The Range of Equal-Time-Delay Ping Rings from Inmarsat-3F1
Duncan Steel, 2014 March 28.
duncansteel.com
As promised (threatened?) in previous posts here and elsewhere I have now used STK plus my own software (simple sums involving planar and spherical trigonometry) to derive sets of ping rings from the Inmarsat-3F1 satellite based on equal time delays (and therefore equal ranges from the satellite to the aircraft) using a proper Earth geometry.
Some explanatory words are needed, obviously, and I adopt a scheme of working through the problem in the order in which I attacked it.
Step 1: The ping time delay and so range at 00:11 UTC on 2014/03/08
First, I do not know the actual ping time delays (and hope that Inmarsat will release them soon), and therefore I must try to back-engineer them. In this previous post I have shown the graphic issued by the Malaysian Government on March 15th and commented on why it is wrong in detail: the satellite was not directly above the equator, and was not at an altitude of 35,800 km, and was not at the nominal longitude of 64.5 degrees. Nevertheless the Inmarsat engineers apparently did an analysis based on the satellite being located in that position, and came up with an elevation angle from the aircraft to the satellite of 40 degrees on that basis. That datum (40 degrees) allows me to estimate the ping time delay and so the range (satellite-aircraft) at that time (00:11 UTC on 2014/03/08).
I assume (rightly or wrongly: if I am wrong then someone else can pick up the thread and put this bit right, and that is why I state my assumptions) that the Inmarsat engineers themselves derived their 40 degree range ring based on the aircraft being above the equator (rendering a specific value for the Earth radius of 3,678.137 km) and at an altitude of 35,000 feet (10.668 km). The satellite distance from the Earth’s centre was taken to be (35,800 + 6,378.137 km), and it’s then a simple problem of solving a plane triangle with the large angle being 130 degrees = 90 + 40 degrees. However, I would like to see how the results vary with the angle and so I solve the triangle for integer values for that elevation angle from 35 to 45 degrees. Here are the results:
|
Elevation angle aircraft to satellite (degrees) |
Range aircraft to satellite (km) |
Speed of light time delay (milliseconds) |
|
35.0 |
38187.733 |
127.381 |
|
36.0 |
38105.002 |
127.105 |
|
37.0 |
38023.504 |
126.833 |
|
38.0 |
37943.258 |
126.565 |
|
39.0 |
37864.281 |
126.302 |
|
40.0 |
37786.588 |
126.042 |
|
41.0 |
37710.196 |
125.788 |
|
42.0 |
37635.120 |
125.537 |
|
43.0 |
37561.376 |
125.291 |
|
44.0 |
37488.978 |
125.050 |
|
45.0 |
37417.941 |
124.813 |
Note that I have assumed that the radio signal propagates back-and-forth between the satellite and the aircraft at the speed of light (i.e. I am assuming no ionospheric retardation at the radio wavelengths used). The time delay given is for one leg only (i.e. up, or down); the accumulated time delay would be twice the figures given, considering only time-of-flight of the signals.
The essential result needed here is the aircraft-satellite range. For an elevation angle at the nominal 40 degrees precisely the result is 37,786.588 km. That is the value I will use in subsequent parts of this post.
However, all physical measurements have some uncertainty. I am informed – although I cannot be sure of this – that the uncertainty in the time delay measurements is ±0.3 milliseconds. Looking at the final column in the table above that would encompass elevation angles a bit below 39 degrees, and a bit above 41 degrees. Taking into account that there must have been some rounding done in arriving at a value of 40 degrees, one might take the range of possible elevation angles to be 38 to 42 degrees, and a range of ranges as 37,943 down to 37,635 km. On the other hand, this pertains to an uncertainty of 0.3 milliseconds applied to a one-leg time delay rather than the two-leg roundtrip. Regardless, I will take the above values (the range being between 37,635 and 37,943 km) as being the limits to be employed.
