Clear skies and cloudy conditions should be discriminated between before performing DLR parametrization, which is achieved by the synthetical analysis of DSR, DLR and CBH from the MPL.

Following the method initiated by Crawford and Duchon (1999), we calculate two quantities reflecting DSR magnitude and variability based on 1 min observed DSR (DSRobs) and calculated clear-sky DSR (DSRcal) values. DSRcal is calculated by the C model of Iqbal (1983), in which direct and diffuse DSR are parametrized separately. Direct DSR (DSRdir) is calculated as follows. 1DSRdir=S0τrτwτoτaτg, where S0 is solar constant and τr, τw, τo, τa and τg are transmittances due to Rayleigh scattering, water vapor absorption, ozone absorption, aerosol extinction, and absorption by uniformly mixed gases O2 and CO2, respectively. Diffuse radiation is estimated as the sum of Rayleigh and aerosol scattering as well as multiple reflectance. Total ozone column (DU) is provided by a Brewer spectrophotometer. Values of w (cm) are from Vaisala RS92 radiosonde profiles in AL and Global Positioning System measurements in NC and NQ. They are used to create linear regression relationships to collocated ground level e (hPa) measurements, which are then used to estimate w from 1 min measurements of e. The Ångström wavelength exponent and Ångström turbidity are from CE318 sun photometer observations in NC and AL, while in NQ we adopt the same value as that in AL because the altitudes of the two sites are similar. The climatic value of single-scattering albedo retrieved from long-period CE318 observation in Lhasa is 0.90 (Che et al., 2019), which is used in three stations. This is reasonable because of high altitude and extremely low aerosol loading over the TP. Surface albedo is 0.25 and 0.22 in Al and NQ according to in situ measurements (Liang et al., 2012). In NC, it is 0.183 (Zhao et al., 2011).

DSRcal values are first scaled to a constant value of 1400 W m-2 for each minute of each day. We adopt this value according to Duchon and O'Malley (1999) and Long and Ackerman (2000). Afterwards, DSRobs values are scaled by multiplying the same set of scale factors. Finally, the mean and standard deviation of the scaled DSR in a 21 min moving window (±10 min centered on the time of interest) are used for cloud screening. The selection of the width of 21 min is empirical but a consequence of having a reasonable time span for estimating the mean and variance (Duchon and O'Malley, 1999). Clear-sky DSR should satisfy three requirements: (1) the ratio of DSRobs to DSRcal is within 0.95 to 1.05; (2) the difference between scaled DSRobs and DSRcal is less than 20 W m-2; (3) standard deviation (δ) of scaled DSRobs in a 21 min moving window is less than 20 W m-2.

Temporal variability in DLR is also used for cloud screening according to Marty and Philipona (2000) and Sutter et al. (2004). Here, δ of scaled DLR (scaled to 500 W m-2) in a 21 min moving window is used for this purpose. A cloud-free sample is determined if δ is less than 5 W m-2.

Since both DSR and DLR experience difficulties in detecting clouds in the portion of the sky far away from the sun (Duchon and O'Malley, 1999) or high-altitude cirrus clouds (Dupont et al., 2008), coincident MPL backscatter measurements are used to strictly select clear-sky samples. There should be a cloud element somewhere in the sky when the MPL identifies cloud; it is thus required that no clouds are detected by the MPL in a 21 min moving window, otherwise it is defined as cloudy.

Given the fact that these methods are complementary to each other to some extent (Orsini et al., 2002), we use the following strategy to guarantee a proper selection of clear-sky samples. If DSR, DLR and MPL measurements at the time of interest synchronously satisfy these specified clear-sky conditions, the sample is thought to be taken under unambiguously cloud-free conditions; on the contrary, the measurements are made under unambiguously cloudy conditions if any method suggests cloudy conditions. Our following clear-sky and cloudy DLR parametrizations are based on measurements under unambiguously cloud-free (8195 min) and cloudy conditions (69 318 min), respectively.

