The Space Speedometer
Using some simple algebra and satellite imagery to determine ship speed
Since we’ve all got boats on the brain thanks to the cargo ship that’s been stuck in the Suez Canal for the last five(?) days, I decided to focus this piece on maritime goings-ons. Plus, a reader mentioned they liked niche stories so hopefully this is niche enough, dear reader.
Satellite imagery is great and all, but it has one main drawback: it only shows the earth as it is at one point in time. At the risk of sounding like a dunce, satellite imagery only shows images of things on earth. What if we want to learn about the invisible but all-important qualities of things on earth? Qualities like an object’s weight, height, or, as we’ll see today, speed.
As it turns out, some phenomena observed in satellite images can be measured and analyzed, allowing us to figure out otherwise unknown information about the attributes of an object at a point in time.
Today, we’ll see how, using nothing more than some satellite pictures and basic algebra, we can measure the speed of particularly fast and heavy ships. This method can be used for estimating arrival times or forecasting voyage plans, but may also crack the door open to figuring out classified information, like the cruising speeds of top-secret military submarines, aircraft carriers, and other sensitive vessels.
But first, we need some background.
This duck is making a Kelvin Wake. As long as an object is moving through sufficiently deep water, it will always make a wake bounded at an angle of (theoretically) 38.94 degrees.
If the Kelvin waves themselves are large enough (say, emanating from a cargo ship or aircraft carrier), they’re actually visible and measurable from space. I measured the Kelvin wakes for a series of vessels and, using two equations I found in these two papers, figured out their speeds at the time they were pictured in Sentinel-2 satellite imagery.
But first, I validated the process against speed data displayed for ships on MarineTraffic.com. Take the vehicle carrier Prominent Ace, for example. It was seen sailing northeast to Livorno, Italy in a Sentinel-2 image on March 19.
Note the ship’s color scheme and length match up nicely with the details on the Prominent Ace’s MarineTraffic page.
MarineTraffic said the ship was sailing at 19.5 knots around the time the image was taken:
To determine the ship’s speed from imagery, you really only need two things: the number of wake waves seen in an image and the distance over which those waves were observed. After trying a few examples, I found it best to count a few waves (think less than 10, but ideally only the ones you can count clearly) as close to the ship’s stern as possible. Anything more than that introduces the possibility that winds or currents could affect the wake and screw up the speed calculation. As one astute reader pointed out, measuring a wake over a long distance could also introduce the possibility that the ship was accelerating (i.e. that it’s speed wasn’t constant over a long distance) and therefore had variable speed and wave measurements.
At any rate, I zoomed in on the Prominent Ace’s wake and started counting:
(I also played with the gain and gamma on the Sentinel-2 screenshots to make the waves clearer. You can find the original image here).
Now for some algebra. To determine the speed from a Kelvin wake, you first need to know the wake’s wavelength, which can be done with this formula:
The squiggly v is the wavenumber (i.e. the number of waves you observed divided by the distance over which you observed them) and the lambda is the wavelength. I broke out what I remembered from 9th grade algebra1 to solve for lambda, namely dividing 1/(3/207), where 3 is the number of wake waves I observed in 207 meters’ worth of the Prominent Ace’s wake. This gives a wavelength of 69 meters.
Now that we have the wavelength, we can plug it into the actual Kelvin Wake speed formula, which is:
λ = (2πV^2)/g
where g is gravitational acceleration at the earth’s surface (9.8 meters per second),
π is pi, λ is the wavelength from the first equation, and V is the speed of the ship. From there, it was a relatively simple matter of plugging all the right numbers in, remembering, uh, PEMDAS, solving for V, and multiplying that by 1.94 to change the answer from meters per second to knots.
For the Prominent Ace, I found that it was sailing at 20.165 knots, which matches up pretty darn closely to the 19.5 knots measured on MarineTraffic. Perhaps my measurements were a bit off or perhaps the wind affected the MarineTraffic speed - but either way, it’s well within what I would consider an acceptable range. For the record, I’m considering all speeds in this piece to be within a plus or minus one knot margin of error off the vessel’s true speed.
