Observational evidence of temperature trends at two levels in the surface layer

X. Lin, R. A. Pielke Sr., R. Mahmood, C. A. Fiebrich, and R. Aiken Department of Agronomy, Kansas Climate Center, Kansas State University, Manhattan, KS, USA CIRES and ATOC, University of Colorado, Boulder, CO, USA Department of Geography and Geology and Kentucky Climate Center, Western Kentucky University, Bowling, KY, USA University of Oklahoma, Oklahoma Climatological Survey, Norman, OK, USA Kansas State University, Northwest Research Center, Colby, KS, USA


Introduction
Physical properties of the atmosphere and dynamic processes mix heat vertically and horizontally, yielding the highest temperatures, on average, at the surface with marked seasonal and spatial variations (IPCC, 2013;Karl et al., 2006). The thermal structure near the surface is affected by various surface forcings (e.g., radiation absorbed and emitted, turbulent mixing, and vegetation interaction) which result in the near-surface lapse rate varying considerably with location and season as well as with atmospheric humidity (Stone and Carlson, 1979;Karl et al., 2006;Mahrt, 2006;Pielke Sr. et al., 2007). In the entire troposphere, climate models indicate a distinct height-dependent temperature response to surface temperature increases (refers to air temperature at a screen height near ground surface) Santer et al., 2005;Karl et al., 2006;Thorne et al., 2011;Seidel et al., 2012;Mitchell et al., 2013). Most of these heightdependent temperature studies focused on tropospheric temperature trends by using radiosonde and satellite observations and climate models , however, the Published by Copernicus Publications on behalf of the European Geosciences Union. 828 X. Lin et al.: Observational evidence of temperature trends near-surface temperature lapse rate has rarely been studied in the surface layer of the atmosphere.
Natural internal climate variability and noise in the data make the detectability of long-term temperature trends in the surface boundary layer difficult. One reason is that the boundary layer typically changes from a convective turbulent regime, with a gain of sensible heat (daytime), to a thermodynamically stable, long-wave radiationally cooled regime (nighttime) with a loss of sensible heat Baldocchi and Ma, 2013). The high-quality two-height surface observations in the Oklahoma Mesonet (Shafer et al., 2000;Lin et al., 2007), however, provide for the first time an accurate observational network to extract the temperature trend signal at two heights in the surface layer. The temperature observations are synchronized in time, fixed in height, and situated in relatively flat terrain, thus providing a unique opportunity to evaluate near-surface temperature trends and thus the lapse rate trends.
This study is the first observational investigation of twoheight, near-surface temperatures to examine lapse rate trends and variability over more than a decade period, a 17year timescale from 1997 to 2013, which substantially increases the signal-to-noise ratio for trend analysis (Santer et al., 2011) compared to a decade observation . In this study, our objective is to provide observational evidence for near-surface lapse rate and temperature trends over 1997 to 2013 in Oklahoma.

Climate stations and data
We selected stations from the Oklahoma Mesonet, which is a world-class network of environmental monitoring stations. As reported in 2009, the National Research Council (NRC) recommended the Oklahoma Mesonet as the "gold standard" for statewide weather and climate networks (https: //www.mesonet.org/, accessed on 4 May 2015). For twoheight temperatures, quality-controlled hourly observations from the Oklahoma Mesonet were used. They include air temperatures at 1.5 and 9.0 m, relative humidity at 1.5 m, wind speeds (WS) at 2 and 10 m, global incoming solar radiation (SR), and precipitation. The uncertainties in observations prior to 1997 in the Oklahoma Mesonet were due to an incomplete thermometer's processing algorithm (a delay time required in HMP35C temperature sensors for temperature and humidity measurements) (Shafer et al., 2000) so our study period was from January 1997 to December 2013. Stations that experienced relocation and missing high-level (i.e., 9 m) temperature measurements were excluded leaving a total of 44 Oklahoma Mesonet stations selected ( Fig. 1)  The US Historical Climatology Network (USHCN, version 2.5) consists of 44 high-quality stations in Oklahoma and the data quality of monthly average temperatures has been rigorously examined (Menne et al., 2009) (Fig. 1). These 44 USHCN stations have long been commonly selected for use in evaluating climate changes on the global, regional, and state scales and thus the USHCN temperature is considered as a reference temperature change when evaluating climate change. It was assumed that both the 44 USHCN stations and the 44 Oklahoma Mesonet stations are representative of the Oklahoma state region in this study.

