Numerous hydrology, water resources, agriculture, and forestry related studies and applications require quantification of evapotranspiration (ET) across a range of spatial and temporal scales [Anderson et al., 2007; Long et al., 2011; McCabe and Wood, 2006]. Satellite remote sensing provides an unprecedented opportunity to derive surface and atmospheric variables over large areas, which are unattainable from ground-based measurements (e.g., weighing lysimeter, Energy Balance Bowen Ratio (EBBR) systems, and eddy covariance (EC) systems) and meaningful in ET modeling over large heterogeneous areas. In this context, a number of satellite-based land surface flux models have emerged since the 1980s by incorporating remotely sensed variables and routinely observed meteorological data [Kalma et al., 2008]. Among these models, the triangle model is unique in interpreting the relationship between the Normalized Difference Vegetation Index (NDVI)/fractional vegetation cover (fc) and surface radiative temperature (Trad) to deduce evaporative fraction (EF, ratio of latent heat flux (LE) to available energy (Rn-G)) over large areas [Carlson et al., 1994; Gillies and Carlson, 1995; Jiang and Islam, 2001; Price, 1990; Sandholt et al., 2002]. This type of model has advantages in utilizing information from visible, near-infrared, and thermal infrared bands to deduce EF without largely depending on ground observations.
 There are, however, several common issues associated with triangle models that have not been adequately addressed. First, triangle models have consistently underestimated [e.g., Choi et al., 2009; Jiang and Islam, 2003; Wang et al., 2006] or overestimated EF/ET [e.g., Batra et al., 2006; Jiang and Islam, 2003; Jiang et al., 2009] compared with ground-based measurements. However, reasons for these deviations have not been fully investigated or appropriately interpreted from a standpoint of model physics and scale effect. Second, most triangle models are combined with moderate- or low-spatial-resolution satellite sensors; that is, the National Oceanic and Atmospheric Administration-Advanced Very High Resolution Radiometer (NOAA-AVHRR) [Batra et al., 2006; Jiang and Islam, 2001; Sandholt et al., 2002], Moderate Resolution Imaging Spectroradiometer (MODIS) [Tang et al., 2010; Wang et al., 2006], and Meteosat Second Generation satellite (MSG)-Spinning Enhanced Visible and Infrared Imager (SEVIRI) [Stisen et al., 2008], for estimating EF over large areas. However, triangle models are rarely applied with high-spatial-resolution images; for example, Landsat Thematic Mapper (TM)/Enhanced Thematic Mapper Plus (ETM+). Determining effective techniques to use data from various sensors has been the focus of considerable research [McCabe and Wood, 2006].
 Third, a recurring issue for application of derived satellite data is whether techniques for one scale are appropriate to another [Carlson et al., 1995]. Methods of addressing spatial and temporal disparities between landscape heterogeneity and sensor and model resolution seem to be limited, because an adequately developed theory of scale dependence or scaling in hydrology does not yet exist [Beven and Fisher, 1996]. Particularly in surface flux estimation, little work has been performed to investigate differences in model outputs between using easily obtained moderate- or low-spatial-resolution sensors and relatively infrequent high-spatial-resolution sensors. These issues remain unresolved and affect surface flux estimation in the operational ET estimation and hydrological communities [McCabe and Wood, 2006].
 Fourth, there is another significant scale issue intrinsic in triangle models: domain scale effect. It is referred to as the dependence of model outputs on the size of the domain where the model is applied or on the size of the usable image [Long et al., 2011]. There are two limiting edges constituting envelopes of the NDVI/fc-Trad space in triangle models. They play a paramount role in determining the magnitude of EF. The upper envelope (here the x axis represents NDVI or fc and the y axis represents Trad) is referred to as the warm edge, pixels on which are taken as surfaces with the largest water stress for a range of NDVI/fc. In contrast, the lower envelope is called the cold edge, pixels on which represent surfaces without water stress; that is, evaporating and transpiring at potential rates. EF for a pixel at a specific NDVI/fc interval is deduced by weighting the extreme Trad values within the interval in terms of the Trad of the pixel. To that end, the warm and cold edges are essential to configuring the triangle space by providing important boundary conditions of the contextual NDVI/fc-Trad relationship and subsequently to determining EF for pixels within these limiting edges. Normally, one focuses primarily on a single size of the domain of interest; for example, the Soil Moisture-Atmosphere Coupling Experiment (SMACEX) site of ∼670 km2 in central Iowa [Choi et al., 2009], the Heihe River basin ∼38,000 km2 in northwestern China [Tang et al., 2010], and the Southern Great Plains site of ∼140,000 km2 [Batra et al., 2006; Wang et al., 2006]. Areas beyond a study site are rarely taken into account. Nevertheless, determination of warm and cold edges of the NDVI/fc-Trad space may depend on the size of the domain being studied. Alternatively, thermal band(s) of a variety of satellite sensors have varying capacity to discriminate the thermal properties of the land surface and therefore to derive Trad. In other words, resolution of Trad retrievals may also influence the definition and determination of limiting edges: The resolution dependence implies that varying spatial resolutions of satellite images are likely to generate varying EF for a given study site.
 A multitude of significant studies on examining resolution scale effects of satellite-based ET modeling has been performed to improve our understanding of the spatial scaling behavior of ET and its relation to controlling factors on the land surface. Carlson et al.  investigated resolution dependence of triangle models by linearly aggregating Trad of high spatial resolution derived from the NS001 multispectral scanner (5 m) to mimic low-spatial-resolution data, with resolutions of 20, 80, and 320 m. They observed successive movement of the warm edge toward the cold edge with increasing pixel size, but concluded that the objectively determined warm edge coincided with the domain of soil moisture availability isopleths and therefore the triangle with its warm edge was not substantially changed. Gillies et al.  indicated that scale issues may influence ET retrievals from triangle models because low-resolution data would not be able to define limiting edges [Gillies and Carlson, 1995]. Batra et al.  and Venturini et al.  showed that the NDVI-Trad space and EF estimates were similar for triangle models applied to MODIS and AVHRR sensors. Brunsell et al.  examined scaling behavior of ET from a triangle model at different resolutions (e.g., 1 km, 2 km, 4 km, and 8 km) using wavelet multiresolution analysis combined with low-pass filters and entropy theory. A similar study was performed using a range of satellite sensors (i.e., Landsat, MODIS, and Geostationary Operational Environmental Satellites (GOES)) to quantify which spatial scales are dominant in determining the ET flux [Brunsell and Anderson, 2011]. Deviated from the previous studies, this study focuses primarily on domain scale effects and resolution scale effects in EF estimation by triangle models from a perspective of model physics.
 The objectives of this study were to (1) evaluate reasons for overestimation and underestimation of EF by triangle models from a standpoint of model physics; (2) examine utility of triangle models using high-spatial-resolution satellite imagery; (3) explore domain and resolution dependencies of triangle models; and (4) develop a framework to constrain those dependencies for EF estimation. Section 2 introduces fundamentals of triangle models and development of a trapezoid model to address the scale dependencies, followed by a description of study site, data collection, and variable derivation in section 3. Sections 4 and 5 provide a systematic analysis of the domain dependence and resolution dependence of triangle models, respectively. Major findings of this study are given in section 6.