1 Introduction

The patterns of climate variation in high-elevation regions may be substantially different from those derived from low elevation observations (Seidel and Free 2003). Hence characterization of climate change in high-elevation regions is of utmost interest for understanding the global climate change and for assessing the impacts of climate change on regional environments and economies (Beniston 2003). Against the background of the global average surface temperature increase (especially since about 1950) (Solomon et al. 2007), and the rapid retreat of mountain glaciers around the world (particularly in the Himalaya, Andes and Alps) (Solomon et al. 2007), huge efforts have been made to explore the features and impacts of climate change in high-elevation regions (Beniston et al. 1997; Beniston 2003; Rangwala and Miller 2012). However, despite numerous studies in the past decades (Beniston and Rebetez 1996; Diaz and Bradley 1997; Liu and Hou 1998; Liu and Chen 2000; Vuille and Bradley 2000; Pepin and Losleben 2002; Vuille et al. 2003; Pepin and Seidel 2005; Diaz and Eischeid 2007; Appenzeller et al. 2008; Pepin and Lundquist 2008; You et al. 2008, 2010; Liu et al. 2009; Lu et al. 2010), our understanding of whether elevation-dependent warming commonly occurs in high-elevation regions, and whether high-elevation regions are warming faster than their low elevation counterparts, remains uncertain (Rangwala and Miller 2012).

While several studies reported positive evidence (Beniston and Rebetez 1996; Diaz and Bradley 1997; Liu and Hou 1998; Liu and Chen 2000; Liu et al. 2009), other studies found no evidence (Pepin and Seidel 2005; You et al. 2008; You et al. 2010) or even negative evidence (Vuille and Bradley 2000; Pepin and Losleben 2002; Lu et al. 2010) of elevation dependency in surface warming based on the surface observations, though most climate models have found enhanced warming in the high-elevation regions (Giorgi et al. 1997; Fyfe and Flato 1999; Snyder et al. 2002; Chen et al. 2003; Kotlarski et al. 2012). The elevation dependency was only statistically confirmed for the minimum temperature anomalies (1979–1993) in the Swiss Alps (Beniston and Rebetez 1996), while it was graphically displayed for minimum temperature (1951–1989) for the stations located in the latitudinal band 30°N–70°N (Diaz and Bradley 1997), and for mean and minimum temperature for the Tibetan Plateau and its surroundings as a whole (Liu and Chen 2000; Liu et al. 2009). However, it should be pointed out that no statistically significant altitudinal dependency of temperature increase has been demonstrated for either the Tibetan Plateau (You et al. 2008, 2010) or the Tibetan Plateau and its surroundings as a whole (Liu and Hou 1998).

This uncertainty is generally attributed to inadequacies in observations at high altitudes (Rangwala and Miller 2012; Ohmura 2012), data incompatibility (Ohmura 2012), methodological choices (Liu et al. 2009) and region-specific conditions (Beniston et al. 1997; Liu et al. 2009; Ohmura 2012). However there may be some fundamental aspects that have been overlooked. Although it is known that the warming is not only affected by altitude but also by latitude within a high-elevation region (Beniston and Rebetez 1996), few studies have considered the interacting effect of altitude and latitude on altitudinal amplification; and no studies have sought to separate out the altitudinal warming components from the warming rates at individual stations over the region. In this study we focus on the examination of both altitudinal amplification and regional amplification in the high elevation regions across the globe with new methodology based on annual mean temperature series (1961–2010) from 2,367 stations around the globe (Fig. 1).

Fig. 1
figure 1

Distribution of 2,367 stations used for this study around the globe

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We begin with analyzing the relationship between the station warming rates and station altitudes for the high-elevation regions selected (Fig. 2; Table 1) using the simple linear regression method as in previous studies (Beniston and Rebetez 1996; You et al. 2008, 2010), and perform a brief attribution study on the mixed results from the analysis, with the aim of showing the necessity of the extraction of altitudinal warming components from the station warming rates for the detection of altitudinal amplification within a high-elevation region. We focus on the examination of altitudinal amplification in the eight high-elevation regions using the new function equations. At the same time, we employ a paired region comparison method to address the question whether high-elevation regions are warming faster than their low elevation counterparts.

