The MODIS LST products, including both the MOD11C1 and MYD11C1 LST products in Collection 6 from 2003 to 2019 (available at https://ladsweb.nascom.nasa.gov/, last access: 1 March 2020), were used to help the generation of Tdm. The MODIS LSTs were retrieved with a refined generalized split-window algorithm, and their accuracies are mostly within 1.0 K over homogeneous surfaces (Zhengming and Zhao-Liang, 1997; Duan et al., 2019; Wan, 2014). The MOD11C1 and MYD11C1 LST products cover the global land surfaces four times per day with a spatial resolution of 0.05∘. At low-latitude and midlatitude regions, MOD11C1 LSTs are obtained around 10:30 and 22:30 (local solar time), and MYD11C1 LSTs are around 01:30 and 13:30 (local solar time) with a time interval of around 1.5 h. At high-latitude regions, due to the convergence of satellite orbit (Fig. A1), the overpass times possess a significant shift from those at low-latitude and midlatitude regions (Østby et al., 2014). More details on the time shift and its impact on the estimation of Tdm with the IADTC framework are provided in Sects. 3.1.3 and 5.2.

Surface air temperatures (SATs) are used to drive the ATC model for the reconstruction of under-cloud LSTs (see Sect. 3.1). We employed the SATs from the MERRA2 reanalysis dataset (the specific collection name is inst1_2d_lfo_Nx, obtained from https://disc.gsfc.nasa.gov/datasets/M2I1NXLFO_V5.12.4/summary, last access: 9 June 2020) from 2003 to 2019 (Gelaro et al., 2017; GMAO, 2015). The spatial and temporal resolutions of these reanalysis SAT data are 0.5×0.625∘ and 1 h, respectively.

The in situ LST measurements from 133 globally distributed stations (Fig. 1) were used to validate the IADTC framework at site level (see Sect. 3.2.1) as well as to evaluate the GADTC product (see Sect. 3.2.2). They include seven SURFRAD (Surface Radiation Budget Network) sites (Augustine et al., 2000) and 126 FLUXNET sites from FLUXNET2015 datasets (Pastorello et al., 2020). These two datasets have been widely used for validating satellite-derived LSTs due to their extensive distribution, rigorous quality control, and long-term availability (Guillevic et al., 2018; Martin et al., 2019; Duan et al., 2019).

The OADTC framework consists of two steps to generate Tdm (Hong et al., 2021): (1) reconstruction of instantaneous under-cloud LSTs with an ATC model to ensure the availability of four valid LSTs at the four daily overpass times and (2) simulation of diurnal LST dynamics using a four-parameter DTC model and estimation of Tdm. This study improved the OADTC framework using a more advanced ATC model as well as by optimizing the estimation of Tdm with the DTC model. The generation of global gap-free Tdm with this improved framework (termed the IADTC framework) includes four steps (Fig. 2): data preprocessing (Sect. 3.1.1), under-cloud LST reconstruction with an advanced ATC model (Sect. 3.1.2), linear interpolation of MODIS overpass time (Sect. 3.1.3), and Tdm estimation with a DTC model (Sect. 3.1.4).

The GADTC products were validated from the following two aspects: (1) validating the IADTC framework indirectly with single-source in situ measurements at the site level and (2) validating the GADTC products directly by comparing with in situ measurements. These two aspects complement each other and allow us to assess the applicability of the IADTC framework and the accuracy of the generated GADTC products. The direct comparison of the GADTC product with in situ measurements (SURFRAD and FLUXNET measurements for this study) provides information on the accuracy of the IADTC framework, especially over homogeneous areas (Guillevic et al., 2018). However, such direct validations can be affected by uncertainties beyond the IADTC framework, e.g., a mismatch of spatial scale between satellite and in situ measurements, different observation angles, and uncertainties from the LST retrieval algorithm (Ermida et al., 2014; Guillevic et al., 2014; Li et al., 2014). Therefore, direct comparisons may not fully reflect the true accuracy of the IADTC framework. To address this issue and assess the applicability of IADTC framework, we validated the IADTC framework indirectly by driving it with in situ measurement and then using hourly measurements for validation. This strategy avoids the mismatch issue of multi-source data and can, therefore, better reflect the accuracy of the IADTC framework (Hong et al., 2021).

