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The complementary relationship for estimating evapotranspiration (ET) is a simple approach requiring only commonly available meteorological data; however, most complementary relationship models decrease in predictive power with increasing aridity. In this study, a previously developed Granger and Gray (GG) model by using Budyko framework is further improved to estimate ET under a variety of climatic conditions. This updated GG model, GG-NDVI, includes Normalized Difference Vegetation Index (NDVI), precipitation, and potential evapotranspiration based on the Budyko framework. The Budyko framework is consistent with the complementary relationship and performs well under dry conditions. We validated the GG-NDVI model under operational conditions with the commonly used remote sensing-based Operational Simplified Surface Energy Balance (SSEBop) model at 60 Eddy Covariance AmeriFlux sites located in the USA. Results showed that the Root Mean Square Error (RMSE) for GG-NDVI ranged between 15 and 20 mm/month, which is lower than for SSEBop every year. Although the magnitude of agreement seems to vary from site to site and from season to season, the occurrences of RMSE less than 20 mm/month with the proposed model are more frequent than with SSEBop in both dry and wet sites. Another finding is that the assumption of symmetric complementary relationship is a deficiency in GG-NDVI that may introduce an inherent limitation under certain conditions. We proposed a nonlinear correction function that was incorporated into GG-NDVI to overcome this limitation. As a result, the proposed model produced much lower RMSE values, along with lower RMSE across more sites, as compared to SSEBop.

According to the U.S. Geological Survey (USGS) Famine Early Warning Systems Network [

McMahon [

One approach to estimating ET with ground-based methods is the complementary relationship proposed by Bouchet [

Granger [

lacked a theoretical background and proved that the symmetric condition is only true when the temperature is near 6˚C. Hence, the author developed a new complementary relationship with the psychrometric constant and the slope of the saturation vapor pressure curve. Later, Crago [

Prior studies show that the complementary relationship is not symmetric with wet environment evapotranspiration (ETW) and that the GG model can be successfully applied to a wide range of physical and surface conditions. Specially, the modified GG model (Anayah [

While these findings are good within the realm of complimentary methods (or ground-based methods), some of the more commonly used ET estimation methods now use remote sensing data. If the complementary relationship and the corresponding methods, such as the model proposed by Kim [

Biggs et al. [

Radiometric land surface temperature-based methods use the fact that ET is a change of state in water that uses energy in the environment for vaporization and reduces surface temperature [

(METRIC) [

As an alternative, FWESNET (USGS) has produced ET measurements from MODIS using the operational Simplified Surface Energy Balance (SSEBop) model [

Despite the general consensus of using SSEBop for estimating ET, a detailed study of SSEBop conducted by Senay et al. [

The facts provided in the previous discussion indicate a need to further validate both Kim [

GG-NDVI is the most updated model using the original GG model. GG-NDVI uses historical annual Normalized Difference Vegetation Index (NDVI) data and precipitation to improve the ET estimates of the modified GG model proposed by Anayah [

The first complementary relationship was proposed by Bouchet [

ET + γ Δ ETP = ( 1 + γ Δ ) ETW (1)

where ET, ETP, and ETW are in mm/day, γ is the psychrometric constant (kPa/˚C), and ∆ is the slope of saturation vapor pressure-temperature (kPa/˚C) relationship. Thereafter, [

ETW = α Δ γ + Δ ( R n − G s o i l ) (2)

where α is a coefficient equal to 1.28, R n − G s o i l is net radiation (mm/day), and G s o i l is soil heat flux density (mm/day). Note that soil heat flux density is negligible compared to net radiation when calculated at daily or monthly time-scale [

ET is then estimated as a fraction of ETW using Equation (3):

ET = 2 G G + 1 ETW (3)

where G is the relative evaporation parameter derived from [

G = ET ETP = 1 1 + 0.028 e 8.045 D (4)

D = E a E a + R n (5)

where E a is drying power of air (mm/day) given in Equation (6).

E a = 0.35 ( 1 + 0.54 U ) ( e s − e a ) (6)

where U is wind speed at 2 m above ground level (m/s), which is adjusted using the work of Allen [

The performance of the GG model, including the modified GG model proposed later, decreased with increasing aridity. A possible reason is G in Equation (4), which was empirically derived from 158 sites representing wet environments in Canada. To improve the parameter G, GG-NDVI model (Kim [

ET ETP = 1 + P ETP − [ 1 + ( P ETP ) ϖ ] 1 ϖ (7)

where P is precipitation (mm) and ETP is estimated using Penman [

M = NDVI − NDVI min NDVI max − NDVI min (8)

where NDVI_{min} and NDVI_{max} are chosen to be 0.05 and 0.8, respectively. An optimal value for the basin can be derived through a curve fitting procedure that minimizes RMSE between the measured and predicted evaporation ratio [

Li [

ϖ = a × M + b (9)

where a and b are constants that are found for each site.

