The Path of the Missing Malaysian Airliner: What We Know, and How — UPDATED


UPDATED: See end for description of possible northern route

On Saturday, March 15, Malaysian authorities released an analysis of satellite data that dramatically narrowed the possibilities for where missing Malaysia Airlines Flight 370 had gone after it disappeared from radar on March 8. Over the course of the following week, Inmarsat released further information that not only showed where the plane went, but also indicated how it got there. The results are shown on this chart. We still don’t know if the plane headed north or south, but if it went north, it made landfall near the western India-Bangladesh border and proceeded along the Himalayas to Central Asia. If it went south, it passed over western Indonesia and out over the southern Indian Ocean.

How are we able to determine this? The procedure requires a bit of explanation. Inmarsat is a communications satellite in geosynchronous orbit over the Indian ocean. That means it remains in the same place in the sky, like it’s sitting on top of an invisible pole. Because it’s so high up, it has a straight line-of-sight to virtually the entire eastern hemisphere. That’s great for radio communications: if you can see it, you can send it a message, and it can send that message along to anyone else in the eastern hemisphere, or to a base station that can then relay it to anywhere in the whole world.

Every hour, Inmarsat sends out a short electronic message to subscribers that says, “Hey, are you out there?” The message contains no information as such; the satellite just wants to find out if that particular subscriber is out there in case it wants to talk. Kind of like picking up your telephone just to see if there’s a dial tone. On the morning of Saturday, March 8, MH370 replied seven times to these pings, saying, in effect, “Yup, I’m here.” The line was open for the plane to communicate with the outside world. But the system that generates the messages themselves, called ACARS, had been shut off. So nothing else was communicated between the satellite and the plane.

All the same, those pings tell us something important about MH370: they allow us to narrow down its location. Because light travels at a certain speed, and electronics take a certain amount of time to generate a signal, there’s always a length of time between the satellite’s “Hey!” and the airplane’s “Yo!” The further away the plane is, the longer it takes to say “Yo!,” because it has to wait for the signal from the satellite to travel that extra distance.

Imagine you and I are in a darkened room. You have no idea where I am, except you know that I’m holding one end of a taut, 20-foot rope, and you’re holding the other. Therefore I must be 20 feet away. You don’t know where I am, exactly, but you know that I must like somewhere along a circle that’s 20 feet in radius, with you at the center:

kids with string

Now, it happens that in this room there are walls and pieces of furniture, so you’re able to rule out certain spots based on that, so instead of a whole circle, you have pieces of circle, or arcs.

MH370 was in an analogous situation. When Inmarsat pinged it at 8.11am, the amount of time it took the plane to reply allow us to calculate its distance from the satellite, just as if it was holding a taut piece of string. Instead of furniture, factors such as speed and fuel capacity provide other limitations of where it could be, so its range of possible locations is also not a circle but a series of arcs:



Note that these arcs do not represent the path that the plane took, but the range of possible locations at 8.11am. That particular ping tells us nothing at all about how the plane got to wherever it happened to be. So at this point all we know is where it started (it disappeared from Malaysian military radar at 2.15am at a spot between the Malay Peninsula and the Andaman Islands) and where it ended up. It could have taken any of a zillion routes to get from its start point to to its final recorded location somewhere on that last arc.

Remember, however, that Inmarsat received six earlier pings as well, and from them we can narrow down the range of possibilities dramatically. The first was received at 2.11am, just before MH370 disappeared from Malaysian military. Its length indicates that the plane must have been somewhere on the green circle at that moment:


Of course, thanks to radar we happen to know in this case pretty much where the plane really was at this time — around the area of the pink dot.

On Friday, March 21, an Inmarsat spokesman told me that “the ping timings got longer,” meaning that the distance between MH370 and the satellite grew increasingly bigger, and never smaller. That means that at no point during its subsequent travels did MH370 travel any closer to Inmarsat. So from the 2.11am ping data alone, we can rule out every spot within the green arc:


MH370 never traveled anywhere in the shaded area. (Of if it did, didn’t stay there for long; by the end of the hour it had to be outside.) We also know that it never was further away from the satellite than it was at 8.11am, so we can exclude everything east of that, as well. Finally, we can rule out some chunks close to its starting point for other reasons:


So just from the 2.11am and the 8.11am pings, we know that MH370’s route of flight must lie within either of these two broad swaths — one lying to the north, and the other to the south. Bear in mind, the reasoning that we’ve just gone through doesn’t tell us anything about whether the plane went to the north or two the south. Because of the symmetry of a circle, the possible paths are mirror images of one another. However, we’ve vastly reduced the range of flight routes that MH370 could have taken. For instance, a popular theory circulating on the internet posits that MH370 tucked in close behind a Singapore Airlines flight, “SIA68,” in order to hide in its radar shadow:


This new Inmarsat data rules out that possibility. It also rules out the idea that MH370 flew south through the middle of the Indian Ocean to avoid military radar. If the flight went south, it would have had to have gone through Indonesian radar coverage.

