#### Abstract

Understanding errors in surface air temperature (SAT) data and related uncertainties is crucial for climate studies because of their impact on the accuracy of statistical inferences in scientific conclusions. In recent decades, considerable research has focused on the trends and evolution of SAT on the Tibetan Plateau (TP). However, assessment of the uncertainties in SAT change on the TP has not been done adequately, which is of considerable importance for climate research. Using station-observed SAT data from the TP, this study estimates long-term variations and trends of sampling error variances in gridded monthly SAT data over recent decades. Results revealed large sampling error variances in northern and western parts of the TP but small variances in eastern, southern, and central areas. The sampling error variances also exhibited strong monthly variations with maximum errors in winter and minimum values in summer. Furthermore, spatial distributions of the trends of seasonal and annual mean sampling error variances were found distributed unevenly with decreasing trends found mainly in central and southern parts of the TP and increasing trends in northeastern, southeastern, and northwestern areas. Additionally, differences were also found in the trends of seasonal and annual mean sampling error variances on various timescales.

#### 1. Introduction

Increasingly, large numbers of applications and studies require information concerning the errors in surface air temperature (SAT) datasets for estimation of the relevant uncertainties and for reaching meaningful and accurate conclusions regarding climate change [1–4]. Typically, such information is critical in assessment of the optimal global or regional average SAT time series, linear trend, and ranking of extreme climate records [5–7]. In general, uncertainties in observation data can be classified into three groups: (1) observational error, the uncertainties due to station data quality; (2) sampling error, the uncertainties in a grid box mean caused by estimating the mean from a small number of point values; and (3) temporal interpolation error, the uncertainties associated with filling gaps in the station data record [3,8].

Sampling error is fundamentally important because each observational network could have spatial and/or temporal gaps, and such gaps tend to be particularly evident in regions of high latitude and high elevation [9]. In recent decades, a growing number of researchers have investigated the sampling error in climate change [10–15]. In this respect, to estimate the sampling error state in SAT, Jones et al. [1] first systematically calculated the sampling error variances of gridded data by estimating two parameters: the average variance of all the stations in a grid box and the average intercorrelation dimensionless percentage of these stations. Brohan et al. [5] further used this average variance to estimate the errors in regionally and globally observed temperature change. In addition, Shen et al. [8] developed a new theory to estimate sampling error variances, and this method was further used in assessment of sampling error variances in SAT changes in the USA [10]. Based on observations from 731 stations from the China Homogenized Historical Temperature Dataset, Hua et al. [12] indicated that the uncertainties arising from the sampling error could reach 50% of the calculated trend of SAT in China. Furthermore, using a multivariate regression method and two gridded precipitation datasets, Shen et al. [16] reported the sampling errors of the annual quasi-global precipitation reconstructed using an empirical orthogonal function expansion.

The Tibetan Plateau (TP) is nicknamed the “world's roof” because its average elevation is above 4000 m. Results obtained by many studies indicate that statistically significant warming has occurred on the TP over the last century [17–21]. However, in many parts of the TP, the observational network is sparse because of the high elevation, complex topography, and severe weather conditions, making it difficult to quantify the “true” trend of the data from the noise. As a first step in addressing this problem, we recently investigated the sampling error variances in SAT on the TP [11]. However, our previous work focused mainly on the impact of the sampling error on the regional average SAT time series of the TP rather than the variational characteristics of the sampling error itself. Although these issues are important in relation to research and understanding of climate change over the TP, they are yet to be assessed and quantified fully. Thus, the spatial-temporal characteristics and trends of the sampling error in SAT on the TP remain to be clarified. The objectives of this study were to examine the spatial and temporal variations of the annual and seasonal mean sampling error over the entire TP and to analyze the trends of the sampling error on different timescales.

The remainder of the paper is arranged as follows. Section 2 describes the data and the method used for calculating the sampling error. Section 3 presents the results of this analysis, and Section 4 contains the conclusions.