Fig. 2
figure 2

Distribution of stations in the high-elevation regions shown across the globe. The typical high-elevation (≥800 m abs), middle high-elevation (≥500 to  <800 m abs) and low high-elevation (≥200 to  <500 m abs) stations are shown in blue, green and red colors, respectively. Dots stand for significant positive trends, and circles indicates non-significant positive trends

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Table 1 Statistics of station mean, minimum and maximum values for altitude (in km), latitude (in degree) and longitude (in degree) over the high-elevation regions across the globe
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2 Data and methods

2.1 Data

The raw data series consisted of 1,861, 499, 72 and 25 station series from the quality controlled adjusted Global Historical Climatology Network monthly mean temperature dataset (GHCNM version 3) (Lawrimore et al. 2011); the daily mean air temperature data set of the National Meteorological Information Center of China (NMICC), the Historical Instrumental Climatological Surface Time Series of the Greater Alpine Region (HISTALP) (Auer et al. 2008), and MeteoSwiss, respectively. Each of the station series from the GHCNM (version 3), HISTALP and MeteoSwiss had at least 37 years of records (with 12 months of monthly mean temperature in each year) during the period 1961–2010, while each of the station series from the NMICC had 50-year records during the same period.

The annual data series for every station was established based on the monthly data from the GHCNM (version 3), while the annual data series for every station was directly from the HISTALP and MeteoSwiss. Because the monthly data from the GHCNM (version 3) have been quality controlled and adjusted, and the data from the HISTALP and MeteoSwiss have been homogenized, the annual data series derived from them were used for change trend test without further homogeneity test.

The overall quality of the data from the NMICC is fairly good, with nearly half of the temperature series having no missing data, and over 80 % of data present for each temperature series that had missing data. The missing data mainly occurred in the early time (1962–1970) of the study period. The annual time series for each station was established based on two criteria: (1) a monthly value was calculated if no more than 3 day’s data were missing in the month, and (2) an annual value was calculated if 12 months of monthly values were complete in the year. To eliminate the possible effect of artificial shifts caused by relocations of measurement sites or unknown reasons, each station time series of temperature was checked for homogeneity (Wang and Feng 2010; Wang et al. 2012). After rejecting 30 stations with inhomogeneous series, 469 stations were selected for trend test.

The trend slope was estimated using Sen method (Sen 1968), and the level of significance was tested non-parametric (Mann 1945; Kendall 1975) with an iterative procedure (Wang and Swail 2001). Of all the candidate stations, 55 of them have negative trends (whether significant or not), and 5 of them show no trend [Sen’s slope Q TOTAL = 0] as in previous study (Fan et al. 2012). After rejecting the stations showing negative (cooling) trends or no trends, a total of 2,367 stations that have positive (warming) trends (whether significant or not) were finally selected for this study (Fig. 1): of which 1,808, 462, 72, and 25 stations are from the GHCNM (version 3), NMICC, HISTALP, and MeteoSwiss, respectively; and 970 are low lying stations (<200 m abs) and 1397 are high-elevation stations (≥200 m).

The exclusion of the stations showing negative trends is primarily due to the fact that the new function equations for the extraction of warming components of altitude, latitude and longitude (Q ALT, Q LAT and Q LONG, respectively) for individual stations are only applicable to warming trends (See next section for details). For the extraction of cooling components of altitude, latitude and longitude (-Q ALT, -Q LAT and -Q LONG, respectively) from the cooling trends, three other function equations should be used (not shown). Besides, this exclusion should have no substantial influence on the results in this study, because 29 of them are low lying stations, and 26 of them are high-elevation stations, and only four of them are located in the eight high-elevation regions (two in the United States Rockies and other two in the Yunnan-Guizhou Plateau, accounting for 1.68 and 1.10 % of the total stations, respectively). None are in the four paired high and low elevation regions.

