Temperature II: Interpolating Temperatures and Rain

I'm using mostly this guide on Cartographer's Guild. It's extremely helpful. One thing, though, that I've extended is the use of exact numbers for altitude, rain, temperatures, etc, rather than the use of categories. This gives me way more granular control over the result (but also introduces more problems, so YMMV).

For every climatic input, two maps are produced: summer and winter. The seasons are always reversed between hemispheres.

However, the problem (for my purposes) is that the climate undergoes a gradual change between those two points in the year. I approximate this as a sine curve, which is good enough. I've found several places where there is more than one local maximum or minimum, so even the process I'll explain here is a bit simplistic.

Let's take a hypothetical place (I'll just grab a random hex).

>>> h = choice(list(hexes))
>>> h
'98S-7W'
>>> hexes[h].temperature
{'Jul': 32, 'Jan': 57} # F
>>> hexes[h].precipitation
{'Jul': 461, 'Jan': 256} # mm

The basic equation we'll fit is:
\[y = A \sin\left(k x + \phi\right) + D, k = {2 \pi \over \lambda}\]
The amplitude is given by $A = {|t_w - t_s| \over 2}$, and the average is given by $D = {t_w + t_s \over 2}$. Since we're looking at a 12 month year, $\lambda = 12$, and so $k = {\pi \over 6}$.

The last variable, then, is $\phi$. This one is just a bit trickier. The minimum of a sine wave is at $\pi \over 4$. If we think about this in terms of 12 months, this is the 3rd month. Therefore, we need to find a value of $\phi$ such that the minimum of the sine wave matches up with the actual minimum of the data. The simple way to do this is just $\phi = 4 \textrm{ if } t_w \equiv \min(t_w,t_s) \textrm{ else } 10$.

Plugging all that in:
\[y_t(m) = {25 \over 2} \sin\left({\pi \over 6} \left(m - 4\right)\right) + 44.5^\circ\textrm{F}\]
\[y_p(m) = {205 \over 2} \sin\left({\pi \over 6} \left(m - 10\right)\right) + 358.5\textrm{ mm}\]

There are two major improvements for using this equation. First of all, I can get a much more accurate value for the total rainfall in a year (merely adding the two numbers I had gives a laughably incorrect number). The second is that (for purposes of climate) only liquid rainfall counts. That means that if the temperature in a given month is less than 32 (and here, that's the minimum, so we'd still get some freezing rain), then there is no liquid rainfall. It could still fall as snow, of course, but that has no effect on climate in Koppen or Holdridge.

With a wet winter and mild summer (not really that dry), the Koppen classification is Cfb. That's a "maritime mild winter"; think France. The Holdridge classification is perhumid subtropical wet forest, and I'm a bit skeptical of that. It'd be odd if both of those were true. I'm more inclined to believe the Koppen version, but we'll see.

To do: this assumes that, for example, the "winter" value is the lowest value. This is not necessarily the case. Perhaps the map merely shows a snapshot of January, regardless of whether January is the coldest month. So there is room for improvement. However, I took a quick look at data at various latitudes, and outside of a short transition zone around the equator, January is usually the coldest month in the northern hemisphere and the hottest in the southern, and July is the hottest in the northern and coldest in the southern. I didn't expect the transition to be as sharp as it was. Obviously, local conditions will vary. I did not do this analysis for precipitation.