3.2.1. Basic Framework
The proposed new parameterization scheme of EF was conducted by the combination of the
triangle method and the TVX method. The temperature-vegetation index (TVX) method, proposed by Nemani and Running [
52] and Goward et al. [
53], was mainly used to estimate near surface air temperature (
) in previous studies. It is assumed that because of the combined impact of vegetation cover on the average surface thermal characteristics and on the evaporative control of energy portioning, remotely sensed land surface temperature tends to approach air temperature with increasing vegetation cover, and the radiometric temperature of a full vegetated canopy is in equilibrium with the temperature of the air within the canopy [
39,
41]. Other assumptions involved in the TVX method are that both uniform atmospheric forcing and soil moisture conditions must take place [
38]. It should be noted that the uniformity of atmospheric forcing is also the assumption of the
triangle method, and is considered to be fulfilled on clear sky days when both the retrieval of
and VI is possible, while the assumption of uniform soil moisture is just presumably fulfilled in the isopleths of soil surface moisture within the trapezoid space. Thus, it is possible to combine these two
relationships together. In fact, as the study conducted by Sandholt et al. [
54] revealed, the isopleths of soil surface moisture within the
trapezoid space can be regarded as several superimposed TVX lines (
Figure 2).
There have been other significant studies performed to derive further the information contained in the
trapezoid space. Based on the studies conducted by Carlson [
28] and Moran et al. [
55], Long and Singh [
30] concluded that pixels along the same isopleth of soil surface moisture has the same soil surface temperature (
) and also the same surface temperature of the fully vegetated surface (
), and
of a mixed surface is a weighted sum of vegetation and soil temperatures. On the basis of this conclusion and by the combination of the
triangle method and the TVX method, the proposed new parameterization scheme of EF was performed as follows.
Figure 3 is a flowchart of this parameterization scheme. Five steps are demonstrated using different colors.
Step 1 is the calculation of the surface temperature of bare soil (
), which is marked with green color. According to the assumption of the TVX method, for each pixel (
,
) in
Figure 2, the remotely sensed
tends to approach its air temperature (
) with increasing of vegetation cover under the same soil moisture conditions, and the radiometric temperature of a full vegetated canopy (
) is in equilibrium with the temperature of the air within the canopy. Thus,
. Variation in
for the same isopleth of soil surface moisture results essentially from the variation in
, and the remotely sensed
is a weighted sum of vegetation and soil temperatures [
30], which is defined as:
For a mixed pixel, and is the surface temperature of full vegetated canopy and bare soil, respectively. Therefore, the surface temperature of the bare soil () corresponding to the isopiestic line which the given pixel (, ) belongs to can be deduced as the intercept of this function.
Step 2 is the calculation of a simplified land surface dryness index (Temperature-Vegetation Dryness Index, TVDI) proposed by Sandholt et al. [
54], which is marked with yellow color. According to Sandholt et al. [
54], the TVDI for each pixel in the
triangle space can be retrieved as:
where
and
is the maximum and minimum surface temperature of each
class, respectively (
Figure 2). For bare soil (
), TVDI can be defined:
It is clear that there is negative correlation relationship between TVDI and soil surface moisture status. TVDI ranges from 0 to 1 with soil moisture availability decreasing from the wet edge to the dry edge.
Step 3 is the calculation of the Priestley-Taylor parameter of bare soil (
), which is marked with blue color. For unsaturated soil, a parameterization scheme of EF was proposed by Komatsu [
56] from the soil moisture status as follows:
where
is the Priestley-Taylor parameter,
is the surface soil moisture, and
is a parameter that depends on soil type and wind speed. Similar to TVDI,
is also a dimensionless parameter ranging from 0 to 1, which represents relative soil moisture status, but there is negative correlation relationship existing between TVDI and
[
57]. Combing Equations (1) and (9) and replacing
with (
), the parameter
in Equqtion (1) can be written as
The subscript “s” indicates that the parameter is only valid for bare soil.
Step 4 is the calculation of the Priestley-Taylor parameter of full vegetated canopy (
), which is marked with orange color. At the same isopiestic line corresponding to
, the parameter
of the full vegetated canopy (
) is calculated as
, just as Jiang and Islam did in
Section 3.1 [
16,
19]. However, it should be noted that
in Jiang and Islam’s parameterization scheme is calculated by using the constant
as input, while
in our new parameterization scheme is calculated by using
as input, and varies with isopleths of soil surface moisture.
