Burn's Night Is The Coldest

In winter I usually mark off 4 dates as the yearly cycle starts transforming into a more positive outlook. I call these 'Milestones'.

Milestone 1

This is the earliest evening. Most people are unaware that evenings start getting later, before the shortest day. In 2024, evenings in the UK started getting later from December 12.

Milestone 2

This is the shortest day, the winter solstice. Everyone knows about this, however, even after this day, the mornings are still getting later; it's just that the evenings are getting later quicker than the mornings are, so the days start getting longer.

Milestone 3

This is the latest morning. Again most people are unaware of this, but it happens right at the end of the year. In 2024 it happened around December 30 or 31.

Milestone 4

This is the coldest day of the year on average and the topic for this post. It's difficult to calculate this date, because daily temperatures vary wildly from day to day and also across years for equivalent days. Nevertheless, it's fairly easy to see that after the days start getting longer, they continue to also get colder for a while. This is because other environmental factors such as cloud cover, heat loss from the ground, air temperatures or the Jet Stream can continue to drive temperatures on average down faster than the sun adds energy to the atmosphere and land.

Anecdotally, I used to figure the coldest time of the year was at the end of January / beginning of February, so I set Milestone 4 on January 31. Later, however, I thought to myself that perhaps it's mid-way between the winter solstice and spring equinox, because all of these diurnal patterns tend to follow year-long sine waves.

Winter solstice is on December 21, and Spring equinox is on March 21. So, that's 10 days in December + January (31 days) + February (28.24 days) + 21 days of March. This is 10+31+28.24+21=90.24 days. Calculating Milestone 4 after 90.24/2=45.12 days, which, given 10 days at the end of December + 31 days in January leaves 4.12 days. So for the past several years I've been setting it on February 4.

But neither of these techniques are based on actual evidence. What if it's not symmetrical as I've assumed? What if temperatures simply aren't shifted mid-way? To figure that out I need real data.

Real Data

I was involved in an on-line, climate discussion trying to work out how temperatures had changed in the UK over the past decade or so and found an open Statistica page on it:

You can hover over the months to get the actual figures, downloading the raw data requires a subscription. It turns out that for nearly all the months in the year there's an upward trend, but for January there's no observable trend.

But as I was looking at it, I realised that I could use my new understanding of Fourier transforms to obtain a better approximation for the coldest day.

The Winding Principle

At University we covered quite a lot of math in the first year including Fourier Transforms (and Laplace Transforms). I was able to do the math, but I didn't remotely understand how one can isolate the set of harmonic frequencies from waveform data. It took a Hackaday article to help me. I can't do justice to the article, nor the associated animated video explainer, but I can précis the idea as far as the fundamental harmonic goes, which is all we care about here.

Every complex, repeating, sampled waveform can be constructed from a set of sine waves at 1x, 2x, 3x.. the fundamental frequency up to half the sample period just added together. However, if I want to isolate the fundamental frequency that turns out to be pretty easy. All you do is multiply each sample by the sine of the corresponding angle within the waveform and add the results together. If the fundamental is present, then its amplitude at any point will cohere with the sine wave itself, but higher frequencies will 'disappear', because their positive phases will end up getting multiplied by both the positive and negative phases of the reference sine wave. For example, consider an 8-sample wave containing a fundamental and 1st harmonic:

Sample#	0	1	2	3	4	5	6	7	Total
Ref Sine	0.000	0.707	1.000	0.707	0.000	-0.707	-1.000	-0.707	0.000
Fundamental	0.000	0.573	0.810	0.573	0.000	-0.573	-0.810	-0.573	0.000
^^^ x Ref Sine	0.000	0.405	0.810	0.405	0.000	-0.405	-0.810	-0.405	3.240
1st Harmonic	0.000	0.210	0.000	-0.210	0.000	0.210	0.000	-0.210	0.000
^^^ x Ref Sine	0.000	0.148	0.000	-0.148	0.000	0.148	0.000	-0.148	0.000

To fully calculate each harmonic you need to consider the phase of each harmonic. That's because a sine wave at any given phase can be generated by a pair of sine + cosine waves with two respective amplitudes; thus the above technique will only recover the sine wave component. For example, if the Fundamental was shifted by +90º, then the Fundamental * the Ref Sine would still end up with a total of 0, but here the wave * a Reference cosine wave would have an amplitude of 3.240.

Finding The Phase

Therefore, a Fourier analysis of the fundamental can tell us not only its amplitude, but also its phase. And it turns out we can obtain an accurate phase from relatively few samples. The phase is simply obtained from the vector obtained from ∑ waveform data * the Reference sine wave on the x axis ∑  waveform data * the Reference cosine wave on the y axis.

This means that even though all we have are monthly values for the temperature data, we can calculate the actual minima, zero-crossings and maxima at a much higher resolution.

The phase calculated is always relative to the reference angles. For example, if we started the reference angle at 30º and the samples were a sine wave starting at 30º, then the phase would still be 0º. If the reference angle was 0º, and the samples were a sine wave starting at -90º, then the relative phase would be reported as 90º, because the zero-crossing for the sine wave would be at 90º.

The phase therefore tells us the average temperature day and the minimum temperature day will be 90º earlier (or 91.31 days earlier). For UK temperatures, the minimum temperature is therefore reported as Jan 25.5. Ironically, this means that Burn's Night is the coldest.

There's one more aspect of the model that's worth mentioning, which is that the reference phases aren't equidistant, because the months don't all have the same number of days in them (though it's close). Therefore, in this calculation, the reference dates are taken from the mid-point of each month, on the basis that the average temperature for that month represents the temperature half-way through the month.

Minimal Temperature

Temperatures:

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