Evidence of high-elevation amplification versus Arctic amplification

Elevation-dependent warming in high-elevation regions and Arctic amplification are of tremendous interest to many scientists who are engaged in studies in climate change. Here, using annual mean temperatures from 2781 global stations for the 1961–2010 period, we find that the warming for the world’s high-elevation stations (>500 m above sea level) is clearly stronger than their low-elevation counterparts; and the high-elevation amplification consists of not only an altitudinal amplification but also a latitudinal amplification. The warming for the high-elevation stations is linearly proportional to the temperature lapse rates along altitudinal and latitudinal gradients, as a result of the functional shape of Stefan-Boltzmann law in both vertical and latitudinal directions. In contrast, neither altitudinal amplification nor latitudinal amplification is found within the Arctic region despite its greater warming than lower latitudes. Further analysis shows that the Arctic amplification is an integrated part of the latitudinal amplification trend for the low-elevation stations (≤500 m above sea level) across the entire low- to high-latitude Northern Hemisphere, also a result of the mathematical shape of Stefan-Boltzmann law but only in latitudinal direction.

along the entire altitudinal gradient for both 1961-2010 and 1976-2010, and the rate of altitudinal amplification over the last 35 years is 1.46 times that for the last 50 years (Fig. 4a). An analogous method has been used to present the relationship between temperature trends and elevation on and around the Tibetan Plateau in previous  . Bars represent latitude, and trend magnitude is plotted on the y axis according to the 8 latitude ranks of 1715 stations. Error bars denote one standard error around the mean. Pearson correlation coefficient (r) between trend magnitude and elevation (latitude) is shown with two-tailed p value. Significant coefficient, at the 95% confidence level, is set in bold. studies [14][15][16] . However, no statistical significance was shown 14,16 or no significance test was performed 15 . Therefore, while the warming rates appear, to various extents, to be amplified with elevation, it is uncertain whether the relationship is statistically significant. For the NH-Ele-low, the increases in warming rate with latitude are evident across the entire low-to high-latitude Northern Hemisphere over the period 1961-2010 (Fig. 4b). A significant latitudinal amplification trend is found along the entire latitudinal gradient for both entire (1961-2010) and recent (1976-2010) periods, and the rate of latitudinal amplification over the last 35 years is 1.64 times that for the last 50 years (Fig. 4b).
Notably, stepwise regression analysis reveals a significant positive relationship between T t and altitude and latitude for the Ele-high, while a significant positive relationship between T t and latitude for the Ele-low, for which the effect of altitude is not significant (Table 1). It can be seen from the linear models for these two groups of stations that the warming for the Ele-high is not only closely associated with altitude but also latitude, while the warming for the Ele-low is only related to latitude. This indicates that there is not only a significant altitudinal amplification trend of 0.2297 °C km −1 50-yr −1 but also a significant latitudinal amplification trend of 0.2510× 10 −3 °C km −1 50-yr −1 for the Ele-high over the past 50 years, while there is only a significant latitudinal amplification trend of 0.1925× 10 −3 °C km −1 50-yr −1 for the Ele-low in the same period. Moreover, the faster rate of latitudinal amplification for the former than the latter indicates that the latitudinal amplification trend for the former is an enhanced result of the base latitudinal amplification trend for the latter due to increase of altitude. Therefore the warming amplification for the Ele-high is essentially a combination of an altitudinal amplification trend and an enhanced base latitudinal amplification trend.
Furthermore, from the linear models established for the Ele-high, that is, Table 1), it can be derived that Table 1). This implies that the warming for the Ele-high is linearly proportional to the temperature lapse rates along altitudinal and latitudinal gradients. As illustrated in Fig. 5, the effect of energy balance variation on the surface temperature can be amplified with decreasing temperature in the environment, as a result of the functional shape of Stefan-Boltzmann law 17 . This suggests that the high-elevation amplification could be a consequence of the mathematical shape of Stefan-Boltzmann law in both vertical and latitudinal directions.
