S1. Supplementary Material to Compare the Slope Method Based on a Linear Fit of
Temperature to the Curvature Method Based on a Second-Power Polynomial Fit of
Cumulative Temperature
More and more evidence has indicated that climate change will cause uncertainty in food/water security
that hence can adversely influence human livelihoods. In order to develop adaptation strategies to
mitigate the uncertainty, it is essential to accurately quantify the multi-decadal temperature trend.
The daily temperature T is generally controlled by three factors (i.e., annual mean temperature,
seasonal sine-wave amplitude, and daily fluctuation) and can be expressed by the following formula:
T (t ) 
TAM  1 (t )

2 t
 365.25
TSA   2 (t )sin 

   3 (t )

where t is day, TAM is annual mean temperature, TSA is seasonal temperature amplitude, while ε1,
ε2,
and ε3 are annual, seasonal, and daily fluctuations, respectively. Conventionally, the slope method
based on a linear fit of either daily or monthly or yearly temperature is used to determine temperature
change at a specific location.
To determine whether the conventional slope method can accurately reflect true temperature change,
we generate a simplified daily temperature by the following formula:
 2 t
T (t )  15  5 sin 
 365.25

  10 random#  0.5

where random# is a random number from 0 to 1. The day t varies from 1 to 9131 (25 years). This
formula creates a 25-year temperature pattern with 15 degrees mean annual temperature, 5 degrees
seasonal amplitude, and 5 degrees daily random fluctuation. Because the sine-wave has no trend and
the 25-year average of daily random fluctuation is close to zero, this temperature pattern does not have
any temperature change. To simulate a temperature increase caused by climate change, we
superimpose a linear daily temperature increase of 2 x 10-4 degree to the pattern. This gives a 1.83
degrees temperature increase in 25 years (7.3 degrees per 100 years), which has occurred at many
stations in the eastern Tibetan Plateau.
The squares on Fig. 2a are the generated daily temperatures. The thin black line across the middle of
Fig. 2a is the linear regression line of the 25-year daily temperature pattern, while the regression result
with R2 value is shown at the top of the chart. For the slope method, the first derivative (i.e., the
coefficient of the x term) of the regression line is the calculated slope of the temperature pattern. As
the slope of the temperature increase is known (i.e., 2 x 10-4 degree per day), the error of the slope
method can be calculated. But, because of the daily random temperature fluctuation, the calculated
error was not a constant (i.e., the calculated error will change when the random temperature
fluctuation changes). Therefore, we generated 50,000 sets of 25-year daily random temperature
fluctuations to calculate an averaged error of the slope method.
We repeated the above procedure by choosing a range of superimposed linear slopes to test how the
error will vary with the superimposed slope. Again, 50,000 sets of 25-year daily random temperature
fluctuations were generated for each superimposed slope to calculate an averaged error. The results in
Fig. 2b indicate that the error of the conventional slope method increases as the superimposed slope
decreases.
The measured temperature at many stations in the eastern Tibetan Plateau showed a range of
temperature increases from 0.9 to 2.7 degrees per 25 years. Within this range, the errors of the
conventional slope method vary from -41.8% to -13.7%. Even at artificially large slope (e.g., 5.5 degrees
per 25 years), the error of the conventional slope method is still -7%. This suggests that the
conventional slope method is not ideal to calculate the true temperature increase trend.
Much of the error of the conventional slope method was caused by the sine-wave seasonal temperature
amplitude and daily random fluctuation. Cumulative temperature method can suppress the effect of
the seasonal amplitudes and daily random fluctuations. The diagonal thick line in Fig. 2a is the
cumulative daily temperature, which is nearly a straight line because the fixed seasonal fluctuations are
predominantly averaged out. A thin black line embedded at the center of the thick diagonal line is the
trend line based on the best fit of a second power polynomial. The regression result with R2 value is
shown at the bottom of Fig. 2a.
For cumulative temperature, the curvature or second derivative of the second-power polynomial (i.e.,
two times the coefficient of the x2 term) is the calculated slope of the linear temperature increase. As
the slope of the temperature increase is known (i.e., 2 x 10-4 degree per day), the error of the curvature
method can be calculated. Again, because the calculated error was not a constant, the same 50,000 sets
of 25-year daily random temperature fluctuations were used to calculate an averaged error of the
curvature method. The results in Fig. 2b clearly demonstrated that, under a wide range of
superimposed slopes, the error based on the curvature of the cumulative temperature was always much
less than that based on the convention linear slope of temperature.