Step 2: Calculating equal-range rings
In my last post here I showed ping rings that pertain to equal elevation angles of 40 degrees from Earth’s surface (shaped as the WGS84 ellipsoid) to two satellites: one, the actual Inmarsat-3F1 satellite at the time in question time (00:11 UTC on 2014/03/08), and the other a fictional model geostationary satellite I called GEOSAT, located directly above the equator at longitude 64.5 degrees and altitude 35,800 km (i.e. apparently the location of the assumed satellite position in the ‘March 15th graphic’). The intent there was to illustrate the difference in the satellite positions and the resultant ping rings.
Having shown that the mauve/purple ring for the imaginary GEOSAT is wrong, I can now drop it, leaving only the yellow equal-elevation-angle-40-degrees ring for the actual Inmarsat-3F1. Here as a reminder is the ring:
This yellow ring is an equal-elevation-angle ring, whereas what we want is an equal-range (equal time-delay) ring. In order to compute the location of an equal-range ring this is what I did.
I used the known position of the satellite in question, as given in an earlier post:
I then (within my software code) figuratively introduced a platform at latitude λ and altitude 35,000 feet above the WGS84 ellipsoid (the distance from Earth’s centre of which varies with latitude) and, starting from the above longitude, I stepped this platform eastwards in steps of 0.001 degrees until such time as the calculated range from the satellite to the platform edged above the nominal range (i.e. 37,786.6 km derived from the 40-degree elevation angle). This gave me the longitude β at which the range achieved that value at latitude λ. I did this in one-degree steps of latitude from λ=35 degrees South to 35 degrees North, thus deriving values of position (λ, β) all at altitude 35,000 feet above the WGS84 geoid, which I take to approximate mean sea level.
The above involved calculating the Earth radius at each latitude (simple because it’s just the geometry of an ellipse); at latitude 35 degrees that radius (Earth centre to surface) is 6,371.078 km, more than 7 km less than at the equator (which is why the top of Mount Everest is not the point on the Earth’s surface furthest from the centre!).
A slightly more complicated calculation involving a spherical triangle to get the longitude values then followed, but it’s only a few lines of code.
This then results in 71 positions, which are shown as green dots in the following 3D view:
Step 3: Discussion of the yellow and green rings
Let me next show a close-up view of this 3D graphic near the equator so as to elucidate matters:
A couple of comments on that graphic above: (a) One can see that the yellow equal-elevation-angle line as depicted within STK is actually a polygon rather than a smooth curve; and (b) The green dots showing equal-range positions lie slightly to the east of the yellow curve. In part this is because the yellow curve is located on Earth’s surface (the WGS84 ellipsoid) whereas the green dot positions are for a platform at altitude 35,000 feet above that surface; this is shown clearly in the following graphic, which is an oblique view obtained within the STK scenario’s 3D window:
In order to show the gap between the yellow line and the equivalent line delineated by the green dots I shift into the STK 2D window, which has the effect of providing a mapping with no parallax. Here is the resultant graphic/map, with the green dots connected with straight green lines:
Step 4: How far out might the green (equal-range) ring be?
The preceding map shows a part of the rings near the equator. Because the flight path of MH370 is believed to have taken it southwards, into the Indian Ocean, I will shift out there (although be aware that all this is just for illustrative purposes).
The question posed above as the title of this analysis step pertains to the potential error in the equal-range (green) ring. How far off this ring might MH370 have been at the time in question, given the uncertainty in the ping time delays?
In order to address this question, this is what I did. In Step 1 above I noted that, so far as I am aware, the uncertainty in the ping time delays might be such that the elevation angle could be between 38 and 42 degrees, and thus the range (satellite-aircraft) could between 37,635 and 37,943 km. I went back to my software code and re-computed values of latitude and longitude (λ, β) for assumed elevation angles of 38, 39, 40 (the nominal value producing the green dots), 41 and 42 degrees, and thus the associated range values as given in the table in Step 1.
Rather than festoon a map with a huge number of new dots/spots/positions I present below only the values for a latitude of 15 degrees south. The green dot at this latitude is as before, of course, for elevation angle 40 degrees; now, though, there are four red dots added associated with (from right to left) elevation angles of 38, 39, 41 and 42 degrees. I have put back in the mauve line for the fictitious GEOSAT satellite, so as also to show how it relates to the other rings.