Homogeneity tests of temperature time series in the Oklahoma Mesonet
In the USHCN data set, the instrument change adjustments in a climate series "is a regional average" (Quayle et al., 1991;Lin, 2002, 2006). The exact effect at individual stations may vary depending on local environmental or climate factors such as the direction of sunlight and wind speeds around the radiation shields. Temperature data used in the study from the Oklahoma Mesonet are quality controlled and thermometers used in the network have been calibrated every 24 to 60 months. The air temperature at 9 m height was measured by a thermistor in a naturally ventilated radiation shield from 1997 to 2013. Air temperature instruments at 1.5 m height were changed from a naturally ventilated radiation shield into an aspirated radiation shield in late 2008. Therefore, homogeneity tests of monthly temperatures for individual Mesonet stations in both T 1.5 m (temperatures at the 1.5 m height) and T 9 m (temperatures at the 9 m height) series over 1997 to 2013 were evaluated using two methods: standard normal homogeneity test (SNHT) (Alexandersson and Moberg, 1997;Peterson et al., 1998) and multiple linear regression (MLR) (Vincent, 1998;Reeves et al., 2007).  Note that a time series was classified as homogeneous only if the null hypothesis of homogeneity was not rejected at the 95 % level, using both methods to evaluate the single-mostprobable discontinuities (or step changes). The reference series (R i ) was formed by using five nearest stations weighted by squared correlation coefficients (ρ i,j ). In their simplest form, the SNHT and MLR are written as The second part of SNHT's Eq. (1) is the reference series. The x j is a surrounding station series and y i is the candidate station series to be tested. In the MLR's Eq.
(2), the I variable is a binary variable which is zero prior to the change point (c) and one after the occurrence of that change point (c). The e i in Eq.
(2) is the regression residual term. Note that for R i in Eq.
(2), the reference series is the same as the second part of Eq.
(1). The 44 T 1.5 m candidate series were tested against the nearest five USHCN stations, creating the reference series. Three documented change points and five undocumented change points were detected in the T 1.5 m temperature series. Three documented change points were adjusted in this study. For the 44 T 9 m candidate series, the instruments have been consistently operated by naturally ventilated radiation shields from 1997 to 2013. Larger ambient wind speeds at the 9 m height relative to the 1.5 m, reduce radiative errors for T 9 m temperatures (Hubbard and Lin, 2002). When the 44 T 9 m series were tested by using a reference series created from the five nearest Oklahoma Mesonet stations at the 9 m height, only two change points were found which were undocumented. These undocumented changes were not adjusted in our T 9 m temperature series.

Data and trend analysis
The lapse rate is defined as − T z by using the hourly temperatures observed at 1.5 and 9.0 m in units of • C (10 m) −1 . A negative trend in the lapse rate when the surface layer is stably stratified means that the temperature change became steeper (warmer at the higher level and/or cooler at the lower level). When the surface layer is unstably stratified, a negative trend means the temperature change with height has become less. All missing data were not filled or interpolated by estimation and no outlier screening was implemented in the study. When there were three hourly temperatures missing, the daily lapse rate was excluded. The monthly data were The ± values define the 95 % confidence intervals for trends. The shaded region around lapse rate anomalies shows the standard deviation of 44 Oklahoma Mesonet stations. The metadata for dates of thermometer status are shown at the bottom of (b) for changes of thermometer radiation shields at 9 and 1.5 m. excluded when more than 5 days were missing in a month. The air temperatures at two heights for daytime and nighttime were calculated based on the sunrise and sunset hours (rounded into an integral hour) during any calendar day. The mean wind speed of 2 and 10 m heights was used to classify wind regimes as windy (87 % percentile or above, i.e., 5 windiest days in a month) or calm (17 % percentile or below, i.e., 5 calmest days in a month) conditions on a monthly basis.
Monthly anomalies for lapse rates, temperatures, and other climatic variables were departures from monthly climatology for the period from January 1997 to December 2013. The regional time series were aggregated by using an equally weighted station average from each station when the observations were available.
The computation of complementary variables shown in this study is briefly described here. The total energy content of a unit parcel of air (per kg) is provided by the sum of the kinetic energy, latent heat, enthalpy, and gravitational potential energy . Without considering the gravitational potential energy and kinetic energy, the air heat content (H ) was then calculated by (Pielke Sr. et al., 2004;Peterson et al., 2011) where T is the Kelvin temperature (K) and q is the specific humidity (kg kg −1 ). Both the specific heat of air at constant pressure C p (J K −1 kg −1 ) and the latent heat of evaporation L (J kg −1 ) are calculated by a function of ambient humidity and temperature (Stull, 1988). The water vapor pressure deficit (VPD) was calculated using (4) e s and e a are the equilibrium (or saturated) vapor pressure and actual vapor pressure with respect to water obtained from (Wiederhold, 1997), where P is the atmospheric pressure (mb) and e w is the equilibrium vapor pressure (mb); for e s (mb), T is the ambi- ent temperature ( • C); for e a (mb), T is the dew point temperature ( • C). The dew point was calculated from ambient temperature and relative humidity observed in the Oklahoma Mesonet. The pressure P was estimated based on the station elevation. The calculation of reference evapotranspiration (ET o ) used the Penman-Monteith equation (Allen, 2000). All variables in the ET o calculation are either directly available at the stations or were estimated from empirical equations (Allen, 2000). For the trend analysis, the adjusted standard error and adjusted degrees of freedom method was used for evaluating the statistical significance of regional temporal trends and individual station trends at the 95 % or otherwise specified confidence levels Karl et al., 2006). This approach is a modification of the ordinary least squares linear regression to substitute the effective sample size by correcting for the effect of temporal autocorrelation in the anomaly time series or its residual series Karl et al., 2006).