We analyzed the difference in LST values and trends between Tdm_cloud_free (the daily mean LST estimated by the traditional method) and the GADTC products. For the difference in LST values, we present the global spatial distribution of ΔTsb using the GADTC product as the reference (see Sect. 4.3). For the difference in LST trends, the seasonal Mann–Kendall test and Theil–Sen slope were used to diagnose the warming/cooling trend of LST and describe its slope, respectively (see Sect. 4.4). The seasonal Mann–Kendall test is a nonparametric test suitable to detect LST warming/cooling trends and to quantify the associated significance level in LST time series (Hirsch et al., 1982; Hussain and Mahmud, 2019), while the Theil–Sen slope reduces the impact of outliers on LST trend analysis (Sen, 1968; Theil, 1950). We conducted a seasonal Mann–Kendall test for both the Tdm_cloud_free and the GADTC product and compared their Theil–Sen slopes in describing global LST trends.

The validations using the SURFRAD measurements show that the MAE and bias of the ATC model for the day are 4.7 and 4.0 K, respectively, while those for the night are 3.6 and -1.6 K, respectively (Fig. 5a and c). Although the results for the ATC model are less satisfactory, the Tdm accuracies estimated with the IADTC framework are generally acceptable: the MAEs of Tdm_IADTC at the daily and monthly scales are 1.4 and 0.6 K, respectively and the corresponding biases are both -0.2 K (Fig. 6). By contrast, the MAEs of the Tdm_cloud_free are 4.1 and 2.5 K at the daily and monthly scales, respectively; i.e., they indicate a significantly lower accuracy compared to that of Tdm_IADTC.

The proportions of the three scenarios were 0.2 %, 95.0 %, and 4.8 %, respectively. In Scenarios no. 1 and no. 3 under which the accuracies were improved compared with the OADTC framework, the IADTC framework improves the MAE of estimated Tdm by around 0.45 K (from 2.80 to 2.35 K; see Fig. B1a). The accuracy improvement results mainly from two aspects: (1) the IADTC framework reduces the systematic negative bias caused by cases for which the DTC-modeled LSTs are significantly underestimated around sunrise; and (2) the outliers due to failed DTC simulations are avoided. The overall accuracies for all three scenarios show that the IADTC framework improves the bias from -0.38 to -0.18 K, while the MAE improvement is relatively small. The relatively slight increase in the overall accuracy is attributed to the relatively small proportion of Scenarios no. 1 and no. 3 (around 5 %).

The validations using the FLUXNET data are similar to those with the SURFRAD data: (1) the IADTC framework significantly reduces the ΔTsb of Tdm_cloud_free; (2) the MAEs of Tdm_IADTC are 1.1 and 0.5 K at the daily and monthly scales, respectively; and (3) the biases are both close to zero (Fig. 7). The validations again indicate that the under-cloud LSTs reconstructed by the ATC model are systematically positive during the day (the MAE and bias are 3.5 and 2.8 K, respectively) and systematically negative during the night (the MAE and bias are 2.2 and -0.9 K, respectively) (Fig. 5b and d).

The proportion of each scenario is 10.2 %, 82.5 %, and 7.3 %, respectively. Compared with the OADTC framework, in Scenarios no. 1 and no. 3 (the proportion is 17.4 %) under which the accuracies are considerably improved, the IADTC framework improved the MAE of the estimated Tdm by around 0.78 K (from 1.95 to 1.17 K; refer to Fig. B1b). However, for all the three scenarios, the overall MAE and bias improvements of the IADTC framework are around 0.15 and 0.30 K, respectively (Fig. 7).

