C

_{HN}and EF, and consequently minimize the difference between $\theta $ and ${\theta}_{SL}$, and $q$ and ${q}_{SL}$. At the Brookings and Bondville sites, the $\theta $ and $q$ estimates are in fairly good agreement with ${\theta}_{SL}$ and ${q}_{SL}$, respectively, implying that the time series of the atmospheric state variable have some information to constrain C

_{HN}and EF, and retrieve turbulent heat fluxes. At the Desert and Audubon sites, $\theta $ and $q$ estimates are more scattered around the 1:1 line, showing the lack of sufficient information in the sequence of air temperature and humidity to accurately tune C

_{HN}and EF.

^{−1}), 2.52 K (0.0015 kg kg

^{−1}), 1.36 K (0.0012 kg kg

^{−1}), 1.99 K (0.0012 kg kg

^{−1}), 1.04 K (0.0011 kg kg

^{−1}), and 0.95 K (0.0009 kg kg

^{−1}), respectively. As anticipated, the RMSEs of the θ and $q$ estimates decrease as LAI and/or SM increase. Arou (with the highest SM) has the lowest RMSEs for θ and $q$. In contrast, Desert (with the lowest LAI and SM) has the highest RMSEs. As mentioned earlier, in dry sites (e.g., Desert and Audubon), evaporation is mainly controlled by the land surface state variable (i.e., LST). Hence, assimilating sequences of air temperature and specific humidity cannot robustly constrain C

_{HN}and EF, leading to larger errors in the θ and $q$ estimates. Bondville and Brookings are neither as sparsely vegetated as Desert and Audubon nor as densely vegetated as Willow Creek and Arou. The RMSEs of the θ and $q$ estimates at these sites are smaller than those of Desert and Audubon but larger than those of Willow Creek and Arou.

## 5. Conclusions

_{HN}(that scales the sum of H and LE), and evaporative fraction, EF (that represents the partitioning between H and LE).

_{HN}estimates. Additionally, at wet and/or densely vegetated sites, the variations in C

_{HN}estimates were consistent with those of LAI. This consistency weakened in dry and/or sparsely vegetated sites. Similarly, the EF estimates agreed well with the observations in terms of the magnitude and day-to-day dynamics in wet and/or densely vegetated sites (e.g., Arou and Willow Creek), but this agreement degraded in dry and/or sparsely vegetated sites (e.g., Desert and Audubon). The RMSE of EF estimates at Desert (dry barren land) was 0.198, which is 73.2% higher than that of 0.053 at Arou (wet grassland). These results show that the sequences of air temperature and specific humidity have a significant amount of information on the partitioning of available energy between H and LE in wet and/or densely vegetated sites. This information content decreases by the reduction of soil moisture and/or LAI.

^{−2}(55.81 W m

^{−2}), 37.03 W m

^{−2}(74.46 W m

^{−2}), 45.01 W m

^{−2}(82.28 W m

^{−2}), 45.21 W m

^{−2}(77.88 W m

^{−2}), 58.13 W m

^{−2}(89.57 W m

^{−2}), and 60.00 W m

^{−2}(102.21 W m

^{−2}), respectively. The RMSEs of the daily H and LE estimates increased as the site became drier and/or sparser in vegetation density. This is due to the fact that in dry and/or sparsely vegetated sites (e.g., Desert and Audubon), evapotranspiration is mainly controlled by the land surface state variable (i.e., land surface temperature) rather than the atmospheric state variables (i.e., reference-level air temperature and specific humidity). Hence, the coupling between EF and the atmospheric state variables weakens and the estimation of EF from the sequences of air temperature and specific humidity becomes uncertain. In contrast, at wet and/or densely vegetated sites (e.g., Arou and Willow Creek), the evaporative demand is primarily controlled by atmospheric state variables and the VDA system can extract that information to estimate H and LE.

^{−1}. The low RMSEs of the $\theta $ and $q$ estimates at wet and/or densely vegetated sites (e.g., Arou and Willow Creek) indicated that the VDA approach can effectively update the two key unknowns (i.e., C

_{HN}and EF) and obtain their optimal values. In contrast, at dry and/or sparsely vegetated sites (e.g., Desert and Audubon), the VDA system cannot effectively update C

_{HN}and EF, leading to high RMSEs for $\theta $ and $q$. The highest and lowest RMSEs of θ and $q$ estimates occurred at Desert and Arou, respectively. The RMSEs of θ and $q$ estimates at the Bondville and Brookings sites fell within those of dry/sparsely vegetated and wet/densely vegetated sites.

