ERA-Interim reanalysis data were downloaded from the European Centre for Medium Range Weather Forecasts (ECMWF); the data had a time step of 6 h (00.00, 06.00, 12.00, and 18.00 UTC), spatial resolution of 0.25 × 0.25°, and covered a time period from 1 January 1979 to 31 December 2017. Geographical locations of the ERA-Interim grid points spanned 35.5–40.75°N and 93.0–104.5°E, sufficient to cover the entire QLM (Figure 1). A time difference conversion method was applied to match the ERA-Interim temperature data in UTC time zone and observation data in local time. The used output variables are 2 m temperature, surface geopotential as well as temperature and geopotential height at 600, 700, 850, and 925 hPa levels. The ERA-Interim surface geopotential height (H_ERA) was obtained by dividing geopotential by gravity at each grid point. Altitudes corresponding to the four pressure layers (600, 700, 850, and 925 hPa) were approximately 4,000, 3,000, 1,500, and 500 m, respectively. It should be noted that the main reason for the selection of these pressure layers was that the geopotential heights of these pressure layers can reflect true characteristics of mountain climate (Gao et al., 2018), and can cover the altitude range of most stations in the QLM, which is convenient for subsequent correction and comparison.

Observational temperature data and altitude information recorded by meteorological stations were downloaded from the China Meteorological Data Sharing Service System of the National Meteorological Information Center (http://cdc.cma.gov.cn/index.jsp). The quality of observed temperature data was strictly controlled and verified by the data provider. The quality and completeness of observed temperature data are significantly improved after controlling, so it can be applied directly in climate change research. Twenty-four meteorological stations located in the QLM were selected. Observations from these stations included altitude, latitude, longitude, daily mean temperature, daily maximum temperature, and daily minimum temperature. Detailed information on the 24 meteorological stations can be found in Table 1. Briefly, these stations are located at different altitudes ranging from approximately 1,000–3,500 m. The highest station is Station No.15 with an altitude of 3,460 m, and the lowest station is Station No.1 with an altitude of 1,100 m. The geographical distribution of the 24 meteorological stations and ERA-Interim grid points are shown in Figure 1 and Table 1. There are positive elevation differences at 23 stations, only Station No.11 has a negative elevation difference between T _e and T _o . ERA-Interim grid points nearest to each meteorological station were selected for comparison based on the longitude and latitude coordinates of 24 meteorological stations, which can avoid the error caused by multigrid spatial interpolation (Zhao et al., 2020). Four seasons were defined as: spring (March to May), summer (June to August), autumn (September to November), and winter (December to February).

ERA-Interim reanalysis temperature data were corrected using Equation 1: T t = T r e f + Γ × ∆ h (1) Γ 600 − 700 = ( Τ 600 − Τ 700 ) / ( H 600 − H 700 ) (2) Γ 600 − 850 = ( Τ 600 − Τ 850 ) / ( H 600 − H 850 ) (3) Γ 700 − 850 = ( Τ 700 − Τ 850 ) / ( H 700 − H 850 ) (4) Γ 700 − 925 = ( Τ 700 − Τ 925 ) / ( H 700 − H 925 ) (5) Γ O b s = Τ O b s / Η O b s (6) where T _t is the corrected temperature, and T _ref is the temperature to be corrected. △h is the elevation difference between meteorological stations and ERA-Interim grid points. Γ represents temperature lapse rates calculated from observations and ERA-Interim temperatures as well as geopotential heights at different pressures. Γ can be calculated from Equations 2–6. These calculations were repeated to obtain temperature lapse rates for five different sets of pressures, here referred to as Methods I–V (Table 2). ERA-Interim internal lapse rates are fully independent of observations (Gao et al., 2017). Γ_600-700, Γ_600-850, Γ_700-850, Γ_700-925 represent the temperature lapse rates at four pressure levels. T₆₀₀, T₇₀₀, T₈₅₀ and T₉₂₅ represent the temperature at 600, 700, 850 and 925 hPa pressure level, respectively. H₆₀₀, H₇₀₀, H₈₅₀ and H₉₂₅ represent the height at 600, 700, 850 and 925 hPa pressure level, respectively. Γ_Obs is the temperature lapse rate calculated from observations. T_Obs and H_Obs are the temperature and elevation of meteorological stations, respectively.

