# On record-breaking extremes

Fig. 3. The ratio of records for the U-shaped climate to that in a stationary climate, as it changes over time. The U-shaped climate has fewer records than a stationary climate in the middle, but more near the end.

So here is one interesting result: even though the linear trend is zero, the U-shaped climate change has greatly increased the number of records near the end of the period! Zero linear trend does not mean there is no climate change. About two thirds of the records in the final decade are due to this climate change, only one third would also have occurred in a stationary climate. (The numbers of course depend on the amplitude of the U as compared to the amplitude of the noise – in this example we use a sine curve with amplitude 1 and noise with standard deviation 1.)

A second thought-experiment

Next, pretend you are one of those alarmist politicized scientists who allegedly abound in climate science (surely one day I’ll meet one). You think of a cunning trick: how about hyping up the number of records by ignoring the first, cooling half of the data? Only use the second half of the data in the analysis, this will get you a strong linear warming trend instead of zero trend!

Here is the result shown in green:

Fig. 4. The record ratio for the U-shaped climate (blue) as compared to that for a climate with an accelerating warming trend, i.e. just the second half of the U (green).

Oops. You didn’t think this through properly. The record ratio – and thus the percentage of records due to the climatic change – near the end is almost the same as for the full U!

The explanation is quite simple. Given the symmetry of the U-curve, the expected number of records near the end has doubled. (The last point has to beat only half as many previous points in order to be a record, and in the full U each climatic temperature value occurs twice.) But for the same reason, the expected number of records in a stationary climate has also doubled. So the ratio has remained the same.

If you try to go to even steeper linear warming trends, by confining the analysis to ever shorter sections of data near the end, the record ratio just drops, because the effect of the shorter series (which makes records less ‘special’ – a 20-year heat record simply is not as unusual as a 100-year heat record) overwhelms the effect of the steeper warming trend. (That is why using the full data period rather than just 100 years gives a stronger conclusion about the Moscow heat record despite a lesser linear warming trend, as we found in our PNAS paper.)

So now we have seen examples of the same trend (zero) leading to very different record ratios; we have seen examples of very different trends (zero and non-zero) leading to the same record ratio, and we have even seen examples of the record ratio going down for steeper trends. That should make it clear that in a situation of non-linear climate change, the linear trend value is not very relevant for the statistics of records, and one needs to look at the full time evolution.

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