_{HN}as a function of LAI.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. List of symbols

B | Stanton number | [-] |

${B}_{EF}^{-1}$ | inverse background error covariance of $EF$ | [-] |

${B}_{R}^{-1}$ | inverse background error covariance of $R$ | [-] |

C_{HN} | neutral bulk heat transfer coefficient | [-] |

c_{p} | specific heat capacity of dry air | [J kg^{−1} K^{−1}] |

${D}_{1},\text{}{D}_{2}$ | dissipation of mechanical turbulent energy | [m^{3} s^{−3}] |

d | zero-plane displacement height | [m] |

E | evaporative rate from ground | [kg m^{−2} s^{−1}] |

EF | evaporative fraction | [-] |

$f$ | atmospheric stability correction function | [-] |

$G$ | ground heat flux | [W m^{−2}] |

${G}_{*}$ | production of mechanical turbulent energy | [m^{3} s^{−3}] |

g | gravitational acceleration | [m s^{−2}] |

$H$ | sensible heat flux | [W m^{−2}] |

${H}_{top}$ | entrainment sensible heat flux | [W m^{−2}] |

${H}_{v}$ | virtual heat flux | [W m^{−2}] |

$h$ | mixed-layer height | [m] |

J | objective functional | [-] |

$k$ | von Karman’s constant | [-] |

$\kappa $ | empirical constant | [kg m^{−2}]^{−1/7} |

L | Monin-Obhukov length | [m] |

${L}_{v}$ | latent heat of vaporization | [J kg^{−1}] |

$LE$ | latent heat flux | [W m^{−2}] |

$L{E}_{top}$ | entrainment latent heat flux | [W m^{−2}] |

LAI | leaf area index | [m^{2} m^{−2}] |

m | constant | [-] |

$N$ | number of days in the assimilation period | |

${P}_{h}$ | pressure at height h | [Pa] |

${P}_{s}$ | surface pressure | [Pa] |

$q$ | mixed layer specific humidity (equation 5b) | [kg kg^{−1}] |

${q}_{a}$ | specific humidity at the reference-level | [kg kg^{−1}] |

${q}_{h}$ | specific humidity immediately above mixed layer | [kg kg^{−1}] |

${q}_{SL}$ | specific humidity at the bottom of mixed layer (equation B2) | [kg kg^{−1}] |

R | transformation variable | [-] |

${R}_{Ad}$ | downwelling longwave radiation from within the mixed layer | [W m^{−2}] |

${R}_{Au}$ | upwelling longwave radiation from within the mixed layer | [W m^{−2}] |

${R}_{ad}$ | downwelling longwave radiation from above the mixed layer | [W m^{−2}] |

${R}_{d}$ | gas constant for dry air | [J kg K^{−1}] |

${R}_{gu}$ | upwelling longwave radiation from ground into the mixed layer | [W m^{−2}] |

$Ri$ | Richardson number | [-] |

${R}_{n}$ | net radiation at the surface | [W m^{−2}] |

${R}_{s}^{\downarrow}$ | incoming solar radiation | [W m^{−2}] |

${R}_{v}$ | gas constant for water vapor | [J kg K^{−1}] |

${R}_{q}^{-1}$ | inverse error covariance of $q$ | [-] |

${R}_{\theta}^{-1}$ | inverse error covariance of $\theta $ | [K^{−2}] |

$SL$ | surface layer | [-] |

$T$ | land surface temperature | [K] |

${T}_{a}$ | reference-level air temperature | [K] |

${T}_{h}$ | air temperature immediately above the mixed layer | [K] |

$t$ | time | [s] |

$U$ | wind speed at the reference-level | [m s^{−1}] |

${u}_{SL}$ | wind speed at the top of the surface layer | [m s^{−1}] |

${u}_{*}$ | friction velocity | [m s^{−1}] |

${z}_{0h}$ | roughness length scales for heat | [m] |

${z}_{0m}$ | roughness length scales for momentum | [m] |

${z}_{SL}$ | surface-layer height | [m] |

${z}_{ref}$ | reference-level height | [m] |

${z}_{veg}$ | vegetation height | [m] |

$\alpha $ | surface albedo | [-] |

${\delta}_{q}$ | specific humidity inversion strength | [kg kg^{−1}] |

${\delta}_{\theta}$ | potential temperature inversion strength | [K] |

${\epsilon}_{a}$ | atmospheric emissivity | [-] |

${\epsilon}_{ad}$ | effective emissivity above the mixed-layer | [-] |

${\epsilon}_{d}$ | effective mixed-layer downward emissivity | [-] |

${\epsilon}_{u}$ | effective mixed-layer upward emissivity | [-] |

${\epsilon}_{m}$ | mixed-layer bulk emissivity | [-] |

${\epsilon}_{s}$ | surface emissivity | [-] |

${\gamma}_{q}$ | lapse rate of $q$ above the mixed layer | [kg kg^{−1} m^{−1}] |

${\gamma}_{\theta}$ | lapse rate of $\theta $ above the mixed layer | [K m^{−1}] |

