Some Comments on the Final Ping of MH370 by the Inmarsat-3F1 Satellite

Some Comments on the Final Ping of MH370 by the Inmarsat-3F1 Satellite

Duncan Steel, 2014 March 23.
duncansteel.com

Various bloggers have commented on the accuracy (or otherwise) of the arcs said to represent the range of possible positions of MH370 at the time of the final ‘ping’ response received by the geostationary satellite Inmarsat-3F1 at 00:11 UTC on 2014 March 08. In trying to cast some further light on the situation, I have looked into this as described below. I will refer hereafter to that time (assumed rightly or wrongly to be at 00.0 seconds) as being Time P.

I have seen various useful mash-ups made using Google Earth, but that tool employs a spherical Earth model. The difference between the equatorial and polar radii of the Earth is about 21.4 km, which one might consider (again, rightly or wrongly) to be significant here. In my analysis I have used the STK software produced by the wonderful people at Analytical Graphics, Inc., and that incorporates a proper geoid (generally one uses WGS84 within STK).

Next, the writers of numerous reports have assumed that since Inmarsat-3F1 is a ‘geostationary’ satellite it follows that at the time in question the satellite was directly above the equator at its nominal altitude of 35,780 km and longitude 64.5 degrees. It wasn’t. It was more than 400 km from that point, according to my calculations. So, let me explain a few background matters regarding the orbits of ‘geostationary’ (GEO) satellites.

Notes on Geostationary Satellites

First, it is common for people to say that GEO satellites are in 24-hour orbits. They are not. The length of a mean solar day is 24 hours. A GEO orbit needs to have a period equal to the length of time it takes our planet to spin once on its axis, and that is called a sidereal day, and it lasts for a shade less than four minutes below 24 hours. You will often see GEO orbits quoted as having periods of 1436.1 minutes; that’s a sidereal day (to one decimal place).

Second, orbital insertion vagaries, gravitational tugs from the Moon and the Sun, the Earth’s non-uniform gravitational field, solar radiation pressure, solar wind effects and so on all contribute to any GEO satellite not remaining constantly above the equator. If the satellite remained in the equatorial plane then we would say it had inclination zero. In fact Inmarsat-3F1 is listed in various places as having an inclination of about 1.7 degrees. This causes it to wander north and south of the celestial equator on a daily basis.

Third, similar perturbations as those listed in the preceding paragraph cause the shape of a GEO satellite’s orbit to deviate from the ideal circle (meaning eccentricity 0.0; at Time P the orbital eccentricity of Inmarsat-3F1 was 0.0005266). This causes a satellite to shift in a direction parallel to the celestial equator, moving ahead when near perigee and then slipping back again when near apogee.

Those second and third points combined lead to GEO satellites wandering daily about an average point, performing loops as observed from the ground that resemble figures-of-eight aligned with the equator.

The Location of Inmarsat-3F1 at the Time of the Final Ping

At Time P the Inmarsat-3F1 satellite had an orbital inclination of 1.6346 degrees. Further, at that instant Inmarsat-3F1 was at a declination of +0.589 degrees (i.e. the sub-satellite point was at latitude 0.589 degrees north).

Turning to the effect of the eccentricity, at present the perigee altitude of this satellite is 35,763 km, and its apogee altitude 35,809 km. At Time P its altitude was 35,793 km. Rather than the nominal longitude of 64.5 degrees, at that instant the sub-satellite point was at a longitude of 64.47 degrees.

The combined effect of the above figures is that at Time P Inmarsat-3F1 was located about 433 km above (north of) the celestial equatorial plane, and the sub-satellite point in the Indian Ocean was about 65 km north of the equator. If my figures are correct then any plots showing the arcs of possible locations of MH370 at Time P (the time of the final ping response) which have those arcs being symmetric about the equator are wrong. Whether they are significantly wrong is another matter.

The graphic below shows a general 3D view of the geometry at the time of the final ping. Note that I have illuminated the entire Earth equally (i.e. I have not shown the night-time, the terminator, etc.). Note also that the sub-satellite point (the yellow dot) is clearly north of the equator.

F4rr

Drawing the Arcs/Circles at Equal Ranges from the Satellite

I am assuming that all readers are familiar with the graphic issued by the Malaysian Government showing a range of arcs/circles centred on the sub-satellite point, and the arcs highlighted thereupon in red being parts of the circle pertaining to an elevation angle of 40 degrees for a sightline to the satellite. Apparently this was based on data supplied by Inmarsat.

As I write this I am in ignorance of the following matters:

(1)    Whether the sub-satellite point shown in that graphic is the true point at Time P or simply a point on the equator at longitude 64.5 degrees;

(2)    Whether the true altitude above mean sea level (MSL) at Time P was used for the satellite, or the indicated altitude of 35,800 km was used in the calculations resulting in those arcs/circles;

(3)    Whether the geometry used approximates the Earth as being spherical, or a true geoid such as WGS84 was employed;

(4)    Whether the circles/arcs pertain to locations on the surface (and, if so, whether they pertain to heights according to WGS84, MSL, or an assumed sphere) or they refer to loci based on some assumed altitude for MH370 (and, if so, which altitude was used).