Surface temperature-related trends at individual levels
Here we present the first observational investigation of two-height, near-surface temperatures to examine lapse rate trends and variability over more than a decade period. For the period of 1997 to 2013, when trends of surface tem-perature anomalies are evaluated by individual surface temperatures at 1.5 m (T 1.5 m ) and 9.0 m (T 9.0 m ) from Oklahoma Mesonet stations, statistically non-significant trends of +0.065 ± 0.59 • C per decade and +0.281 ± 0.58 • C per decade, respectively were documented ( Fig. 2a and b). However, trends could not be confirmed for either of these two individual surface temperatures over Oklahoma (derived from T 1.5 m and T 9.0 m ) when adjusting the statistical analysis for first-order autocorrelation effects as shown by the adjusted p values in the trend analysis. When we used the USHCN data, the surface temperatures (T USHCN ) again showed a statistically non-significant trend of 0.079 ± 0.58 • C per decade (Fig. 2c) over 1997 to 2013.
In terms of month-to-month variability of these three time series (Fig. 2a to c), the standard deviations over the period studied were 1.63, 1.64, and 1.65 • C for T 1.5 m , T 9 m , and T USHCN , respectively, without any statistical differences. To further examine the change over 1997 to 2013 at a single height, the surface air heat content (H ) (Pielke Sr. et al., 2004;Peterson et al., 2011) was evaluated. Again, we were unable to confirm a statistically significant trend in H although the H showed an apparent "cooling" trend (i.e. −0.737 ± 1.08 kJ kg −1 per decade) (Fig. 2d), which was caused by a decrease in air humidity (Fig. 2e).
The air heat content variability was very similar to the air temperature's month-to-month variability although it was coupled with air humidity (Fig. 2d and e). The temperature difference between measurements at 1.5 m of the Oklahoma Mesonet and USHCN (T 1.5 m − T USHCN ) had an overall standard deviation of 0.17 • C where less variation occurred during the first 10 years, relative to the subsequent 7 years. A slightly positive T 1.5 m − T USHCN difference, observed during the last 3 years, cannot be attributed to the thermometer's exposure changes in the Oklahoma Mesonet because the aspirated thermometers could have a cool-bias compared to nonaspirated thermometers at observing stations. Nonetheless, the overall 0.17 • C standard deviation of T 1.5 m − T USHCN is of the order of uncertainties associated with any current thermometer used in climate monitoring networks (Hubbard and Lin, 2002;Lin et al., 2005). Figure 3 shows lapse rate changes and changes in monthly anomalies for daily, daytime, and nighttime conditions. The lapse rate is defined as − (T 9 m −T 1.5 m ) 7.5 m with values plotted in units of • C (10 m) −1 . There was a substantial and clear seasonality signal in the daily lapse rate time series (Figs. 3a and  4a). The lapse rates in summertime were larger than in the wintertime which indicated that the lapse rate involved interactions with stronger turbulent energy exchanges in summer and relatively weaker turbulent energy exchanges in winter in the surface boundary layer (Figs. 3a and 4a).

