# A Bayesian Three-Cornered Hat (BTCH) Method: Improving the Terrestrial Evapotranspiration Estimation

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## Abstract

**:**

_{EC}) at AmeriFlux sites and ET values from the water balance method (ET

_{WB}) are used to evaluate the BTCH- and EM-integrated ET estimates. Results indicate that BTCH performs better than EM and all the individual parent products. Moreover, the trend of BTCH-integrated ET estimates, and their influential factors (e.g., air temperature, normalized differential vegetation index, and precipitation) from 1982 to 2011 are analyzed by the Mann–Kendall method. Finally, the 30-year (1982 to 2011) total water storage anomaly (TWSA) in the Mississippi River Basin (MRB) is retrieved based on the BTCH-integrated ET estimates. The TWSA retrievals in this study agree well with those from the Gravity Recovery and Climate Experiment (GRACE).

## 1. Introduction

_{EC}) and ET estimates from the water balance equation (ET

_{WB}). Moreover, the 30-year (1982 to 2011) trends of ET product from BTCH and its influential climate factors (e.g., air temperature, normalized differential vegetation index NDVI, and precipitation) are studied by the Mann–Kendall method. Finally, using the BTCH-integrated ET, the 30-year total water storage anomaly (TWSA) in the Mississippi River Basin (MRB) is retrieved, and validated against the Gravity Recovery and Climate Experiment (GRACE) measurements.

_{EC}and ET

_{WB}; (3) analyze the trend of 30-year BTCH-integrated ET product and its climate factors by the Mann–Kendall method; and (4) retrieve the 30-year total TWSA in the Mississippi River Basin (MRB) using the BTCH-integrated ET product.

## 2. Study Regions and Datasets

## 3. Methodology

#### 3.1. Bayesian-Based Three-Cornered Hat (BTCH) Method

_{i}) can be expressed as,

_{t}is the true value of ET. ε

_{i}and σ

_{i}are a zero-mean white noise and error variance of the ith ET product, respectively. L(•) is the likelihood function.

_{j}) can be expressed as,

_{j}and σ

_{j}are a zero-mean white noise and error variance of the jth ET product.

_{t}) is the maximum value of its joint probability distribution,

_{t}in Equation (3), the cost function J is defined as,

_{t}) to zero (i.e., $\delta $J(ET

_{t}) = 0), ET

_{t}can be obtained as,

_{t}= w

_{i}ET

_{i}+ w

_{j}ET

_{j}, then

_{t}can be obtained via ET

_{t}= w

_{1}ET

_{1}+ ··· + w

_{N}ET

_{N}when there are N sets of ET products. The weight of each ET product (e.g., w

_{k}) can be obtained by minimizing a similar cost function (Equation (4)) as,

_{i}) of each ET product can be obtained using the three-cornered hat (TCH) method.

_{t}is not available, the difference between (N-1) ET products and a reference ET product (ET

_{R}) (chosen arbitrarily from N ET products) can be expressed as,

**Y**is the difference matrix with (N-1) rows and M columns (M is the total number of time samples in each ET product). The covariance matrix (

**S**) of

**Y**is defined as [84],

**R**is related to

**S**via,

**J**is the identity matrix, and can be defined as,

**R**is defined as,

_{ij}= cov(ε

_{i}, ε

_{j}). There are N × (N + 1)/2 unknowns in Equation (10) (number of distinct elements of

**R**), but there are only N × (N − 1)/2 equations (number of distinct elements of

**S**). Thus, there remain N ”free” parameters that must be reasonably determined to obtain a unique solution [84]. To determine the N free parameters, the following objective function is minimized based on the Kuhn–Tucker theorem.

**Q**is a diagonal matrix with elements σ

_{11}, ···, σ

_{NN}on its diagonal. These arrays can be calculated by minimizing Equation (13) through the initial condition iterations [86]. The square root of the diagonal elements of

**R**(i.e., σ

_{11}, σ

_{22}, …, σ

_{NN}) represent the relative uncertainty of each ET product [43]. The readers are referred to Long et al. (2014) [42] and Xu et al. (2019) [43] for detailed information on the TCH approach.

#### 3.2. Water Balance Budget Method

_{WB}is the ET (mm/month) estimates from the water budget equation, TWSC is the total water storage change (mm/month), and P and R are the precipitation (mm/month) and runoff (mm/month), respectively. According to Zhang et al. (2010) [87] and Velpuri et al. (2013) [88], ET

_{WB}values can be used as the reference data to validate ET estimates at multiyear scale.

#### 3.3. Long-Term Total Water Storage Anomaly Reconstruction

_{t}and O

_{t}are the predicted and observed values at time step t, respectively.

_{xy,z}is the partial correlation between x and y, given the control z. The variables R

_{xy}, R

_{xz}, R

_{yz}are the correlation between x and y, x and z, and y and z, respectively. More details on the formulation of the partial correlation method is provided by [96] and [97].

