Analysis of Sampling Error Uncertainties and Trends in Maximum and Minimum Temperatures in China

Assessing regional and global climate changes is a valuable topic in meteorological research, and the details of such assessments can enrich understanding of the climate change being studied. The interest in assessing climate changes using surface air temperature (SAT) resides not only in mean temperature (Tmean), but also maximum and minimum temperatures (Tmax and Tmin), because changes in Tmean are caused by changes in either Tmin or Tmax, or both (Karl et al., 1993; Jones, 1995; Brunetti et al., 2000; Braganza et al., 2004; Vose et al., 2005; Li and Yan, 2009; Elguindi et al., 2012).

The obvious warming that has occurred over the past century, observed throughout many parts of the world, has been strongly associated with increases in Tmin and Tmax (Easterling et al., 1997; Henderson and Muller, 1997; New et al., 2000; Bonsal et al., 2001; Fischer and Schär, 2009; Meehl et al., 2009). Changes in long-term variation of Tmax and Tmin over China have also been studied by many scientists (Zhai et al., 1999; Yan et al., 2002; Zhai and Pan, 2003; Tang et al., 2005; Ding et al., 2007; Zhang et al., 2007; Li et al., 2010b; Du et al., 2012). However, not only did many of these studies on extreme temperature exhibit little or no consideration of the uncertainties of their data and conclusions, numerous related studies were also calculated by the original data without any correction for homogenization, which may have further increased the uncertainties. Thus, information about the uncertainties in climate change is urgently needed to systematically evaluate climate change and risks over both short and long timescales (Li et al., 2010a).

The fundamental errors in observation data include sampling errors, observational error, random error, and other errors (Karl et al., 1994; Kent et al., 1999; Barnett et al., 2000; Folland, 2001; Kalnay and Cai, 2003; Brohan et al., 2006; Menne et al., 2009; Shen et al., 2012). A complete assessment of "true" uncertainties is impractical because of incomplete sampling and unknown random errors, and quantitative correction can hardly be made for all the unknowns. Therefore, we focus only on sampling error in the current paper, which always occurs because each observational network has either spatial or temporal gaps.

The main purpose of the paper is not only to quantify the sampling error uncertainties in the Tmax and Tmin of China, but also to investigate the homogenization effect on uncertainties. The remainder of the paper is organized as follows. The next section presents the data and method. Section 3 discusses the results, and section 4 presents a discussion and conclusions.

Two datasets were used in this study. First, the China Homogenized Historical Temperature (CHHT) Dataset (1951-2004) Version 1.0, compiled by the National Meteorological Information Center (NMIC) of the China Meteorological Administration (CMA), which includes monthly mean homogenized Tmax (Tmax-hom) and Tmin (Tmin-hom) data from 731 stations distributed across China, and have been quality-controlled and homogeneity-adjusted (Li et al., 2009). Second, the monthly mean original data of Tmax (Tmax-ori) and Tmin (Tmin-ori) records at 756 stations in China, which were also processed and obtained from the NMIC of the CMA. We excluded 25 stations in the second dataset to maintain the same number of stations as the CHHT dataset. Thus, this dataset was completely the same as the CHHT dataset, except that it comprised the original data without homogenization.

The history of the number of stations from January 1951 to December 2004 is shown in Fig. 1, which shows a sharp expansion in the coverage and density of the observation network during the 1950s. The number of stations was relatively stable until the end of the 1980s, when it started to decrease slowly because of station closures. The spatial distribution of the stations is shown in Fig. 2, which demonstrates stations are sparse over western and northwestern China, but denser in eastern and southern China.

Following Shen et al. (2007, 2012), we first aggregated station data onto grid boxes at a 2.5^° (lat) × 3.5^° (lon) resolution. All 731 stations fell within 138 grid boxes. Figure 2 shows the 2.5^° (lat) × 3.5^° (lon) grid map bounded by (16.50^°-54.00^°N, 74.75^°-134.25^°E). We then excluded 21 one-station and station-void grid boxes at the national borders and high-mountain areas in subsequent calculations. The inclusion of these one-station and station-void grid boxes may have yielded more noise than useful climate change signals. Thus, 107 grid boxes remained for our uncertainty estimation. The analysis was made with temperature data expressed as anomalies (with respect to the 1961-90 climatology base period). The anomaly calculation method was the same as that of (Jones et al., 1997).

Figure 1.  History of the number of observational stations with temperature records in China.

Figure 2.  Station locations.

The calculation follows the method of Shen et al. (2007, 2012).

