Inmarsat concluded that the flight ended in the southern Indian Ocean, and its analysis has become the canonical text of the Flight 370 search. It’s the bit of data from which all other judgments flow—from the conclusive announcement by Malaysia’s prime minister that the plane has been lost with no survivors, to the black-box search area, to the high confidence in the acoustic signals, to the dismissal by Australian authorities of a survey company’s new claim to have detected plane wreckage.

Although Inmarsat officials have described the mathematical analysis as “groundbreaking,” it’s actually based on some relatively straightforward geometry. Here’s how it works: Every so often (usually about once an hour), Inmarsat’s satellite sends a message to the plane’s communication system, asking for a simple response to show that it’s still switched on. This response doesn’t specify the plane’s location or the direction it's heading, but it does have some useful information that narrows down the possibilities.

You can think of the ping math like a game of Marco Polo played over 22,000 miles of outer space. You can’t see the plane. But you shout Marco, and the plane shouts back Polo. Based on how long the plane takes to respond, you know how far away it is. And from the pitch of its voice, you can tell whether it’s moving toward you or away from you—like the sound of a car on the highway—and about how fast.

This information is far from perfect. You know how far the plane was for each ping, but the ping could be coming from any direction. And you how fast the plane is moving toward or away from you. It could also be moving right or left, up or down, and the speeds would sound the same. The task of the Inmarsat engineers has been to take these pieces and put them together, working backwards to reconstruct possible flight paths that would fit the data.

There are two relevant pieces of information for each ping: the time it took to travel from plane to satellite, and the radio frequency at which it was received. It’s important to keep in mind that the transit times of the pings correspond to distances between satellite and plane, while frequencies correspond to relative speeds between satellite and plane. And this part’s critical: Relative speed isn’t the plane’s actual airspeed, just how fast it’s moving toward or away from the satellite.

Authorities haven’t released much information about the distances—just the now-famous “two arcs” graphic, derived in part from the distance of the very last ping. But they’ve released much more information about the ping frequencies. In fact, they released a graph that shows all of them:

This graph is the most important piece of evidence in the Inmarsat analysis . What it appears to show is the frequency shifts or “offsets”—the difference between the normal “pitch” of the plane’s voice (its radio frequency) and the one you actually hear.

The graph also shows the shifts that would be expected for two hypothetical flight paths, one northbound and one southbound, with the measured values closely matching the southbound path. This is why officials have been so steadfastly confident that the plane went south. It seems to be an open-and-shut verdict of mathematics.

So it should be straightforward to make sure that the math is right. That’s just what a group of analysts outside the investigation has been attempting to verify. The major players have been Michael Exner, founder of the American Mobile Satellite Corporation; Duncan Steel, a physicist and visiting scientist at NASA’s Ames Research Center; and satellite technology consultant Tim Farrar. They’ve used flight and navigation software like STK, which allows you to chart and make precise calculations about flight scenarios like this one. On their blogs and in an ongoing email chain, they’ve been trying to piece together the clues about Flight 370 and make sense of Inmarsat’s analysis. What follows is an attempt to explain and assess their conclusions.

What We Know

Although the satellite data provides the most important clues about the plane’s overall flight path, they’re not the only clues available. Authorities have some basic but crucial additional information about the flight that can help to make sense of the satellite math:

1. The satellite’s precise coordinates

The satellite in contact with Flight 370 was Inmarsat’s IOR satellite, parked in geostationary orbit above the Indian Ocean. The satellite is meant to be stationary, but its orbit has decayed somewhat, so that it actually rotates slightly around its previously fixed position. Its path is publicly available from the Center for Space Standards & Innovation.

2. The plane’s takeoff time and coordinates

16:41 UTC from the Kuala Lumpur airport.

3. The plane’s general motion toward or away from the satellite

From radar tracking, we know the plane traveled northeast, away from the satellite, over the first 40 minutes after takeoff, then westward, toward the satellite, until 94 minutes into the flight, when it was last detected on radar. Inmarsat spokesmen have stated that the ping distances got progressively longer over the last five hours of flight, meaning that the plane was moving away from the satellite during that time.

