Research Article | Open Access
Relative Contribution of the Topographic Influence on the Triangle Approach for Evapotranspiration Estimation over Mountainous Areas
Evapotranspiration (ET) is an important component of the water budget. Estimation ET through remote sensing over a mountainous terrain is typically obstructed by topographic effects. In this paper, topographic corrections were applied to ET estimates using the surface-air temperature difference-Normalized Difference Vegetation Index (()-NDVI) triangle method with MODIS data for the Taihu Basin in China. The effect of topography on ET was evaluated over an area with a complex terrain. After applying the topographic correction, the results indicate that the ET decreased with elevation and slope. The slope had a stronger impact on ET than the elevation, which caused the corrected ET to decrease by 90% from 6.8 mm day−1 to 0.6 mm day−1 for slopes over 50°. On average, the corrected ET decreased by 10.4% and 32.1% for north- and south-facing slopes, respectively. The ET corrected using the triangle method strongly depended on the evaporative fraction correction, which can mainly be attributed to the surface temperature correction. We conclude that a topographic correction is necessary when the triangle method is applied to areas with a complex terrain.
Evapotranspiration (ET), which includes water evaporation from soil surfaces and vegetation transpiration, is an important variable in water and energy balances on the Earth’s surface. Understanding the spatial distribution of ET is essential for many environmental monitoring applications, including water resource management, agricultural efficiency, global vegetation analysis, climate dynamics, and ecological applications [1–3]. Although the ET may be accurately estimated from detailed ground observations, direct field measurements only represent local scales over a few hundred square meters. It is difficult to conduct ground observations on a large scale due to the high cost and spatial range discontinuities .
Remote sensing technology can provide land surface parameters, such as surface temperature, albedo, and vegetation indices, which are indispensable to remote sensing-based methods that estimate the area-averaged ET on a regional scale . Several remote sensing-based methods with varying complexities have been developed to map ET at various spatial scales. Typical examples include the simplified empirical method , the surface energy balance-based single- or dual-source models [7, 8], the spatial contexture information-based surface temperature-vegetation index (-VI) triangular or trapezoidal methods [9, 10], and data assimilation techniques .
Among these methods, the -VI triangle method is widely accepted for its simplicity. The method was first introduced by Price , who used the spatial contexture of surface radiant temperature and fractional vegetation cover to estimate ET in the Southern Great Plains, USA. Thereafter, the triangle method was improved and successfully applied to monitor soil moisture water content and droughts [12–17] in addition to ET [18–20]. Notably, Jiang and Islam [10, 19] used the well-known Priestley-Taylor equation  to estimate ET using a triangular relationship constructed for remotely sensed versus the Normalized Difference Vegetation Index (NDVI). This relationship was further developed with other VI, including the Enhanced Vegetation Index (EVI) and fractional vegetation cover (Fr), using various remote-sensing data [5, 22–26].
The triangle method implicitly requires a flat surface and a large number of satellite pixels over an area containing a wide range of soil moisture and fractional vegetation cover . Elevation, terrain slope, and aspect angles variations can interact with the satellite-viewing geometry to produce biases in the land surface parameters retrieved . The biases in the relevant parameters may induce errors in the subsequent ET retrieval. While a large fraction of the Earth’s surface consists of mountainous areas, few satellite products are corrected for terrain-induced angular effects. Gao et al.  evaluated the spatial variation in daily ET for a complex terrain based on the surface energy balance algorithm for land (SEBAL) but did not address ET retrieval errors related to topographic effects. To date, the extent to which the terrain affects ET retrieval using the triangle method and the parameters responsible for the terrain-induced errors are unknown. Clarifying the uncertainties will enhance our confidence using satellite data over rugged areas.
The aim of this study was to address the effects of topography on evapotranspiration estimates from remote-sensing data using the triangle method. Thus, topographic correction techniques were applied to correct the parameters relevant to ET over a complex terrain. We compared the ET estimated with and without topographic corrections to quantify the influence on ET in mountainous areas. Finally, suggestions were proposed for practical use of the triangle method using satellite products without topographic corrections.
2.1. ET Retrieval Using the Triangle Method
The triangle relationship between the remotely sensed surface temperature and vegetation index can be applied to estimate ET using the following equation : where denotes ET (W m−2), is the net radiation (W m−2), and is the soil heat flux (W m−2). denotes the evaporative fraction (EF), which is defined as the ratio of ET to the available radiant energy : where is the slope of the saturated vapor pressure at the air temperature (), is the psychrometric constant (hPa K−1), and is a parameter that accounts for aerodynamic and canopy resistances.