2.2 Computation of warming components

Assume that the temperature T is an intensive quantity (a single-valued, continuous and differentiable function of three-dimensional space), that is,

$$ T = T\left( {x,y,z} \right), $$
(1)

where x, y, and z are the coordinates of the location of interest, then the derivative of temperature as a vector quantity can be defined as,

$$ T_{SD} = \left( {\frac{\partial T}{\partial x},\frac{\partial T}{\partial y},\frac{\partial T}{\partial z}} \right). $$
(2)

According to the Pythagorean theorem, the statement can be expressed as,

$$ \left| {T_{SD} } \right| = \sqrt {\left( {\frac{\partial T}{\partial x}} \right)^{2} + \left( {\frac{\partial T}{\partial y}} \right)^{2} + \left( {\frac{\partial T}{\partial z}} \right)^{2} } . $$
(3)

Similarly, the warming rate (Q TOTAL) (magnitude of positive trend) at a station can be defined as,

$$ \mathop Q\nolimits_{TOTAL} = \sqrt {\mathop Q\nolimits_{ALT}^{2} + \mathop Q\nolimits_{LAT}^{2} + \mathop Q\nolimits_{LONG}^{2} } . $$
(4)

Corresponding to Eq. 4 for the station total warming rate (Q TOTAL), the total effect of altitude, latitude and longitude (E TOTAL) can be defined as,

$$ E_{TOTAL} = \sqrt {(EC_{ALT} \times ALT)^{2} + (EC_{LAT} \times LAT)^{2} + (EC_{LONG} \times LONG)^{2} } , $$
(5)

where EC ALT  × ALT, EC LAT  × LAT and EC LONG  × LONG are the effects of altitude, latitude and longitude, respectively; in which EC ALT , EC LAT and EC LONG are the effect coefficients for altitude, latitude and longitude; and ALT, LAT and LONG stand for altitude, latitude and longitude, respectively. Note that altitude (ALT), latitude (LAT) and longitude (LONG) are all expressed in kilometer (km) here.

Since altitude is commonly expressed in meter, and latitude and longitude in degree,

$$ ALT = {{altitude} \mathord{\left/ {\vphantom {{altitude} {1000}}} \right. \kern-0pt} {1000}}, $$
(6)
$$ LAT = latitude \times 111.317, $$
(7)
$$ LONG = longitude \times {{\pi \times R \times \cos (latitude)} \mathord{\left/ {\vphantom {{\pi \times R \times \cos (latitude)} {180}}} \right. \kern-0pt} {180}}, $$
(8)

where 111.317 (expressed in km) is the distance constant for per degree of latitude, and R is the radius of the earth. Because the distance between two degrees of longitude changes with latitude, the Eq. (8) is used.

According to Eqs. (4) and (5), the following relationships exist,

$$ \frac{{Q_{ALT} }}{{Q_{TOTAL} }} = \frac{{EC_{ALT} \times ALT}}{{\sqrt {(EC_{ALT} \times ALT)^{2} + (EC_{LAT} \times LAT)^{2} + (EC_{LONG} \times LONG)^{2} } }}, $$
(9)
$$ \frac{{Q_{LAT} }}{{Q_{TOTAL} }} = \frac{{EC_{ALT} \times LAT}}{{\sqrt {(EC_{ALT} \times ALT)^{2} + (EC_{LAT} \times LAT)^{2} + (EC_{LONG} \times LONG)^{2} } }} $$
(10)
$$ \frac{{Q_{LONG} }}{{Q_{TOTAL} }} = \frac{{EC_{ALT} \times LONG}}{{\sqrt {(EC_{ALT} \times ALT)^{2} + (EC_{LAT} \times LAT)^{2} + (EC_{LONG} \times LONG)^{2} } }}, $$
(11)