4. Two flight paths investigators think are consistent with the ping data

In addition to the frequency shift graph, the Inmarsat report includes a map with two “Example Southern Tracks,” one assuming a flight speed of 400 knots, the other a speed of 450 knots. Check it out:

Inmarsat

These bits of knowledge allow us to put some basic constraints on what a graph of the ping frequency shifts should look like. We’ll use more precise numbers later; for now, it’s helpful just to have some qualitative sense of what to expect:

5. Frequency shifts that should all be negative

When the plane is moving away from the satellite, the radio signal gets stretched out, so the frequency decreases. This means that the frequency shifts should be negative over most of the flight. Although there was an approximately one-hour period starting 40 minutes after takeoff when radar showed the plane moving westward, toward the satellite, the graph shows that no pings were sent during that time—so actually, all of the shifts on the graph should be negative.

6. Frequency shifts before takeoff that should be near zero

Plotting the satellite’s path in STK, you can see that it moves through an ellipse centered around the equator. Space scientist Steel has created this graphic of the satellite’s motion, including marks for its position when the plane took off and when it last pinged the satellite:

The satellite’s motion is almost entirely north-south, and the plane’s takeoff location in Kuala Lumpur is almost due east of the satellite. This means that the satellite was only barely moving relative to Kuala Lumpur, so the frequency shift for a plane nearly stationary on the ground at the airport would be nearly zero.

7. Frequency shift graph should match map of southbound flight paths

The way the Marc-Polo math works is that, if you assume the plane traveled at some constant speed, you can produce at most one path north and one path south that fit the ping data. As the example flight paths on Inmarsat’s map show, the faster you assume the plane was moving overall, the more sharply the path must arc away from the satellite.

This constraint also works the other way: Since flight paths for a given airspeed are unique, you can work backwards from these example paths, plotting them in STK to get approximate values for the ping distances and relative speeds Inmarsat used to produce them. The relative speeds can then be converted into frequency shifts, which should roughly match the values on the frequency graph. (This is all assuming that Inmarsat didn’t plot the two example paths at random but based on the ping data.) We’ll put more precise numbers on this below.

The Troubled Graph

But the graph defies these expectations. Taken at face value, the graph shows the plane moving at a significant speed before it even took off, then moving toward the satellite every time it was pinged. This interpretation is completely at odds with the official conclusion, and flatly contradicted by other evidence.

The first problem seems rather straightforward to resolve: the reason the frequency shifts aren’t negative is probably that Inmarsat just graphed them as positive. Plotting absolute values is a common practice among engineers, like stating the distance to the ocean floor as a positive depth value rather than a negative elevation value.

But the problem of the large frequency shift before takeoff is more vexing. Exactly how fast does the graph show the plane and satellite moving away from each other prior to takeoff?

The first ping on the graph was sent at 16:30 UTC, eleven minutes prior to takeoff. The graphed frequency shift for this ping is about -85 Hz. Public records show that the signal from the plane to the satellite uses a frequency of 1626 to 1660 MHz. STK calculations show the satellite’s relative motion was just 2 miles per hour toward the airport at this time. Factoring in the satellite’s angle above the horizon, the plane would need to have been moving at least 50 miles per hour on the ground to produce this frequency shift—implausibly high eleven minutes prior to takeoff, when flight transcripts show the plane had just pushed back from the gate and not yet begun to taxi.

On the other side of the frequency graph, the plane’s last ping, at 00:11 UTC, shows a measured frequency shift of about -252 Hz, working out to a plane-to-satellite speed of just 103 miles per hour. But the sample southbound paths published by Inmarsat show the plane receding from the satellite at about 272 miles per hour at this time.

In other words, the frequency shifts are much higher than they should be at the beginning of the graph, and much lower than they should be at the end. Looking at the graph, it’s almost as if there’s something contributing to these frequency shift values other than just the motion between the satellite and the plane.