I repeated the example for the behemoth Cap San Lazaro seen off the coast of Yemen last week. I came up with a speed of 15.3 knots, while MarineTraffic said she was sailing at 14.8 knots. Again, not the exact same but I’ll take it.
With the wind in my sails (heh), I decided to move on to some more interesting vessels. Since I’m no ship spotting expert, I searched around on Twitter for actual experts to identify ships they saw in satellite imagery, which I then found on my own and measured the wakes for. So with many thanks, I dedicate the rest of this article to @OSINT_1, @August20190831, and @lobsterlarryliu.
First up is my personal favorite, a Chinese Navy Type 094 ballistic missile submarine spotted sailing off the coast of Hainan Island. Its top surfaced speed is unknown, but assuming the identification of the sub and my measurements are correct, it can reach speeds of at least 20 knots (20.88 knots, specifically) while sailing on the surface. I found this speed by plugging the five Kelvin waves over 370 meters observed below into the equations above and solving for V.
The Russian frigate Admiral Grigorovich has also been spotted on Sentinel-2 imagery at least twice. Interestingly, in both appearances she was traveling at around the same speed - between 16 and 17 knots. While her top recorded speed is 30 knots, these two instances could indicate that her cruising speed is around 16 knots.
First, she was seen on February 4 steaming toward Pakistan for a naval exercise. With seven Kelvin waves observed over 380 meters, she appears to have been traveling at around 17.8 knots.
Next, she was seen shadowing the US Navy’s USS Eisenhower carrier battle group in the Mediterranean Sea on March 26. This time, I observed five Kelvin waves in the space of 227 meters, bringing Admiral Grigorovich’s speed to about 16.3 knots.
The US carrier group, for their part, appears to have been going slower - between 12 and 14 knots. The faster Russian ship may have been trying to catch up with the USS Eisenhower, the aircraft carrier at the front of the convoy. Or maybe Admiral Grigorovich was just out for a high speed spin in the Mediterranean - who knows.
Last (and fastest), is one of the Chinese Navy’s three new Type 075 helicopter carriers. One of these vessels was seen rocketing toward Shanghai on August 22, 2020.
With eight Kelvin waves spotted in 609 meters, her speed comes out to 21.18 knots. Additionally, this image is so clear that one can actually count the waves extending back over a mile. Unfortunately, they wouldn’t all fit in a screenshot, but I counted 21 waves over 1,610 meters, indicating that, regardless of wake length or number of waves, her speed is constant at a hair over 21 knots.
Defense News lists the Type 075’s top speed as 23 knots, so the ship seen above was likely approaching her top speed. Why she set this blistering pace is unclear to me.
While the Type 075, Admiral Grigorovich, and Type 094 are just three examples, my full dataset for your perusal is here:
If you’d like the Twitter or Sentinel-2 links to any of these ships, just dm me on Twitter @LOActualControl and I’d be happy to get them for you. Finally, if you do choose to use the equations, try calculating the ship speed with the following formula in Excel (it makes things a lot easier than using a calculator each time), where L is the length of wake distance measured and W is the number of wake waves counted:
Speed in Knots
All in all, measuring Kelvin wakes can be used to establish the speeds of heavy and fast ships sailing the open seas. While simple in nature, this technique has some exciting possible applications, such as determining likely ports of call for smugglers’ vessels, figuring out the top speeds of secretive, strategic weapons like the Type 094 sub, or pinpointing when and where military activity has taken place or is likely to take place.
If you use this technique for your own applications or are able to refine or edit it at all, I’d love to hear about it. Drop a comment or shoot me a message on Twitter and we can chat. Happy ship hunting!
Lest you think that you can’t do this, fear not, as I never got above a B in algebra or calculus in high school and stopped taking math as soon as I possibly could in college.