#### 2. Data and Methods

##### 2.1. Data

This study used 103 regular surface meteorological observations from the TP region for the period 1951–2013 with an initial quality control provided by the China Meteorological Administration. Variables included the monthly mean maximum and minimum temperature (Tmax and Tmin, respectively). The evolution of the number of meteorological stations from January 1951 to February 2013 is shown in Figure 1(a). The number of stations started at around 10 in 1951, increased monotonically to around 90 in the 1960s, and then maintained this level until the 1970s. However, because of station closures, this number has decreased gradually since the early 1980s. At the first step in the data analysis, the mean temperature (Tmean) was calculated as the average of Tmax and Tmin to estimate the sampling error variances. The locations of all 103 stations data were plotted within a grid of 54 boxes (25.00–40.00°N, 75.00–106.50°E) with 2.5° × 3.5° resolution (Figure 1(b)). Overall, the density of stations was found lowest in the sparsely populated high mountain and desert areas of the western and northwestern TP and highest in eastern and southeastern parts. Ultimately, only 30 of the 54 grid boxes contained data for our analysis.

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Additionally, to compare the sampling error calculated from observational and reanalysis data, the monthly 2 m air temperature from National Centers for Environmental Prediction, Department of Energy (NCEP-DOE) Reanalysis II [22], is used. For purpose of comparison, the reanalysis data were first gathered for the same time period and grid resolution with observational gridded data.

##### 2.2. Methods

In this paper, we are following the sampling error evaluation method developed by Shen et al. [10]. The main idea of Shen et al. [10] was to decompose the mean square error into the spatial variance and the correlation factor. For clarification, the key mathematical formulations are presented below.

The sampling error variances are given as follows:where is the true average in the grid box, is its estimator, and the angle brackets stand for the operation of the ensemble mean or the expected value. Here, the spatial variances and the correlation factor are used to estimate the standard error of the grid box data.

The spatial variance is estimated bywhere *N* is the number of stations in the grid box and is the *i*th observation in the gird box.

The correlation factor is

The spatial variance is estimated bywhere denotes the set of a moving time window (MTW) and is the number of years of the set. Here, we follow the idea of piecewise stationarity [5], and we use a 5-year MTW.

Note that the correlation factor is estimated by regression rather than being computed directly from eq. (3) because the true is unknown. Thus, for a grid box with *N* observations, the SAT data are treated as a statistical population. The population mean of the station SAT data in the grid box is

Simple random subsampling of *n * stations is obtained from the population 1000 times, and the sample mean of the *n* stations is

The mean square difference between the population mean and the sample mean is

Similar to (4), the 5-year MTW is applied to to obtain the estimated mean square error:

Thus, for each month and for each grid box of *N* station anomalies, the *N*−1 data pairsare used in the regressionto find the correlation factor .

Thus, the error variance for each grid box data from January 1951 to February 2013 is obtained by

The sampling error variances of Tmean were calculated for the 30 grid boxes for each month from January 1951 to February 2013 when a box had data. In this study, we analyzed only the data from 1979 to 2012 because most grid boxes did not have continuous data until the late 1970s.

#### 3. Results

##### 3.1. The Spatial Distribution of Sampling Error Variances

Annual and seasonal mean sampling error variances on the TP averaged over 1979–2012 are shown in Figure 2. For annual mean sampling error variances, it can be seen that large values occur mostly in those grid boxes with few stations and with large spatial variances due to the elevation gradient (Figure 2(a)). The grid boxes with large sampling error variances are distributed mainly over the mountain regions of the northern and northwestern TP and in southern and eastern parts of the TP, e.g., grid box G45 located on the border between the TP and the Sichuan Basin. Small sampling error variances are distributed mainly in eastern, southern, and central parts of the TP where the density of meteorological stations is high. More specifically, the grid boxes with sampling error variances larger than 0.2 (°C^{2}) are G5, G18, G20, G23, G29, and G45, all of which represent areas with sparse spatial distribution of observations and complex topography with large climatic variability. In contrast, because of the relatively dense distribution of observations and the flat terrain, the sampling error variances are small in central parts of the TP.

