FIGURE 1: MEAN DECADAL SEASONAL CYCLE IN THE CET 1722-2016

FIGURE 2: TRENDS IN THE DIFFERENCE BETWEEN SUMMER AND WINTER

FIGURE 3: FULL SPAN OLS TREND FOR EACH CALENDAR MONTH

FIGURE 4: TREND PROFILE FOR EACH CALENDAR MONTH

FIGURE 5: RELATIONSHIP BETWEEN FULL SPAN TREND & THE TREND PROFILE

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1. Temperature data taken at weather stations contain a diurnal cycle, a seasonal cycle, and random natural variations. Along with these they may also contain a long term trend over a period of many years. Typically, the diurnal and seasonal cycles represent more than 80-90% of the total variance in the actual temperature measurements. The remaining 10-20% or so consists mostly of unexplained random variations. In cases where a statistically significant trend is found with OLS linear regression, no more than a small portion of the variance, around 3% or so, can be ascribed to a long term warming or cooling trend.
2. It is for these reasons that the study of global warming over many decades, regression coefficients for long term trends are relatively a very weak feature of the time series that must be teased out of the data net of the greater diurnal, seasonal, and random variations. The large seasonal cycle is removed from monthly mean temperature data by subtracting the corresponding temperature in a reference period. The temperature data thus deseasonalized are published as “temperature anomalies”. Global warming research is carried out with these data sets. This procedure contains the assumption that the seasonal cycle is constant and unchanging across the full span of the time series being studied.
3. This study is an investigation of the validity of this assumption. Figure 1 is a graphical representation of the seasonal cycle. The seasonal cycle is represented as a plot of the decadal mean temperature for each calendar month from January (labeled as “1”) to December (labeled as “12”). The GIF image cycles through 25 decades of decadal mean seasonal cycles in the CET from 1772 to 2016. The display clearly shows that the shape of the seasonal cycle graphic changes over time and implies that the seasonal cycle is not constant across time. The changes may seem small and insignificant but they should be understood in relation to long term trends in temperature which are typically less than 5% of the diurnal range per century.
4. The inconstancy of the seasonal cycle is confirmed in Figure 2 where long term trends in the differences in temperature among calendar months is presented. The results show that there are statistically significant long term trends in the temperature differences between summer months and winter months. This difference confirms that the seasonal cycle is changing as the climate warms and that the constancy of the seasonal cycle assumed in the construction of temperature anomalies is not valid.
5. The reason for the differences in seasonal cycles seen in Figures 1&2 is presented in Figure 3.  It shows that when the long term trends are computed separately for each calendar month we find that the warming trend is stronger in the winter months than in the summer months. These differential trends imply a gradual narrowing of the the their temperature differences as for example between January and July as shown in Figure 2.
6. Figure 4 is a GIF image that cycles through the twelve calendar months displaying the full span OLS linear trend for that calendar month as well as a graphic for a different approach to the study of temperature trends proposed in this work that we shall refer to as a Trend Profile. Rather than one full span OLS trend over the full sample of N observations, the Trend Profile tracks changes in temperature through the full span of the time series as a time series of temperature trends in a moving 30-year window. The window moves through the temperature time series one year at a time computing 30-year trends.
7. The trend profiles in the GIF image of Figure 4 show that an observed full span warming trend is the net result of multiple 30-year warming and cooling trends with magnitudes that are more than an order of magnitude larger than the full span trends.
8. A great variety of shapes are seen in the Trend Profiles of the twelve calendar months. We can also see in these images that the differences among the calendar months are much more intense and complex than just differences in the numerical values of the full span trend seen in Figure 3. The Trend Profile procedure provides a a great deal of more information. More insight is thus gained into the trend behavior of the time series into the differences among the calendar months.
9. The relationship between the full span OLS trend and the Trend Profile is displayed in Figure 5. It shows that the mean of thee moving 30-year trends is a good and possibly more robust estimate for the overall full span trend and that this estimate is related the the ratio of warming and cooling trends as well as their magnitudes in the Trend Profile. In short, the argument is presented that the trend profile provides more information than a single OLS full span trend line.
10. The temperature trend information presented and understood in terms of trend profiles for each calendar month contains more information and offers greater statistical validity and reliability than a single OLS trend line drawn through the deseasonalized temperatures for all twelve calendar months.
11. IN CONCLUSION WE FIND THAT THE USE OF TEMPERATURE ANOMALIES TO FACILITTE THE COMBINATION OF THE TEMPERATURES OF THE TWELVE CALENDAR MONTHS IS AN INADEQUATE, ERRONEOUS, AND UNNECESSARY INNOVATION AND THAT TEMPERATURE TRENDS ARE BEST UNDERSTOOD IN TERMS OF THE TWELVE TREND PROFILES FOR THE TWELVE CALENDAR MONTHS.

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