To incorporate Equation (7) into the modified GG model, Kim [

G n e w = ET ETP = 1 + P ETP − [ 1 + ( P ETP ) ϖ ] 1 ϖ (10)

Note G n e w is the updated definition of relative evaporation, G, which includes the Budyko hypothesis and the vegetation index. To estimate G n e w , ETP is required and can be estimated using Equation (11) [

ETP = Δ γ + Δ ( R n − G s o i l ) + γ γ + Δ E a (11)

Having found G n e w from Equation (11) and estimated ETW from Equation (2), we can estimate ET of the proposed model from Equation (12).

ET = 2 G n e w G n e w + 1 ETW (12)

The SSEBop algorithm (Senay et al. [

ETf = T h − T s d T = T h − T s T h − T c (13)

Here, ETf is between 0 and 1, with negative ETf values set to zero; Ts is surface temperature derived from MODIS LST; Th is hot reference value representing the temperature of hot conditions; and Tc is the cold reference value derived as a fraction of maximum air temperature [

ET is estimated using Equation (14) as a fraction of reference ET.

ET = ETf × k ET o (14)

where ET_{o} is reference ET, which is calculated from the Penman-Monteith equation [_{o} into the level of maximum ET experienced by an aerodynamically rougher crop. A recommended value of k for the United States is 1.2.

First, we used the SSEBop ET data set from the USGS Geo Data Portal (http://cida.usgs.gov/gdp/, last accessed on May 23, 2016) for the period 2000-2007 covering the United States. Second, ET data from GG-NDVI were generated using meteorological data and NDVI. Meteorological data required are temperature, wind speed, precipitation, net radiation, and elevation (pressure). Among these, net radiation (Rn) was calculated using the equations recommended by Allen [

We collected the level 4 meteorological data including latent heat flux (LE) from 76 AmeriFlux stations (Oak Ridge National Laboratory’s AmeriFlux website, http://ameriflux.ornl.gov/, last accessed on Nov 23, 2015) then, we excluded those stations with actual vegetation type different from the MODIS global land cover product (MOD12) at any of surrounding 500 m by 500 m spatial resolution. Also, we further excluded those stations with fewer than half a year of measurements during 2000-2007. As a result, 60 stations were used in this study as shown in

We defined the climate class of each site using the aridity index of the United Nations Environment Programme (UNEP) proposed by Barrow [

This study was conducted in two phases. Phase 1 is the validation stage in which comparisons are made between the SSEBop model and measured ET to assess the accuracy of the remote sensing method to estimate ET. In Phase 2, a comparison of estimated ET from GG-NDVI with observed data will be performed

to identify the weaknesses of the GG-NDVI model, especially relative to the complementary relationship, and appropriate corrections will be proposed.

Capturing inter-annual variations of ET estimates is important. Although such variations are not significant when water is unlimited, estimating these variations in water-limited conditions is essential for water resources management. In this phase, ET has been estimated from both SSEBop and GG-NDVI and compared against measured monthly ET data from 2000 to 2007.

Year | AmeriFlux mean (mm/month) | R-square | RMSE (mm/month) | ||
---|---|---|---|---|---|

SSEBop | GG-NDVI | SSEBop | GG-NDVI | ||

2000 | 43 | 0.82 | 0.79 | 16 | 15 |

2001 | 44 | 0.54 | 0.58 | 23 | 20 |

2002 | 41 | 0.73 | 0.67 | 19 | 16 |

2003 | 42 | 0.68 | 0.65 | 21 | 17 |

2004 | 42 | 0.68 | 0.60 | 18 | 18 |

2005 | 42 | 0.37 | 0.57 | 28 | 18 |

2006 | 41 | 0.61 | 0.55 | 20 | 18 |

2007 | 34 | 0.40 | 0.40 | 18 | 17 |

All years | 44 | 0.65 | 0.61 | 19 | 18 |

to the predefined cold boundary (Tc), which brings ETf closer to 1.0, resulting in a corresponding ET that is close to the maximum ET.