Surface lapse rate trends and seasonality
The statistically significant trend of the daily lapse rate was −0.18 ± 0.03 • C (10 m) −1 per decade, and this daily lapse rate trend is the average of daytime (−0.16 • C (10 m) −1 per decade) and nighttime (−0.20 • C (10 m) −1 per decade) lapse rate trends as expected; all at the 99.9 % confidence levels. The nighttime lapse rate not only showed a larger trend than in the daytime but also varied significantly more ( Fig. 3b and d).
In Fig. 3b, the metadata inventory of thermometer changes suggests that there could be systematic biases which might compromise trend analysis. In addition to the routine qualitycontrol and instrument calibrations of T 1.5 m and T 9 m , we conducted multiple lines of data examination in T 1.5 m and T 9 m for their fidelity: (1) data homogeneity tests were conducted and documented change points were adjusted; (2) the T 1.5 m − T USHCN time series (Fig. 2f) showed that no systematic biases existed in T 1.5 m due to instrument changes late in 2008; and (3) the relatively flat lapse rate anomalies from 2009 to 2013 did not support a systematic bias caused by changes from naturally ventilated thermometers to aspirated thermometers (Fig. 3b). Therefore, it is unlikely that changes from naturally ventilated to aspirated thermometers in 2008 and 2009 contribute to the lapse rate trends.
That the lapse rate trend is statistically significant is initially surprising, since the individual two-height temperatures have no significant trends ( Fig. 3a and b). We explained how this can occur in Appendix A (see Figs. A1 and A2). Results in Fig. 3 indicated that the temperature difference between T 9.0 m and T 1.5 m had a statistically significant increasing trend. Considering the statistically non-significant trends in T 1.5 m and T 9.0 m ( Fig. 3a and b), we infer that the near-surface vertical temperatures at 9 m were warming faster than temperatures at the screen level (1.5 m) in the surface boundary layer. However, it is possible that cooling (which is within the range of statistical uncertainty) at the 1.5 m level could account for the increased temperature difference (T 9 m −T 1.5 m ). The −0.18 • C (10 m) −1 per decade of lapse rate trend with a 7.5 m height difference is equivalent to a warming trend +0.135 • C per decade of the (T 9 m minus T 1. tainty that has not yet been accounted for in the use of surface temperature trends to diagnose and monitor global warming. The seasonality shown in the daytime lapse rate was clearer than in the nighttime lapse rate (Figs. 3 and 4), suggesting that strong turbulent mixing controlled the daytime mixing layer but as expected, there was stabilized surface air (weak turbulence) in the nocturnal boundary layer (Stone and Carlson, 1979;Stull, 1988;Karl et al., 2006;McNider et al., 2012). Thus, the nighttime lapse rate clearly consistently varied much more than the daytime lapse rate over 1997 to 2013 (Figs. 3a and 4a). Figure 4 indicates that part of the daytime was unstably stratified in the surface boundary layer, however, for most of the time over a 24 h period, the lapse rates show a stable surface boundary layer for all months in the Oklahoma region. During the spring season, the daytime lapse rates were relatively suppressed while nighttime lapse rates were suppressed during the fall season in Oklahoma (Fig. 4a). All daytime, daily, and nighttime lapse rates showed a change between the averages of the first 10 years and the last 10 years (Fig. 4b).

Spatial distributions of station's lapse rate trends
To examine spatial aspects of lapse rate changes, the lapse rate trends in 44 individual stations are shown in Fig. 5 for daily, daytime, and nighttime lapse rates. All but one station lapse rate trend showed a decrease irrespective of whether they were the daily, daytime, or nighttime analyses. About 16, 36, and 23 % of all stations showed statistically non-significant trends for the daily, daytime, and nighttime time series, respectively. The majority of stations showed significant decreasing trends, especially for daily lapse rates (Fig. 5). The histogram of individual trends for nighttime indicated trends were more negative relative to daily and daytime lapse trends (meaning the higher level temperature increased more (or decreased less) than the lower level temperature). Across Oklahoma, the lower latitude region showed more negative lapse rate trends. When dividing all of Oklahoma into northern and southern areas by a 35.4 • N line in latitude, the average lapse rate trends in the southern area were significantly more negative than the aver-