Table 3 and Figure 2 show the variability in five temperature lapse rates over a 12-month period across the QLM; these lapse rates are Γ_600-700, Γ_600-850, Γ_700-850, Γ_700-925, and Γ_Obs. The four ERA-Interim temperature lapse rates (Γ_ERA) were different from the temperature lapse rate based on observations (Γ_Obs) with Γ_Obs was lower than Γ_ERA across the whole period in general. The five temperature lapse rates initially increased, and then decreased across the 12-month period, with the largest values in May and June (Figure 3). The lowest temperature lapse rates were found in winter.

Table 4 shows the biases between observed and original ERA-Interim temperature as well as corrected ERA-Interim temperature for 24 meteorological stations during the period 1979–2017. The bias between observed and original ERA-Interim temperature at Station No.11 was positive (2.83°C), which indicated that ERA-Interim data were higher than observation data at Station No.11. Biases at Station No.11 remained positive after correction, but biases were smaller than the uncorrected ERA-Interim; the smallest bias for corrected ERA-Interim was found using Method Ⅰ (0.08°C). Biases between the original ERA-Interim temperature and observed temperature at Stations No.12, 13, and 15 were negative (−1.33, −1.05, and −3.86°C, respectively), which indicated that ERA-Interim data were lower than observation data. There were also clear biases for the five sets of corrected ERA-Interim temperature at these stations, and these biases were also negative. The largest reductions in bias were obtained with Method Ⅰ, in which they were reduced to −0.45, −0.34, and −0.90°C at Stations No.12, 13 and 15, respectively. Biases between original ERA-Interim and observed temperatures were also negative at Stations No.8, 20, 21 and 24 (−2.06, −2.54, −2.57 and −3.02°C, respectively), which again indicated that ERA-Interim data were lower than observation data. The largest reductions in bias for these four stations were found by correcting ERA-Interim using Method Ⅴ, and biases were reduced to −0.03, −0.27, −0.05, and −0.48°C, respectively. Biases for corrected ERA-Interim at these four stations using Methods Ⅰ‒Ⅳ were larger and remained negative. For all other stations, biases for uncorrected ERA-Interim were negative, but those for corrected ERA-Interim (using all five methods) were positive. Generally, the corrected results obtained via Method Ⅴ were better than those obtained with the other four methods, which was mainly because the temperature lapse rates of Method Ⅴ were calculated using observation data, while those of the other four methods were calculated using ERA-Interim data. Biases for corrected ERA-Interim using Method Ⅰ were generally smaller than when using Methods II‒IV; this may be because the temperature lapse rates of Method Ⅰ were calculated with ERA-Interim height and temperature in the 600–700 hPa range, and the geopotential height in this range is higher than that at the other three pressure levels used. Meteorological stations in the QLM are mainly at elevations above 2,000 m; therefore, temperature lapse rates may be not applicable to the correction of ERA-Interim temperature data for those areas where elevations are below 1,500 m.