Interestingly, on March 19, the website Antara News reported that “Indonesian Defense Minister Purnomo Yusgiantoro said the Indonesian military radar placed in the country’s western-most city of Sabang did not detect an airplane flying over Indonesian territory.” 

On March 22, 2014, CNN reported that China, India, Pakistan, Myanmar, Laos, Kyrgyzstan and Kazakhstan have told investigators that “based on preliminary information, their nations had no radar sightings of missing jetliner.”

So far, we haven’t talked about what we can deduce from the remaining five Inmarsat pings, the ones received at 3.11, 4.11, 5.11, 6.11, and 7.11. It should be possible, based on the presumed speed of the plane and the distance between the successive arcs, to make some reasoned guesses about how the plane traveled from one to the other. I haven’t seen the data yet—I’m working on it—but earlier this week the Washington Post published a map that showed what appeared to be the results of just such analysis as applied to the southern route, carried out by the NTSB:

Southern crop

This appears to be why the nations assisting the investigation have poured so many assets into searching that particular stretch of southern ocean. If MH370 took the southern route, it would have had nowhere to land, so it must have crashed and its debris must still be floating somewhere in this area.

Of course, the information we glean from Inmarsat data about MH370’s flight route is, by itself, symmetrical around an axis that runs from the spot on the ground underneath Inmarsat to the point where the aircraft was last observed. So assuming that the NTSB’s interpretation of the southern route was only based on factors of speed and arc spacing, it should be applicable in mirror form to the northern route as well. I’m working on that right now.

UPDATED 3-23-14: Okay, I feel a little slow on the uptake on this one, but it turns out that if you flip the NTSB’s guesstimated southern route, you come up with a northern route that looks pretty much like this one published in the Daily Mail (I know, I know):

17M-Missing plane search MAP.jpg

Basically, you make landfall in the vicinity of Bangladesh, skirt along the border between India and Nepal, then cut across northeastern Pakistan and Afghanistan before winding up in Uzbekistan or Kazakhstan. This may be why Malaysia recently asked Kazakhstan if it could set up a search center there.

Kazakhstan would not be a bad place to try to hide an airplane. It is larger than Western Europe with a population of just 17.7 million. Its expansive, sparsely populated steppe and desert terrain make it perfectly suited as a touchdown spot for Soyuz space capsules. The country gained independence from the Soviet Union in 1991 but its communist-era ruler,  Nursultan Nazarbayev, remains in power. He is a close ally of Putin, and two days after MH370 disappeared told the Russian premier “that he understands the logic of Russia’s actions in Ukraine,” according to Reuters.



311 thoughts on “The Path of the Missing Malaysian Airliner: What We Know, and How — UPDATED”

  1. Correct me if I’m wrong, but with the time/return & doppler data, you cannot construct a straight line flight at any speed within the aircraft’s envelope in any direction other than between south-southwest and southeast

  2. To solve this problem…

    We also need the last lat/lon/alt/time record obtained by the radars. I know, it won’t happen. But I hope we could at least get the final position. That would be a big help.

    I’m starting to glean more from the Annex I Doppler graph. I am especially interested in the period between 01:07 and 03:40…a 2 ½ hour gap in the Inmarsat data, except for 3 rapid fire aircraft initiated emergency transmissions, probably triggered by an event in one of the engines, such as a rapid decrease in ambient pressure, fire, or something. Whatever triggered it, the message would have contained ambient pressure (altitude). We need verify and to squeeze Boeing and RR for those data, if it exists.

    The military radars tracked the plane for the first half of the 01:07 -03:40 period. Some reports have said that the radar was lost at 02:15, others a bit later. It would be nice to know if either radar tracked the plane between 02:25 and 02:30, the period when we can see from the Doppler data that the plane was in trouble and turning from a generally east bound direction to either north or south direction (moving from a direction at 02:25 with relatively high radial velocity away from the satellite, to one nearly tangent to the radial).

    The final period between 06:40 and 08:19 is also intriguing. Whatever happened during the 01:07-03:40 period, we know things settled down between 03:40 and 06:40, indicated by the smoothly increasing Doppler obtained from 4 sequential hourly Pings. There should have been another ping at 07:40, but it was not received. The next transmission was at 08:11, but we do not know for sure if it was another Ping or an aircraft initiated emergency message, like the 3 around 02:30. If it was an emergency initiated message at 08:11, and the rep rate on the first 3 emergency messages was ~2 minutes, then the 08:11 and 08:19 transmissions might have been 2 of 4 or more emergency messages…one complete and one partial. Bottom line…I suspect that the point of impact might have been at 08:11 or earlier, and the aircraft was trying to send messages every 2 minutes for at least 8 minutes after impact.