That is, the ratio of altitudinal (latitudinal or longitudinal) warming component versus station total warming is equal to the ratio of altitudinal (latitudinal or longitudinal) effect versus the total effect. In other words,

$$ Q_{ALT} = \tfrac{{Q_{TOTAL} \times EC_{ALT} \times ALT}}{{\sqrt {(EC_{ALT} \times ALT)^{2} + (EC_{LAT} \times LAT)^{2} + (EC_{LONG} \times LONG)^{2} } }}, $$
(12)
$$ Q_{LAT} = \tfrac{{Q_{TOTAL} \times EC_{LAT} \times LAT}}{{\sqrt {(EC_{ALT} \times ALT)^{2} + (EC_{LAT} \times LAT)^{2} + (EC_{LONG} \times LONG)^{2} } }}, $$
(13)
$$ Q_{LONG} = \tfrac{{Q_{TOTAL} \times EC_{LONG} \times LONG}}{{\sqrt {(EC_{ALT} \times ALT)^{2} + (EC_{LAT} \times LAT)^{2} + (EC_{LONG} \times LONG)^{2} } }}. $$
(14)

So the Q ALT, Q LAT and Q LONG can be separated out from the Q TOTAL at individual stations within a high-elevation region using Eqs. 12, 13 and 14, respectively.

2.3 Estimation of effect coefficients

Besides the development of above functional equations, another crucial point for separating out Q ALTs from the Q TOTALs at individual stations within a high-elevation region is to determine the EC ALT, EC LAT and EC LONG in the region. Here we define the EC ALT, EC LAT and EC LONG as the negative values of temperature gradients for altitude, latitude and longitude (G ALT, G LAT and G LONG, respectively).

To guide the estimation of G ALT, G LAT, and G LONG within a high-elevation region, we first estimated the base G ALT, G LAT, and G LONG on global scale. The base G ALT was calculated based on the data from the 643 typical high-elevation stations (≥800 m abs), where the noise (like the urbanization effect) is assumed to be small or negligible, while the base G LAT and G LONG were computed based on the data from the 970 low lying stations (<200 m abs), where the altitudinal effect is negligible. The base G ALT, G LAT and G LONG on the global scale and the G ALT, G LAT and G LONG for every region were estimated with the stepwise regression method based on the 50-year mean temperature and the station metadata (the absolute values of altitude, latitude and longitude) at the individual stations within a specific area: a latitude zonal band, longitude zonal band or a high-elevation region. Specifically, the base G ALT was calculated for the latitude zonal bands in the Northern Hemisphere (NH) (0–40°N and 40–51°N for the west NH (0–180°W) and east NH (0–180°E), respectively) and in the Southern Hemisphere (SH) (0–45°S for the SH as a whole), because the high-elevation stations selected are only distributed in the south of 51°N and the north of 45°S, with 601 and 14 stations in the NH and SH, respectively. The GR ALT for the 754 middle elevation stations (from ≥200 to <800 m abs) was also computed for the NH and SH, respectively. The base G LAT was estimated for the latitude zonal bands (0–20°N, 20–40°N, 40–60°N and 60–85°N; and 0–20°S, 20–40°S, 40–60°S and 60–80°S), while the base G LONG was estimated for the longitude zonal bands (0–90°W, 90–180°W, 0–90°E, 90–180°E for the NH and SH, respectively).

3 Relationship between station warming rates and station altitudes

Our analysis shows that a significant warming trend is found for all the eight regions tested, with the greatest warming in the Tibetan Plateau, and the weakest in Yunnan-Guizhou Plateau (Table 2). However, mixed relationships between station warming rates (Q TOTALs) and station altitudes are detected among these regions, with significant altitude amplification of Q TOTALs in three regions, but no altitude amplification of Q TOTALs in five regions (Table 3). It is noteworthy that a significant altitude amplification of Q TOTALs is detected in the Southern Tibetan Plateau but not in the Northern Tibetan Plateau (Table 3) when we divide the entire Tibetan Plateau into these two sub-regions along the dividing ridges of Tanggula Mountains and east Bayan Har Mountains.