Let be the true average of the SAT field over a grid box: Here, T(r_i, t) is the SAT anomaly field over a grid box Ω whose area is ‖ Ω ‖; r_i is the position vector of the station in the grid box; and t is time.

The estimator of the spatial average , denoted by (t), is In the above equation, T_i(t)=T(r_i, t) is a sampling anomaly datum of the station at r_i, and N is the number of stations in the grid box.

Following (Shen et al., 2007), we use spatial variance σ _s² and a correlation factor α _s to estimate the sampling error variance for each grid box. The estimation of the mean square error (MSE) is based on the following equation: where is spatial variance, T_j(t) is a sampling anomaly datum of the station at r_j, and is the correlation factor; and <·> is the ensemble mean [for details of the derivation of Eq.(3), see (Shen et al., 2007)].

We assume the piecewise stationarity (Folland, 2001) for the spatial variance σ _s², and use a 5-yr moving time window (MTW) smoothing for the temporal mean.

The estimator of spatial varianceσ _s² is estimated by where the 5-yr MTW(t) is centered around year t, and ‖ MTW(t)‖ denotes the number of years in the MTW(t), and τ is year limited to ‖ MTW(t)‖. In the 5-yr time window, N may vary from year to year, and thus ‖ MTW(t)‖ may be less than five. Therefore, we need at least three years of data. For 1953, the MTW has only three years of data: MTW=[1951, 1952, 1953], so ‖ MTW(1953)‖=3. For spatial variance, N≥ 4 is chosen as the minimum number of stations within a box because the regression estimates of α _s needs at least four stations.

Instead of applying Eq. (4) directly, the correlation factor α _s is calculated via least squares regression. N stations are considered a statistical population. Thus, the estimator of the population mean of the station temperature anomalies _N (t) can be calculated as the average of the anomalies of all stations in the grid box at time t: Simple random sampling of n(n≤ N) stations is obtained from the population, and the estimator of sample mean of those n stations _n (t) is computed as follows: where T_{n, i} is the ith station's anomaly temperature in the sub-sample network of size n. The estimator of mean-square differences between the population mean and the sample mean is used as an initial estimate of the sampling error and is computed by where MTW(t) denotes the 5-yr moving time window at time t, τ is year limited to ‖ MTW(t)‖, and S₁₀₀₀ stands for the set of 1000 simple random samples of size n.

We then apply a regression procedure using the following data: The least squares regression between these data pairs estimates the α_s value. This regression is performed for every grid box of N≥ 4. Then, we populate the _s and _s² onto the grid boxes with less than four stations.

Finally, the sampling error variance of the SAT grid box data for a given box and a given month is computed by where N is the number of stations in the grid box.

The uncertainties of the linear trend are the standard error of the regression coefficient. The statistical significance was determined by the Mann-Kendall test (Sneyers, 1990).

The sampling error variances of Tmax and Tmin were calculated for 107 grid boxes for each month from January 1951 to December 2004 when a box had data. The spatial distribution of sampling error variances of Tmax-hom and Tmax-ori for the given months (January and July 1960) are shown in Fig.3. This figure indicates that the sampling error variances in each map are inversely proportional to the number of stations. Both the Tmax-hom of January and July (Figs.3a and c) has large sampling error variances over northern and western China where there are few stations, while the sampling error variances are reduced where observations are denser over eastern and southern China. A secondary effect is the sampling error variances in January (Fig.3a) are larger than July (Fig.3c) in most grid boxes. The large error variances in winter are attributed mainly to strong local temperature inhomogeneity, especially in the high-latitude and high-elevation areas over northwestern and southwestern China. The spatial distribution of sampling error variances of Tmax-ori in January (Fig.3b) are basically the same as those of Tmax-hom (Fig.3a). However, the error variances of Tmax-ori in July (Fig.3d) are much larger than those of Tmax-hom (Fig.3c), especially in the grid boxes over southwestern, northwestern and northeastern China.

Figure 3.  Spatial distribution of the sample error variances [units: (^°C)²] of Tmax: (a) Tmax-hom in January 1960; (b) Tmax-ori in January 1960; (c) Tmax-hom in July 1960; and (d) Tmax-ori in July 1960.

Figure 4.  Spatial distribution of the sample error variances [units: (^°C)²] of Tmin: (a) Tmin-hom in January 1960; (b) Tmin-ori in January 1960; (c) Tmin-hom in July 1960; and (d) Tmin-ori in July 1960.