For the triangle method, the parameter was calculated based on a triangular space using several types of expression, including a -VI scatterplot, -albedo scatterplot, and surface-air temperature difference versus VI scatterplot . The -VI triangle method requires a flat area , which yields large uncertainties when applied over areas with irregular topography. Because air temperature varies greatly over the screen, the surface-air temperature difference is the true temperature variable for the triangle method over the topographic terrain [9, 10]. In this study, based on the surface-air temperature difference versus the NDVI space, the upper and lower for each NDVI interval were established, and we generated a linear interpolation for each NDVI interval between the lower and the highest . The for a pixel at is as follows :
The global minimum and maximum are, respectively, for the driest bare soil pixel and for the pixel with dense vegetation and the largest NDVI. The was then linearly interpolated for each NDVI interval (NDVIi) between and , and the for each NDVIi was obtained from the lowest pixel at the particular NDVI interval. The value in each NDVI interval was interpolated between the lowest pixel and highest pixel . Consequently, the value for each pixel can be determined using the spatial context of the and NDVI.
is the net flux of four radiative components, including downward and upward shortwave radiation as well as downward and upward longwave radiation, which can be expressed as follows: where and are the downward and upward shortwave radiation (W m−2), respectively, and and are the downward and upward longwave radiation (W m−2), respectively. Equivalently, where is the surface albedo, is the emissivity of the air, is the surface emissivity, is the air temperature (K) at the screen level, and is the Stefan-Boltzmann constant (5.67 × 10−8 W m−2 K−4).
The soil heat flux varies with the time of day, and it can be parameterized as a function of NDVI and using the following equation :
Instantaneous ET estimates are calculated at the satellite overpass time and then extrapolated to daily values by assuming a constant daytime evaporative fraction. The ET in a time series should be reconstructed based on ET retrieval for a clear day. In the Taihu watershed, the potential ET (based on Peman-Monteith equation) and daily ET correlate well (, ). The ET gaps in the year were filled using this relationship; the annual ET was then obtained by summing daily ET in the Taihu watershed.
2.2. Topographic Correction
The ET can be determined using , , , and , all of which can be estimated independently using remotely sensed data in accordance with (1)–(5). Among the variables in these equations, shortwave radiation , longwave radiation , and change substantially with the terrain features. The variability in elevation, surface orientation (slope and aspect), and shadows cast by topographic features create strong local radiation gradients. In this study, was corrected based on a topographic correction of the albedo, and was corrected based on a topographic correction of the and . The evaporative fraction EF was corrected based on a topographic correction of the because the terrain effects on NDVI can generally be ignored . Considering the effect of elevation on , was corrected using the DEM and the lapse rate (0.0065°C/m), which indicates that the air temperature decreases 0.65°C with a 100 m elevation increase. Terrain affects the satellite-retrieved and values . For the MODIS land surface temperature product, only primary terrain information is considered in the bidirectional reflectance distribution function (BRDF) ; it ignores the terrain angular effect on retrieval. Consequently, the angular effects on the combined and were accounted for in the terrain correction, which may be corrected using the following equation : where is the corrected land surface temperature and is the angle between the satellite view path and is normal to the terrain element. The emitted radiance angle can be geometrically determined using the following equation: where is the local slope angle, is the sensor zenith angle, is the sensor azimuth angle, and is the aspect angle of the terrain element.
The terrain-induced angular effect on can be corrected based on the cosine method as follows : where denotes the corrected surface albedo.
3. Study Materials and Data Processing
3.1. Study Area
The Taihu Basin in China was used as a case study due to its topographic features and data availability (Figure 1(a)). Lake Taihu is in the area bordered by the Yangtze River, Hangzhou Bay, and Qiantang River. The Taihu Basin covers 36500 km2 and extends from 119.1°E to 122.2°E longitude and 30.0°N to 32.1°N latitude. It is a heterogeneous area characterized by mixed farming, forest, and grass lands. The area surrounding Lake Taihu is dominated by a monsoon climate. The annual average air temperature is 14.9–16.2°C, and the annual precipitation is between 1000 and 1400 mm. Farmland is the dominant land cover type in the Taihu Basin, which accounts for 59.8% of the total basin area; woodland area accounts for 13.4%, building land area accounts for 13.5%, water body area accounts for 13.0%, and grassland area accounts for 0.3% of the total basin area.