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It can be seen that the spatial distributions of the seasonal mean sampling error variances are similar to that of the annual mean (Figure 2(b)–2(e)) and the seasonal values of error variances vary seasonally with the regional average of error variances are 0.092, 0.095, 0.128, and 0.244 (°C^{2}) for spring, summer, autumn, and winter, respectively. Large variations in seasonal mean error variances are often observed in the northern, northwestern, and border regions of the TP. The seasonal patterns also show that errors in winter are much larger than in summer because the winter SAT has much larger variances than the summer SAT. Furthermore, the spatial distribution of sampling error variances in spring is similar to autumn. Overall, the spatial distribution of sampling error variances over the TP is determined mainly by the distribution of the observational network and by the topographic characteristics of the TP and its surrounding areas.

##### 3.2. Interannual Variation in Sampling Error Variances

The variations of annual and seasonal mean sampling error variances of each grid box with data (Figure 1(b)) over the TP from 1979 to 2012 are shown in Figure 3. For the annual mean error variances, marked interannual variations can be observed in southeastern, northern, and northwestern parts of the TP (Figure 3(a)). For example, the value of G45 ranges from 0.323 (°C)^{2} in 1991 to 0.763 (°C)^{2} in 2010, whereas grid boxes in eastern and southern parts of the TP (e.g., G34 and G26) exhibit weak interannual variability with values in the range 0.018–0.040 (°C)^{2}. Overall, most of the grid boxes show that sampling error variances increase with time. For seasonal mean error variances (Figure 3(b)–3(e)), the evolution characteristics of spring and summer are consistent with the annual mean variations, whereas the variations of sampling error variances in autumn and winter are different from those in spring, summer, and annual, showing strong variation for most grid boxes. Comparison of the variation of annual and seasonal mean sampling error variances reveals that sampling error variances in winter contribute significantly to the annual mean series.

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The multiyear mean of monthly sampling error variances for all grid boxes over the TP is shown in Figure 4. It can be seen that the maximum monthly sampling error variances are found in winter (November–February), with monthly values of 0.263, 0.187, 0.211, and 0.257 (°C)^{2} for January, February, November, and December, respectively. The minimum sampling error variances are found in summer, with monthly values of 0.070, 0.063, and 0.063 (°C)^{2} for June, July, and August, respectively (Figure 4(a)). In addition, Figure 4(b) presents the multiyear monthly variation in sampling error variances for all grid boxes. Overall, the maximum monthly sampling error variances for all grid boxes occur mainly in winter. It is also worth noting that G45 has the largest sampling error variances in each month. Because of the considerable size of the sampling error variances in this grid box, it is important to note that the associated uncertainties should be considered carefully when estimating a linear trend or a regional average series for this area.

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##### 3.3. The Annual and Seasonal Trends of Sampling Error Variances

Based on the long-term sampling error variances estimated from the 103 observation stations on the TP, we further calculated the trends in annual and seasonal mean sampling error variances during the entire study period (Figure 5). It can be seen that the trends of annual mean error variances are distributed unevenly (Figure 5(a)). Trends of decrease are found mainly in central-eastern and western parts of the TP, and the area with significant trends covers eastern and southwestern parts of the region. In contrast, the areas with significant trends of increase are found in northeastern, southeastern, southern, and northwestern parts of the TP.

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The spatial distribution of seasonal trends of sampling error variances show that the central-northeastern, northwestern, and southern parts of the TP are characterized by a significant increase of sampling error variances in all seasons (Figures 5(b)–5(e)). In spring, trends of increase are found in southern and northern parts of the TP, whereas summer presents remarkable trends of increase over most of the TP, except in several grid boxes in western, southern, and eastern areas. A general trend of increase is evident on the northern TP, while a trend of decrease is found on the southern TP. Moreover, a trend of increase is found in winter on the eastern TP, while a decreasing trend is evident in central-western parts of the TP.

To further estimate the trends of sampling error variances on different timescales, a running window trend analysis was performed on the trends of annual and seasonal mean sampling error variances (Figure 6). The trends were calculated for different timescales starting from 1979 to 2003 and ending in 2012, with at least a 10-year span. As illustrated in Figure 6(a), the annual mean sampling error variances over the TP show significant decrease before 1993 and then begin to increase during the 1990s within a time window of less than 23 years. Then, there is a significant increase after 2001.