  3. My guess is that the last ping was triggered by an engine flameout following fuel exhaustion.

    @airlandseaman – where are you getting the times & doppler data from? I didn’t think it had been publicly released.

  4. Still, why? Suicide mission makes no sense – if that, why not crash it at once? If mechanical failure, why deliberate deviation from flight path? The only explanation that makes any kind of sense is the money one…pilot(s) suborned, coerced or greedy, plane stolen for parts, evades radar by following another plane closely, undetected because transponders turned off, landed in, say, Kazakhstan, pilots shot, passengers sold off for body parts, kept alive until sold, told they’re being ransomed to keep them docile. This is so horrific that, if it were known, would devastate the airline industry and lead to huge international repercussions. Therefore even the possibility of it must be covered up. Inmarsat requested to come up with a plausible reason to direct the search away, as a duty to avoid international panic. Frankly, even if they do find “wreckage” and declare it to be from MH370, I’m not sure I’d believe them. Best for them if nothing is ever found.

  5. I would agree that the only straight, constant speed trip is to the SSE. It’s the non-uniform trips that interest me, especially since this plane wasn’t doing anything uniform in the first 2 hours.

    The peak Doppler at 2:25 looks to me like an aggressive dive. Any vertical speed would have the greatest impact on the Doppler contribution, considering that the sat is almost straight up. It would take a much lower vertical descent rate to achieve the same doppler contribution as horizontal movements.

    Around 3:40, by my calculation, both the plane (according to the southern route) and satellite had just crossed the equator. This is shown on the graph as a contribution of 110 Hz, but since there would be very little relative motion between the plane and the satellite, this would seem to be the baseline. A Doppler below this would indicate that the plane was approaching the satellite (even if it was parked.)

    At 3:40, flight SQ68 would have been closing in on India’s east coast. While I am unable to do the precise math, it appears this route at a certain speed could create the 3:40 Doppler as well. The satellite is moving south at this time at 86km/h. The plane need only have been heading more westward than a radial path to counteract this and achieve a constant distance.

    The lack of a 2:40 ping probably stems from the fact that the ground station had just heard from the plane.

    The lack of a 7:40 ping is much more intriguing. This suggests that either the ground station gave up, the plane stopped responding, or the signal was blocked. It would useful to know if the satellite actually attempted this ping.

  6. I wonder if Inmarsat has Ping Doppler and range from SQ68? It would not be at the same time within the hour as MH370, but it would be an interesting comparison.

    JS I agree that a dive would cause Doppler.

    The radial range is/should be the mathematical integral of radial speed and therefore much more useful than radial Doppler, not least because it cannot be fooled by short-term changes in course or altitude. I wish we had proper data with error bars.

  7. I would like to verify my interpretation of the Annex I Doppler data. As I understand it:
    1. The vertical axis, “Burst Frequency Offset (Hz)”, is related to the sum of the satellite and aircraft Doppler (defined on page 1 of Annex I).
    2. The first ACARS message at ~00:30 establishes the system bias = 87 Hz. Thus, the total s/c + aircraft Doppler can be derived by subtracting 87 Hz from all measurements, assuming zero bias drift.
    3. The aircraft is known to have departed around 00:41 on a heading of 25 degrees, which would have been slightly away from the sub-satellite point. This establishes the sign (radial direction) for the vertical scale. It means increasing positive values greater than 87 Hz correspond to increasing radial velocity away from the sub-satellite point.
    4. The apparent sign inversion, positive numbers corresponding to a lower Doppler frequency, is probably due to the the way the raw carrier frequency was processed in an IF stage, or similar. This is common in radio receiver architecture.

  8. It would be an interesting exercise to see if it would be possible to create a zig-zag route to the north that would meet both doppler and position criteria.
    Also, there are no error bars on the Doppler chart, I wonder how good the fit really is.

  9. I am skeptical about interpreting raw offset frequencies as indicative of a “dive”. A turn will change the Doppler much more than a dive. Even in a steep decent, the L/D would be >>1.

  10. So if the plane started a dive at 2:22, still in the steepest part at 2:25, and pulling out by 2:29, could this have accomplished two things:

    1) descend from FL450 to FL295 to intercept SQ68, both vertically and horizontally since it was a few miles ahead of MH370,

    and 2) shaken the ground radar at the same time?