Table 2 Monotonic trends of annual mean temperature during 1961–2010 over the high-elevation regions across the globe
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Table 3 Relationships between station warming rates and station altitudes in the high-elevation regions across the globe
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In high-elevation regions temperature change depends strongly on altitude, latitude and longitude (Beniston and Rebetez 1996). Due to the effects of these three factors, especially the positive effects of altitude and latitude, on temperature change in the high-elevation regions tested (See next section for details), the mixed results could be attributed to region-specific interactions between altitude and latitude (positive, negative or no interaction) and the relative magnitudes of altitude and latitude effects (comparable to each other or one is much larger than the other) across these regions. As exemplified by the Tibetan Plateau and the Loess Plateau, a significant negative spatial correlation between the station altitudes and station latitudes (SCOALLA) is detected across the Tibetan Plateau, whereas a significant positive SCOALLA is observed in the Loess Plateau (Fig. 3). This indicates that the altitude effect could be cancelled out over the former but enhanced over the latter by the latitude effect. Therefore the altitude amplification of Q TOTALs can be detected within the latter but not within the former. At the same time, a significant negative SCOALLA occurs across the Northern Tibetan Plateau (as in the entire Tibetan Plateau), whereas no significant SCOALLA appears across the Southern Tibetan Plateau (Fig. 3). This suggests that the altitude effect could be cancelled out over the Northern Tibetan Plateau (as in the entire Tibetan Plateau) while no obvious correlation between altitude and latitude exists across the Southern Tibetan Plateau. Hence the altitude amplification of Q TOTALs can not be detected for the Northern Tibetan Plateau either while an altitude amplification of Q TOTALs has been observed for the Southern Tibetan Plateau. Nevertheless, as long as the total warming rates (Q TOTALs) are involved, altitude amplification cannot be detected in any high-elevation region, except by chance. The separation of altitudinal components from the warming rates at individual stations is therefore a prerequisite for a precise analysis of altitudinal amplification within a high-elevation region.

Fig. 3
figure 3

Exemplification of the effect of SCOALLA on altitudinal dependency of station warming rates in the high-elevation regions. SCOALLA indicates spatial correlation between station’s altitudes and station’s latitudes as shown in the left panels for the Tibetan Plateau, Loess Plateau, Northern Tibetan Plateau, and Southern Tibetan Plateau, respectively. The altitudinal dependency of station warming rates is shown in the right panels for the related regions, respectively. Pearson correlation coefficients (r) are shown with two-tailed p values in parentheses. The significant coefficients, judged using 95 % CI, are set in bold

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4 Altitudinal warming amplification

With the Q TOTALs at individual stations, and the EC ALT, EC LAT and EC LONG for every region (Table 4), we perform the extraction of Q ALT (Q LAT or Q LONG) from QTOTAL for every station within each region using the functional equation(s) for them. As a result, we find a highly significant altitude amplification trend for all the regions except for the Mongolian Plateau, where a marginally significant altitude amplification trend is detected (Fig. 4). When the two sub-regions of the Tibetan Plateau are considered together, the greatest amplification appears in the Southern Tibetan Plateau while the weakest in the Mongolian Plateau among these regions. The averaged 50-year trend of 0.19 (±0.09) [mean (±SD)]°C per km of altitude is found for the eight regions. Note that if the small amplification trend in the Mongolian Plateau is set aside, the average amplification rate will be slightly greater [0.21 (±0.08) °C per km of altitude]. Judged from the averaged warming components of altitude, latitude and longitude (Q ALTAV, Q LATAV, and Q LONGAV, respectively) in the eight regions (Table 5), the relative contributions of altitude, latitude and longitude account for 13.5 (±7.8) %, 72.9 (±23.4) % and 13.6 (±20.9) %, respectively.

