Swell Height[1] is a deceptively difficult thing to measure. The first thing to realize is that this isn’t an objective number — it doesn’t correspond one-to-one with any specific thing in reality. Instead, it’s a number that should be close to the height of an average wave in some swell. I know how squishy and odd that sounds but stick with me because the metric is worth it in the end.

So how to get it? You might think, “I’ll get a boat, drive out a ways, drop anchor, sit there for a bit and keep track of the highest and lowest points the ocean surface touched on the anchor line.”

Nope. That’ll get you the crest of the tallest wave minus the trough of the lowest wave. Odds are good that there was no single wave that was close to as tall as this measurement. Not quite what we want for swell height.

“Well ok. I’ll sit out there for an hour and write down the heights of all the crests and troughs of all the waves that pass. I’ll then calculate each individual wave height and take the average.”

Here’s what measuring the crests and troughs would look like. Courtesy of CDIP — Link

Closer —you’ve now got a good set of crest-and-trough data through time. The problem here is that you’ll have recorded quite a few very small waves. Including these in the average will result in a very small swell height — one that doesn’t capture the intention of the metric.

“Alrighty, I’ll throw out the smallest two-thirds of the waves and then take the average of the rest.”[2]

Random but yes — it turns out this tends to give us a number that corresponds well to the close-to-the-height-of-an-average-wave-in-some-swell idea we set out to capture. Unfortunately there’s one more nasty wiggle that you’ve forgotten about — there will most likely be multiple swells in the water.

“Uhhhh. How can I tell which swell the wave I’m measuring is part of? They are all jumbled together.”

They sure are. How should we unmelt this gnarly bar of wax that dripped into our car vents and cup holders?

MATH

Math is how. We use the Fourier Transform[3]. We could spend all day on this puppy but we’re going to say only that it allows us to convert our crest-and-trough data into this[4] —

DATA

Here’s a plot of a single row of the above data—

Courtesy of CDIP — Link

This plot shows the period (on top) vs. the energy density. You can see that in this case we have a high energy spike at a 15 second period and a bunch of energy in the (less interesting for surfers) 8–2 second band.

In addition to period vs. energy we can use the above plot to get height — we would integrate under the whole curve to get the height for all swells combined or under a part of the line to get the height for a specific period swell[5].

Note that picking a part of the plot that represents “the 15 second period swell” is not well defined — it could be a very narrow band just around 15 seconds, but it also could include the somewhat gentle slopes that go from 18 seconds to 12 seconds. How this is chosen varies among surf forecasts[6].

Fun Sidenote: We can also use the fourier transform to get period vs. swell direction which allows us to generate really cool plots like this —