Figure 1(b) shows a digital elevation model (DEM) of the Taihu Basin. More than 80% of the area in the basin is flat with a 40 m average elevation. The mountainous area is located in the western and southwestern parts of the region, where the highest elevation is over 1500 m. A mountainous area in the southwest Taihu Basin was selected to illustrate the effects of topography (Figure 1(b)). The selected area, which is part of the Tianmu Mountains, extends from 30.0° to 31.0°N latitude and 119.5° to 120.5°E longitude and covers most of the mountainous areas in the Taihu Basin. The zonal vegetation is distributed vertically and includes evergreen broad-leaved and artificial bamboo forests below a 700 m elevation, mixed evergreen and deciduous forests at elevations between 850 and 1100 m, deciduous forests below a 1380 m elevation, and deciduous dwarf forest over a 1380 m elevation. Coniferous forests are spread sporadically at elevations between 300 and 1100 m.
To validate ET estimates using remote sensing, ET was estimated as a residual variable based on the principle of water balance in the subbasin. The Xitiaoxi subbasin, which is a closed subbasin upstream of the Taihu watershed, was used to validate the retrieved ET. The total monthly rainfall data from 10 rain gauge stations were collected. Monthly runoff data from the Fanjiacun hydrological station were obtained, which represent the total surface runoff in the Xitiaoxi subbasin. The data were provided by the Hydrological Bureau of Zhejiang Province.
3.2. Data Processing
The standard MODIS products from the Terra (EOS AM-1) satellites were used to estimate the regional ET. To avoid a cloud effect, we used a clear day on September 26 in 2006 (DOY 269) to examine the topographic effects on the ET estimates. This date included the growing season for the cropland and forest ecosystems, wherein the NDVI formed a gradient in the basin, which is implicitly required by the triangle method. The sensor zenith ranged from 0.5° to 13.8° with a 4.7° average on September 26, 2006, over the Taihu Basin, which implies that the remote-sensing image was screen under the point of vertical satellite direction, and the geometric image distortion was minimal this day.
The MODIS products were acquired from the Land Processes Distributed Active Archive Center (LP DAAC) (https://lpdaac.usgs.gov/). The selected products included MODIS geolocation (MOD03), the atmospheric profile product (MOD07), the surface reflectance (MOD09) and land surface temperature/emissivity product (MOD11_L2), all of which were generated using the Product Generation Executive (PGE) code Version 5. MOD03 contains the solar zenith and azimuth angles as well as satellite zenith and azimuth angles. MOD07 includes the air and dew point temperatures. MOD09 provides the surface reflectance through seven reflective bands and the values corrected for radiometric and atmospheric effects. MOD11_L2 provides the land surface temperature and surface emissivity in bands 31 and 32.
The NDVI values were retrieved from the surface reflectance in the red and near infrared (NIR) bands (MOD09). The 500 m data in the NIR band was resized to 250 m using a nearest-neighbor algorithm and again resized using the 250 m data in the red band to generate the 250 m NDVI. Likewise, the 1 km LST and 5 km products were resized to a 250 m resolution using a nearest-neighbor algorithm. To estimate the pixel-by-pixel ET and EF, both the dry and wet edges in the -NDVI space must be determined. For a given NDVI, the surface-air temperature difference increases gradually due to water stress in the surface soil from a minimum value at the wet edge to a maximum value at the dry edge, whereas the EF correspondingly decreases from a maximum to minimum value. In general, the wet edge is identified with the lowest . In the Taihu Basin, the remained stably within 0.5 K in DOY 269, 2006. The dry edge was determined using an automatically iterative method proposed by Tang et al. . The linear relationship was determined as . The values were subsequently derived from the dry and wet edges using an interpolation scheme (3) .
The surface net radiation was estimated using (4b). The downward shortwave radiation was calculated using (A.2). The surface albedo was produced from surface reflectance in seven reflective bands using the narrowband-to-broadband conversion approach proposed by Liang . The air emissivity was estimated from the air and dew point temperature data using (A.5) . The broadband emissivity was generated from the narrowband emissivities in bands 31 and 32 . In addition, the soil heat flux was estimated from the NDVI using (5). For data processing, the parameters used to estimate and were resampled at a 250 m resolution before they were used in the estimation.