The comparison between the GADTC products and in situ measurements (SURFRAD and FLUXNET datasets) shows that the MAEs of the GADTC products are 3.0 and 2.6 K at the daily and monthly scales, respectively, and the mean bias on both scales is -1.5 K (Fig. 8). The MAE and bias are larger than those of the IADTC framework at site level (Fig. 6). This is thought to be due to inconsistencies between MODIS cloud-free observations and in situ measurements, i.e., errors of MODIS cloud-free observations propagating into the GADTC products. The mismatch in spatial resolution between the GADTC products and in situ measurements can also lead to lower accuracies.

The validation with the SURFRAD measurements show that the MAE of the GADTC products is 2.2 and 1.6 K at the daily and monthly scales, respectively, and the bias is around -1.6 K at both scales (Fig. 8a and d). These accuracies of daily mean LST are generally on a par with those of instantaneous LSTs in studies comparing instantaneous MODIS cloud-free observations and SURFRAD measurements (Duan et al., 2019; Martin et al., 2019). Across the different SURFRAD sites, the MAEs of the GADTC products are relatively similar (around 2.2 K; see Table 1).

The direct comparison between the GADTC products and FLUXNET measurements shows that the MAEs are 3.1 and 2.8 K at the daily and monthly scales, respectively, and the bias at these two timescales is -1.5 K (Fig. 8b and e). Compared with the validations over the SURFRAD sites, the accuracies over the FLUXNET sites decrease slightly, and the standard deviations increase. The relatively larger errors at several FLUXNET sites (e.g., AU-Wac, SJ-Adv, and CH-Fru sites, with MAEs larger than 8.0 K; refer to the red ellipse in Fig. 8e) partly account for the lower accuracy. The relatively large errors at these sites might be related to the erroneous in situ measurements as well as the high spatial heterogeneity around these sites. However, the accuracies at most FLUXNET sites are acceptable.

The validations over the FLUXNET sites show that the MAEs vary from 2.6 to 4.8 K and depend on land cover type. Relatively lower accuracies of the GADTC products (MAE larger than 3.5 K) are observed over IGBP land cover types OSH (open shrublands) and SNO (snow and ice) (Table 2). This may be related to unusually large measurement errors and the relatively high spatial heterogeneity at some sites as well as the limited number of sites representing a particular land cover type. For example, the accuracy assessment over the SNO land cover type is performed with a single site, and there are only three sites of the OSH land cover type (e.g., the RU-Cok with MAE as large as 4.6 K).

The validations based exclusively on in situ LST measurements (Fig. 6) show that the IADTC framework can reduce the sampling bias (ΔTsb, i.e., Tdm_cloud_free-Tdm_true) significantly, especially at the monthly scale. ΔTsb directly affects the value of Tdm and may further influence the LST trend. Therefore, based on the GADTC products, we analyzed the global distribution of ΔTsb (calculated by Tdm_cloud_free-Tdm_IADTC) at the monthly scale (Sect. 4.3.1) and compared the LST trend differences between monthly averaged Tdm_cloud_free and Tdm_IADTC to study the impact of ΔTsb on LST trends (Sect. 4.3.2).

To determine the threshold for the first criterion (i.e., the threshold for the DTRfour; see Fig. 2), we analyzed the variations in the error of Tdm_ATC_four depending on DTRfour using SURFRAD and FLUXNET data (Fig. 11). The assessments show that the errors of Tdm_ATC_four generally increase with DTRfour. The linear fitting lines show that the error of Tdm_ATC_four is relatively low when DTRfour is small. In other words, the direct average of the four LSTs per daily cycle (Tdm_ATC_four) is a good estimate of Tdm when the DTRfour is small. Based on the linear fits in Fig. 11a, b, and c, we therefore chose the DTRfour threshold of the first criterion to be 5.0 K.