Figure 1 shows an example of clear-sky discrimination results based on our method. DSRobs presents a smooth temporal variation from sunrise to about 14:00 (LT), being consistent with DSRclr. Similarly, DLR also varies very smoothly during the same period when 21 min standard deviations of DLR are <5 W m-2. Both facts suggest sunny and cloudless skies. This inference is supported by the MPL that suggests no cloud is detected overhead. Contrarily, abrupt changes of 1 min DSRobs and DLR are evident during 14:00–17:00 LT, and we can see DSRobs occasionally exceeds the expected DSRclr, indicating the frequent occurrence of fair-weather cumulus clouds. The MPL detects a persistent thin cloud layer at 4 km aboveground, which agrees with DSR and DLR measurements very well.

Given synoptic cloud observations are very limited and temporally sparse, various parametrizations using DSR or DLR data have been developed to estimate CF (e.g., Deardorff, 1978; Marty and Philipona, 2000; Dürr and Philipona, 2004; Long et al., 2006; Long and Turner, 2008). Because of good agreement between clear-sky DSRobs and DSRcal calculated by the Iqbal C calculations (Iqbal, 1983; Gubler et al., 2012), with a mean bias of 1.7 W m-2 and root-mean-square error (RMSE) of 10.7 W m-2 (not shown), we use the Deardorff (1978) method to calculate CF from DSRobs and DSRcal. The method is based on a fairly simple cloud modification to DSR as follows. 2CF=1-DSRobsDSRcal CF (no unit) has values ranging from 0 to 1. To avoid the error caused by abrupt DSR variation, the 21 min mean DSR value rather than its instantaneous measurements are used here.

A total of 11 clear-sky DLR (DLRclr) parametrizations (Table 2) are evaluated based on 1 min DLR measurements under unambiguously cloud-free conditions. To compare the performance of these 11 models, the RMSE and the coefficient of determination (R2) are shown by a Taylor diagram in Fig. 2a. Relatively smaller RMSEs (generally <15 W m-2) and larger R2 values (>0.95) are derived for the Brutsaert (1975), Konzelmann et al. (1994), Dilley and O'Brien (1998), and Prata (1996) models. This is likely because these parametrizations were developed in cool and dry areas, for example, in England (Brutsaert, 1975), Greenland (Konzelmann et al., 1994) and dry desert region in Australia (Prata, 1996). The climate in those areas is likely similar to that over the TP to some extent, so those parametrizations are expected to perform well. The higher RMSE (>37 W m-2) and the lower R2 (∼0.7) are derived for the Swinbank (1963) and Idso and Jackson (1969) models. This can be partly explained by the fact that only Tis used in these two methods. Previous studies suggest substantial uncertainty (RMSE >37.5 W m-2 and R2<0.75) if the water vapor effect on DLRclr is not accounted for (Duarte et al., 2006). Since w is very low over the TP and thereby DLR is highly sensitive to variation in w in that case, much more attention should be paid to the water vapor effect on the parametrization of DLRclr.

The coefficients in 11 parametrizations (Table 2) were originally calibrated and determined in different geographical locations; therefore, they may not be the optimal values for the TP. Thus we make use of 1 min clear-sky DLR samples to locally calibrate the parameters of these parametrizations. We use the 10-fold cross-validation method to determine the parameters. This is a widely used method to estimate the skill of a regression model on unseen data. It is expected to result in a less biased or less optimistic estimate of the model skill than other methods, such as a simple train–test split (James et al., 2013). All the data are randomly divided into 10 groups of approximately equal size; the coefficients are computed by using nine groups as a training set, and the remaining group is used as validation. This procedure is repeated 10 times to get the representational value of coefficients (with the lowest test error).

