The Locations of Inmarsat-3F1 when Pinging MH370

The Locations of Inmarsat-3F1 when Pinging MH370

Duncan Steel, 2014 March 24.
duncansteel.com

This post contains some further comments from me regarding the pinging of MH370 from the Inmarsat-3F1 satellite.

The graphic issued by the Malaysian Government is in error

In a previous post here almost 24 hours ago I made various observations with regard to the final ping of MH370 by the Inmarsat-3F1 satellite at 00:11 UTC on 2014 March 08 (‘Time P’). It was on the basis of that successful ping that the Malaysian Government, apparently using information supplied by Inmarsat via SITA, issued a graphic that showed a set of corridors or rings/arcs along which MH370 might have been at that time, based on the time-delay of the pings. That graphic, repeated in many newspapers and on many websites around the world, is fundamentally flawed because it is based on an assumption that the satellite was directly above the equator at that instant in time.

Here is that graphic:

MH370_last_ping_corridors

I now draw in the equator by connecting the dash marks at each side of the graphic:

MH370_last_ping_with_equator

It is clear that the sub-satellite point has been placed on the equator, and the rings have been calculated/drawn concentric with that point, making them symmetric about the equator. As I indicated in my preceding post, at the time in question (Time P) the satellite was located north of there, directly above a latitude of 0.589 degrees, about 65 km north of the equator. It is that which should be the centre of the rings/arcs.

Further, the graphic states that the satellite was 35,800 km above sea level. In fact at Time P it was at an altitude near 35,793 km; and at that time is was above a longitude of 64.47 degrees east, rather than its nominal 64.5 degrees.

Later in this post I will give some indication of the amount which the above considerations have placed the graphic/map from the Malaysian Government in error. For the time being one might consider simply that the rings/arcs should be displaced about 65 km northwards.

The locations of Inmarsat-3F1 at other pertinent times

Various commentators have mused on the possibility of determining constraints on earlier positions of MH370 through pings from Inmarsat-3F1 that might have occurred in earlier hours than Time P. As I would maintain that it is important to take into account the actual position of the satellite at the times of pings, I have calculated the locations of Inmarsat-3F1 at a dozen different times, these being times of interest as tabulated in the Wikipedia article on the disappearance of MH370.

These calculated positions, as in the table below, are all based on using the SGP4 integrator within STK with the latest orbital elements (i.e. equivalent to the TLEs) used by download from Analytical Graphics, which means via the Center for Space Standards and Innovation or Space-Track and that in turn means the latest unclassified-satellite orbits from the US military.

Inmarsat-3F1 positions at various times

Time
(UTC)

Latitude North (degrees)

Longitude East (degrees)

Altitude above MSL (km)

2014/03/07 16:41:00.000

1.185

64.549

35801.4

2014/03/07 17:01:00.000

1.279

64.546

35802.7

2014/03/07 17:07:00.000

1.306

64.545

35803.0

2014/03/07 17:19:00.000

1.356

64.542

35803.7

2014/03/07 17:21:00.000

1.364

64.542

35803.8

2014/03/07 17:22:00.000

1.368

64.542

35803.9

2014/03/07 17:30:00.000

1.399

64.541

35804.3

2014/03/07 17:37:00.000

1.424

64.539

35804.6

2014/03/07 18:11:00.000

1.529

64.534

35806.0

2014/03/07 18:15:00.000

1.539

64.533

35806.2

2014/03/07 22:30:00.000

1.187

64.491

35801.4

2014/03/08 00:11:00.000

0.589

64.471

35793.3

 

If any reader should wish to have positions for the Inmarsat-3F1 at other times, please contact me via my website and I will do my best to oblige.

In order for readers to visualise precisely what the satellite is doing in terms of its sub-satellite point during each orbit, I have produced the following graphic. It starts, in terms of time, with the take-off of MH370 (the first line in the table above) and ends one orbit/one sidereal day later. Tick marks are spaced by 15 minutes, and go counter-clockwise from the upper right as shown. Obviously only the first 7.5 hours of this loop (through to Time P with the final ping response, as in the last line in the preceding table) is of direct interest here, but I show the complete loop so that readers can understand more of the overall process.