The second criterion uses the ΔDTR to filter cases for which Tdm is significantly underestimated. To determine the optimal threshold for ΔDTR, we analyzed the MAE differences between Tdm_ATC_four and Tdm_ATC_DTC (i.e., the MAE of Tdm_ATC_four minus the MAE of Tdm_ATC_DTC) and their dependence on ΔDTR for SURFRAD and FLUXNET data (Fig. 11d and e). The assessments show that ΔDTR is generally less than 10 K, and the accuracy of Tdm_ATC_DTC is better than that of Tdm_ATC_four. However, with the increase of ΔDTR, the overall accuracy of Tdm_ATC_four can be superior to Tdm_ATC_DTC. For SURFRAD data, the overall accuracy of Tdm_ATC_four is better than that of Tdm_ATC_DTC once ΔDTR exceeds 22.0 K (i.e., the ΔDTR threshold is 22.0 K), while this threshold is 13.0 K for FLUXNET data. With the further increase of ΔDTR, the accuracy of Tdm_ATC_DTC can be even lower than that of Tdm_ATC_four, e.g., by up to 2.0 K in Fig. 11d and e. In other words, Tdm can be estimated more accurately with Tdm_ATC_four than Tdm_ATC_DTC once ΔDTR is relatively large (i.e., Scenario no. 3).

Note that the optimal threshold of ΔDTR for the SURFRAD data (22.0 K) differs from that for the FLUXNET data (13.0 K). Here, we set the ΔDTR threshold as 20.0 K, which is close to that determined for the SURFRAD data, mostly because of the following factors: (1) the SURFRAD sites have been managed uniformly by NOAA (National Oceanic and Atmospheric Administration) for over 15 years, and the associated radiance measurements have been consistently quality-controlled (Augustine et al., 2000); and (2) the land cover types of the SURFRAD sites are not limited to vegetation. We acknowledge that using a single threshold of 20.0 K may not be optimal for all climate zones and land cover types across the globe, but use of a single threshold effectively avoids outliers due to failed simulations while keeping the simplicity in the global generation of Tdm products.

With the thresholds given as above, we provide the percentage of each scenario within each 10∘ latitude zone (Fig. 12). In low-latitude and midlatitude regions, the percentage of Scenario no. 2 (i.e., DTRfour≥5.0 & ΔDTR <20.0 K) reaches over 80 %, indicating that the IADTC framework mainly uses the DTC-modeled results to estimate Tdm in those regions. With the increase of latitude, the percentage of Scenario no. 1 (i.e., DTRfour<5.0 K) gradually increases, mostly due to a decrease in DTR with the weakened incoming solar radiation over higher-latitude regions. The percentage of Scenario no. 1 reaches around 60 % in the Arctic and Antarctic, which echoes well with the small ΔTsb in high-latitude regions (Fig. 9). The percentage of Scenario no. 3 (i.e., DTRfour≥5.0 & ΔDTR ≥20.0 K) remains relatively stable at around 10 % over most regions across the globe, but it can increase to 20 % in the equatorial zone (10∘ N–10∘ S) and Antarctic, which indicates the relatively poor performance of the DTC model over these regions. The lower performance of the DTC model in the equatorial zone may be related to the high cloud percentage, while over the Antarctic, it reflects the expected difficulties over polar regions (see Sect. 5.2 for more discussions).

GADTC products uncertainties arise from four main sources: (1) MODIS data quality or LST retrieval error, (2) cloud cover and contamination; (3) overpass time drift and linear interpolation, and (4) uncertainties associated with the IADTC framework. These four uncertainty sources can affect the under-cloud LST reconstruction with the ATC model as well as the diurnal LST dynamics modeling with the DTC model and, consequently, affect the accuracy of the GADTC products. In addition, these uncertainties can influence each other via error propagation. In the following, we discuss the four error sources and their effect in more detail.

The ATC and DTC models use cloud-free LST observations to estimate Tdm. Therefore, retrieval errors of MODIS LSTs affect the results of ATC and DTC models and the accuracies of the estimated Tdm. Figure A2a shows that the quality of MODIS LSTs in the equatorial regions is lower than that in the other regions. This suggests that GADTC products should have larger uncertainties in equatorial regions where, consequently, the IADTC framework may need further improvements.