The coefficient values derived from the nonlinear least-squares fitting of the DLRclr parametrizations (Table 2) over the TP are presented in Table 3. For each fitted parametrization, we calculated the RMSE and R2, and the results are shown in Fig. 2b. When using the parametrizations with the locally fitted parameters, the accuracy of the parametrization relative to the published values is obviously improved. Most RMSEs are <10 W m-2 except the parametrization proposed by Swinbank (1963) and Idso and Jackson (1969) that still produce the worst results (with R2 of 0.71 and RMSE of 15 W m-2) even after the parameters are locally calibrated. This is probably because e is not considered in these two methods.

The Dilley and O'Brien (1998) parametrization, which is initially developed by considering the adaptation of climatological diversities, is expected to be able to fit the measurements in tropical, midlatitude and polar regions. This expectation is verified by its wide deployment in DLRclr estimations in different climate regimes and altitude levels, for example, in tropical lowland (eastern Pará state, Brazil) and mild mountainous area (Boulder, United States; Marthews et al., 2012; Li et al., 2017). The present study confirms that the Dilley and O'Brien (1998) parametrization is the best clear-sky parametrization over the TP. The locally calibrated equation is as follows. 3DLRclr=-2.53+158.10×T273.166+106.40×46.50×eT2.5012 The RMSE and R2 of Eq. (3) are ∼3.8 W m-2 and >0.98, respectively, which are substantially lower than those in previous studies over the TP; for example, the RMSE was 9.5 W m-2 (Zhu et al., 2017). The Dilley and O'Brien (1998) parametrization was suggested to give the most reliable estimates of DLRclr over the TP (Zhu et al., 2017). Note that the parameters here differ quite a lot from their values (Zhu et al., 2017), as shown in Eq. (4). 4DLRclr=30.00+157.00×T273.166+97.93×46.50×eT2.5012 Figure 3 compares instantaneous clear-sky DLR data from measurements with calculations by Eq. (3) of this study and by Eq. (4) from Zhu et al. (2017). The former performs very well as shown by an overwhelmingly large number of data points falling along or overlapping the 1:1 line. By contrast, the latter overestimates DLR by 25 W m-2 (10 %). This difference is not very likely due to different DLR measurements used to produce Eqs. (3) and (4) giving the following considerations. First, this systematic overestimation is much larger than the expected uncertainty in DLR measurements (2.5 % or 4 W m-2; Stoffel, 2005). More important, comparison of cloudy DLR parametrizations between this study and Zhu et al. (2017) showed good agreement (not shown). Note that only 1 h CG3 DLR observations are used for clear-sky discrimination in Zhu et al. (2017). This method was shown to be very likely contaminated by the thin high cloud (Sutter et al., 2004). This certainly would produce an overestimation of clear-sky DLR parametrization since larger DLRs are associated with potential residual clouds relative to real clear-sky DLRs.

Parametrizations of cloudy-sky DLR (DLRcld) are based on estimated DLRclr coupled with the effect of cloudiness or cloud emissivity, which depends primarily on CF as well as other cloud parameters, like CBH and cloud type (Arking, 1990; Viúdez-Mora et al., 2015). Four parametrizations (Table 4), which modify the bulk emissivity depending on CF, are assessed and locally calibrated in this section.

DLRclr is estimated according to Eq. (3). The fitted values of the coefficients (using 10-fold cross validation) of the four cloudy parametrizations are presented in Table 4. The RMSEs and R2 values of original and locally fitted parametrizations over the TP are presented in Fig. 4.

Relative to clear-sky conditions, cloudy parametrizations using the given parameters have higher RMSEs (generally exceeding 35 W m-2) except the one developed by Jacobs (1978) (RMSE of 18 W m-2). R2 was generally smaller than 0.9. RMSE values decrease significantly in Maykut and Church (1973) and Sugita and Brutsaert (1993) as locally calibrated parameters are used. Relatively smaller and almost no RMSE improvements are found for the methods developed by Konzelmann et al. (1994) and Jacobs (1978), respectively.