As for the Ele-high, the relationship of T t with altitude and latitude was also analyzed for the Arctic stations. However, no significant relationship between T t and altitude and/or latitude is found for these stations (Table 1), suggesting there is no altitudinal amplification or latitudinal amplification within the Arctic region. Despite this fact, however, further analysis shows a significant positive relationship between T t and latitude for the NH-Ele-low (Table 1), indicating a significant latitudinal amplification trend across the low-to high-latitude Northern Hemisphere. This suggests that the Arctic amplification is an integrated part of the latitudinal amplification trend for the NH-Ele-low. Similarly, from the linear models established for the NH-Ele-low, that is, Table 1), it can be derived that Table 1. Linear models showing relationships of temperature trends with altitude and latitude, of temperature trends with base temperatures and of base temperatures with altitude and latitude for the four groups of stations tested. The models are established based on temperature trends (T t , in °C 50-yr −1 ), base temperatures (the base period 1961-1990 means, T b , in °C), altitudes (x 1 , in km), and latitudes (x 2 , in km) at individual stations using the stepwise regression or simple linear regression method according to models where c is the distance constant (111.317 km degree −1 ) for each degree of latitude. Multiple/ simple correlation coefficients (R/r) are given with two-tailed p values, and significant coefficients, at the 95% confidence level, are set in bold (see Supplementary Table S1 for details on the descriptive statistics of the four groups of stations). is in fact another format of the model Table 1). This implies that the warming for the NH-Ele-low is linearly proportional to the temperature lapse rate along the latitudinal gradient. This result suggests that the latitudinal amplification trend on the hemispheric scale, as well as the Arctic amplification, is probably a consequence of the functional shape of Stefan-Boltzmann law, but only in the latitudinal direction.
Turning to the examination of the sensitivity of altitudinal amplification and latitudinal amplification trends to altitude and available stations, as shown in Supplementary Table S3, the altitudinal amplification and latitudinal amplification trends are only detected for the first 17 and 16 altitude extents sampled, respectively. Afterwards, neither altitudinal amplification trend nor latitudinal amplification trend can be detected further despite even larger average temperature trends. At the same time, as shown in Supplementary Fig. S1, it can be seen that the rates of the last three altitudinal amplification trends and the last two latitudinal amplification trends detected drops dramatically compared to that just before despite increasing warming trends. This suggests that, with the increase of average altitude, and the decrease of altitude (latitude) extent, there should be a critical altitude (latitude) extent for the detection of altitudinal (latitudinal) amplification trend; and an even larger critical altitude (latitude) extent for an accurate estimation of altitudinal (latitudinal) amplification trend.
Why is there such a threshold effect? As shown in Supplementary Fig. S2, with the omission of lower elevation bands, the average altitude increases, whereas the number of available stations decreases sharply. Predominately controlled by this decrease, the warming signal quantity (WSQ) diminishes dramatically at the same time. Consequently, the altitudinal and latitudinal warming signal quantity (WSQ alt and WSQ lat , respectively) decreases sharply. When the WSQ alt (the number of stations) falls below the critical value of about 100 °C 50-yr −1 (about 120), the rate of altitudinal amplification trend becomes abnormally small until no altitudinal amplification trend can be detected anymore. When the WSQ lat (the number of stations) falls below the critical value of about 75 °C 50-yr −1 (about 95), no latitudinal amplification trend can be detected further. This indicates that the threshold effect is essentially a reflection of the threshold effect of warming signal quantity (the number of stations).
For the NH-Ele-low, the evaluation of the sensitivity of latitudinal amplification trends to latitude and available stations show a similar result (Supplementary Table S4 and Supplementary Fig. S3). The critical warming signal quantity (the critical number of stations) for the detection of latitudinal amplification trend is about 915 °C 50-yr −1 (580). Therefore, no latitudinal amplification trend can be detected within the Arctic region.
For the detection of latitudinal amplification trend, the value of the critical warming signal quantity obtained from sampling the NH-Ele-low is obviously larger compared with that revealed from sampling the Ele-high. This is due to the substantial difference in the areas across which the sampled stations are distributed. The land area above 2.2 km a.s.l is about 6.3 million km 2 according to the global pattern of land area outside Antarctica per altitude in 100 m steps a.s.l. 18 , while the land area in the north of 47.5 °N is over 35 million km 2 . Whereas a general comparison can only be made using signal intensity (per unit area signal quantity) rather than signal quantity. Nevertheless, from these two sampling experiments, it can be derived that even if the warming is very strong for a high-elevation region (or a high-latitude region), a certain number of stations are still required for the detection of altitudinal amplification and latitudinal amplification trends (or latitudinal amplification trend) and for an accurate estimation of altitudinal amplification and latitudinal amplification trends (or latitudinal amplification trend), the number of stations required is even larger.

