**4. Discussion**

### *4.1. Uncertainty of Actual Evapotranspiration Estimation by the CR Approach*

Using only routine meteorological variables (air temperature, relative humidity, wind speed, and net radiation) in the CR approach can better estimate daily or monthly *E* in the frozen ground regions of the QTP according to the present study; however, the uncertainty of the CR approach itself, physical variable calculations, and key parameter values are still non-negligible for estimating *E* on different spatiotemporal scales.

For the K2006 model, characterizing variables such as *Epa* and *Epo* is difficult. Although using the Penman equation to calculate *Epa* is widely accepted, determining *Epo* is controversial because a fixed analog of the Priestley-Taylor coefficient *αe* usually cannot vary with climate, which means that the value of *αe* cannot truly reflect the interaction between the land and atmosphere [65,66]. We also need to know atmospheric status–such as air temperature–under well-supplied water conditions at the land surface before determining *Epo*, despite some progress [61,62] that has been made in obtaining air temperature under well-supplied water conditions. However, usual iterative solving exists for unsolved or anomalous solutions, which makes it difficult to promote the above method as a universal approach.

For generalized complementary functions, in addition to the above common problems, the definition of boundary constraint conditions is another one important problem. Han [65] noted that boundary constraint conditions determine the domain of definition and the analytic formula of complementary functions. Many of the latest debates [67,68] on the CR-based functions have focused on boundary constraint conditions, which is controversial for the CR approach. The above-mentioned problems reflect a grea<sup>t</sup> lack of understanding of the evaporation process on different spatiotemporal scales for frozen ground regions on the QTP, where the current CR function forms seem to perform inferiorly in the specific regions described by Wang [42].

For the CR principle, an important prerequisite of the CR approach is that at large and homogeneous land surfaces, the influence of air advection could be negligible, so atmospheric evaporation demand is totally caused by feedback of the land surface. Morton [26] pointed out that the CR principle should be applied at spatial resolutions larger than 1 km and temporal resolutions longer than five days, since large-scale weather fronts may bring air masses over the land with a moisture signature decoupled from the underlying surface, which thus may temporarily disrupt the dynamic equilibrium of air humidity and surface fluxes in the land–atmosphere system. Although previous studies [28,38,42,69] have applied the CR approach to hourly or daily *E* estimations, the theory of the CR principle on short timescales still needs to be improved.

### 4.1.1. Influence of Parameter Values on Actual Evapotranspiration Estimation

Determining the parameter values of CR functions is urgen<sup>t</sup> for the application of each CR-based model to estimate *E*, as well as for the development of a CR approach. The K2006 and H2018 both have two parameters, *αe* and *b*; the B2015, S2017, and C2018 all have one parameter, *αe* (note: parameter *c* in the B2015 model is an adjustable parameter, *c* = 0 often in most conditions, so, here we did not discuss parameter *c*). To determine the impacts of parameter values on evapotranspiration estimation, we discuss the following two problems: (i) difference in simulated results by each CR-based model with parameter calibration on different time lengths and (ii) parameter sensitivity for actual evapotranspiration estimation.

Here, we first investigated the influence of calibrated parameter values by different time lengths (whole year and warm season) on simulating daily *E* during the warm season. For the cold season, due to higher uncertainty of observations, we did not discuss the results here. Table 4 compares the parameter values and simulated NSE values during the warm season by the calibration period of one whole year and the corresponding warm season, respectively. The results indicated that the time length of parameter calibration did not have much of an impact on simulated *E* during the warm season, parameter *αe*

was very close between the two calibration periods, and the differences in parameter *b* was slightly larger. According to our results, the simulated *E* was insensitive to the variations in parameter *b* in the K2006 and H2018 model, because even if the difference in parameter *b* was large, like the K2006 model at NAMORS, the simulated daily *E* between the two calibrated parameter values were still approximate. However, variations in parameter *αe* in all five CR-based models exerted more of an influence on the simulated results. The above results indicated that parameter calibration—especially *<sup>α</sup>e*—in CR-based models at specific sites is more important to simulate daily *E* compared with different time lengths of calibration periods in the frozen ground regions of the QTP.