Movement of the sub-satellite point of Inmarsat-3F1 over
the orbit/sidereal day starting from the time of take-off of MH370.

E6 small

From the above graphic/map one can deduce that it is likely that the rings on the graphic issued by the Malaysian Government is are mis-placed through another source of error: if Inmarsat/SITA/ assumed that the satellite longitude was its nominal slot at 64.5 degrees east then the rings were placed slightly too far eastwards.


Locations of the rings at different times

As indicated above the graphic/map issued by the Malaysian Government was incorrect in that it was based on the Inmarsat-3F1 satellite being directly above the equator at the time of the last ping from MH370. Obviously, then, one might like to know the correct locations of such rings.

In my preceding post on this subject I described how I have used STK to set up a simulation of the situation on 2014 March 07/08 in terms of the actual positions of the Inmarsat-3F1 satellite and the rings for equal elevation angles (of 35, 40 and 45 degrees) to that satellite from a platform I located at an altitude of 35,000 feet above mean sea level (MSL). The Malaysian Government graphic, apparently based on information from Inmarsat via SITA, highlighted in red the ring at elevation angle 40 degrees, a matter to which I will return later in this post.

Immediately what I have done is to prepare graphics/maps (i.e. 2D representations) showing the rings (defined as above) centred on the positions of the Inmarsat-3F1 satellite at the following times:

(A)     2014 March 07 16:41:00.000 – the time when MH370 took off from Kuala Lumpur; at that time the satellite was at a declination (equivalent to the latitude of its sub-satellite point) of about 1.2 degrees, as shown in the table given earlier in this post.

(B)      2014 March 08 00:11:00.000 – Time P, the time of the last ping receipt from MH370 by the satellite, which was then at a declination of about 0.6 degrees.

(C)      2014 March 08 01:36:00.000 – This time was chosen because it was the next instant when the satellite was crossing the equator, and so was at declination/latitude zero degrees (as was implicitly assumed in the calculations leading to the graphic/map issued by the Malaysian Government and shown at the start of this post). In the diagram on the preceding page this time is that at which the left-hand side of the yellow loop crosses the equator moving south, at a longitude near 64.45 degrees.

If we now examine the three maps on the next page we can see that the rings move southwards by a noticeable but not huge amount. In terms of steps in latitude, the two differences between the three sets of rings are close to being equal (stepping from latitudes of 1.185 to 0.589 to 0.000 degrees, each step being equivalent to near 65 km on Earth’s surface). From the table given previously, plus the graphic on the preceding page, one can see that there is also a longitude shift, but it is smaller, below about 0.1 degrees. As aforementioned, this is another source of error in the placement of the rings in the graphic issued by the Malaysian Government.

Rings centred on Inmarsat-3F1 for elevation angles of 35, 40 and 45 degrees from a platform at altitude 35,000 feet at the three times indicated 

Dplot

Precision of the locations of the rings

In my previous post I discussed how the precision of the time delay measurements leading to the ring locations might be estimated, in the absence of further information, and I came up with an answer of about one millisecond at the most. A blog to which I have been contributing (but see also this related blog) has rendered the information that the precision is ±300 microseconds (0.3 milliseconds). I am gratified!

There is an implication of this, in terms of interpreting the above rings. An uncertainty of ±0.3 milliseconds translates into an uncertainty of about two degrees each way in the rings above; that is, the dark blue ring at 40 degrees should be interpreted as meaning that it is likely (based only on the time-delay information from the Inmarsat-3F1 ping) that MH370 was located within a band at 38–42 degrees elevation angle (i.e. two degrees each side of that dark blue ring) at the time of the central map above. It should be easy to visualize this based on the three sets of rings being spaced by five degrees each.