Figure 4 is the same as Fig.3, but for Tmin. Here, both the sampling error variances of Tmin-hom and Tmin-ori (Figs. 4a and b) in January are larger than those of Tmax (Figs. 3a and b). The grid boxes with large uncertainties are more widely distributed over western China. The sampling error variance of Tmin-hom (Fig.4c) in July also shows a consistent featurein which the values of grid boxes take on a northwest-southeast gradient distribution. However, comparison of the Tmin-ori for July and Tmin-hom for January suggests that the error variances of Tmin-ori in summer months do not display the same northwest-southeast gradient distribution as Tmin-hom in winter. Instead, the error variances show a considerable large-small-large pattern from southeastern to northwestern China, suggesting the original data may raise large uncertainties.

Figure 5 shows the monthly national average Tmax-hom and Tmax-ori time series with their uncertainties calculated by area-weighted averaging. Both the Tmax-hom and Tmax-ori time series have an initial decreasing or slight increasing trend, indicating no significant trends during the early period. After the initial variation, the series begin to increase from the 1980s, and continue to increase until around 1990. Figure 5 also clearly demonstrates that uncertainties of Tmax-hom (left column of Fig.5) are much larger in the winter than the summer. However, the uncertainties of Tmax-ori (right column of Fig.5) are not only larger than those of Tmax-hom, but also opposite to the distribution of uncertainties of Tmax-hom during the year. This means the uncertainties of Tmax-ori are large during April-September but small during October-March. These features also suggest that, although inhomogeneity does not impose large bias in the average SAT time series, bias can lead to significant uncertainties.

Figure 5.  Time series of average Tmax-hom (left column) and Tmax-ori (right column) anomalies (relative to the 1961-90 climatology) in China: January-December. The shaded area is the 2-sigma error margin.

Figure 5.  (Continued)

Figure 6.  Time series average Tmin-hom (left column) and Tmin-ori (right column) anomalies (relative to the 1961-90 climatology) in China: January-December. The shaded area is the 2-sigma error margin.

Figure 6.  (Continued)

Figure 7.  Annual mean temperature anomalies (relative to the 1961-90 climatology) for China based on Tmax and Tmin: (a) Tmax-hom; (b) Tmax-ori; (c) Tmin-hom; and (d) Tmin-ori. The error bars represent the 95% confidence interval. The black curve is the 10-yr moving average.

Figure 6 is the same as Fig. 5, but for Tmin. The warming trends in Tmin-hom (left column of Fig. 6) and Tmin-ori (right column of Fig. 6) can be observed in all months of the year. The warming trends during winter and spring are greater than those of summer and autumn. For the 1950s-1970s, the evolution of SAT in each month is slightly different: the SAT series during March to May and December to February mainly show large negative anomalies, whereas those for June to August and September to November show positive or small negative anomalies. Since the 1980s, both the Tmin-hom and Tmin-ori series in each month have been mainly positive. The uncertainties in the Tmin-hom series show similar feature contrasts to those of Tmax-hom (Fig. 5), with the largest value in winter. However, the uncertainties in the Tmin-ori series are large in winter and summer and small in spring and autumn.

The annual mean Tmax and Tmin time series for mainland China are shown in Fig. 7. This figure shows that both the annual mean Tmax-hom and Tmax-ori (Figs. 7a and b) time series have no significant trend before the mid-1980s, but then start to warm up gradually after the late 1980s, before an accelerated warming process since the early 1990s. The obvious warming in the annual mean Tmin-hom and Tmin-ori time series (Figs. 7c and d) takes place mainly from the early 1980s, and the series mainly show negative anomalies before the 1980s, especially in the mid-1950s and late 1960s. The uncertainties in all four series decrease gradually in the 1950s-1970s and reach minimum values during the late-1970s to mid-1980s, but then increase again in the later period (1990s and 2000s). The large-low-large pattern in uncertainties, as shown in Fig. 7, is primarily due to station closures during the 1990s (see details in Fig. 2).