    It seems awfully coincidental that these signals, radar loss, and SQ68’s path interception all happen in the same time and place. Was this really the edge of the military coverage, or was the military radar fooled by the plane’s actions?

    The descent rate would be about 2250 fpm, or about 40km/hr. My hunch is that from that location, a 40km/hr vertical speed would easily counteract any lateral speed as Doppler goes. Is 2250 for 7 minutes reasonable?

  11. A descent rate of about 5500 fpm would counteract a lateral speed of 500 mph (at that Location roughly). If the transmissions were triggered by the fast descent (or if the descent was caused by the same emergency that caused the Transmission) then the time Frame is maybe only slightly bigger than 4 minutes.

  12. @Lee
    Why do you assume that I think there is a conspiracy with Inmarsat and the UK government? I don’t, but I do find it odd that they don’t do as Jeff and others request and release the data. There sounds like there is nothing proprietary in the math seeing as it is from the 19th century. The satellite is 20 years old and the idea of industrial espionage and the loss of any proprietary info there is a little ridiculous as well. So why not put the data out there? What are they afraid of? A mistake? Invalid assumptions? If someone helps to find an error does that not help? I noticed earlier posts by someone named Arthur T. He was self described as a mathematical physicist. I had to look up what that was. Anyway, he seemed to have some qualms about things. And maybe he is the kind of person that should be peer reviewing the findings?

    On a different level it has been my experience that some pretty smart and experienced people will get so locked on a single idea that they miss or overlook things. By their account and any logical look it should be as they say it is. It should work, but it doesn’t. They will argue and insult because they are told there is something wrong with what they have done. They have the blinders on so firmly that they can not even consider they possibility there is something they missed. But they have. As for this case and until there is debris from the plane firmly in somebody’s hands then what does it hurt to speculate? Whether it be variation on the interpretation of Inmarsat data or imagining things with slimmer possibilities.

    In final you seem to really want me to come up with some sort of scenario. Conspiracy or otherwise. So here goes… Let’s suppose the plane was a zombie plane. But only the plane was a zombie. Perhaps a virus get’s introduced or some sort of cascading electronic glitch get’s into the AIMS system or electronics bay of that plane. Knocks out communication. Scrambles both navigation and attitude control. Disables the manual flight control inputs and override for the fly by wire. The plane makes heading and altitude changes until finally the auto pilot/FMS lock into a southerly heading and stays there until fuel is exhausted. However, for the entire time both crew and passengers are powerless to do anything. This solves:
    Fire-Something aloft does not fly for 7 hours.
    Decompression-Passed out crew and passengers don’t make turns.
    Suicide-Unlikely someone intent on dying waits 7 hours.
    Terrorists-Like to be public and messy not mysterious.
    Hijacking- Where are the ransom notes?
    Insurance- Value of used plane is not enough to cover financial troubles of an international carrier.
    Are you satisfied Lee?

    My apologies.

  13. Did anyone out there grab the SQ68 Flight Aware data before it disappeared? If so, please post. I would like to see an overlay with the following MH370 positions derived from the Butterworth radar data:

    02:02 279 degrees/89 NM (5 41′ 28″N; 98 55′ 28″ E)
    02:22 295 degrees/200 NM (6 52″ 03″ N; 97 20″ 21″ E)

    These two radar fixes give a MH370 average vector for 02:02 to 02:22 of:
    118 NM, 354 kts, 306.5 deg

  14. Response to JS: “My hunch is that from that location, a 40km/hr vertical speed would easily counteract any lateral speed as Doppler goes.”

    You’re right:

    An aircraft on the green circle that is going 450 knots (833 kph) directly toward the center of the circle, is getting closer to the satellite at a rate of 89 kph. For the general case where the aircraft is not moving radially, multiply this by the sine of the off-tangent angle. For example if the aircraft is moving 10 degrees off-tangent it is approaching the satellite at 15 kph.

    An aircraft moving along a tangent of the green circle and descending at 40kph, is moving away from the satellite at 39.8 kph.

    Satellite 4-F1 apparently has an inclination of 3 deg (of course with a period of 24h). It remains at approximately constant altitude and thus has a maximum lateral speed of 490 kph mostly in the N-S direction. At maximum speed, the satellite will be moving closer to the green circle (on the ground) at about 53 kph.

    Note that this means the green circle moves North and South by about +/- 330 km each day. I do not know the 24 h phasing of satellite 4-F1 but I’m sure Inmarsat must know. This could have interesting consequences. For example if the green circle is moving South during the Malaysian night, then a NW-bound aircraft might appear to stay on the green circle even when it is moving slightly closer to the “nominal” location of the center of the circle.