${\lambda}_{1},\text{}{\lambda}_{2}$ | lagrange multipliers | [-] |

${\mathsf{\Psi}}_{\mathrm{h}}$ | stability function for heat | [-] |

${\mathsf{\Psi}}_{\mathrm{m}}$ | stability function for momentum | [-] |

${\mathsf{\Psi}}_{\mathrm{q}}$ | stability function for water vapor | [-] |

$\rho $ | air density | [kg m^{−3}] |

$\sigma $ | Stefan-Boltzmann constant | [W m^{−2} K^{−4}] |

$\theta $ | Mixed layer potential temperature (equation 5a) | [K] |

${\theta}_{a}$ | reference-level potential temperature | [K] |

${\theta}_{SL}$ | potential temperature at the bottom of mixed layer (equation B1) | [K] |

$\phi $ | mechanical turbulence dissipation parameter | [-] |

$\xi $ | stability parameter | [-] |

## Appendix B. Monin–Obukhov Similarity Theory (MOST)

^{−1}°C

^{−1}), ${R}_{v}$ is the gas constant for water vapor (461 J kg

^{−1 o}C

^{−1}), g is the gravitational acceleration, $\rho $ is the air density, and ${c}_{p}$ is the specific heat capacity of air. Sensible (H) and latent heat fluxes (LE) are obtained respectively from Equations (1) and (3). ${u}_{*}$ is the friction velocity that is related to the wind speed measurements at the reference level ($U)$ via [77]:

- Guess a reasonable value for ${u}_{*}$.
- Substitute ${u}_{*}$ from step 1 in B6 to estimate L.
- Estimate $k{B}^{-1}$ from B10.
- Substitute the ${C}_{HN}$ estimate from the VDA approach and obtained $k{B}^{-1}$ from step 3 in B8 to find ${z}_{0h}.$
- Substitute ${z}_{0h}$ from step 4 in B9 to find ${z}_{0m}$.
- Substitute ${z}_{0m}$ from step 5 and L from step 2 in B7 to find ${u}_{*}$.
- Repeat steps 2–6 until the algorithm converges (i.e., the difference between ${u}_{*}$ estimates from the last two iterations becomes smaller than 0.01 m/s). This step gives us ${u}_{*}$, ${z}_{0h},{z}_{0m}$, and $L$ estimates for a given ${C}_{HN}$ value.
- Substitute ${u}_{*}$ and $L$ from step 7 in B1 and B2 to estimate ${\theta}_{SL}$ and ${q}_{SL}.$

## Appendix C. Euler–Lagrange Equations

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**Figure 1.**Idealized profiles of the potential temperature ($\theta $) and specific humidity (q) in the atmospheric boundary layer (ABL), and corresponding fluxes.

**Figure 2.**Locations of the four study sites (i.e., Audubon, Brookings, Bondville, and Willow Creek) in the United States.

**Figure 4.**Time series of estimated (solid lines) and observed (open circles) EF values. Precipitations are shown by blue bars.

**Figure 5.**Scatterplot of half-hourly sensible heat flux (H) estimates versus observations at the six investigated sites.

**Figure 6.**Scatterplot of half-hourly latent heat flux (LE) estimates versus observations at the six investigated sites.

**Figure 7.**Time series of sensible heat flux (H) estimates from the VDA (solid lines) and open loop (blue dashed lines) approaches at the six study sites. Observed H values are shown by open circles.

**Figure 8.**Time series of latent heat flux (LE) estimates from the VDA (solid lines) and open loop (blue dashed lines) approaches at the six study sites. Observed LE values are shown by open circles.

**Figure 9.**Scatterplot of half-hourly potential temperature (θ) estimates from VDA versus ${\theta}_{SL}$ at the six experimental sites.

**Figure 10.**Scatterplot of half-hourly specific humidity ($q$) estimates versus ${q}_{SL}$ at the six experimental sites.