Wind influences on lapse rate trends
Daytime and nighttime lapse rate trends demonstrate different properties largely due to the diurnal solar cycle, wind speed and its interaction with the land surface (Pepin, 2001;Karl et al., 2006;Mahrt, 2006;McNider et al., 2012). Wind strongly influences turbulent mixing and surface boundary layer depth (Stull, 1988;Pepin, 2001). Figure 6 shows the lapse rate trend and variations under windy and calm conditions. There was no significant lapse rate trend observed under windy daytime conditions (Fig. 6c). The most negative lapse rate trend, −0.40 ± 0.03 • C (10 m) −1 per decade, was found under calm nighttime conditions (Fig. 6f). Both the trend magnitude and variation of lapse rate under calm nighttime conditions were the largest among all classified lapse rates as shown in Fig. 6. Since the stable nocturnal boundary layer is very sensitive to local radiative effects from atmo-spheric CO 2 and water vapor, wind speed, surface roughness, and soil heat capacity McNider et al., 2012), slight changes to the surface layer structure from these local effects could explain part of the observed trends. The observed slight increase in wind speed could have resulted in the 9 m level being above the nocturnal cool level more often later in the observational period, thus a positive temperature trend would be seen in the data due to this effect.

Trends of related climate variables in Oklahoma over 1997 to 2013
The MODIS Land Cover product (MOD12Q1) was used for the year 2005 (Friedl et al., 2010) to classify all 44 Oklahoma Mesonet stations into 34 grassland stations and 10 cropland stations to examine possible effects of land use and land cover on lapse rates (Fig. 1). Figures 7 and 8 showed that there were no statistical differences among respective lapse rate trends between grassland and cropland stations.
Due to the complexity of the surface vertical temperature profile variations (Stone and Carlson, 1979;Pepin, 2001;Mahrt, 2014), here we simply presented a monthly smoothed anomaly time series of climatic variables including solar radiation (SR, W m −2 ), water vapor pressure deficit (VPD, kPa), mean wind speed (WS, m s −1 ), precipitation (mm), and reference evapotranspiration (ET o , mm month −1 ), and their correlations with the monthly lapse rate time series (Fig. 9). Only mean wind speed and reference evapotranspiration showed significant trends; both of which were increasing ( Fig. 9d and f). In terms of correlation with lapse rates, solar radiation, reference evapotranspiration, and vapor pressure deficit showed significant correlations with values of −0. 55, −0.46, −0.18, and 0.35, respectively. In summary, for related climate variables, it is understandable that solar radiation is the most correlated due to its strong role on turbulent sensible heat flux from the ground surface associated with vertical temperature gradients and stability. The wind speed did play a role for lapse rate changes in the surface boundary layer (Pepin, 2001;Pielke Sr. et al., 2007;McNider et al., 2012;Baldocchi and Ma, 2013). Precipitation changes can provide information about soil moisture changes and its effect on variations of the daytime surface energy budget and heating of atmospheric temperatures (McNider et al., 2012;Baldocchi and Ma, 2013). Nevertheless, the mechanism of decreased lapse rates and latitudinal gradients of surface lapse rate trends observed in Oklahoma from 1997 to 2013 warrants further study and longer observation data in the future.

Summary and concluding remarks
Our study has the following major findings. First, using the lapse rate (defined as the difference in temperature at two levels) trends can be diagnosed with more statistical confidence than considering temperature trends from each level separately. Second, trends of surface temperature depend on the height at which the measurements are made. A greater warming at the 9 m level, or larger cooling at the 1.5 m screen level would explain such an observation. This is important as the surface temperature is used to diagnose and model global warming (IPCC, 2013). Using just the 1.5 m level trends would provide a different magnitude of trend than if obtained from the temperatures at 9 m (at least in Oklahoma and this may be true elsewhere). Third, the near-surface lapse rate trends were altered by wind speed. Fourth, lapse rate trends in southern Oklahoma were significantly more negative than further north in the state. Our study suggests a positive temperature trend at 9 m could be due in part to a change in wind speed during the time period such that the 9 m level more often remains above the nocturnal cool layer later during the observing period.
Finally, since land surface temperatures are often not taken at the same height above the ground, if the magnitude of long-term trends depends on the height of the measurement, it further complicates the ability to accurately quantify global warming using a global average surface temperature trend from a single height of observation at each location used in the construction of the global assessment (IPCC, 2013). This research should provide impetus for building additional or vertical expansion of current in situ observational infrastructure for a more robust understanding of climate change.