Cloud percentage can also impact the accuracies of the GADTC products. In regions with a higher cloud percentage, e.g., the equatorial regions (Fig. A2b), more under-cloud LSTs need to be reconstructed with the ATC model. However, errors of reconstructed under-cloud LSTs are larger than those of cloud-free LSTs. Therefore, regions with a higher cloud percentage are also associated with larger errors from ATC modeling and, consequently, DTC modeling and the estimated Tdm. In polar regions, the cloud detection algorithm has larger uncertainties due to the spectral similarities between clouds and snow (Østby et al., 2014; Westermann et al., 2012), which introduces additional uncertainties to the GADTC products.

The impact of the overpass time drift mainly occurs over high-latitude regions where the time drift is larger. On the one hand, the cloud-free observations used for solving the free parameters of the ATC model come from significantly different times within a daily cycle (Fig. A1), which affects the under-cloud LST reconstruction. On the other hand, approximately 30 % of the Tdm over high-latitude regions were estimated with the DTC modeling results (i.e., Scenario no. 2; refer to Fig. 12), and the time drift can affect the shape of the DTC curve and, therefore, the estimated Tdm. Temporal normalization methods can adjust the LST observations at fluctuated overpass time to the fixed time, which can eliminate the uncertainties in the under-cloud LST reconstruction and diurnal LST dynamics modeling (Ma et al., 2022; Z. Liu et al., 2019; Duan et al., 2014).

The uncertainties of the GADTC products derived with the IADTC framework mainly include three parts: the reconstruction error of the ATC model, the fitting error of the DTC model, and the choice of the two thresholds. First, the currently used ATC model reconstructs under-cloud LSTs during the day (night) with small positive (negative) biases (Fig. 5), even though information on under-cloud air temperature has been incorporated (Z. Liu et al., 2019). Additionally, the errors in the ATC model can affect the determination of scenarios and, consequently, the way to calculate the Tdm. Second, the DTC model assumes clear-sky conditions and is less capable of simulating under-cloud LST dynamics accurately, which introduces additional uncertainties, especially under some complex situations (Hong et al., 2021). Third, the two fixed thresholds for DTRfour and ΔDTR were determined empirically (Fig. 11): the threshold for DTRfour may introduce additional uncertainty over high-latitude regions with small DTRs, while the threshold for ΔDTR may still miss some cases with unrealistic modeling results.

It is difficult to distinguish and quantify the individual contributions of these four uncertainty sources to the estimated Tdm, as they can affect the ATC and DTC modeling individually and interactively. We are therefore unable to provide a quality control flag for each pixel of the GADTC products. The validations have shown that the accuracies of the GADTC products are generally acceptable over most areas across the globe. However, there are relatively larger uncertainties over equatorial and polar regions, where further validations of the GADTC products and an optimization of the IADTC framework are required.

Further improvements of the GADTC product can focus on the following three aspects:

More extensive validation and inter-comparison of the GADTC products. The GADTC products have been evaluated with FLUXNET and SURFRAD datasets, which include in situ measurements from most climate zones. However, the number of sites is very limited in regions where the uncertainties of the GADTC products are largest (e.g., equatorial and polar regions; refer to Fig. 1). It is therefore hard to validate the IADTC framework as well as its improvements over these regions, e.g., the use of a multi-type ATC model instead of a single-type ATC model. The current in situ data are also insufficient to verify the accuracies of the GADTC products over these regions. It is therefore necessary to obtain more in situ measurements over these regions to validate the accuracy of IADTC framework as well as the GADTC product more completely. Furthermore, reanalysis data, which provide long-term spatiotemporally seamless LSTs and have been widely used in relevant studies (Simmons et al., 2017), can be used to assess the GADTC products (Trigo et al., 2015).