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The features of the seasonal trends are found similar to those of the annual trends, except for summer (Figures 6(b)–6(e)). In spring, there is a moderate trend of decrease in the 1980s. Then, the trend is positive in the early 1990s before decreasing again during the mid-1990s. In summer, the trend of decrease is weak but significant before 1983 within the time window at 10–15 years. However, the trend reverses after the mid-1980s, and a larger increase is found during the final two decades. The trends of autumn and winter are most similar to the annual series. For instance, both decrease before the mid-1980s within the time window of less than 15 years and then increase significantly until 1991, following which they decrease remarkably during the 1990s before increasing again after 2001.

##### 3.4. Calculation of the Sampling Error from Reanalysis Data

According to eq. (5), the “true” average SAT anomaly of a grid box is estimated by the spatial average of the station with data, which may not be true in reality. Here, we simply treat the NCEP-DOE reanalysis II data as the “true” SAT field because the output of general circulation models (GCMs) can yield a good correspondence between the model results and the observations. Thus, using the annual mean results as an example, we will analyze the spatial distribution and trends of sampling error variances which are calculated based on the reanalysis data. The spatial distribution of annual mean sampling error variances calculated based on reanalysis data is very similar to that determined from observation data of monthly SAT (Figures 2(a) and 7(a)). The most remarkable feature is that sampling error variances exhibit strong northwest-southeast gradient across the TP which result from the observation station distribution and climate variability. It should, however, be noted that the error variances of some gird boxes over southeastern and eastern TP, calculated from reanalysis data, have larger error variances than those of observation data.

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For the trends of the annual mean sampling error variances estimated from the reanalysis data, as shown in Figure 7(b), the spatial distribution fits in with the trends of the observation data, which are distributed unevenly. In general, trends are negative over central-eastern and western parts of the TP with significant trends covering most parts of eastern and western TP, while the trends are positive and significant for the southern and eastern regions especially northeastern TP.

#### 4. Conclusions

Using observed SAT data collected from 103 surface observation stations on the TP during the past 34 years (1979–2012), we analyzed the spatial distribution, interannual variation, and trends of sampling error variances of SAT in the TP during different periods. The main conclusions derived from this study are as follows.

Large sampling error variances were found in northern and western parts of the TP where there are few meteorological stations and large spatial variances due to the elevation gradient, and small sampling error variances were found in eastern, southern, and central parts of the TP where the observational network is denser. Large interannual variations were found in southeastern, northern, and northwestern parts of the TP, while small interannual variations were found in eastern and southern areas. Overall, most of the grid boxes showed an increase in sampling error variances with time from seasonal to annual timescales.

Sampling error variances also showed strong monthly variation with maximum errors in winter and minimum errors in summer. The spatial distributions of trends of seasonal and annual mean sampling error variances were found distributed unevenly. Trends of decrease were found mainly in central and southern parts of the TP, whereas areas with significant trends of increase were found in northeastern, southeastern, and northwestern regions.

A running window trend analysis showed significant decrease in annual mean sampling error variances before 1993, which began to increase during the 1990s within a time window of less than 23 years. Seasonally, the features were found similar to those of the annual trends, except in summer, which showed a weak trend of decrease before the early 1980s and a trend of increase after the mid-1980s.

The sampling error variances estimated from both observational and reanalysis gridded data are under the same spatial distributions and trends being only slightly different over the eastern part of TP.

#### Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

#### Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

#### Acknowledgments

This research was jointly funded by the National Natural Science Foundation of China (41775072, 91537214, and 41405069), the National Key R&D Program of China (2018YFC1505702), the Outstanding Young Talents Project of Sichuan Province (19JCQN0002), the Key Foundation of the Education Department of Sichuan Province (16ZA0203), the Scientific Research Foundation of Chengdu University of Information Technology (KYTZ201517, J201516, and J201518), and the Scientific Research Foundation of Key Laboratory of Meteorological Disaster (KLME), Ministry of Education (KLME201803).