According to

is more frequent than with SSEBop in both dry and wet sites. The averages of RMSE across 24 dry sites for GG-NDVI and SSEBop are 19 mm/month and 22 mm/month, respectively. For 36 wet sites, GG-NDVI and SSEBop showed an average RMSE of 17 mm/month and 20 mm/month, respectively. These results indicate that GG-NDVI ET estimates improve with wetness, which is similar to the previous studies of Hobbins [

Based on these results, we could conclude that GG-NDVI is a reliable approach for estimating ET, the novelty of GG-NDVI being that the Fu equation can be used to define relative evaporation in the original GG model using NDVI. This approach showed a reasonable match between GG-NDVI and the 60 AmeriFlux sites. However, GG-NDVI may not predict ET accurately when the vegetated cover changes significantly or is dense. For example, at Brooking in South Dakota, the mean RMSE of GG-NDVI was 42 mm/month, compared to 18 mm/month with all sites, and NDVI has a large seasonal vegetation cover as shown in

to Yang et al. [

As described earlier, GG-NDVI performed slightly better than SSEBop in both dry and wet climate conditions, and GG-NDVI increased the predictive power with increasing humidity. One interesting finding is that RMSE from GG-NDVI increases slightly with the relative evaporation parameter as shown in

Within the complementary relationship, increasing G means that climate is becoming wetter and ET is closer to ETW. When ET equals to ETW, surface has access to unlimited water as shown in

ET = 2 G n e w G n e w + 1 × f ( G ) × ETW (17)

where f(G) is the correction function. We expect the correction function to be nonlinear, similar to an exponential function, since the magnitude of the difference between ET and ETW decreases exponentially as shown in

f ( G ) = α e β ⋅ G (18)

Regression analysis found that α is 0.7895 and β is 0.9655. Hereafter, the GG-NDVI model with the proposed correction function given as Equation (17) is called the Adjusted GG-NDVI model.

To determine the accuracy of Adjusted GG-NDVI, comparisons were made between the results from the Adjusted GG-NDVI and GG-NDVI and between measured ET data and ET values from SSEBop. These comparisons are shown in

ET + ETP = 2 f ( G ) ETW (19)

where the value of 2f(G) can vary between 1.64 and 3.04 as G varies based on site-specific conditions. The new formulation of the Adjusted GG-NDVI model described in Equation (19) clearly shows that the relationship between ET and ETP is not symmetric with respect to ETW, further confirming the earlier conclusions that the hypothesis of Bouchet [

ET model | RMSE (mm/month) | ||
---|---|---|---|

Minimum | Mean | Maximum | |

GG-NDVI | 7 | 18 | 48 |

SSEBop | 8 | 20 | 48 |

Adjusted GG-NDVI | 7 | 15 | 34 |

with appropriate corrections.

ET estimation models using the complementary relationship are able to estimate ET in most instances. In particular, the model proposed by Anayah [

The first phase of the analysis showed that the GG-NDVI model with the Budyko framework and relative evaporation was found to work reasonably well. Validation with 60 AmeriFlux sites indicated similar levels of accuracy for both SSEBop and GG-NDVI. R-square between GG-NDVI and measured ET ranged from 0.40 to 0.79, overall RMSE of GG-NDVI ranged between 15 and 20 mm/month, and GG-NDVI showed lower RMSE than SSEBop every year. Furthermore, the occurrences of RMSE less than 20 mm/month with GG-NDVI were more frequent than SSEBop. Based on these results, we concluded that GG-NDVI is a reliable approach for estimating ET.

The second phase of the analysis showed that the predictive power of GG-NDVI decreased with relative evaporation possibly due to the use of the symmetric complementary relationship in estimating ET. In order to identify the true relationship between ET and ETP with respect to ETW, an exponential correction function was proposed. This phase demonstrated that the inclusion of relative evaporation with a correction function greatly improved the performance of the Adjusted GG-NDVI. For example, 68% of Adjusted GG-NDVI sites had RMSE less than 15 mm/month compared 43% with GG-NDVI.

In essence, this study strengthens the idea that the use of vegetation cover information in the complementary relationship has increased ET estimation power. More importantly, this work showed that the symmetric relationship typically assumed with the complementary relationship may not be valid. Instead, the results show that the symmetrical relationship needs to be updated with a nonlinear correction function as proposed here. A key strength of this study is that the latest proposed version of the GG model, Adjusted GG-NDVI, overcomes limitations of both relative evaporation as proposed by Granger [

Kim, H. and Kaluarachchi, J.J. (2018) Developing an Integrated Complementary Relationship for Estimating Evapotranspiration. Natural Resources, 9, 89-109. https://doi.org/10.4236/nr.2018.94007

AA: The Advection-Aridity model by [

CRAE: The Complementary Relationship Areal Evapotranspiration model by [

ET: Evapotranspiration

ETP: Potential Evapotranspiration

ETW: Wet Environment Evapotranspiration

GG: Granger and Gray Model by [

GG-NDVI: The ET model developed by [

METRIC: Mapping EvapoTranspiratrion at high Resolution with Internalized Calibration model by [

Modified GG: The ET model developed by [

NDVI: Normalized Difference Vegetation Index

NGG: The Normalized GG model by [

SEBAL: The Surface Energy Balance Algorithm for Land model by [

SSEB: The Simplified Surface Energy Balance model by [

SSEBop: The Operational Simplified Surface Energy Balance model by [

RMSE: Root Mean Square Error