Discussion
Pepin and Lundquist 19 have analyzed the relationship between T t and altitude for the world's 1084 high-elevation stations (> 500 m above sea level) as a whole, but found no strong correlation between T t and altitude. As Pepin where F i is the ith energy flux (i = 1 to n), T is temperature, σ is Stefan-Boltzmann constant, and x is a given energy flux in a general sense. Suppose a change in x causes changes in energy fluxes exactly in the same manner in warmer and colder climates, and the energy flux changes (∑dF i /dx) are the same for these climates, the temperature sensitivity (dT/dx) at lower temperature is larger than at higher temperature, as a result of T 3 in the denominator 17 . For instance, the temperature sensitivity at 0 °C is 25% smaller than at − 25 °C but 30% larger than at 25 °C. and Seidel 20 described in an earlier study, the main trend analyses for the entire study period 1948-2002 were restricted to the period 1948− 1998, and the median trend for the 1084 high elevation stations is 0.13 °C per decade over the period 1948-1998, with 444 (41%) of the stations showing significant positive trends. In the current study, however, the median trend for the 910 high-elevation stations is 0.25 °C per decade over the period 1961-2010, with 786 (86.4%) of the stations showing significant positive trends. A significant relationship between altitude and T t is uncovered for these stations in the last 50 years (r = 0.200, p < 0.0001). Meantime, analysis for the period 1961-1998 reveals no strong relationship between altitude and T t for the 910 high-elevation stations (r = 0.037, p = 0.263), even though the median trend for them is 0.17 °C per decade in this period, with 421 (46.3%) of the stations showing significant positive trends. Therefore, the failure in quantifying altitudinal amplification trend in the previous study 19 is primarily due to the weaker warming over 1948-1998 than 1961-2010, or because the decade of 2000's, when warming was stronger than for any the previous decades of the instrumental record 21 , was not covered.
Further, it is worth noting that when the relationship of T t with altitude and latitude is tested for the Ele-high, a significant positive relationship can be detected for every time period longer than 30 years starting from 1961 during the period 1961-2010. For instance, a significant positive relationship is detected for 1961-1995 (R = 0.456, p < 0.0001) and 1961-2005 (R = 0.469, p < 0.0001), with the altitudinal amplification trend of 0.0307 °C km −1 per decade and 0.0330 °C km −1 per decade, respectively. The rates of altitudinal amplification trends are 0.33 and 0.28 times smaller than that (0.0459 °C km −1 per decade) for 1961-2010. This is because both the effects (signals) of altitude and latitude and the interacting effect (signal) of altitude and latitude are taken into consideration when T t is regressed against these two variables, whereas only the effect (signal) of altitude is taken into consideration when T t is regressed against altitude alone.
Climate in mountainous regions is proposed to be controlled by four principal factors, i.e., altitude, latitude, continentality, and topography 22 . Altitude and latitude are proved to be major factors in determining the geographical pattern of temperature change in the Alps 23 . The current study confirms further that both altitude and latitude are key factors in shaping the global-scale high-elevation amplification, and that latitude is the dominant factor of Arctic amplification. Notably, our results suggest that the Stefan-Boltzmann law could be a key mechanism that induces the amplifications of warming at high-elevations and in the Arctic region.
Several physical processes have been suggested to explain Arctic amplification 4 . It is widely accepted that changes in the surface albedo associated with declining sea ice and snow cover enhance warming in the Arctic 2,4,5,24,25 , but other processes may play a part. For example, it has been suggested that changes in cloud cover and atmospheric water vapor content [26][27][28] , and changes in atmospheric heat transport 1 may be more important for Arctic amplification than the snow and ice albedo feedbacks. Using an energy balance model, Izumi et al. 29 reveal that surface downward clear-sky longwave radiation (influenced by atmospheric water vapor, moist static energy transport, and CO 2 concentration) is the most important component driving high-latitude amplification, though surface albedo also plays a significant role in this regard. The distinct latitudinal amplification trend across the low-to high-latitude Northern Hemisphere observed in this study suggests that much of the present Arctic warming is likely a result of the functional shape of Stefan-Boltzman law in latitudinal direction. From the view of response scale, this result is in agreement with the notion that the recent Arctic amplification is mainly originated from the large-scale processes such as enhanced northward heat and moisture transports 1,30 rather than the local-scale surface albedo feedback mechanisms 3 .
Snow-albedo feedback has also been shown to play a significant role in generating the high-elevation amplification 31,32 . However, its relative importance remains uncertain compared with other processes such as changes in cloud cover, atmospheric water vapor, and aerosols. For instance, it has been suggested that increases in downward longwave radiation caused by increasing atmospheric water vapor, combined with increases in absorbed solar radiation caused by decreases in snow cover extent, are partly responsible for recent large warming trend over the Tibetan Plateau 33 . In a global study, Ohmura 17 found that 11 out of 18 regions show a larger warming rate at the summit of mountains during the last 30 to 40 years. He ascribed the high-altitude amplification to two primary processes: the increasing diabatic process in the mid-and high troposphere as a result of the cloud condensation, and the amplifying process in the effect of the energy balance variation on the surface temperature as a result of the functional shape of Stefan-Boltzmann law. In the current study, we find a distinct relationship of the high-elevation warming with the temperature lapse rates along altitudinal and latitudinal gradients. This indicates that much of the high-elevation warming is probably a result of the functional shape of Stefan-Boltzmann law in both vertical and latitudinal directions. However, the question what processes (energy flux terms) are the most important components in the surface energy balance requires further investigation.  36 , and MeteoSwiss, respectively. The procedure for establishing the series of annual mean temperature from the raw daily and monthly data was the same as in the previous study 13 . Each of the station series had at least 37 years of records (with twelve months of monthly mean temperature in each year) during the period 1961-2010.