**Figure 11.**Mean diurnal cycle of estimated (solid lines) and measured (symbols) turbulent heat fluxes from the VDA approach for the six study sites.

Site | Year | DOY | Latitude | Longitude | Vegetation Type | SM^{*} | LAI^{*} | Elevation (m) |
---|---|---|---|---|---|---|---|---|

Arou, China | 2015 | 170–259 | 38.0473°N | 100.4643°E | Grassland | 0.36 | 3.57 | 3033 |

Willow Creek, WI | 2005 | 170–259 | 45.8059°N | 90.0799°W | Forest | 0.22 | 5.67 | 520 |

Brookings, SD | 2009 | 176–265 | 44.3453°N | 96.8362°W | Grassland | 0.29 | 1.72 | 510 |

Bondville, IL | 2005 | 182–271 | 40.0062°N | 88.2904°W | Cropland | 0.16 | 2.24 | 219 |

Audubon, AZ | 2006 | 170–259 | 31.5907°N | 110.5092°W | Grassland | 0.12 | 0.54 | 1469 |

Desert, China | 2015 | 170–259 | 42.1100°N | 100. 9900°E | Barren land | 0.03 | 0 | 1000 |

^{*}LAI and SM represent respectively the mean leaf area index (m

^{2}m

^{−2}) and soil moisture (m

^{3}m

^{−3}) over the modeling periods. WI: Wisconsin, AZ: Arizona, IL: Illinois, and SD: South Dakota.

**Table 2.**The magnitudes of the initial conditions for $h$, ${\delta}_{\theta}$, and ${\delta}_{q}$, as well as ${\gamma}_{\theta}$ and ${\gamma}_{q}$ for the study sites.

Site | $\mathit{h}\left({\mathit{t}}_{\mathit{o}}\right)$ (m) | ${\mathit{\delta}}_{\mathit{\theta}}\left({\mathit{t}}_{\mathit{o}}\right)$ (K) | ${\mathit{\delta}}_{\mathit{q}}\left({\mathit{t}}_{\mathit{o}}\right)$ (kg kg ^{−1}) | ${\mathit{\gamma}}_{\mathit{\theta}}$ (K km ^{−1}) | ${\mathit{\gamma}}_{\mathit{q}}$ (kg kg ^{−1} km^{−1}) |
---|---|---|---|---|---|

Arou | 400 | 4.5 | −2.9 $\times $ 10^{−3} | 5.7 | −1.2 $\times $ 10^{−3} |

Willow Creek | 400 | 4.4 | −3.2 $\times $ 10^{−3} | 5.5 | −1.5 $\times $ 10^{−3} |

Brookings | 400 | 4.0 | −2.0 $\times $ 10^{−3} | 4.5 | −0.5 $\times $ 10^{−3} |

Bondville | 400 | 4.0 | −4.0 $\times $ 10^{−3} | 4.5 | −3.0 $\times $ 10^{−3} |

Audubon | 400 | 2.8 | −4.4 $\times $ 10^{−3} | 3.0 | −4.0 $\times $ 10^{−3} |

Desert | 400 | 2.4 | −4.4 $\times $ 10^{−3} | 3.0 | −4.5 $\times $ 10^{−3} |

**Table 3.**Estimated neutral bulk heat transfer coefficient (C

_{HN}) values by the VDA approach at the six study sites, and corresponding LAI values.

Site | DOY | C_{HN} | LAI |
---|---|---|---|

Arou | 170–199 | 0.0102 | 2.97 |

200–229 | 0.0325 | 4.41 | |

230–259 | 0.0261 | 3.35 | |

170–199 | 0.0245 | 5.87 | |

Willow Creek | 200–229 | 0.0242 | 5.84 |

230–259 | 0.0222 | 5.29 | |

Brookings | 176–206 | 0.0054 | 1.93 |

207–237 | 0.0102 | 2.15 | |

238–265 | 0.0048 | 1.07 | |

Bondville | 182–211 | 0.0130 | 2.23 |

212–241 | 0.0150 | 2.24 | |

242–271 | 0.0140 | 2.24 | |

Audubon | 170–199 | 0.0031 | 0.27 |

200–229 | 0.0029 | 0.57 | |

230–259 | 0.0033 | 0.77 | |

Desert | 170–199 | 0.0022 | 0 |

200–229 | 0.0020 | 0 | |

230–259 | 0.0010 | 0 |

**Table 4.**MAE and RMSE of EF estimates from the VDA approach at the six study sites. Mean soil moisture (m

^{3}m

^{−3}) and LAI (m

^{2}m

^{−2}) over the modeling periods are also presented.