Table 4 Effect coefficients of altitude, latitude and longitude in the high elevation regions across the globe
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Fig. 4
figure 4

Relationships between altitudinal warming rates and station altitudes in the high elevation regions across the globe. Dots represent altitude-related warming rates (°C per 50-year), and dark cyan lines indicate linear regression lines. Pearson correlation coefficients (r) are shown with two-tailed p values in parentheses. The significant coefficients, judged using 95 % CI, are set in bold, with the marginally significant coefficient in italic bold

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Table 5 Average warming components for altitude, latitude and longitude over the high-elevation regions across the globe
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The altitude warming amplification in the high-elevation regions is consistent with the modeled and observed warming amplification in the free troposphere (Santer et al. 2005; Thorne et al. 2011), indicating a decrease of the surface mean temperature lapse rate over the last 50 years, consistent with the model results (Santer et al. 2005; Kotlarski et al. 2012). The fact that the upward shift in the height of the freezing level surface and hence the significant decline of snow covered area in the mountainous regions (Diaz et al. 2003; Bradley et al. 2009) could be a result of the altitude warming amplification within these regions. Compared with altitude, latitude has much lager contribution to the sum of Q ALTAV, Q LATAV and Q LONGAV in every high-elevation region (Table 5), suggesting a greater effect from latitude than altitude. This verifies that the effect of the positive or negative SCOALLA on the altitudinal dependence of station warming rates (Fig. 3) is closely associated with the latitude effect. Although the longitudinal effect is not significant in over half of the regions, it does occur in the Tibetan Plateau, Loess Plateau and Yunnan-Guizhou Plateau. The magnitudes of longitude effect in these three plateaus (Table 5) coincide inversely with their distances (about 1,200, 800 and 500 km, respectively) to the Pacific Ocean in the southeast. This suggests that the longitude effect in these three regions is to some extent negatively linked to their distance from the ocean under the condition of strong southeast summer monsoon and northwest winter monsoon in China.

5 Regional warming amplification

The analysis of temperature trends for the paired regions shows that the warming for the sampled North Tibetan Plateau, East Loess Plateau, Southeast Rockies and Alps is clearly greater than their low-lying counterparts (Fig. 5; Table 6). It is noteworthy that the warming for the sampled North Tibetan Plateau is also much greater than the North China Plain (both of which are located at the same latitudes as well). The warming is also much greater over the sampled North Tibetan Plateau than those for the Southeast Rockies (USA), Alps, East lower region of SE Rockies, and East lower region of Alps, even if the latter are located at higher latitudes than the former (though at very different longitudes) (Table 6). In addition, the warming over the Southern Tibetan Plateau is much greater than the Loess Plateau though the former is located at lower latitudes than the latter, and, similarly, the warming over the Northern Tibetan Plateau is greater than the Mongolian Plateau (Table 3). These results suggest that the warming in a high-elevation region is usually greater than its lower elevation counterpart(s) at the same latitudes or even those at the somewhat higher latitudes.

Fig. 5
figure 5

Monotonic trends of annual mean temperature over the paired regions shown across the globe. Left hand axis shows temperature anomalies relative to the 1961–1990 average. Significant trend, judged using 95 % CI, is set in bold (see Table 6 for details)

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Table 6 Monotonic trends of annual mean temperature in the paired regions
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6 Discussion