The 90 m Shuttle Radar Topography Mission (SRTM) data was used to generate the 250 m DEM data using the cubic convolution algorithm . The 250 m DEM was used to generate the local slope and aspect angles of the terrain area. Equation (6) was used to correct the data, and (8) was applied to correct . The corrected values were used to derive , EF, and, subsequently, ET. The , EF, and ET values were also generated using uncorrected values. The ET values with and without the terrain correction were then compared to examine the topographic effects. The and EF values with and without the terrain correction were also compared to investigate the major sources of the topographic effects on ET. Finally, the average values of the parameters before and after a topographic correction were calculated for the 8 slope classes in the terrain area. The relative error (RE) was used to evaluate the topographic influence in this study. The relative error can be expressed as follows: where and are the mean values of the parameters before and after the topographic correction, including the net radiation, evaporative fraction and ET.
4. Results and Discussion
The actual daily ET on DOY 269, 2006, for the Taihu Basin was derived using the triangle method with a topographic correction. The spatial variation for the estimated ET is shown. The effect of topography on ET and the contributing factors is analyzed.
Based on the principle of water balance, precipitation in the watershed equals the sum of the ET and runoff when the variation in watershed soil water content on an annual scale is ignored. Therefore, we can compare the sum of the estimated subwatershed ET and runoff with precipitation and validate watershed ET in the Xitiaoxi subwatershed. Based on meteorological and hydrological station, the observed annual total precipitation in 2006 was 1214.12 mm; the corresponding observed annual runoff depth was 514.89 mm. The estimated annual ET by remote sensing data was 825.09 mm before topographic correction and 649.30 mm after topographic correction in the subwatershed. Based on the principle of water balance, the water budget difference is the error for the ET retrieved from remote sensing if the observed precipitation and runoff data are highly accurate. The in 2006 was 125.86 mm and −49.93 mm before and after topographic correction, respectively. The average relative error of the retrieved ET was 18.0% and −7.1% before and after topographic correction, respectively, compared with ground-based water components in 2006. The validation results indicate that the ET retrieved using the triangle method before the topographic correction was overestimated by 18.0%, while the relative error of the retrieved ET after the correction was reduced by 7.1%, which demonstrates remarkably improved accuracy in the estimated ET.
Notably, direct validation of topographic effect is yet unavailable and the present validation with the water balance approach is applicable at an annual scale. The annual ET estimation may be influenced by the temporal upscaling strategy and the triangle method, in addition to topographic effect. Given the same upscaling strategy and ET estimation approach, the annual difference between the estimated ET values with and without the correction was attributable to topographic effect.
4.2. Topographic Features of the Selected Mountainous Area
The selected mountainous area accounts for 9% of the Taihu Basin (Figure 1(b)). The aspect angle, elevation, and slope in the selected area range from 0.11° to 360°, 5 m to 1509 m (244 m on average) and 0° to 69.29° (7.88° on average), respectively. Most of the area (over 84%) is located below a 400 m elevation, and less than 1% of area is higher than 1200 m (Figure 2(a)). More than 73% of the area has a slope less than 20° (Figure 2(b)). The frequency distribution of the aspect angle is similar across eight different direction bins and averages 12% (Figure 2(c)); it was greater from 45° to 180° than 180° to 315°. The relationship between elevation and slope (Figure 2(d)) indicates that the slope increases with elevation below 800 m and then decreases with elevation until approximately 1200 m. The mean slope is greater than 20° above a 1200 m elevation, which includes less than 0.1% of the selected area. Based on these statistics, the selected area includes a wide range of elevations and slopes with a nearly constant aspect angle distribution for the topographic analyses.
4.3. Effects of Topographic Correction on the ET
The ET spatial distribution over the Taihu Basin is shown in Figure 3. Lake Taihu and other water bodies exhibited the highest ET values, averaging 6.5 mm day−1. Elsewhere, the ET values range from 4.5 mm day−1 to 6 mm day−1 in flat regions in the north and southeast due to the farmland and grassland distribution. Before topographic correction, most of the ET values over the mountainous areas were, on average, lower than the values over flat regions, except for high values (more than 6 mm day−1) in the southwestern Taihu Basin area (Figure 3(a)). This observation indicates that the ET estimated using the triangle method may include a large bias over mountainous areas. After the topographic correction, the ET values over the mountainous areas were much lower than that before the topographic correction (Figure 3(b)). Furthermore, the ET values before correction showed an increasing trend with elevation, and the maximum ET values reached 6-7 mm day−1 in the selected region (Figure 3(c)). In contrast, the ET values reduced to 3 mm day−1 on average after the topographic correction (Figure 3(d)). Moreover, there were obviously differences in ET distribution for different aspect angles of the terrain. The ET values for the south-facing slopes were lower than for the north-facing slopes, and this was alleviated after the correction.