Step 5 is the calculation of the Priestley-Taylor parameter of a mixed pixel (
), which is marked with red color. Along each isopiestic line, because of the invariance of soil moisture, the parameter
is only determined by the variation of vegetation cover, and increases from
to
with the increasing of
. After the lower and upper bounds of
values for each isopiestic line have been determined, the
value for each pixel within this isopiestic line is interpolated as follows:
Similar to the parameterization scheme developed by Jiang and Islam [
21,
24], the proposed new parameterization scheme was also performed based on an extension of Priestley-Taylor equation. The parameter
for each pixel within the trapezoid space was interpolated linearly from the established upper and lower bounds in both of these two schemes. What is different is that the interpolation procedure in Jiang and Islam’s parameterization scheme was performed for each
interval, while in the proposed new parameterization scheme it was performed for each isopiestic line of soil surface moisture. As for the determination of the limiting bounds, there is not much difference in the determination of the upper bound (
in Jiang and Islam’s scheme and
in our scheme). Both of them are calculated as
. However,
in Jiang and Islam’s parameterization scheme is calculated by using the constant
as input, while
in our new parameterization scheme is calculated by using
as input. Therefore,
is a constant in the traditional approach, while
varies with isopleths of soil surface moisture in the NPS. Besides, because
increases with the increase of
, the pixels with more stressed soil moisture conditions have a lower value of
. Therefore, compared with the constant
in Jiang and Islam’s scheme, the variable
in the new parameterization scheme is more reasonable, because it accounts for the impact of soil moisture conditions partly. In Jiang and Islam’s parameterization scheme, the determination of the lower bound (
) for each vegetation class
was interpolated linearly as
. According to the research conducted by Stisen et al. [
23], the weakness of this parameterization of
, where evaporative fraction is not zero along the observed dry edge, is that it does not allow for the presence of water stressed full cover vegetation. In our proposed new scheme, the lower bound (
) for each isopiestic line was determined under bare soil conditions based on the TVDI proposed by Sandholt et al. [
54] and the parameterization scheme proposed by Komatsu [
56]. This is an alternative way to bypass the water stress involved in the mixed surface (vegetation/bare soil), and thus can overcome the weakness involved in the parameterization of
. Another great advantage of the new parameterization scheme is that both the maximum surface temperature of the bare soil and fully vegetated canopy is required in the parameterization scheme proposed by Jiang and Islam [
21,
24], while only the maximum surface temperature of the bare soil (
) is indispensable in the new parameterization scheme, and thus the reliance of the
triangle method on the dry edge has been reduced significantly.
3.2.2. Estimation of Near Surface Air Temperature
In the proposed new parameterization scheme, the isopiestic lines of soil surface moisture were assumed to be several superimposed TVX lines [
54]. Besides remotely sensed
and VI, near surface air temperature (
) is another variable needed to determine the TVX lines. A simple and operational algorithm was developed by Zhu et al. [
49] to retrieve the instantaneous
for the Southern Great Plains (SGP) of the United States of America. They find that the systematic errors caused by near surface air temperature (
) retrieved from MOD07_L2 product and
retrieved from MOD06_L2 product are in completely different directions. This means that these errors can balance each other out by the combination of
and
, especially when the absolute values of the errors caused by them are similar to each other, which is just the case in our study. Therefore, the instantaneous
under clear sky conditions was estimated as the average of near surface air temperature (
) retrieved from MOD07_L2 product and land surface temperature (
) retrieved from MOD06_L2 product. The validation results are shown in
Figure 4. Specific statistical measures used in the validation are shown in
Table 3. The coefficient of determination (R
2), MAE, RMSE, RRMSE and B are 0.94, 2.28 K, 3.02 K, 0.01 and 1.72 K, respectively.
of the same 21 days in the SGP region was also estimated by Sun et al. [
50] using an empirical method proposed by Zaksek and Schroedter-Homscheidt [
58] with a R
2 of 0.91, and RMSE of 3.1 K. Therefore, the accuracy of the algorithm developed by Zhu et al. [
49] applied in the SGP is acceptable.