Table 5 shows seasonal biases between observed temperature and original ERA-Interim temperature, and for the corrected ERA-Interim temperature data for the QLM. In general, all five correction methods showed significantly reduced biases compared to those between observed and original ERA-Interim temperature. Biases for corrected ERA-Interim temperature data with four methods, and the correction result obtained using Method Ⅳ were markedly lower. The mean seasonal biases of the original ERA-Interim and the four correction methods based on the Γ_ERA were 0.77, 0.71, 0.67, 0.68, and 0.24°C, respectively. Corrected biases for spring and autumn were generally smaller than for summer and winter. For example, with Method Ⅴ, biases in spring, summer, autumn, and winter were 0.03, 0.55, 0.07, and 0.30°C, respectively. Correlation coefficients (r) between observations and original ERA-Interim as well as corrected ERA-Interim were all above 0.9 (Table 6). RMSE improved (decreased) for the corrected ERA-Interim temperature data (Table 7). RMSE for corrected ERA-Interim temperature data decreased for Stations No. 5–24 indicating that temperature lapse rates effectively corrected ERA-Interim temperature at these stations. Method Ⅰ performed best at Station 11, reducing RMSE by 91%. Methods Ⅱ, III, and Ⅳ performed best at Station No.17, reducing RMSE by 90.3, 90.2, and 90.3%, respectively. Method Ⅴ performed best at Station No.19, reducing RMSE by 91.6%. However, RMSE increased for corrected ERA-Interim temperature data at Stations No. 1-4 compared to those for non-corrected ERA-Interim temperature data. This may be due to low elevations of these four stations, which were below 1600 m. Therefore, temperature lapse rates may be unsuitable for correcting ERA-Interim temperature data at these stations. The average RMSE for corrected ERA-Interim temperature data across the 24 meteorological stations using our correction methods were 0.98, 0.94, 0.91, 0.92, and 0.70, respectively for Methods I to V, representing reductions in the mean RMSE of 50.2%, 52.2%, 53.9%, 53.5%, and 65.3%, respectively, compared to the uncorrected ERA-Interim data. In general, Method Ⅴ performed best. Correction Methods Ⅰ‒Ⅳ performed better than Method Ⅴ at Stations No. 11, 12, 13, 14, 15, 17, 20, and 24. These eight stations were at high elevations; stations No.11, 15, and 20 were located at altitude above 3,000 m; 13, 14, and 17 were at were located at altitude above 2,000 m, and 12 and 24 were at were located at altitude above 1500 m. Methods Ⅰ‒Ⅳ may therefore be more suitable for correcting ERA-Interim temperature data at high-elevation meteorological stations.

Figure 4 shows the inter-annual variations and warming trends over the QLM from 1979 to 2017. The ERA-Interim and corrected temperature can successfully capture trends in the observations, including an increasing trend in observed temperature, original ERA-Interim temperature, and corrected temperature during the period 1979–2017 (Table 8). The largest increasing rate was found in summer (0.55°C/decade), and the smallest increasing rate was found in winter (0.34°C/decade). The increasing trends shown by both the original ERA-Interim temperature and corrected temperature data underestimated the warming trend exhibited by observation data. The internal variability and increasing trends were similar for Methods Ⅰ‒Ⅳ, while Method Ⅴ most closely matched the warming trend of the observed temperatures. In general, Method Ⅴ reflected the best correction results, showing the smallest biases at a seasonal scale, and the best reflection of warming rates in the QLM.

Our results indicated that temperature lapse rates exhibited bigger values in warm periods and smallest values in cold periods, which is mainly because the cold air invasion in winter has a more dramatic effect on the temperature at low altitudes than at high altitudes. (Jiang et al., 2016). Qing et al. (2018) also found that temperature lapse rates during warmer months are bigger than that in colder months in the QLM, which is consistent with this study. In this study, the absolute value of Γ_Obs was lower than that of Γ_ERA, and the lowest temperature lapse rates were detected in winter, which is consistent with previous research (Jiang et al., 2016).