## 4. Results and Discussions

#### 4.1. ET Product from the BTCH Method

_{EC}). As indicated, ET estimates from the BTCH method are closer to the observations as compared with other ET products. Since GFET is upscaled based on FLUXNET tower observations, it is also close to the AmeriFlux ET observations. The ET estimates from BTCH have higher R and lower RMSD as compared with those of EM and individual algorithms over the three land cover types. For DBF, ENF, and Crop, the RMSDs of ET estimates from BTCH are respectively 11.25, 15.13, and 17.25 mm/month, which are 31.28%, 20.16%, and 22.05% lower than those of 16.37, 18.95, and 22.13 mm/month from EM.

_{WB}values are also used to assess the accuracy of ET estimates from BTCH, EM, and ten individual models at the basin scale in Figure 5. As indicated, BTCH performs better than EM and other models. ET products from BTCH and Noah28 are closest to the ET

_{WB}estimates, followed by VIC412 and GFET. For the 30-year comparisons (1982 to 2011), the average RMSD of ET product from the BTCH method is 24.64 mm/year, which is 53.46% lower than the RMSD of 52.94 mm/year from the EM method.

#### 4.2. Relationships between BTCH-Integrated ET and Climate Factors

#### 4.3. Interannual Variations of BTCH-Integrated ET Estimates

#### 4.4. Long-Term Reconstruction of GRACE Total Water Storage Anomaly

#### 4.5. Time Window of BTCH Method

_{WB}) and ET from CLM4 (ET

_{CLM4}) are used as the reference data to quantify different integration results. The results of EM are also shown for comparison. As indicated, the BTCH-integrated ET estimates are closer to observation, implying that BTCH outperforms EM. ET estimates from BTCH1 have the lowest RMSD and standard deviation, followed by those of BTCH2, BTCH3, BTCH6, and BTCH12. The seasonal variations in uncertainties of ET estimates can be efficiently used by the BTCH method to reduce the uncertainties in the integrated ET product.

## 5. Conclusions

_{EC}) stations and ET derived from the water balance method (ET

_{WB}). The results show that BTCH is able to capture the seasonal variations of ET estimates more accurately than the ensemble mean (EM) method. The BTCH-integrated ET estimates have the RMSD of 14.54 mm/month, which is 24.07% lower than that of the RMSD of 19.15 mm/month from EM.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**The land cover types and location of the 15 AmeriFlux stations over the contiguous United States (CONUS). DBF, EBF, and ENF represent deciduous broadleaf forest, evergreen broadleaf forest, and evergreen needleleaf forest, respectively.

**Figure 2.**Spatial distribution of 30-year averaged evapotranspiration (ET) from the Bayesian-based three-cornered hat (BTCH) method.

**Figure 3.**Maps of 30-year averaged normalized differential vegetation index (NDVI) and precipitation over the contiguous United States.

**Figure 4.**Comparison of monthly BTCH-integrated, ensemble mean (EM), and ten parent evapotranspiration (ET) datasets with flux tower observations. There are 144, 336, and 300 samples for the deciduous broadleaf forest (DBF), evergreen broadleaf forest (ENF), and cropland, respectively.

**Figure 5.**Comparison of annual BTCH-integrated, ensemble mean (EM), and ten parent evapotranspiration (ET) dataset with ET from water balance method from 1982 to 2011. There are 360 samples over the contiguous United States.

**Figure 6.**Weight (%) of each evapotranspiration (ET) product in the Bayesian-based three-cornered hat (BTCH) method.

**Figure 7.**The spatial pattern of trends in air temperature (Ta), normalized differential vegetation index (NDVI), precipitation (P), and evapotranspiration (ET) over the contiguous United States from 1982 to 2011. The black points represent 95% level of significance.

**Figure 8.**Plots of 30-year-averaged evapotranspiration (ET) against air temperature (Ta), precipitation (P), and normalized differential vegetation index (NDVI) over the 12 river forecast centers (RFCs). The red line represents the fitted linear regression.

**Figure 9.**Partial correlation coefficients (R) between the annual evapotranspiration (ET) and air temperature (Ta), precipitation (P), and normalized differential vegetation index (NDVI).

**Figure 10.**Annual variations of BTCH-integrated evapotranspiration (ET), soil moisture, and precipitation anomalies for 12 river forecast centers (RFCs) from 1982 to 2011.

**Figure 11.**Time series of the total water storage anomaly (TWSA) retrievals over the Mississippi River Basin (MRB) from 1982 to 2011 (top). Comparison of the TWSA estimates with those of the Gravity Recovery and Climate Experiment (GRACE) from 2003 to 2011 (bottom).

**Figure 12.**Monthly variations of evapotranspiration (ET) estimates from Bayesian-based three-cornered hat (BTCH) for different time windows, and ensemble mean (EM). BTCH1, BTCH2, BTCH3, BTCH6, and BTCH12 denote 1-month, 2-month, 3-month, 6-month, and 12-month time windows, respectively. The black circles are observations from eddy covariance flux towers.