The results of the Mann-Kendall test on monthly, seasonal and annual Tmax-hom, Tmax-ori, Tmin-hom and Tmin-ori series with their uncertainties are presented in Table 1. The most important Tmax-hom and Tmax-ori increases are detected in January, February, April, May and June-five months that have statistically significant positive trends. The results demonstrate a statistically significant trend at 0.147^°C(10yr)^-1 0.061^°C(10yr)^-1 and 0.153^°C(10yr)^-1 0.062^°C (10 yr)^-1 for annual Tmax-hom and Tmax-ori respectively, because all the months show a warming trend, especially February. The analysis of Tmax-hom by season indicates an increase of 0.279^°C(10yr)^-1 0.136^°C(10yr)^-1, 0.157^°C(10yr)^-1 0.087^°C (10 yr)^-1, 0.048^°C (10 yr)^-1 0.067^°C(10yr)^-1 and 0.105^°C(10yr)^-1 0.089^°C(10yr)^-1 for winter, spring, summer, and autumn respectively, which are smaller than those for Tmax-ori. As in the case of Tmin-hom and Tmin-ori, the trends are larger than Tmax and indicate a pronounced monthly, seasonal and annual warming, with a statistically significant positive trend. The difference between the trends of homogenized and original series could mainly be attributed to the effects of local inhomogeneities compensation for each station when calculating the averages (Li and Yan, 2009).

Sampling error uncertainties, linear trends and their uncertainties and the influence of homogenization on uncertainties in temperature change were quantitatively estimated. The following conclusions can be drawn from the analysis:

(1) The sampling variances of homogenized data are larger in the high-latitude and high-elevation areas over northern and western China in winter and smaller in low-latitude and low-elevation areas over southern and eastern China in summer. The sampling error variances of original data were found to be more irregular and larger than those of the homogenized data.

(2) The sampling error uncertainties are large in the 1950s-1960s and 1990s-2000s, and small in the 1970s-1980s. The uncertainties are not yet large enough to alter conclusions about upward or downward trends of SAT in China.

(3) The national average annual mean Tmax-hom and Tmin-hom over China increased by 0.147^°C (10 yr)^-1 0.061^°C(10yr)^-1 and 0.283^°C(10yr)^-1 0.053^°C(10yr)^-1 respectively, with the seasonal averages in winter, spring, summerandautumnrisingrespectivelyby:0.279^°C(10yr)^-1 0.136^°C(10yr)^-1and0.489^°C(10yr)^-1 0.106^°C(10yr)^-1; 0.157^°C(10yr)^-10.087^°C(10yr)^-1and0.319^°C(10yr)^-1 0.076^°C(10yr)^-1;0.048^°C(10yr)^-10.067^°C(10yr)^-1and 0.156^°C(10yr)^-10.057^°C(10yr)^-1;and0.105^°C(10yr)^-1 0.089^°C(10yr)^-1and0.233^°C(10yr)^-1 0.082^°C(10yr)^-1. The warming trends of the original data were found to be larger than those of the homogenized data.

Although the sampling errors of both the homogenized and original data were assessed in this study, there are still issues that need to be resolved. First, in general, the sampling error uncertainties in SAT data are larger in winter than in summer (Jones et al., 1997; Brohan et al., 2006; Shen et al., 2007, 2012). However, a question arises as to why the sampling errors calculated by the original data in our paper are large in summer and even comparable to the sampling errors in winter. To answer this question, we can decompose the sampling error variance into two parts: the correlation factor and spatial variance, according to Eq. (10). By doing this, the correlation factor differences between the homogenized data and the original data were found to be small (not shown), while the differences in spatial variance between the two datasets were very large, especially from April to September (not shown). Thus, the large summer sampling error variances in the original data (Figs. 5 and 6) can mainly be attributed to the same large variation in spatial variance. However, a further question to be posed is why did the spatial variance increase significantly? To answer this question, we need to employ Eq. (4). Because the spatial variance is determined by temperature within the grid box, we therefore believe that the increase in spatial variance is at least partly due to the homogenization. We further examined the two datasets carefully and found that the differences between the homogenized data and the original data are relatively small in winter, whereas large difference values can be found from April to September. These features indicate that the data without homogenization can lead to large spatial variance and further contribute to large uncertainties. Thus, homogenization can remove large uncertainties in the original records resulting from the various non-natural changes in China.

Another key issue is to quantitatively compare the difference in uncertainties between extreme temperature and mean temperature. A recent study by Hua et al. (2012) explored the uncertainties in Tmean over China. A comparison of our Tmax and Tmin uncertainties with those of Hua et al.(2012) ^a indicate that the uncertainties of Tmean, either at grid scale or national-average scale, are smaller than Tmax and Tmin. However, because of different temporal timescales and different numbers of stations, it is not fair to directly compare the values of the uncertainties of Hua et al. (2012) ^a with ours at both temporal and spatial scales. Hence, a prospective study is needed to evaluate the differences in uncertainties between Tmax, Tmin and Tmean.