Investigators searching for the missing
Malaysian Airlines flight were ebullient when they detected what sounded
like signals from the plane’s black boxes. This was a month ago, and it
seemed just a matter of time before the plane was finally discovered.

But now the search of 154 square miles of ocean floor
around the signals has concluded with no trace of wreckage found.
Pessimism is growing as to whether those signals actually had anything
to do with Flight 370. If they didn’t, the search area would return to a
size of tens of thousands of square miles.

Even before the black-box search turned up
empty, observers had begun to raise doubts about whether searchers were
looking in the right place. Authorities have treated the conclusion that
the plane crashed in the ocean west of Australia as definitive, owing
to a much-vaunted mathematical analysis of satellite signals sent by the
plane. But scientists and engineers outside of the investigation have
been working to verify that analysis, and many say that it just doesn’t
hold up.

### A Global Game of Marco Polo

Malaysia Airlines flights are equipped with
in-flight communications services provided by the British company
Inmarsat. From early on, the lynchpin of the investigation has been
signals sent by Flight 370 to one of Inmarsat’s satellites. It’s
difficult to overstate the importance of this lonely little batch of
“pings.” They’re the sole evidence of what happened to the plane after
it slipped out of radar contact. Without them, investigators knew only
that the plane had enough fuel to travel anywhere within 3,300 miles of the last radar contact—a seventh of the entire globe.

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.

### What’s the Frequency?

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:

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.

### Cracking the 'Doppler Code'

Exner, an
engineer who’s developed satellite and meteorology technologies since
the early 1970s, noted that the measured frequency shifts might come not
just from each ping’s transmission from plane to satellite, but also
from the ping’s subsequent transmission from the satellite to a ground
station that connects the satellites into the Inmarsat network. In other
words, Exner may have found the hidden source that seems to be throwing
off the frequency graph.

Inmarsat’s
analysis is highly ambiguous about whether the satellite-to-ground
transmission contributed to the measured frequency shift. But if it did,
a ground station located significantly south of the satellite would
have resulted in frequency shifts that could account for the measured
shifts being too large at the beginning of the graph and too small at
the end. And sure enough, Inmarsat’s analysis states that the ground
station receiving the transmission was located in Australia.

It’s possible to check the theory more precisely. Public records of Inmarsat ground stations show just one in Australia: in Perth. Using STK, you can precisely chart the satellite’s speed relative to this station, and, using the satellite-to-ground signal frequency
(about 3.6 GHz), you can then factor the satellite-to-ground shifts out
of the frequency graph. Finally, you can at last calculate the true
satellite-to-plane speed values.

The results
seem to be nearly perfect. For the first ping, you wind up with a
satellite-to-plane speed of about 1 mile per hour—just what you’d expect
for a plane stationary or slowly taxiing eleven minutes before takeoff.
This finding seems to provide a basic sanity check for interpreting the
graph, and led Exner to declare on Twitter,
“Doppler code cracked.” He produced a new graph of the frequency
shifts, shown below. The gently sloping blue line shows the shifts
between the satellite and the ground station in Perth, while the dotted
red line shows the newly calculated satellite-to-plane shifts:

Why Inmarsat’s Analysis Is Probably Wrong

If this
interpretation—based on the work of Exner, Steel, Farrar, and myself—is
correct, it would allow independent experts to fully review Inmarsat’s
analysis, verify its work and check to see if Inmarsat might have missed
any important clues that could further narrow down the plane’s
whereabouts.

The problem
is, although this interpretation matches two basic expectations for the
frequency graph, it still doesn’t match Inmarsat’s example flight
paths. The new frequency values, calculated by Exner, show the flight’s
speed relative to the satellite as only about 144 miles per hour by the
last ping, but Inmarsat’s example flight paths show a relative speed of
about 272 miles per hour.

It’s
possible these outside experts have still erred or missed some crucial
detail in their attempts to understand the Inmarsat analysis. But that
just means that Inmarsat’s analysis, as it has been presented, remains
deeply confusing, or perhaps deeply confused. And there are other
reasons to believe that Inmarsat’s analysis is not just unclear but
mistaken. (Inmarsat stands by its analysis. More on that in a minute.)

Recall that
the Marco-Polo math alone doesn’t allow you to tell which direction
pings are coming from. So how could Inmarsat claim to distinguish
between a northern and southern path at all? The reason is that the
satellite itself wasn’t stationary. Because the satellite was moving
north-south, it would have been moving faster toward one path than
another—specifically, faster toward a southbound track than a northbound
one over the last several hours of the flight. This means that the
frequency shifts would also differ between a northbound and southbound
path, as the graph shows with its two predicted paths.

But this is
actually where the graph makes the least sense. The graph only shows
different predicted values for the north and south tracks beginning at
19:40 UTC (presumably Inmarsat’s model used actual radar before this).
By this time, the satellite was traveling south, and its southward speed
would increase for the rest of the flight. The frequency shift plots
for northern and southern paths, then, should get steadily further apart
for the rest of the flight. Instead, the graph shows them growing closer. Eventually, they even pass each other: by the end of the flight, the graph shows the satellite traveling faster toward a northbound flight path than a southbound one, even though the satellite itself was flying south.

One ping
alone is damning. At 19:40 UTC, the satellite was almost motionless,
having just reached its northernmost point. The graph shows a difference
of about 80 Hz between predicted northbound and southbound paths at
this time, which would require the satellite to be moving 33 miles per
hour faster toward the southbound path than the northbound one. But the
satellite’s overall speed was just 0.07 miles per hour at that time.

Inmarsat
claims that it found a difference between a southbound and northbound
path based on the satellite’s motion. But a graph of the frequency
shifts along those paths should look very different from the one
Inmarsat has produced.

### Losing Faith

Either Inmarsat’s analysis doesn’t totally make sense, or it’s flat-out wrong.

For the
last two months, I’ve been trying to get authorities to answer these
questions. Malaysia Airlines has not returned multiple requests for
comment, nor have officials at the Malaysian Ministry of Transportation.
Australia’s Joint Agency Coordination Centre and the UK’s Air Accidents
Investigation Branch, which have been heavily involved in the
investigation, both declined to comment.

An Inmarsat
official told me that to “a high degree of certainty, the proponents of
other paths are wrong. The model has been carefully mapped out using
all the available data.”

The
official cited Inmarsat’s participation in the investigation as
preventing it from giving further detail, and did not reply to requests
for comments on even basic technical questions about the analysis.
Inmarsat has repeatedly claimed that it checked its model against other
aircrafts that were flying at the time, and peer-reviewed the model with
other industry experts. But Inmarsat won’t say who reviewed it, how
closely, or what level of detail they were given.

Until
officials provide more information, the claim that Flight 370 went south
rests not on the weight of mathematics but on faith in authority.
Inmarsat officials and search authorities seem to want it both ways:
They release charts, graphics, and statements that give the appearance
of being backed by math and science, while refusing to fully explain
their methodologies. And over the course of this investigation, those
authorities have repeatedly issued confident pronouncements that they’ve
later quietly walked back.

The biggest
risk to the investigation now is that authorities continue to assume
they’ve finally found the area where the plane went down, while failing
to explore other possibilities simply because they don’t fit with a
mathematical analysis that may not even hold up.

After all,
searchers have yet to find any hard evidence—not so much as a shred of
debris—to confirm that they’re looking in the right ocean.

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