    In summary: a) Doppler due to aircraft climb rate can be much larger than the expected Doppler due to horizontal motion of the aircraft. b) Satellite motion will induce significant Doppler for aircraft North of the satellite center position c) Satellite motion can shift the green circle plus or minus 330 km.

  15. Update: apparently Inmarsat 4-F1 has a period of 23.93 h rather than 24 h. This means that the day-night phasing of the N-S motion changes throughout the year, making it harder to look up on-line. If anyone can find out what its N-S motion phasing was on March 8th I would be interested in knowing. It might explain how an aircraft “drafting” SQ 68 could appear to be father North than it really was.

    Getting the ping data (i.e. green circles) for SQ 68 would be even more useful.

  16. @jeff – Since doppler is based, simply speaking, on nothing more than relative velocity at the precise moment of the ping, it should be possible create many scenarios where a plane traveling north away from the satellite looks like a plane traveling south. It would need to speed up towards the satellite every hour, on the hour, to match the precise moment of ping. Just enough to compress the signal to a degree that matches the compression seen in a southward traveling plane. Then turn and head north again after the ping. That requires some crazy assumptions — human intent, for one, and very good timing, for two, and the ability to forecast the future (thus knowing ahead of time that inmarsat technicians would use this method to track them). Altogether unlikely I think. 🙂

  17. I located the Flight Aware data fro SQ68 and MH370. Hard data stopped at 01:02. Also computed the positions for MH370 at the two known Butterworth radar fixes at 02:02 and 02:22. The last MH370 ADS-B position was at 01:02.

    Bottom: SQ68 was 100 NM NW of MH370 at 02:22. Both were headed in the same direction. It looks like a close match in direction, but with MH370 trailing by 100 NM means there is no way MH370 could have caught up when SQ68. SQ68 was already at altitude (30,000 ft) and cruising at 487 kts.

  18. @airlandseaman

    SQ68 on 3/8/14:

    02:02 6.6576N/98.2648E heading 312°
    02:22 8.2750N/96.4248E heading 311°

    Whatever fancy math you’re wielding try it on KLM flight 838. At 5:11 it was at 20.4548N/86.9526E heading 320° just off the coast from Bhubaneshwar. When I was speculating about a northern route, that is the flight I suspected. Get over the Indian interior and then fly over Nepal and the Himalayas right into Western China. Not that I doubt Inmarsat’s conclusions, but I do wish they would publish their data so these crazier scenarios could be definitively ruled out. It would be an interesting exersize and would prevent a cult of conspiracy theorists from growing up around this incident.

  19. @ airlandseaman: oops, I guess you posted again while I was finding my copy. KLM836 (not 838) was behind MH370 at 2:15. Might MH370 have been able to stay at a slower speed for a while at let it catch up? Bhubaneshwar would be further from the satellite that MH370 position at 2:15 and it could have then curved NW over Nepal (higher and faster to get over the mountains) and then moved more directly away from the satellite after that (lower and slower to avoid radar). I would love to know what airspeeds would be required to match the doppler shifts on that route.

  20. @Hal I don’t think that’s necessarily true.

    First of all, the satellite was moving south for most of the trip. So it is merely a matter of what degree the plane moves away from it.

    Second, we don’t have the actual distances yet because, I don’t know, we wouldn’t understand milliseconds or something. But let’s assume that we did and we had 12-13 arcs.

    For each reasonable point on each distance arc, we could have an unlimited number of combinations of heading, speed, and vertical speed. Of those, there would also be an unlimited number of them that satisfy the Doppler for that ping. The only narrowing we could do is eliminate the ones that exceed the plane’s abilities. We might further narrow them by selecting smooth paths rather than jagged ones. But still, the possibilities are endless. North or south. I would probably start with headings at 10 degree increments, speed at 10kt increments, and vertical at 100fpm increments.

    From that alone, I would have 180,000 different vectors at every single point on each distance ring that I could not completely exclude based on the plane’s performance alone. Still, cake for a computer to calculate dopplers from. The hard part would be identifying meaningful combinations from the different rings that would make up a route.

    Notice that out of the last 6 pings, the selected northern route STILL satisfied 2 of the Doppler values, and even better than the southern route. That strikes me as a pretty high number by coincidence. A path selected at random with respect to the doppler values ends up matching at a third of the points?

    This leads me to the conclusion that even very minor heading or altitude changes would yield an entirely different graph. I don’t think a plane would have to do acrobatics to get a matching Doppler plot. A simple landing approach up north might look the same as a continuing cruise down south. Look at the spikes for evidence – Inmarsat calls it a turn, some of us call it a dive, but it shows that more drastic Doppler values are possible without crashing the plane.

  21. Latest News: Flight 685 miles northwest of earlier estimate.

    Huh?….. Assumptions not written in stone then.