 

6 thoughts on “The Locations of Inmarsat-3F1 when Pinging MH370”

  1. Mr. Steel,
    Please excuse my ignorance. I am a recently retired airline pilot with over 22,000 hours, all in swept wing aircraft ranging from Navy fighters to flying Delta’s international routes. Obviously, the MH 370 flight has captured my interest. Also, CNN’s airline pilot experts have been nothing short of an embarrassment.
    As it stands right now there seems to be confusion as to whether the aircraft flew around the north end of Sumatra.
    I won’t bore you with the details, but I am intimately familiar with the integration of the Flight Management System, Mode Control Panel and autopilot. Human intervention is the only possible explanation. However, evasive radar maneuvers seem to be at the crux of the aircraft’s actual route of flight.
    Only you can answer my question regarding the Inmarsat “arcs”. I’ve been trying to reverse engineer the aircraft’s flight path based on the current position of the four recent sonar ping locations. If one were to draw a line from that point on that last handshake arc through the tangents to the five previous handshake arcs wouldn’t that give investigators a firm answer as to where the aircraft turned south?
    Thank you for your help.
    Sincerely,
    PJ Falten

    1. Thanks PJ, your expertise welcome.

      Others will also likely comment on what you have written, but I’d just note that what you ask at the end: (a) Is based on the belief that the aircraft did indeed crash in that location, and as of yet there is no definitive evidence of that available (i.e. you are accepting media reports and the comments of the Australian PM: none of which are acceptable as definitive evidence!); and (b) Beware that in trying to “reverse-engineer” something you are not actually applying a circular argument!

      That sounds negative and dismissive when I re-read it, but I am just trying to countenance caution in jumping to end points without carefully pondering what has gone into one’s analysis. Your input invaluable, and I encourage you to keep at it; just with continued self-assessment of the assumed input parameters.

  2. The orbit of the INMARSAT satellite (inclined at 1.66° to the equator) which leads to a latitude-dependent Doppler shift with a 24 hour period, the velocity and orientation of the plane (the moving platform Doppler shift is a function of position, speed and heading), and the short-term stability of the carrier frequency from the plane’s transmitter, including considering possible variations in the power supply voltage. There will be a cluster of velocity-dependent positional solutions for each ping, which can be evaluated for a unique consistent solution for a plane’s route, if one exists. If the transmitted frequency varies at source and/or if the plane’s route and speed varied erratically then there may be no unique solution.

    1. Thanks for your comment.
      However:
      (1) The satellite does not have a 24-hour period; geostationary [sic] satellites have orbital periods of one SIDEREAL day.
      (2) In my latest post (as I write) I have given Doppler shifts for a range of southerly (speed-dependent) aircraft routes, and those shifts include the actual shift component due to the satellite at the times in question.
      (3) Yes, we know that we could likely solve for the aircraft’s actual route if we had available the ping time delays (and the measured Doppler shifts at those times would be helpful, although they can be deduced from the graph issued on 24th March). But those time delays have not been made available, and this is disgraceful behaviour on the part of all concerned.

      Duncan Steel

  3. It occurs to me that the IOR Inmarsat S/C, with a significant inclination, could provide the means to solve for a crude absolute path, not just hemisphere ambiguous LOPs and Doppler. If one used a single “Grand Solution” for all the equations of motion, for all 12 observation sets, in a LSF/Kalman filter type solution, you get math not unlike what is used for older LEO navigation satellites like Transit. IOW…A single moving satellite is essentially like having 12 quasi independent satellites from which we can triangulate. Oversimplified, but it helps to visualize what is going on. Of course, the geometric dilution associated with a geo s/c moving a few degrees is much greater than what you get with a 15 minute Transit pass, but potentially informative. Note that this is apparently the information used by Inmarsat to conclude the aircraft went south, not north. But there may be more that can be squeezed out.

  4. regarding the 300 us radio spec…I am pretty sure that this uncertainty is not a random number. I believe the radio delay is very deterministic, and thus constitutes another source of near constant bias error, thus a common mode error that can be eliminated. The +87 Hz pre-flight bias allows the radio bias to be removed from all the other observations.

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