Obviously it is best to spell out precisely what one is doing/has done, and so I state here just what I have assumed in my own calculations using STK and other simple tools.

  • I have assumed MH370 to be at a cruise altitude of 35,000 feet (10.668 km) above MSL.
  • I have used the true sub-satellite point (latitude 0.589 degrees north, longitude 64.47 degrees east) at Time P.
  • I have drawn three rings (as shown in the preceding graphic) that pertain to the apparent elevation angle of the Inmarsat-3F1 satellite being 35, 40 and 45 degrees as viewed from a putative platform at altitude 35,000 feet above MSL.
  • I have assumed that times-of-flight for the radio signals can be derived directly from the relevant ranges (i.e. no significant ionospheric retardation at the frequencies used).

Note that these rings will not be true circles, because Earth is not spherical. These rings/curves are shown more clearly in the following 2D/map view, also produced in STK.

F1rr

The Range of the Ranges to MH370

I must now admit to another piece of ignorance: I do not know precisely how the arcs at elevation angle 40 degrees shown in the graphic issued by the Malaysian Government, and repeated in hundreds of media sites worldwide, were decided upon. Background knowledge and understanding led me immediately to them being derived from time-delay data (rather than direct measurements of angle), but whether that is through simple time-of-flight ping returns from a transponder on MH370 or through a range determination from TDMA offsets from the centres of allocated time slots I do not know; having worked with Link 16 and the like in the past I tend towards the latter (TDMA offsets).

Regardless, there are things that can be said about the 40 degree arcs even if one does not know in detail how they were derived. Where I live (in New Zealand) one often sees in the local newspapers items that state things such as “the accident occurred 32 kilometres north of Las Vegas.” What actually happened is that a road accident took place and the police spokesman in reporting to the local media said that it was about 20 miles of out of town. In NZ the reporters have converted a one-figure estimate (about 20 miles) to two-figure accuracy (32 km). In fact if you went and measured the distance from the Las Vegas city limits you might find that it was 15 miles, or perhaps 25; those are both “about 20” in this context.

So, the 40-degree elevation angle. I would, based on being a physicist, anticipate that a single time-delay datum was available, and when converted into a distance from the satellite based on time-of-flight it rendered a value corresponding to a position on-or-near Earth’s surface for which the elevation angle was calculated to be somewhere between 35 and 45 degrees, and so a nearest-round-figure value of 40 degrees was given for it. This may well have represented good sense on the part of the person or persons who derived it, as they recognized the inherent uncertainties in the derived figure.

As to whether it is based on a spherical Earth or the WGS84 geoid, sea-level or a jetliner cruise altitude, I know not. What I do know is that the many maps showing the arcs and claiming that MH370 must have been somewhere on those lines at that time were ill-informed.

In my maps/graphics I’d say that, following my interpretation as above, it seems likely (based solely on the information available from Inmarsat) that at Time P the aircraft in question was somewhere between the green and the light-blue curves, of course with additional limits being imposable by other factors such as the maximum speed of a Boeing 777.

How Precise is the Range Measurement?

Various people have asked somewhat plaintively for information about the precision of the range measurement: however that time delay is measured, how accurate/precise is it?

I would argue that my discussion above enables one, in principle, to reverse-engineer the precision (or imprecision = uncertainty) of the time delay measurements.

If one looks back at my first graphic you will see that there are green, dark blue and light blue dots on the far-eastern edges of their respective curves (elevation angles of 45, 40 and 35 degrees respectively). These dots/positions I inserted by finding, through trial-and-error within STK, the latitudes and longitudes of the points (at altitudes of 35,000 feet) which have such elevation angles and also lie at an azimuth of 90 degrees measured from the sub-satellite point in the Indian Ocean (the yellow dot) and as viewed from the actual satellite far above. These three points I labelled Q, R and S (green, dark blue, light blue). Relevant information pertaining to them is as follows.

Point

Latitude (degrees)

Longitude (degrees)

Elevation Angle to Satellite (degrees)

Azimuth from Sub-Satellite Point (degrees)

Range from Satellite (kilometres)

Q

0.757

103.324

45

90

37,411

R

0.810

107.809

40

90

37,780

S

0.877

112.346

35

90

38,181

 

The essential outcome required here derives from the final column. The differences between the ranges are 369 km (R minus Q) and 401 km (S minus R). A differences in range of 400 km translates into a difference in time-of-flight of 1.33 milliseconds. Thus I would imagine that the precision of the time delay measurements, however they are made, is about one millisecond; and the range determinations are good to a few hundred kilometres.

 

One thought on “Some Comments on the Final Ping of MH370 by the Inmarsat-3F1 Satellite”

  1. The ranges are calculated using the free space propagation time delay between the spacecraft L band antenna and aircraft L band antenna. There are a number of corrections and bias sources that need to be removed, including:

    1. C band up and down link delays
    2. s/c transponder delays
    3. aircraft radio ping response delay (<300 usec by spec)

    The good news is that all these bias delays can be accurately estimated by exploiting the pre-flight knowledge of what the total delay was while not moving on the airport ramp 1 minute before takeoff. From that one ACARS transmission, and precise knowledge of the s/c orbit, the bias can be removed for all positions.

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