The altitude temperature gradient (G ALT) in a high-elevation region is dominated by the height temperature gradient (G H) in the free troposphere, and modulated by local conditions. The G ALT and G H typically change between about −9.8 °C per km (the dry adiabatic lapse rate) and about −4.0 °C per km (the saturated adiabatic lapse rate, i.e. the maximum moisture adiabatic lapse rate) (Rolland 2003). Comparatively, the estimates of base G ALT [−4.88(±0.94) °C per km] on global scale and the averaged G ALT [−5.00(±1.38) °C per km] for the high-elevation regions as a whole are close to the maximum value of moisture adiabatic lapse rate rather than the value of dry adiabatic lapse rate. This suggests that the altitude effect, and hence the warming amplification within the high-elevation regions are most likely associated with the effective moist convection. Quantitatively, based on the regression equation between altitude warming rate (y) and altitude (x) established for each region (Fig. 4), the altitude warming rate (Q ALT) at the highest observation site (Table 1) can be easily estimated. For instance, it is 1.26, 0.78 and 0.24 °C for the Tibetan Plateau, Alps and South American Andes, respectively. Furthermore, if the equation is still robust at the highest point of each region, the Q ALT for the Tibetan Plateau, Alps and South American Andes will reach 2.38 °C (at 8.848 km abs), 1.04 °C (at 4.810 km abs) and 1.45 °C (at 6.962 km abs), respectively. This kind of signals is of great importance for climate impact assessment. In addition, it should be noted that the G ALT at the highest point of the Tibetan Plateau or the South American Andes may be close to the dry adiabatic lapse rate with the decrease of temperature, i.e. the altitude effect coefficient (EC ALT) could be higher with the increase in altitude. Therefore the Q ALT could be even greater than 2.38 or 1.45 °C in the highest area of the Tibetan Plateau or the South American Andes theoretically.

Compared with the base latitude temperature gradient [−0.55(±0.41)oC per degree] on global scale, the averaged latitude temperature gradient [−0.81(±0.19) °C per degree] for the high-elevation regions is much smaller, indicating an enhancing effect of altitude on the decrease of latitude temperature gradient in these regions. Therefore a generally larger latitude effect coefficient should exist in a high-elevation region relative to its low lying counterpart(s). Since the larger latitude effect coefficient can lead to a greater latitudinal warming rate (i.e. a latitudinal amplification) and the increase of station total warming rate at the same time, the greater warming in a high-elevation region relative to its lower elevation counterpart(s) can be attributed to not only the altitude amplification but also the latitude amplification.

The warming of 1.87 °C per 50 years in the Tibetan Plateau is the greatest among the eight high-elevation regions, though second to the warming of 2.23 °C per 50 years in the Northern Polar Area (Table 7), the greatest warming at the higher northern latitudes. However despite some apparent differences between the Tibetan Plateau and Northern Polar Area in terms of the patterns of seasonal warming (Table 7), and the percentages of warming components for altitude, latitude and longitude (Table 5), there exists an essential similarity between them: the greatest (weakest) seasonal warming coincides exactly with the lowest (highest) seasonal mean temperature (Table 7). The analysis of correlation between the temperature anomalies (1961–2010) further reveals that while a highly significant correlation is found between the Tibetan Plateau and Northern Polar Area, the significant correlation is also detected between the Tibetan Plateau and Loess Plateau (LP), and the Northern Polar Area and the High Latitude Area (Table 8). This suggests that the warming amplification in the Tibetan Plateau (the Northern Polar Area) could be an extension of the temperature increase of the neighboring lower altitude (latitude) region rather than mediated by local processes. We speculate that the air temperature gradient, which exists between the surface and the atmosphere, and between the equator and northern polar area is the common driving force leading to the Tibetan Plateau (Arctic) amplification. While the interaction between equator-to-pole air temperature gradient and poleward atmospheric heat transport may have played a major role in leading to the Arctic amplification (Graversen et al. 2008; Bekryaev et al. 2010), the interaction between the surface-atmosphere gradient and upward heat transport has likely played a substantial part in causing the Tibetan Plateau amplification. At the same time, it should be noted that other mechanisms may have also made some contribution in causing the amplification on regional scale (Graversen et al. 2008; Screen and Simmonds 2010; Serreze and Barry 2011) and the snow and ice feedbacks might become the dominant mechanism for a future Arctic amplification (Graversen et al. 2008) or Tibetan Plateau amplification.