Site | EF | SM | LAI | |
---|---|---|---|---|

MAE | RMSE | |||

Arou | 0.039 | 0.053 | 0.36 | 3.57 |

Willow Creek | 0.067 | 0.079 | 0.22 | 5.67 |

Brookings | 0.065 | 0.083 | 0.29 | 1.72 |

Bondville | 0.073 | 0.091 | 0.16 | 2.24 |

Audubon | 0.146 | 0.178 | 0.12 | 0.54 |

Desert | 0.152 | 0.198 | 0.03 | 0.00 |

**Table 5.**MAE and RMSE of half-hourly H and LE estimates from the VDA and open loop approaches at the six experimental sites.

Site | Method | H (Wm^{−2}) | LE (Wm^{−2}) | ||
---|---|---|---|---|---|

MAE | RMSE | MAE | RMSE | ||

Arou | VDA (Open-loop) | 27.02 (205.92) | 36.18 (242.09) | 63.75 (219.33) | 90.04 (247.15) |

Willow Creek | VDA (Open-loop) | 35.52 (188.52) | 44.52 (223.67) | 75.29 (179.98) | 97.31 (239.42) |

Brookings | VDA (Open-loop) | 40.57 (105.89) | 54.48 (138.58) | 83.21 (212.36) | 104.03 (239.00) |

Bondville | VDA (Open-loop) | 46.25 (118.67) | 63.56 (158.02) | 85.13 (156.58) | 105.82 (188.92) |

Audubon | VDA (Open-loop) | 59.54 (107.75) | 74.45 (127.97) | 86.96 (140.09) | 112.64 (166.11) |

Desert | VDA (Open-loop) | 60.67 (77.78) | 80.19 (103.76) | 72.73 (95.96) | 117.31 (156.28) |

Six-site average | VDA (Open-loop) | 44.93 (134.09) | 58.89 (165.68) | 77.85 (167.38) | 104.53 (206.15) |

**Table 6.**MAE and RMSE of daily H and LE estimates from the VDA and open loop approaches at the six experimental sites.

Site | Method | H (Wm^{−2}) | LE (Wm^{−2}) | ||
---|---|---|---|---|---|

MAE | RMSE | MAE | RMSE | ||

Arou | VDA (Open-loop) | 18.07 (172.66) | 25.43 (197.61) | 43.99 (231.71) | 55.81 (237.78) |

Willow Creek | VDA (Open-loop) | 30.05 (157.65) | 37.03 (197.65) | 56.93 (179.33) | 74.46 (200.28) |

Brookings | VDA (Open-loop) | 32.05 (100.47) | 45.01 (123.03) | 57.79 (185.88) | 82.28 (207.54) |

Bondville | VDA (Open-loop) | 31.97 (108.15) | 45.21 (134.12) | 55.34 (152.44) | 77.88 (172.74) |

Audubon | VDA (Open-loop) | 46.16 (74.73) | 58.13 (91.77) | 71.77 (125.67) | 89.57 (145.45) |

Desert | VDA (Open-loop) | 50.08 (67.07) | 60.00 (80.95) | 68.74 (90.23) | 102.21 (132.62) |

Six-site average | VDA (Open-loop) | 34.73 (113.46) | 45.14 (137.52) | 59.09 (160.88) | 80.37 (182.74) |

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## Share and Cite

**MDPI and ACS Style**

Tajfar, E.; Bateni, S.M.; Heggy, E.; Xu, T.
Feasibility of Estimating Turbulent Heat Fluxes via Variational Assimilation of Reference-Level Air Temperature and Specific Humidity Observations. *Remote Sens.* **2020**, *12*, 1065.
https://doi.org/10.3390/rs12071065

**AMA Style**

Tajfar E, Bateni SM, Heggy E, Xu T.
Feasibility of Estimating Turbulent Heat Fluxes via Variational Assimilation of Reference-Level Air Temperature and Specific Humidity Observations. *Remote Sensing*. 2020; 12(7):1065.
https://doi.org/10.3390/rs12071065

**Chicago/Turabian Style**

Tajfar, Elahe, Sayed M. Bateni, Essam Heggy, and Tongren Xu.
2020. "Feasibility of Estimating Turbulent Heat Fluxes via Variational Assimilation of Reference-Level Air Temperature and Specific Humidity Observations" *Remote Sensing* 12, no. 7: 1065.
https://doi.org/10.3390/rs12071065