The corrected temperatures obtained in this study using Method Ⅴ were better than those obtained using the other four correction methods in terms of the increasing trend in observed temperature, which was similar with previous studies (Gao et al., 2012; Gao et al., 2017). The temperature lapse rates in Method Ⅴ were calculated using observed temperature data, the correction results were tested with observation data, and the method depends critically on the density of ground observation stations; therefore, Method Ⅴ may outperform the other methods. However, although Method Ⅴ performed well, mountainous areas have few meteorological stations, therefore this method cannot be widely promoted. Gao et al. (2012) also found that temperature lapse rates calculated from observations were not reliable when observations contained outliers. Methods Ⅰ‒Ⅳ, which were based on Γ_ERA, were more flexible and independent of meteorological stations, may be applicable to correcting temperature data specifically for high-elevation stations. Furthermore, this method should be easy to extend, and will be particularly useful in mountainous areas with few observation stations. The most important advantage is that the method based on Γ_ERA is fully independent from the observed data. Thus, it provides a tool for correcting ERA-Interim temperature data for any high mountainous areas where no observations exist (Gao et al., 2018; Luo et al., 2019; Fan et al., 2021). Gao et al. (2017) indicates that the correction method using ERA-Interim internal vertical lapse rates can capture the inter-annual variations for the plateau-wide climatology very well, which is similar with this study. The bias reflects the specific difference of temperature before and after the correction, while the correlation coefficient mainly reflects the similarity between the site observations and the ERA-Interim temperature changes. The correction method based on temperature lapse rate mainly corrects the ERA-Interim temperature according to the height difference between meteorological stations and reanalysis grid points. Therefore, it mainly corrects the systematic errors between ERA-Interim temperature and observed values, which has little influence on the temperature trend. This is the reason why the bias of some sites decreased after correction, but the correlation coefficient did not change much. The correction skills are weak in winter except for Method V (Table 5, Table 8). In addition to the altitude difference, the snow cover and glaciers may also be the main reasons for the bias in winter (Zhao and He, 2022a), so the elevation correction method may not be suitable for the bias correction in winter. In addition, after comparing biases, RMSE, and r of Methods Ⅰ‒Ⅳ based on Γ_ERA, we concluded that the corrected results using Method Ⅲ were superior to those using Methods Ⅰ, Ⅱ, and Ⅳ. The seasonal bias, RMSE, r, and mean station biases of Method Ⅲ were 0.67°C, 0.91°C, 0.905, and 0.67°C, respectively (Table 9). Hence, Method Ⅲ should be the most suitable for the correction of ERA-Interim temperature data in the Qilian Mountains. Gao et al. (2012) found that using (global climate) model showed a convincing performance when compared to measured data of the twelve stations, again especially for those in higher elevations, which is similar with this study.

Although the five correction methods can improve the accuracy of ERA-Interim temperature data, they do not consider the characteristics of terrain or geographical location. Errors between the observed and ERA-Interim temperature data were not only caused by the differences between the station and ERA-Interim grid cell elevation, but also by other factors, such as the large-scale error of ERA-Interim, glacier, and errors introduced by the assimilation system and the interpolation method (Dee, 2005; Dee and Uppala, 2009). Therefore, it may not be sufficient to correct ERA-Interim temperature using only the temperature lapse rate; additional influencing factors need to be considered in future research. The 24 meteorological stations considered in this study were situated at different altitudes and were widely spaced geographically. Discrepancies exist in direct comparisons of temperature lapse rates established using observation data and those established using ERA-Interim data. The 2881 ERA-interim grid points shown in Figure 1 span a large area, covering not only the QLM, but also areas adjacent to the QLM. Some of these grid points were located in the interior of the Tibetan Plateau, which increased terrain complexity and may lead to different temperature lapse rates. Moreover, due to the spatial location and altitude of the 24 meteorological stations are different, the calculated temperature lapse rates cannot represent a single station, but instead, represent a generalized area. Whether the representativeness is significant or not is worthy of a further study. Finally, in the formula T _t =T _ref +Γ×△h, the T _ref uses the ERA-Interim 2 m temperature values corresponding to the 24 stations, and the system error for T _ref is transferred, contributing to the final correction error. In addition, temperature lapse rate changes could be more variable from surface to within boundary layer and in the presence of clouds, which may affect the overall bias correction.

The aims of the error corrections performed in this study were to verify the reliability of the method and to establish a foundation for future downscaling research. For example, with DEM data for the entire QLM, we can correct all 2881 ERA-Interim grid points using Method III, and obtain temperature data from all grid points. ERA-Interim temperature data can be downscaled to higher resolution, contributing to a high-resolution data set for climate change research in this region (Gao et al., 2018; Fan et al., 2021). We focused on ERA-Interim data in this study, which is the third-generation reanalysis product of the ECMWF. However, future studies can extend the methods used here to other reanalysis products such as ERA5 and ERA5-Land (Wang C. et al., 2019; Liu et al., 2021; Zhao and He., 2022a). In addition, it would be worth trying to use radiosonde vertical profiles to calculate lapse rates and to verify the same in this study. Moreover, temperature lapse rate may vary in different atmospheric conditions and slightly from one geographical location, especially in the boundary layer and in the presence of cloud.