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Appendix A: How do two individual heights show no statistically significant trends, but the difference or the lapse rate does?
One might question how measurements from two individual heights can show no significant trends but the difference does. To evaluate this, we first generated two monthly temperature anomaly series, representing measurements at 9 m height (m 1 ) and 1.5 m height (m 2 ) with a length of 360month values (i.e., 30 years). The correlation coefficient between m 1 and m 2 was preset at 0.97, which was a typical value for the monthly T 9 m and T 1.5 m series in this study. The simulated m 1 and m 2 were generated by introducing fields of random month-to-month temperatures that were normally distributed with a mean of zero and a variance of one. Secondly, the initial trends and noise values in m 1 and m 2 were added to produce the s 1 and s 2 series as where trend 1 and trend 2 are initial trends imposed on the series, which have four combinations of a non-trended series and a linear trended series. These four trend combinations were [0.00 0.00], [0.00 0.12], [0.12 0.00], and [0.12 0.12] • C per decade. The n 1 and n 2 are normally distributed noise and n 2 's power level was set four times larger than the power level in n 1 because it was assumed that surface temperatures at T 1.5 m may have larger non-climatic and localclimatic noise than T 9 m . In terms of noise level, the normally distributed noise n 1 had a zero mean and 0.2 of standard deviation.
The third step was to run simulations 1000 times to generate 1000 pairs of s 1 and s 2 series for the four trend combinations individually, resulting in 1000 difference series of s 1 -s 2 for each set of trend conditions. Figure A1 illustrates an example result out of running 1000 simulations when trend 1 and trend 2 were [0.12 0.00] • C per decade. This example shows that two individual temperatures (s 1 and s 2 ) can show no statistically significant trends but the difference (s 1 -s 2 ) does (Fig. A1).
Finally, trend analyses were conducted for the s 1 , s 2 , and s 1 -s 2 series. The results indicate that there were about 600 chances out of 1000 simulations, where two trends of s 1 and s 2 were not significant but the s 1 − s 2 trend was significant, that is the [001] status shown in Fig. A2b and c, under the combination of trends imposed by [0.00 0.12] and [0.12 0.00] • C per decade. When both trends were zero or both trends were 0.12 • C per decade, there was a rare chance to have a significant s 1 -s 2 trend ( Fig. A2a and d).
In summary, a differential process (s 1 -s 2 ) is able to robustly suppress noise common to the s 1 and s 2 series relative to the difference signal (s 1 -s 2 ). Therefore, an improved signal-to-noise ratio series of s 1 -s 2 could show a statistically significant trend, but two individual s 1 and s 2 series do not show statistically significant trends.  Figure A1. An example of two non-significant trends in s 1 (a) and s 2 (b) temperature time series individually but differentiating them, s 1 -s 2 temperature series (c) shows a significant trend. This is one realization example taken from the simulations; s 1 and s 2 were constructed with trend values of 0.12 and 0.00 • C decade −1 , respectively.  Figure A2. These figures illustrate the frequency of outcomes (shown as the y axis) for four combinations of initial trends for series s 1 and s 2 . The eight possible combinations (shown as the x axis) are represented by 3-bit binary numbers: the first bit represents the s 1 trend status; the second bit represents the s 2 trend status; and the last bit represents the s 1 -s 2 trend status. Each trend status has two possibilities of either a non-significant trend (0) or a significant trend (1). For example, the 001 in the x axis stands for a combination of a non-significant trend (0) in s 1 , non-significant trend (0) in s 2 , and significant trend in s 1 -s 2 (1). Initial trends of s 1 and s 2 were imposed as (a) 0.00 and 0.00; (b) 0.00 and 0.12; (d) 0.12 and 0.00; and (d) 0.12 and 0.12 for each corresponding set of 1000 realizations. The trend units are • C per decade. The y axis represents the number out of 1000 simulations for eight combinations of the s 1 , s 2 , and s 1 -s 2 trend status.