  22. ” Latest News: Flight 685 miles northwest of earlier estimate.
    Huh?….. Assumptions not written in stone then ”

    AND … this comes within a day of Thai and Japanese satellites discovering a floating junkyard in the general area first targeted a week ago. I had a laugh this morning hearing Miles O’Brien saying he thought the Thai ‘objects’ “looked aerodynamic” … as seen from satellite, no less. Apparently, someones else have discounted all the junk seen over the past ten days from satellites of AU, China, France, and now Thailand and Japan.

    Someone commenting here a couple days ago speculated about whether AU has got good over-the-horizon radar capabilities…. I suspect we now have our answer.

  23. I found it interesting that in some of the early satellite photos that the wing sized objects didn’t look very wing shaped. At the time I thought maybe it was just the poor resolution throwing things off. Or has Boeing come up with some new rectangular wing design?

  24. The new analysis verifies that they don’t know what the speed was. Moreover, the CNN report stating that it was going faster than previously thought, and thus ended up 1000km NE, is impossible. The lower the speed, the more the path clocks back to the east. The faster it went, the more it will clock to the west. They have is backwards. You can plainly see that on Page 3 of Annex I. Therefore, the latest report must be carefully scrutinized. It probably did go much further NE,but it did so because the speed was lower, not higher.

  25. Kudos to Jeff and a number of others here who are attempting to analyze the data and information concerning MH370 that is in the public domain. This blog has turned into the nuts concerning the incident, as far as I am concerned.

    The Inmarsat data set as a whole and its analysis bother me, as confirmation bias lurks large here, and the entire search effort is largely based upon the data and its analysis.

    The lack of additional data from primary radar sources and the other national security assets of the nations concerned also bothers me. Even the remote-sensing satellite data has generally suffered delays in its production, not all of which can be accounted for by time-in-analysis.


    1. From the outside, the Inmarsat data and the analysis by the Brits and the fact that it has not been peer reviewed (as far as we know) leaves any conclusions open to confirmation bias, does it not?

    2. Are there any positive indications that any other sources of data, classified or unclassified, have been withheld by the national security assets of any nation? Has anyone covered this aspect of the investigation somewhat adequately other than the NYT (Mar 26)?

    3. The principles of Transformational Grammar and the processes of “translation, distortion and deletion” are quite obviously producing disseminated information (produced from the data) by government authorities that is more ‘surface structure’ (less true)-oriented as opposed to ‘depth structure’ (more true)-oriented. Noam Chomsky has yet to really chime in (he is getting older), but have you (Jeff) come across any other transformational grammarians that have properly analyzed the government information?

    4. Who are the transnantional czars/organizations for both the investigation and the search? Are they adequately empowered, or is it really more national security and safety assets working away independently in their own silos?

    5. Is anyone leveraging one of the crowd-sourcing petition platforms in an effort to seek higher-quality information from government authorities on MH370?

    Thanks to anyone whom can provide some answers to my queries. And please do keep up the good work; this is a mystery that demands answers, with the alternative being only a form of madness.

    (Jeff: I am a good friend of Sue Brooks and John Leary; I am sure that they would want me to say ‘hello!’)

  26. The new math is more flawed that the original. It is simple vector math, really. If the speed assumption is reduced, the whole path moves to the east and ends up on the final LOP further north. If the speed was higher, it moves to the west. The new statements that the plane was traveling faster, burning fuel faster and ended up 1100 km NW on the final LOP at 08:11 or 08:19 is obviously impossible. Even a fool can see that you cant run out of gas an hour sooner and still fly for the same time, no matter where you end up. It’s completely ridiculous as reported.

  27. Some have suggested that we should interpret the breaking news to mean that the aircraft was travelling faster only before radar loss, and by implication, slower than previously believed after radar loss, thus ending up 1100 km further NE. I could believe that. But if this is the case, it reveals something important. It means they are estimating speed from an assumption about the total fuel at TO and the integrated burn rate over the path. This in turn means they must be assuming a high altitude after 02:22 (or 03:40?). As I have noted, a slower speed at a lower altitude can give you about the same fuel exhaustion time. If the altitude assumption is wrong, the speed is also wrong.

  28. Response to Michael D and JS:
    “An aircraft on the green circle that is going 450 knots (833 kph) directly toward the center of the circle, is getting closer to the satellite at a rate of 89 kph.”

    No. The Elevation of the Inmarsat for this region is about 50 degrees. So vrel = 833 kph * cos(50) = 535 kph.

    “An aircraft moving along a tangent of the green circle and descending at 40kph, is moving away from the satellite at 39.8 kph.”

    No. 40 kph * sin(50)= 30 kph.