**Figure 13.**Comparison of evapotranspiration (ET) estimates (1982 to 2011) from the ensemble mean (EM) and Bayesian-based three-cornered hat (BTCH) methods for different time windows.

**Figure 14.**The uncertainties of evapotranspiration (ET) products from the ensemble mean (EM) and Bayesian-based three-cornered hat (BTCH) methods with different time windows over the contiguous United States.

ID | Site Name | Period | Location (N°, W°) | Elevation (m) | Land Cover |
---|---|---|---|---|---|

US-Bar | Bartlett Experimental Forest | 2004–2005 | 44.06, -71.28 | 272 | DBF |

US-MOz | Missouri Ozark | 2004–2006 | 38.74, -92.20 | 219 | DBF |

US-MMS | Morgan Monroe State Forest | 2000–2004 | 39.32, -86.41 | 275 | DBF |

US-Oho | Ohio Oak Openings | 2004–2005 | 41.55, -83.84 | 230 | DBF |

US-Blo | Blodgett Forest | 2000–2006 | 38.89, -120.63 | 1315 | ENF |

US-SP3 | Donaldson | 2000–2004 | 29.75, -82.16 | 50 | ENF |

US-Fuf | Flagstaff Unmanaged Forest | 2005–2006 | 35.08, -111.76 | 2180 | ENF |

US-Me2 | Metolius Pine | 2003–2005 | 44.45, -121.55 | 1253 | ENF |

US-NR1 | Niwot Ridge | 2000–2003 | 40.03, -105.54 | 3050 | ENF |

US-Wrc | Wind River | 2000–2006 | 45.82, -121.95 | 371 | ENF |

US-ARM | ARM Southern Great Plains Main | 2003–2006 | 36.60, -97.48 | 314 | Cropland |

US-Bo1 | Bondville | 2000–2006 | 40.00, -88.29 | 219 | Cropland |

US-Ne1 | Mead Irrigated | 2001–2005 | 41.16, -96.47 | 361 | Cropland |

US-Ne2 | Mead Irrigated Rotation | 2002–2005 | 41.16, -96.47 | 362 | Cropland |

US-Ne3 | Mead Rainfed | 2001–2005 | 41.18, -96.44 | 363 | Cropland |

Dataset | Data Source | Spatial, Temporal Resolution | Spatial, Temporal Coverage | References |
---|---|---|---|---|

P & Ta | NCEI | 4.16 km, monthly | CONUS, 1979–2012 | [74,75] |

R | USGS | ~2°, daily | CONUS, 1982–2011 | [78] |

TWSA | GRACE | ~1°, monthly | Global, 2003–2016 | [76] |

SM & ET | CLM4 | 12.5 km, monthly | CONUS, 1980–2014 | [79] |

NDVI | AVHRR | 0.05°, daily | Global, 1981–2018 | [80] |

SE ET | GFET | 50 km, monthly | Global, 1982–2011 | [10,81] |

RSM ET | GLEAM | 25 km, daily | Global, 1982–2017 | [37] |

NLDAS-2 ET | Noah28 | 12.5 km, monthly | CONUS, 1979–2013 | [69,70] |

SAC | ||||

Mosaic | ||||

VIC403 | ||||

NLDAS-testbed ET | CLSM25 | 12.5 km, monthly | CONUS, 1979–2013 | [79] |

Noah36 | ||||

NoahMP36 | ||||

VIC412 |

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## Share and Cite

**MDPI and ACS Style**

He, X.; Xu, T.; Xia, Y.; Bateni, S.M.; Guo, Z.; Liu, S.; Mao, K.; Zhang, Y.; Feng, H.; Zhao, J.
A Bayesian Three-Cornered Hat (BTCH) Method: Improving the Terrestrial Evapotranspiration Estimation. *Remote Sens.* **2020**, *12*, 878.
https://doi.org/10.3390/rs12050878

**AMA Style**

He X, Xu T, Xia Y, Bateni SM, Guo Z, Liu S, Mao K, Zhang Y, Feng H, Zhao J.
A Bayesian Three-Cornered Hat (BTCH) Method: Improving the Terrestrial Evapotranspiration Estimation. *Remote Sensing*. 2020; 12(5):878.
https://doi.org/10.3390/rs12050878

**Chicago/Turabian Style**

He, Xinlei, Tongren Xu, Youlong Xia, Sayed M. Bateni, Zhixia Guo, Shaomin Liu, Kebiao Mao, Yuan Zhang, Huaize Feng, and Jingxue Zhao.
2020. "A Bayesian Three-Cornered Hat (BTCH) Method: Improving the Terrestrial Evapotranspiration Estimation" *Remote Sensing* 12, no. 5: 878.
https://doi.org/10.3390/rs12050878