Table 7 Mean temperature and temperature trends within the Tibetan Plateau (TP), Loess Plateau (LP), Northern Polar Area (North of 60°N; NPA) and High Latitude Area (55–60°N; HLA) during 1961–2010
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Table 8 Correlation between temperature anomalies (1961–2010) for each pair of regions
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The altitudinal amplification is a common feature of the high-elevation regions across the globe. However it does not imply that this phenomenon occurs across all high-elevation regions on the earth. It is likely that if the signal-to-noise ratio is not big enough, then the altitudinal amplification would be undetectable. The signal from altitude effect can be contaminated by the noise from specific factors such as temperature inversion (Beniston and Rebetez 1996; Ohmura 2012) and urbanization effect (Pepin and Lundquist 2008; Ren et al. 2008) characteristic of low elevation stations. Similarly, the high-elevation region does not always have a faster warming than its lower counterpart. This can also be attributable to multiple factors, especially the higher urban and industrial effect at low-altitude locations (Pepin and Lundquist 2008). For instance, if the lower region is more strongly affected by such an effect, leading to an additional increase of 0.35 and 0.80 °C per 50 years (Ren et al. 2008) for the high and lower elevation regions, respectively, then the same or even faster warming may be detected in the lower region. However, these cases can not offset the common occurrence of warming amplification in the high-elevation regions. Therefore the warming amplification in the high-elevation regions is an intrinsic feature of global warming in recent decades.

7 Conclusions

In the present study, particular focus has been given to the development of the function equations for the extraction of warming components of altitude, latitude and longitude from the total warming rates at individual stations within a high-elevation region. This is primarily due to that if the total warming rates are directly used for analysis, altitude amplification can hardly be detected in any high-elevation region. The separation of altitudinal warming rates from the total warming rates at individual stations is a prerequisite for the precise analysis of altitudinal amplification within a high-elevation region.

With the new equation(s), we have performed the extraction of altitudinal (latitudinal or longitudinal) warming rate from total warming rate for every station within each region. As a result, a significant trend of altitudinal amplification is found for the eight high-elevation regions tested. The 50-year trend of 0.19 (±0.09) °C per km of altitude is observed for these regions as a whole, and the relative contributions of altitude, latitude and longitude account for 13.5 (±7.8) %, 72.9 (±23.4) % and 13.6 (±20.9) %, respectively. This indicates that the surface mean temperature lapse rate has decreased at a rate of about 0.2 °C per km over the last 50 years. The greater latitudinal effect compared with the altitudinal effect confirms that the positive or negative SCOALLA could have substantial effect on the altitudinal dependence of the station warming rates.

The paired region comparison shows that the warming for the sampled North Tibetan Plateau, East Loess Plateau, Southeast Rockies and Alps is clearly greater than their low-lying counterparts. The warming for the sampled North Tibetan Plateau is also much greater than the North China Plain at the same latitudes. This indicates that the warming in a high-elevation region is usually greater than its lower elevation counterpart(s) at the same latitudes.

Comparatively, the averaged altitude temperature gradient for the high-elevation regions is close to the moist adiabatic lapse rate rather than the dry adiabatic lapse rate. This suggests that the altitude effect, and hence the warming amplification within these regions are most likely associated with the effective moist convection. The smaller latitude temperature gradient for these regions indicates an enhancing effect of altitude on the decrease of latitude temperature gradient, and hence a generally larger latitude effect coefficient in a high-elevation region relative to its low lying counterpart(s). Therefore the greater warming in the high-elevation region can be attributed to not only the altitude amplification but also the latitude amplification. The results of this study suggest that warming amplification in high-elevation regions is an intrinsic feature of recent global warming.