Methods
The overall quality of the data from these five sources is fairly good. The monthly data from the GHCNM version 3.2.0 have been quality controlled and adjusted, and the data from the HISTALP and MeteoSwiss have been homogenized. So the annual data series derived from these three sources were used for trend estimation without further homogeneity test. For the annual time series derived from the daily data collected from the NMICC and Scientific RepoRts | 6:19219 | DOI: 10.1038/srep19219 the RMS, each annual time series was checked for homogeneity using RHtests V3 37 . After removing the stations with inhomogeneous series, 464 and 360 station series from these two sources were used for trend estimation.
The trend for each station was estimated from the anomalies (relative to the 1961-1990 average) using the least squares best-fit method. Of all the stations used for this study, 2348 (84.4%) and 395 (14.2%) show significant positive, and non-significant positive trends, respectively; while 6 (0.2%) and 32 (1.2%) show significant negative, and non-significant negative trends, respectively (see Supplementary Table S2 for details on the trends for each of the six groups of stations used in this study).

Comparison of temperature trends between high-and low-elevation (latitude) stations.
A method based on the principle of paired region comparison 13 was used for this end. It was performed for the Ele-high and the Ele-low located in the same latitudinal band (3.40°N/S− 63.25°N/S); and for the Lat-high and the Lat-low with the same range of altitudes (0− 500 m a.s.l). The high-(low-) elevation stations were first grouped into 100-m-wide elevation bands starting at 500 m (0 m). The band anomaly values were then produced by simple averaging of individual station anomaly values within each band. Afterwards, the regional anomaly values were computed by re-averaging of the individual band anomaly values, and the regional trend was calculated as the slope of simple linear regression. A similar procedure was used for the comparison of temperature trends between the Lat-high and the Lat-low. The main difference is that the high-(low-) latitude stations were first grouped into 2-deg-wide latitude bands starting at 60°N (0°N).
Analysis of elevation-dependent warming and latitude-dependent warming. This was performed for the Ele-high and the Ele-low as a whole, as well as for the Lat-high and the Lat-low as a whole, using a method similar to the elevation band method 15,16

Analysis of relationships of T t (T b ) with altitude and latitude and of T t with T b . The relation-
ship of T t (T b ) with altitudes (x 1, in km) and latitudes (x 2, in km) was tested according to linear model of fit respectively. The relationship between T t and T b was tested according to model of fit T t = b 0 T b + c. Latitude in km = latitude in degree × c, where c is the distance constant (111.317 km degree −1 ) for each degree of latitude. The reason we have taken the regression coefficients (b 1 and b 2 ) in the linear model (T t = b 1 x 1 + b 2 x 2 + c) as the rates of altitudinal and latitudinal amplification trends is that (1) the goodness of fit of the linear model is comparable to that of the non-linear model, judged from the multiple correlation coefficients (R); and (2) the statistical properties of the linear estimates are easier to determine relative to the non-linear estimates.
Evaluation of the sensitivity of altitudinal amplification and latitudinal amplification trends to changes of altitude and number of available stations. This was carried out by repeatedly sampling the data of the Ele-high by omitting segments progressively from the lower limit of altitudinal gradient in 100 m steps, and computing the altitudinal amplification and latitudinal amplification trends as well as the related descriptive statistics for each altitude extent sampled (i.e., each subset of stations from the Ele-high). The altitudinal amplification and latitudinal amplification trends were estimated using the model of fit T t = b 1 x 1 + b 2 x 2 + c as above. A similar method was used for the evaluation of the sensitivity of latitudinal amplification trend to changes of latitude and number of available stations (see Supplementary Table S4 for details on the difference).
To properly assess the effect of altitude (latitude), we proposed a new measure "warming signal quantity (WSQ)". The WSQ for each altitude (latitude) extent was defined as the product of the average warming rate (°C) × the number of the stations for the altitude (latitude) extent. Meanwhile, three steps were taken to compute the altitudinal warming signal quantity (WSQ alt ) and latitudinal warming signal quantity (WSQ lat ) for each altitude extent where both altitudinal and latitudinal amplification trends can be detected. First the altitudinal warming component (T talt ) and latitudinal warming component (T tlat ) were computed as the products of the rate of altitudinal amplification trend × mean altitude, and the rate of latitudinal amplification trend × mean latitude, respectively. Second, the relative contributions of altitude and latitude (RC ALT and RC LAT , respectively) were estimated using the formula RC ALT = T talt /(T talt + T tlat ), and RC LAT = T tlat /(T talt + T tlat ), respectively. Finally, the WSQ alt and WSQ lat were produced as the products of WSQ × RC ALT , and WSQ × RC ALT , respectively.