**Table 4.** Comparison of simulated results by different CR-based models using two different calibrated parameter values during the warm season. One is calibrated by a whole year period, another is calibrated by a corresponding warm season.

### 4.1.2. Sensitivity Analysis of CR-Based Models to Parameter Values

Then, we tested the sensitivity of five CR-based models to parameter values at each site, by adding increments from −50 to 50% at an interval of 10% to optimized parameter values for each CR-based model. Figure 8 displays the RMSE of the simulated *E* values for the in situ measurement. It is clear that variations in *b* values combined with constant *αe* values have little impact on evapotranspiration estimation, and a 50% variation in the *αe* value combined with constant *b* values led to increased RMSE values within 0.5 mm d−1. The K2006, B2015 and H2018 models exhibited lower sensitivity to variations in parameter values, and the H2018 model had the lowest sensitivity among all CR-based models.

We also noticed that larger RMSE values for the S2017 and C2018 models when the optimized *αe* value increased more than 20% because there were many abnormal *E* values, which indicated that the S2017 and C2018 models were more sensitive to parameter values, especially when the *αe* value increased more than 20%. Brutsaert [70] found that *αe* was closely related to the aridity index (AI, AI = *Epa*/P); accordingly, based on the global distribution of the parameter *αe*, the *αe* values were mainly from 0.8 to 1.2 around the world, except in extremely arid and wet regions. Assuming *αe* values were within the above scope in the present study–which means *αe* mainly varied from about −10% to 20% of its optimized values for all five CR models at each field site–according to Figure 8, the averaged RMSE increased by 0.14 mm d−1, 0.12 mm d−1, 0.15 mm d−1, 0.09 mm d−<sup>1</sup> from the RMSE of the *E* estimated when using optimized *αe* values at TGL, XDT, BJ, and NAMORS, respectively.

**Figure 8.** Sensitivity analysis of CR-based models to the parameter *αe* and *b* at (**a**) TGL, (**b**) XDT, (**c**) BJ, and (**d**) NAMORS. Δ*αe* represents the increment to the optimized *αe* values for all five CR-based models and Δ*b* represents the increment to the optimized *b* values for K2006 and H2018 model. Suffix "(*α*)" and "(*b*)" represents simulated actual ET with perturbations of *α* (with constant *b*) and *b* (with constant *α*), respectively.

Thus, the parameter *αe* value is very important to the accuracy of *E* estimation according to our study; certainly, the smaller deviation of optimized *αe* value would exert little impact on the accuracy of *E* estimation. Furthermore, the K2006 and H2018 models were completely insensitive to deviations of parameter *b* values.

#### *4.2. Comparison with Previous Studies on the QTP at a Single Point Scale*

Here, we compared estimation results of monthly *E* by five CR-based models in the present study with the latest two improved remote-sensing ET models that have been validated for the QTP. One is an improved MOD16 model by Yuan [71], referred to as "MOD16\_Yuan" in this study; the another one is an improved SEBS model by Han [72], referred to as "SEBS\_Han" in this study. Both obtained better accuracy with in situ measurements than that in previous results. Due to shared use observation data with Yuan's work at NAMORS in 2009 and with Han's work at BJ in 2013, the above two field sites during the corresponding period were chosen for comparison. Note that there are some differences

in the observed *E* between this study and the above two works, which may be caused by different data procedure processes, such as quality control and gap-filling approaches.