Equation (5) shows the best cloudy-sky parametrization over the TP by combining the clear-sky parametrization of Dilley and O'Brien (1998) with the cloud modulation correction scheme of Jacobs (1978). 5DLRcld=1+0.23×CF×59.38+113.70×T273.166+96.96×46.50×eT2.5012 The RMSE and R2 are ∼18 W m-2 and ∼0.89, respectively. The RMSE here is close to the 15 W m-2 obtained in different-altitude areas in Switzerland (Gubler et al., 2012) and slightly lower than the 23 W m-2 obtained in mountainous area in Germany (Iziomon et al., 2003). Compared to previous studies over the TP (RMSE of 22 W m-2 in Zhu et al., 2017), our cloudy model produces better results.

In order to validate the newly developed DLR parametrizations, clear-sky and cloudy-sky DLR parametrizations are validated against DLR measurements at Lhasa. The results are shown in Fig. 5. Compared to the existing parametrizations, Eqs. (3) and (5) produce the smallest bias (both less than 2 W m-2) and RMSE (Eq. 3 bias is less than 5 W m-2, and Eq. 5 bias is less than 25 W m-2). This independently demonstrates that the improved DLR parametrizations can be used in other stations over the TP.

Since clouds behave approximately as a blackbody, the most relevant cloud parameter (besides CF) to DLR under overcast skies (DLRovc) is CBH (Kato et al., 2011; Viúdez-Mora et al., 2015). Firstly, CBH defines the temperature of the lowest cloud boundary, which through the Stefan–Boltzmann law drives the cloud emittance; secondly, DLR emitted by the atmospheric layers above a cloud is totally absorbed by the cloud itself (clouds are thick enough). Radiative-transfer-model simulation has suggested that CBH under overcast conditions is an important modulator for DLR. The cloud radiation effect (CRE), the difference between DLRobs and DLRclr, decreases with increasing CBH at a rate of 4–12 W m-2 that depends on climate profiles (Viúdez-Mora et al., 2015). This indicates that overcast DLR parametrization would be improved if CBH were considered.

A close relationship between CRE and CBH under overcast conditions over the TP is presented in Fig. 6. Compared to Viúdez-Mora (2015) results derived in Girona, Spain, a midlatitude site with low altitude, CRE over the TP is generally lower by 5–10 W m-2. This is likely because clouds over the TP with the same CBH as that at Girona have relatively lower temperature, thereby producing lower radiative effect on DLR. CRE generally decreases as CBH increases. The result agrees with the expectation since CBH influence on DLR should decrease as CBH increases as a result of increasing water vapor effects on DLR. According to Fig. 6, CRE is about 70 W m-2 for clouds <1 km and decreases to ∼40 W m-2 for clouds at 3–4 km over the TP. The decreasing rate of CRE with CBH is estimated to be -9.8 W m-2 km-1 over the TP, which agrees with model simulations (Viúdez-Mora et al., 2015).

Since the CBH effect on overcast DLR is apparent, we introduced a modified parametrization to consider the CBH effect on DLR under overcast conditions. A linear correlation is firstly established based on the measured CBH and the ratio of observed DLR (DLRovcobs) and calculated DLR by Eq. (5) (DLRovccal) under overcast conditions in Fig. 6. Since we can see that DLRovccal is equal to DLRclr times 1.23 (because CF is equal to 1 in Eq. 5), we derived a CBH-corrected DLRovc parametrization as follows. 6DLRovc=1.23×DLRclr×(1.0746×CBH), where CBH is given in kilometers. The bias and RMSE of Eq. (6) between measurements and calculations are -2.15 and 19.79 W m-2, respectively, which are significantly lower than those of Eq. (5) (10.3 and 21.4 W m-2) in overcast conditions. The result indicates a remarkable improvement in the estimation of DLR under overcast conditions by introducing CBH to the DLR parametrization; therefore, the introduction of such instruments as ceilometers to measure CBH is highly significant for studying clouds' impacts on DLR.