It is obvious that the spread of possible locations of MH370 (at that time) is dominated by the accuracy or inaccuracy of the ping delay time measurements rather than the refinement of the simulation geometry: the red dots are spread wider than the gap between the yellow line (or the mauve line) and the green line/dots.
(Below) The region around 15 degrees South from the STK 3D window
(Below) The region around 15 degrees South from the STK 2D window
Conclusion
If the ping time delay uncertainties are as much as ±0.3 milliseconds then that uncertainty/imprecision limits the determination of the location of MH370 at the times of the pings via Inmarsat-3F1. Under that circumstance the real shape of the Earth makes only a minor difference to the ping ring determinations, in a comparative sense. If, however, the ping time delay uncertainties are rather smaller, as some have suggested to me, then analysis of the ping time delays aimed at deriving possible routes taken by MH370 must include the detailed shape of the Earth (plus, of course, the locations of the satellite at the times of the pings).
Hi Duncan.
Would you consider releasing your code you used to calculate the green dots along the 40 degree last ping arc?
Or releasing a spreadsheet of lat/lon values for the green dots?
My plan is to take the last known radar contact position and then create a circle around that with a radius equal to the maximum distance the aircraft could’ve flown from the last radar contact until the time of the 1st ping after that.
Then take each of your green dots and for a range of assumed ground speeds and assuming a great circle route work backwards to see which end up back inside the circle above.
So for example assuming a ground speed of 400kts that would give a range along the last ping arc showing which would result in a great circle route starting inside the circle.
Then I’d like to try a similar exercise using the same ground speed ranges but with a range of magnetic headings, e.g. from 210 degrees to 150 degrees taking into account magnetic variation along the route.
Lastly once/if Inmarsat release all the ping data then the starting circle can be used to place northern and southern bounds to the first ping arc after the last radar contact.
Thanks
While I agree with Duncan with regards to the ping radius, in that it is not materially affected by altitude, I would point out that altitude changes would have a significant impact on the Doppler values.
Basically, a sudden dive of X km/hr would contribute to the doppler shift much more than a radial movement at X km/hr.
The extent of this is not entirely clear, because I don’t fully understand the Doppler compensation performed by either the satellite or the plane. However, it seems to me that a descent to the SE could mimic a level flight due E at 0 latitude, for example.
Yes, I agree with you JS.
Perhaps the most important thing about the Doppler data is differentiating between a northerly and a southerly route (and I intend, later today, to repeat my latest analysis – Doppler shifts for various southerly routes – for possible northerly routes so as to confirm that they can be excluded on the basis of Doppler data). In the end (?!) if the ping time delays were all made available by Inmarsat then a near-unique solution for the route would be achievable, the Doppler data then being solely a check on its validity.
Best,
Duncan
Duncan,
Have you considered the effect of a drop in altitude of the plane from 35000 ft to 12000 ft causing 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. I believe, this fact is being overlooked both by Inmarsat engineers and in your simulation.
The answer to the specific question is yes. This difference is essentially the difference between the green and the yellow arcs shown in this post (i.e. http://www.duncansteel.com/archives/419 ). Yes, the yellow one applies to sea level whilst the green one applies to 35,000 feet, but clearly the equivalent arc for 12,000 feet will be between the two (and thus not much different from either!) Hope that clarifies it.
That is, the aircraft altitude makes very little difference in this situation (location of the ping arcs), because the satellite altitude is so much greater.
Regards,
Duncan
Hi//
Great analyses. My concern is timing and time base.
Does anyone have the doppler frequency data? Leslie mentioned a , ‘…table of burst frequency offsets…’. Could you point me towards that?
Where did the +/- 0.3 ms error come from? Where is it measured from? Is it measured from the satellite to the aircraft? From the aircraft, through the satellite, to the GES? Round trip?
How vital is the satellite’s time base to these analyses? The satellite was launched in April of 1996, which probably means 1995 or earlier technology. These were not the golden days of lateration.
I would like to know the ping time and accuracy between the satellite and the aircraft – and how that accuracy is determined.
best regards,
Bob
Sir – amazing work. The satellite information you’ve provided previously is what I needed to at least get a sense of how many routes would be allowed by the Doppler data.