The net radiation and evaporative fraction were the key parameters used to estimate the regional ET. The and EF between the data before and after the topographic correction were compared over the selected region (shown in Figure 4). The before correction ranged from 200 to 400 Wm−2, the high value was generated for the south and southwest areas (Figure 4(a)); however, a slight decrease was observed after the topographic correction, especially in the high-altitude mountainous areas (Figure 4(b)). For the EF, a distinct decease was observed in the mountainous area after the correction (Figures 4(c) and 4(d)); the pattern clearly differed between the north- and south-facing slopes. Previous research shows that the cosine correct method for remote sensing images overcorrects when the slope is greater than 30° . However, the overcorrection phenomenon only slightly influences the selected region because the area dominated by a high slope (>30°) was <1%.
A comparison of the daily ET values before and after the topographic correction over the selected region is shown in Table 1. The corrected daily ET decreased by 7.0%–37.3% for different aspect angles. The relative error for the aspect angles between 180° and 360° (19.5%–37.3%) was greater than for the aspect angles between 0 and 180° (7.0%–12.7%); the largest error was observed for aspect angles between 225° and 315°.
Figure 5 shows the ET response to elevation and slope at different aspect angles over the selected area in the Taihu Basin. The daily ET before correction distinctly increased with elevation and slope (Figures 5(a) and 5(c)). The ET was lower after the correction than before. After the correction, the ET slightly increased with elevation, ranging from 4.1 mm day−1 to 8.0 mm day−1 for aspect angles from 0 to 180° (north facing) (Figure 5(b)), while for other aspect angles, the ET tended to decrease below a 800 m elevation and increase over 800 m, ranging from 0.5 mm day−1 to 4.6 mm day−1. Furthermore, the ET magnitude and range varied with the aspect angle. Figure 5(d) shows that the corrected ET dramatically decreased with the slope, especially for aspect angles from 180° to 360°. ET decreased by 90% from 6.8 mm day−1 to 0.6 mm day−1 at slopes over 50°. The lower ET values were observed for south-facing slopes, which may be due to a soil moisture deficiency and net radiation . The topographic effect is vegetation dependent [38, 39]. Gao et al.  indicated different responses of ET estimates to elevation in south-facing woodland and grassland. The grassland ET estimate was lower than in the woodland at elevations ranging from 1200 m to 1600 m, mainly due to the lower average LAI in the grassland. The low ET values at elevations ranging from 600 m to 800 m could be attributed to the coniferous forest distribution in the selected region.
4.4. Environmental Influence on ET Estimates
Based on (1), the regional ET was estimated using the , , and EF. The was estimated using (4a) and (4b). Among the variables in the equations, the surface temperature () and albedo were sensitive to topographic angular effects. The and albedo increased by 0.22%–1.01% and 0.79%–4.35%, respectively, after the topographic correction (Table 2). The relative errors of the two parameters for the aspect angles 180° to 360° were also greater than for the other aspect angles; these observations differ from the results in Wen et al. , where the corrected forest albedo increased in north-facing pixels and decreased in south-facing pixels. The and albedo increases induced increases in the upward longwave and shortwave radiation, which induced a decrease in the daily . Thus, after the topographic correction, the decreased by 0.63%–3.17%, and the decreased by 0.78%–2.99% (Table 2).
The EF was estimated using (2) and (3); the EF is relevant to the NDVI and . Based on Table 2, the NDVI is not distinctly different for the four terrain aspect angles with a 0.73 average. The corrected EF decreased by 9.3%–48.8% for the different aspect angles. Of the parameters in Table 2, the largest relative error was observed for the aspect angles 225°–315° because the sensor azimuth angle ranged from 90° to 105°, which yielded the lowest in (7) among the aspect angles.