Figure 9a exhibits the estimation results of monthly *E* by the CR approach and MOD16\_Yuan model at NAMORS. Statistical results showed that in situ measured annual *E*values ranged from 467.2 to 481.6 mm, and the annual *E* values simulated from the five CR-based models were 495.5 mm (K2006), 455.6 mm (B2015), 371.1 mm (S2017), 418.4 mm (C2018), and 452.8 mm (H2018), respectively. For the MOD16\_Yuan model, the annual *E*values reached 539.7 mm which was overestimated by 12.1% compared with in situ measurements. The largest bias occurred in June when the positive bias could reach 36 mm. The K2006 model overestimated the annual *E* by 6.1% with smaller positive deviations than the MOD16\_Yuan model. The B2015 and H2018 models obtained better accuracy with smaller negative bias–approximately 20 mm on an annual scale–and both two CR-based models performed better than the K2006 and MOD16\_Yuan models. However, the S2017 and C2018 models both performed poorly due to large negative biases, especially the S2017 model, with a bias of approximately 100 mm on an annual scale.

**Figure 9.** Comparison of monthly observed *E* values (filled area enclosed by calculated results from this study and another corresponding work) and simulated *E* values (solid lines) of CR-based models with improved MOD16 evapotranspiration model by Yuan [71] at (**a**) NAMORS; with improved SEBS model by Han [72] at (**b**) BJ.

Figure 9b is the same as Figure 9a, but for BJ. Due to the lack of observed *E* data in some months, only available data during the corresponding month were analyzed. The simulated monthly *E* values by the five CR-based models were very close to each other– which could also capture the monthly variations in observed E–but a larger positive bias occurred in August. The SEBS\_Han model performed well in August; however, it significantly overestimated monthly *E* values in May, June, and July, with a larger positive bias than that of the five CR-based models. Thus, the overall performance of the CR approach is still better than that of the SEBS\_Han model in the present study.

Both the MOD16\_Yuan and the SEBS\_Han models have solid physical foundations, and the MOD16\_Yuan model based on the original MOD16 model takes soil moisture and soil texture into soil evaporation estimation, and simultaneously optimizes canopy transpiration estimation. The SEBS\_Han model introduces the description of the form drag caused by subgrid-scale topographical obstacles, the effective roughness lengths for momentum, and sensible heat transfer into the SEBS model. The above two models both improved the description capability of remote sensing models for the physical process of ET. For the CR approach, from another aspect of feedback of atmospheric evaporation demand to land surface moisture conditions, with a few unknown parameters and routine meteorological variables, the CR approach has also been demonstrated to have comparable accuracy with current sophisticated ET models in the present study.

### *4.3. Perspectives from the Present CR-Based Model Evaluations*

This study provided a relatively comprehensive assessment about applicability of the CR-based models to frozen ground regions on the QTP. A performance comparison of five

CR-based models was evaluated at four observation field sites. After local calibration, the *E* estimated by all five CR-based models captured daily variations; however, consistent with the findings of previous studies, it is difficult to capture daily variations during cold seasons (Figure 5). The reason for poor performance during cold seasons is still unclear. Theoretically, variable *Epo* (approximated by the Priestley-Taylor equation) in the CR approach assumes evaporation for an extensive saturation with minimal advection; however, during cold seasons, air advection is usually stronger than that in warm seasons. Thus, greater deviations may be derived from the conduction of *Epo* by only local air temperature and humidity in cold seasons, and the degree of land–atmosphere coupling during the cold season is usually weaker than that in warm seasons. The above conditions for cold seasons hardly satisfy the requirements of the CR approach.

Han [65] pointed out that accurate terrestrial ET estimation depends on precisely determining the land surface and atmospheric status; however, the deficiencies of CRbased models are focused only on the atmospheric status and neglect the land surface status, and the influence of land surface status on evaporation processes may not be fully captured through changes in atmospheric status alone, especially at small spatiotemporal scales. The CR principle may be an alternatively valuable approach when there is a lack of land surface information in the past, but for now, remote-sensing technologies make obtaining information on land surface status significantly easier. The CR approach also needs to take land surface information into account. Currently, some studies [70,73,74] have introduced shallow soil moisture, the vegetation index, or AI into the CR approach when applied at large-scale regions. Some other studies [75,76] also adhere to using only atmospheric status information, developing a free-calibrated CR approach when estimating large-scale ET. Further assessments about two different CR approaches need to be explored in the future.