    Obviously you did’t take into account the earth being a sphere. Your calculations perhaps would be correct if the Indian ocean was a plane.

  29. If the plane was going faster than previously assumed then it would have gone further. (X=VxT) we know the time of flight from the satilite pings.

  30. @airlandseaman. With regard to your request for verification posted March 27, 2014 at 1:49 PM, I got my definition of “Burst Frequency Offset (Hz)” from the body of “INFORMATION PROVIDED TO MH370 INVESTIGATION BY UK AIR ACCIDENTS INVESTIGATION BRANCH (AAIB)”: “The Inmarsat technique analyses the difference between the frequency that the ground station expects to receive and that actually measured. This difference is the result of the Doppler effect and is known as the Burst Frequency Offset.”.

    I interpreted this explanation simply as the Doppler-shifted carrier frequency it expected for the assumed track/speed minus that actually received.

    I did not see any reference in this document of how it constructed this expected track/speed. The same for whether it used these observed Doppler shifts directly to derive radial speed components on the tangential plane to the earth’s surface at a point on their related circle of positions (COPs). I believe this could be quite useful constructing tracks for different assumed speeds given the ping data and the last known position.

  31. As noted on Don Lemon tonight, the problem now is that the debris is no longer of much use for the purpose of locating the aircraft. Finding the debris is still very important for other reasons, but they need to refocus on the best possible Inmarsat range solutions (high confidence dimension in the search area box) and altitude assumptions (a dimension that is 100X less certain).

    IOW, the focus really has to be on error analysis at this point. They must reduce the size of the aircraft search box (not necessarily the debris search area). The error box is for the aircraft location is a very long skinny arc now, not a box. They need to recognize this. They need to put a sharp pencil on the range error analysis, which should show that it is much much smaller than the long altitude assumption dependent arc length.

  32. Can someone please explain why there are no satellite images of the aircraft itself as it was flying in daylight for a couple of hours over the Indian Ocean in the morning on March 8?

  33. This might be a really stupid question, but what would happen if the pilot had flown the plane directly away from earth, out into the atmosphere, and just kept going?

  34. But if Ockham applies. The path is projected through a lot of Sumatra airspace coming in range of Java and then continue towards Xmas. To suggst that there is no or lack of radarcapability (we talk about 4 civilian airports + militair which could/should have already been put on the alert by malaisian militairy after 2:30. ) among malaysian neighbors is a too short turn for me I don’t buy.
    The next assumption then must be a deliberate west route 510 km west out of sumatra. This comes only with a south route on purpose.

  35. @Ben
    There comes a point where the atmosphere is too thin for engines on that plane to operate. As well as being too thin for the wings to generate enough lift. Or something like that.

  36. I watch with amazement all the projections of final positions based on refined credible evidence, etc. It seems to me it all comes down to the position, altitude, and track speed vector at the point primary radar contact was lost and the circle of position at 8:11. MH370’s ending position keeps floating around in my mind as I mentally vary speed, either through changing power settings or altitudes.

    How would I sort this out if I had the data? The potentially available facts include:
    1. The position, altitude, and track speed vector at point where radar contact was lost
    2. Circle of position (COP) data for each complete ping (and ACARS message?):
    a. Value(s) that define or allow the computation its locus of points on the earth’s surface
    b. The un-shifted frequency of inbound transmission
    c. The observed Doppler shift of MH370’s transmission
    d. The height of the satellite above the surface

    I would also like have available winds aloft data over the area at each ping time.

    Step 1 — Compute the radial track velocity for each ping.

    The MH370’s speed towards or away from the satellite must be adjusted to reflect the fact that the relevant track speed vector is tangential to the earth’s surface at the COP. This conversion is relatively straight forward, though.

    Step 2 — Compute the implied track speed (and, in some cases, direction) between adjacent COPi and COPj (the latter ping).

    For simplicity, it is assumed that the earth is flat over the area of interest and all references lie on this plane. This constraint can be removed at the expensive of more complicated mathematics, but the general concept will be unchanged. An illustration is available at:

    At time ti we know COPi, the locus of points on which MH370 was located, and its radial speed component Vri. At time tj we know COPj and its radial speed component Vrj. We also know distance and direction of the satellite at tj from its position at tj.

    The “constant” V between ti and tj can be computed via an iterative computation:

    bi = ArcCos(Vri / V)
    bj = ArcCos(Vrj / V)
    aj = bi – bj
    Dij = funct(di, dj, Ds, SBj, …)
    V = Dij / (tj – ti)

    For the fixed satellite position case, this track segment and velocity can begin at any point on COPi as well as its mirror. For the varying satellite position case, Dij and V will vary with SBj, the satellite bearing to Pj. This model may also help resolve any remaining north-south ambiguity.