The large correction in the EF was mainly attributed to the correction; however, the correction varied over a small range. To explain this phenomenon, the -NDVI triangle space was analyzed (Figure 6). The two boundaries of the LST/NDVI feature space are limiting conditions for the surface fluxes. The dry edge represents the minimum ET with unavailable of soil moisture for different vegetation index. The wet edge that corresponds to lower temperatures represents the potential ET with unlimited soil water availability for different vegetation indices. An implicit assumption of the triangle method is that the ET primarily depends on soil moisture and vegetation cover. This assumption requires a heterogeneous area with a full range of possible soil moisture and vegetation index values as well as simultaneous relatively uniform atmospheric forcing .
The triangle approach requires a full range of soil moisture and vegetation index for reliable determination of the dry and wet edges in study area. This is usually hard to satisfy for a relatively small area, for example, the selected mountainous area. Instead, the whole basin was used to determine the edges to satisfy the underlying assumption (Figures 6(a) and 6(b)). Furthermore, the dry and wet edges were determined for the case with topographic correction, and were applied to the case without correction. The reasons are the MODIS and product was only corrected for the selected region, which only slightly influences the -NDVI triangle space in the entire Taihu Basin; the EF was controlled by the relationship between the and NDVI (the dry and wet edges). When the edges were stable, the EF changes were only generated by the corrected ; thus, the relative influence of the topographic correction on the ET estimate can be evaluated. The -NDVI spatial distribution in the selected area (Figures 6(c) and 6(d)) composed part of the region, so the dry and wet edges in the entire Taihu Basin were used in the selected area (Figures 6(a) and 6(b)).
MODIS products are typically underestimated over mountain areas . Figure 6(c) shows that the before correction is lower than the wet edge when the NDVI is near 0.8, which produced an overestimated EF. However, the corrected values increased with the corresponding NDVI values. Most corrected values were distributed between the dry and wet boundaries (Figure 6(d)). The abnormal phenomenon in Figure 6(d) shows that a high value correlated with a high NDVI; the -NDVI space seemed inverse to the variation trend in the selected region. However, the local spatial distribution cannot represent the -NDVI relationship throughout the entire region. The relationship between and NDVI was controlled by the soil moisture conditions . In the selected region, the available soil moisture decreased with a rising elevation and slope, which corresponds to a lower ET and higher surface temperature for different vegetation indices. The scatters of lower ET should be close to the dry edge in the -NDVI space.
Based on (3), the EF estimate depended on the normalized difference temperature index . Using the dry and wet boundaries in the -NDVI triangle space (Figure 6), decreased with an increasing NDVI. In the topographic area with high NDVI values, the values were small compared with the absolute values but large compared with , which can cause large variations in the EF correction. In the selected area, the ranged from 6.5% to 29.3% for different aspect angles (averaging 8.7% for the aspect angles 0°–180° and 21.3% for the aspect angles 180°–360°). The ET topographic correction was similar to the EF and is attributed to the correction. Thus, the topographic correction was more important to the ET estimate over a complex terrain, especially using the triangle method.
4.5. Influence of Topographic Parameters
The and EF responses to elevation and slope for different aspect angles over the selected Taihu Basin area are shown in Figures 7 and 8. The corrected decreased with increased elevation (Figure 7(b)), which is the opposite of estimates before the correction was applied (Figure 7(a)). Moreover, the degree to which decreased for aspect angles from 180° to 360° was greater than for other aspect angles. The mean varied with elevation from 328 W m−2 to 372 W m−2 after the correction was applied. The varying trend for net radiation with elevation was determined using the combined effects of net shortwave and longwave radiation. On one hand, the air density and water content in the atmosphere tend to decrease with increased elevation, which produces an increase in atmospheric transmissivity and direct solar radiation. However, diffuse solar radiation decreases with an elevation increase, thus limiting the increase in total shortwave radiation . On the other hand, the air temperature at the screen level and decrease with increased elevation, which simultaneously reduces atmospheric downwelling and surface upwelling longwave radiation . Eventually, the net radiation decreased slightly with increased elevation. The EF response to elevation was similar to ET. The EF values for the south-facing slopes were lower than for the north-facing slopes; the mean EF varied with elevation from 0.05 to 0.8 after the correction.
Figure 8 shows that, after the correction, both the and EF dramatically decreased with the slope, especially for aspect angles from 180° to 360°. The corrected decreased by 40% from an average of 358 W m−2 to 225 W m−2 for slopes greater than 50°. The corrected EF decreased with the slope instead of increasing, which characterized the estimate before the correction. Based on Figures 8 and 2(d), most areas with large slopes (15° on average) range in elevation from 600 m to 800 m, which indicates that the low EF and ET values between the elevations at 600 and 800 m are mainly due to the large terrain slopes. The south-facing slope surfaces receive more sunlight, which heats the surface and may enhance turbulent mixing of air mass. More energy was then partitioned into a sensible heat flux, which decreases the latent heat flux (ET). Gao et al.  also indicated that the corrected ET for south-facing woodland and grassland tended to decrease with increasing elevation.