Minor (or not?) item – you list the Earth’s radius in the 3,600 km range. Is that a transposition error, and did it make it into the final output?
Please keep up the blog – this is fascinating for so many of us who are technically skilled, but not astronomers or aviators. Your site is one of the few with actual scientific data and insight that we can use. Thanks again.
Since shortly after the ping arcs of the last ping was released, I was the first one to strongly advocate looking at all the previous ping arcs, together with speed vectors and time, to reduce the size of the search area (see history avherald.com). I kept on insisting that they not only release all the ping data but also verify the ping arcs with other known 777 a/c in the region.
Then, finally, the information was released that they also have data for burst frequency offsets. My immidiate reaction to that was:
– Verify tracks based on burst frequency offsets with last radar data, return trip delays of ALL then pings, maximum range etc.
– Verify it all with real ping tracks flown with aircraft under the same conditions (satellite, equipment) as MH370 tracks
I have found your site now and am happy to see I was not alone. Still have the following questions / concerns that you might want to look into:
• Can’t we have all the data, test data and margins of error?
• Could the satellite movements / asymmetries in the satellite receivers and transmitters influence the calibration results “on the same day”?
• Did they really have “test” 777’s flying near the Southern arcs positions under exactly the same ping conditions as MH370 or any other positions with false positives?
• Why start the tracks from the last known (radar) point? It is far from sure that this has the location of the “turn”.
• According to a previous communications from Immarsat (perhaps by the CEO so probably BS) all the signals where showing the plane moving away from the satellite. But this DOES NOT match with the table of burst frequency offsets…
• What are the possible tracks with non-constant speeds, directions, and start points?
• If the plane made several turns after the loss of transponder signal, couldn’t this have happened several times after the last radar point as well?
• Doesn’t the burst frequency offsets analysis suggest a change of course about 4 hours before the end? Explain
• Why not use / publish the distance (return trip delay) at the different point as well to verify: are these also hourly and matching with the first radar points?
• Time discrepancies between burst frequencies offsets table and pings?
• What happended to the partial handshake?
What can I say? I agree, I agree, I agree. Thanks for saying all those things.
Duncan,
I have really appreciated your input on the tmfassociates.com comment thread and am using your locations and satellite velocity vectors in my modeling (although I wish I had more time to work on my model). Your analysis above is excellent and confirms my suspicion that the +/- 0.3ms error in ping response (if this is two way trip error then 0.15ms / leg) exceeds the error introduced by spherical/ellipsoid model used for earth geometry calculations. In any event, I am trying to develop a model that allows input of assumptions for average speed, instantaneous speed, instantaneous heading, altitude, average wind speed and instantaneous wind speed/direction to attempt to calculate some realistic paths.
Without the ping RTT data though, this model is likely a pointless exercise.
MH370 Doppler reconstruction
A contributor (hamster3null) on PpRuNe.org has given this link
https://docs.google.com/spreadsheet/ccc?key=0AivpyP95UrcBdHdfU0c3ZEVXMzVxSkZ4SjJiNWZrNVE#gid=0
(on the post here http://www.pprune.org/8405984-post8573.html)
Thanks Annette:
From my perspective (looking only at the satellite part of the problem) the reality is that satellite orbits are changing all the time. And I do mean all the time. The orbit is not precisely the same from one hour to the next.
What I do is to use the SGP4 integrator within STK with 60-second integration intervals. What that actually means is that the orbit of the satellite is updated for every one-minute step after the last updated orbit available (from actual measurements) within the USSPACECOM database, and that newly-determined orbit is usually less than 3 days old. Thus STK steps forward in one-minute steps from the last epoch at which USSPACECOM updated the orbit, with corrections being made for gravitational perturbations by the Moon, Sun, planets, and Earth’s non-uniform gravitational field.
And when I am calculating satellite positions within STK, any time which is not a round minute (i.e. contains seconds, or even briefer steps) has an interpolation applied to derive the elements at that precise instant.
That means that my values for the satellite are more accurate: UNLESS I have made an error somewhere (and that is not unknown!).
Regards,
Duncan Steel