    I suspect that if this analysis is performed starting with the last know position and adding track segment by track segment, one will have a more likely ending position than one making assumptions speeds and altitudes beyond the last known position. At minimum, it will confirm speed assumptions that are critical to where MH370 crosses the 8:11 COP.

  37. For the locus of possible locations of the plane for a given ping response time to be determined accurately, the altitude of the plane needs to be known accurately. For example, a drop in altitude of the plane from 35000 ft to 12000 ft will cause the radius of equal ping time ring (at the same altitude of the plane) to get shorter, considering the same range corresponding to the hypotenuse of the right triangle, with any radius of the equal ping time ring and the straight line connecting the center of earth to the satellite forming the right-angled edges. This amounts to a drop in altitude of 23000 ft (7.0104 km), which causes the vertical right-angled edge in the right triangle to grow by that much amount. This results in some reduction in the radius of the equal ping time ring, since the hypotenuse length of the right triangle remains unchanged. The Inmarsat data assumes that the plane was flying at 35000 ft, when the ping responses were received, which may not be necessarily true. Is there any way to determine the altitude of the plane from the satellite data?

    This needs to be considered in correlation with some circumstantial evidence originating in the southern Maldives island of Kuda Huvadhoo, that the islanders there might have possibly seen the plane flying low over their island at 9:15 am Malaysia Time (6:15 am Maldives time), which is within an hour from when the last partial ping handshake was detected from the plane. This was reported at the following website of a news outlet from Maldives:

  38. Jeff,
    Once the plane turned south it probably continued on a fixed heading at fixed speed.
    Therefore the distances between the intersections of the flight path and the ping arcs should be constant.(ie the distance travelled per hour) A flight path which gives this result passes through 30 S 100 E (approx) which would be a good place to start looking.
    Rod Craig

  39. @pdcurrier and @neo –

    Both the Doppler and the COP radius change based on altitude. If the plane is gliding (descending) then the Doppler will be exaggerated.

    Because there are three variables – descent rate, heading, and ground speed, there are basically an infinite number of solutions. It could be descending at heading 0 over the Himalayas at 200kts, or at level altitude heading 180 at 400kts, for example. At each point on the ring there are multiple solutions.

    The only constraints we can put on it are the aircraft’s performance limitations, since it obviously can’t fly straight up or stand still or at 1000kts. Those constraints eliminate most of each ping circle but not the number of solutions remaining.

    The fact that two weeks without ANY sign of floating debris suggests to me that they are looking in the wrong place. In my opinion, Christmas and Kazakhstan should be on the table right now, or at least after the black box pings are expected to stop. At that point there is no longer any greater urgency to look in one place vs. another.

  40. @Neo and @JS.

    I read somewhere that the accuracy of the round trip ping time was accurate to, I believe, 300 microseconds and, in one example case, would amount to around a one degree error. On the surface, in the theatre of this drama, this would amount to an error approaching 75-100 nm. A quick back of the envelop calculation shows an error at 35000 ft (from sea level) of around 10 nm. This is materially less than this other expected error from the measurement of the ping round trip. That said, the assumed altitude should be factored into an analysis having actual data.

    I am more concerned about a rapid change of altitude at the time of a ping since it throws off the Doppler shift by a much greater percentage of radial velocity. This will impact the modeled track speed and implied track angle between pings. However, I suspect that rapid changes in altitude preceded the relevant ping times. Therefore, unless the flight was constantly changing altitude, any change will probably occur between ping samples or occur with gradual rates with minimal effects.

    With regards to the Maldives sightings, even if they were contemporaneously logged before the first MH370 news reports, this location is too close to the satellite to be reachable from the 8:11 COP in an hour.

    I have presented my thoughts to Inmarsat regarding the use of radial speed vectors in modeling to gain extra clarity regarding MH370’s path and speed following the last primary radar contact. I believe that it was forwarded to the right department once I clarified that I was not trying to obtain data or enter into a discussion. I can only hope that, if the data is available and they find the approach reasonable, it will speed the locating of the wreckage.

  41. is there any consideration of military antisubmarine listening posts hearing the impact of MH370 or hearing the 37kHz beacon from the black box?


  42. I have been reading all of the information and comments on this site and can conclude that all are highly intelligent. With this said and along with all of this information, has anyone been able to determine a resonable location of the missing plane?

  43. I spent the bulk of my engineering career designing missile tracking systems. Precise Doppler information was a key ingredient and all systems used phase-locked loops (we called them coherent) systems.

    If the Inmarsat system is not phase locked (in the signal round trip)all Doppler data is merely speculation. Who knows the technology used in the Inmarsat system?

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