Based on Figures 5, 7, and 8, the topographic parameters (elevation, slope, and aspect angle) strongly influence the ET, , and EF estimates. Among these parameters, the aspect angles affect the ET magnitude and trend, especially for north-facing and south-facing slopes. The ET, and EF variations were more sensitive to slope than to elevation.
In this study, the cosine correction method was applied to ET estimates through topographic correction on the net radiation and evaporative fraction. This correction method for net radiation does not involve a complex radiative transfer model. Nevertheless, the cosine method does not consider diffuse irradiance from atmospheric and terrain sources, and it has an overcorrection problem [44, 45]. Assuredly, an improvement in the -NDVI triangular space will significantly improve the ET estimate accuracy, particularly its spatial distribution across large heterogeneous areas.
We should emphasize that our purpose was to evaluate the relative contribution of the topographic influence using the triangle approach to evapotranspiration estimates over mountainous areas. The ET estimates after the topographic correction was more reasonable than before the topographic correction; in particular, they exhibited realistic spatial variability across mountainous areas. In addition, the results for the influence of terrain on ET variations showed similar findings as in other studies [28, 43, 44, 46].
Remote sensing provides land surface characteristics at fine temporal and regional spatial scales to retrieve parameters relevant to land processes. Evapotranspiration cannot be measured directly from satellite observations, but it can be estimated by retrieving the evaporative fraction and net radiation from remote sensing data. The evaporative fraction was obtained from MODIS data using the triangle method. The net radiation was also retrieved from MODIS data. Considering the terrain and satellite angles, we performed topographic corrections for the triangle method to estimate the regional ET. The topographic parameters strongly influence ET. The corrected daily ET values decreased with elevation and slope; the relationships were more sensitive to slope than elevation. Furthermore, the ET magnitude and range varied with the aspect angle. The corrected ET decreased by an average of 10.4% and 32.1% for north- and south-facing slopes, respectively, over the selected Taihu Basin area. The topographic correction for ET was consistent with the EF correction, which strongly depends on the surface temperature correction. Although the topographic corrections were performed to evaluate ET retrieval over a complex terrain area, more carefully designed examinations or experiments are necessary to more comprehensively understand the relevant issues.
A. Estimation of Net Radiation Using MODIS Data
The downward shortwave radiation can be expressed as follows : where is the atmospheric clear sky shortwave transmission factor, is the solar constant at the top of the atmosphere, which is approximately 1367 W m−2, and is the solar zenith angle. For this study, we used the parameterization scheme developed by Zillman : where and is the water vapor pressure (hPa).
The longwave radiation can be expressed using the Steffan-Boltzmann equation: where is the air emissivity, is the surface emissivity, is the air temperature (K) at screen level, is the land surface temperature (K), and W m−2 K−4 is the Steffan-Boltzmann constant.
The value was calculated using the following nonlinear formula : where and denote the emissivity in bands 31 and 32, respectively. The value was estimated using the scheme given by Prata : where . The water vapor pressure can be estimated using the dew point temperature according to the Clausius-Clapeyron equation: where is the latent heat of vaporization (2.5 × 106 J kg−1), is the gas constant for water vapor (461 J kg−1 K−1), K, and is the dew point temperature (K) at screen level.
B. Daily Estimation
Bisht et al.  proposed a sinusoidal model to estimate the diurnal cycle of net radiation for clear sky days, which closely follows the framework of Lagouarde and Brunet’s  model of the diurnal cycle of surface temperature. The sinusoidal model is given as follows: where is the maximum value of estimated during the day and and are the local times at which becomes positive and negative, respectively. For a given study day, the satellite overpass time (overpass), sunrise time , and sunset time are known, so the value of can be obtained from the corresponding value of the INR estimate.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
This work was jointly supported by a Key Program of Nanjing Institute of Geography and Limnology of the Chinese Academy of Sciences (NIGLAS2012135001) and the National Natural Science Foundation of China (no. 40901162). MODIS data were obtained from the website of http://ladsweb.nascom.nasa.gov/.
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