Christmas Island Revisited

Christmas Island Revisited

A great deal has been written about the terminus of MH370 by "experts" of all persuasions, some very good and some not so good. The prevailing theory among the cognoscenti is that the plane travelled pretty much directly South into the Southern Indian Ocean, SIO.  Problems with this scenario arose almost immediately. There is no plausible motive or causation hypothesis, and no debris from the aircraft has been found there. That did not deter the expenditure of tens of millions of dollars searching the ocean floor in that region. As we all know, nothing has been found. What most people do not realize is that the SIO hypothesis is flawed - the major flaw is the assumption relative to the fixed AP flight dynamics after the turn South. This turn is often referred to as the final major turn, FMT.

This fixed AP flight dynamic assumption has taken on a life of it's own. Too much has been invested to retreat from it. A few people, Dr. Victor Iannello and Jeff Wise among them, have proposed viable alternatives. However, they have not been seriously regarded by the people advising or conducting the search operation. The reality is the plane could be virtually anywhere on the last range ring. A competent analyst can create a set of flight dynamics to support a location North or South of the equator. While these flight paths might seem contrived, they are certainly not far removed conceptually from the flight dynamics used by the ATSB and "consensus" IG.  The observables suggest that MH370 was actively piloted prior to the FMT.  To assume that everything was "hands off" after the FMT is logically unsustainable (for me).

I have taken a somewhat more holistic approach to the problem. An approach that invokes the notions of motive, lack of radar sightings, and lack of debris. When one considers the totality of information, some of it nuanced, it becomes clear that the SIO terminus is tenuous at best. Neither pilot displayed any suicidal tendencies. Suicide is usually not the result of a spontaneous decision. It evolves over a long time period (i.e. the German Wings case), and warning signs are almost always evident.  It is more likely that Captain Shah simply wanted to make a political statement or negotiate a political concession from the Malaysian government. Shah had several landing locations available with a more Northern flight path depending on the status of "negotiations" going on in Malaysia between the government and his co-conspirators - Banda Aceh, Cocos, Christmas Island, Bandung,... Certainly flying into the SIO toward the current search area provided no opportunity to land the aircraft. Flying along the Southern coast of Sumatra/Java offers a diverse set of landing opportunities.

With that as background, I have constructed a revised flight path below which satisfies all the BTO and BFO data after and including the 19:40 handshake. The revisions from earlier postings are based on:

• Carefully derived BTO range rings using a WGS84 ellipsoid and correct sub-satellite centers. My original rings were made in haste using a spherical earth and centered at the nominal sub-satellite location over the equator.

• More care relative to speed and heading selection to better match the BTO and BFO data without the need to include arbitrary altitude variations.

• The conclusion that the PIC intended to land from North to South instead of South to North. This change was the result of considering surface winds from the Southwest at that time.

• The conclusion that the PIC did a flyby South of the Island to verify a clear runway before turning North for the final approach.
• It has been pointed out to me by a very sincere source that it is also possible that the PIC turned North to land at Bandung, and the intention was not to land on Christmas Island.

The Inmarsat BTO and BFO data were used in accordance with the vector diagram below.

The revised flight path is shown below along with the supporting data in spreadsheet form. The spreadsheet continues to be shown in two parts. The upper part calculates Doppler residuals based on the known observables. These residuals would be the same for any flight path. The lower part of the spreadsheet shows the aircraft location, speed, and heading used to match the residuals. The area inside the white circle will be the subject of additional discussion below.

Note that no altitude variations were used to reconcile the BFO data with the aircraft position, speed, and heading.

Edit 2/23/2016 to table below:

A 23:15 data point was added at the request of Oleksandr. No other changes were made.

In the table data above D2 refers to the aircraft and satellite Doppler, and D1 refers to the Doppler compensation produced by the AES. The heading lag is associated with the turn North and the Doppler change associated with this lag is quantified in an earlier post in this blog. A three degree per second turn was assumed with an inertial update interval of 0.25 seconds. Two values are shown for the 00:10 Doppler - one with no ROC or heading lag, and one with an ROC of +2 meters/sec OR a heading lag of 0.7 degrees (either accounts for the additional 7Hz of Doppler).

The aircraft speed at 19:40 was based on a calculation made in another post in this blog "Mid-flight Speed - MH370" below.

The conclusion is that the aircraft ran out of fuel during or shortly after the turn North, and went into the ocean almost due East of Christmas Island. Please take a look at "Coincidences - Are You a Believer ??" elsewhere in this blog.

The graphic below shows an enlarged version of the white circle area in the first graphic above.

The area inside the circle (with the exception of the 19:40 location) represents residual confusion. The radar data presented by the Malaysian government showed a last contact at a range of 200nm on a 295 degree radial. This contact was time tagged at 18:22 UTC. This location and time are not compatible with reaching the 18:25 UTC range ring location. The aircraft could not fly that fast. Other analysts have concluded that the radar data presentation was made in haste, and that the labeling was incorrect. They have used overlays to infer a last radar contact range of 250nm on a 285 degree radial toward Mekar.  An "official" correction to the radar data has never been made. It is also true that the GM400 radar at Butterworth has a manufacturer specified detection range of slightly more than 200nm, and that the radar data showed by the Malaysians did not come from a GM400 display. Until revised and officially certified data is available, I won't speculate on the flight path prior to 19:40.

A final comment is that the flight path associated with my analytics lies too far West to have been the sighting described by Kate Tee.  Kate could have seen the plane, but at a distance of some 70nm.

Of course, the usual disclaimers apply. While I believe the calculations above are correct, history has shown that making errors is virtually inevitable.

Update 20 November 2016

The flaperon found on Reunion Island has been confirmed by the French to have come from 9M-MRO (MH370). This finding and confirmation allows drift models to be created in an attempt to narrow down the likely terminus of the aircraft on the last ping ring. The most credible such model has been created by  Geomar, a Research Institute in Germany.

Dr. Jonathan Durgadoo and Prof. Dr. Arne Biastoch from GEOMAR Helmholtz Centre for Ocean Research Kiel used a state-of-the-art ocean model in combination with observational data. This provides a coherent realistic dataset for their drift analyses to determine the possible origin of the flaperon. To do so, they release virtual particles around La Réunion and compute their trajectories back in time. "Of course it does not make much sense just to track only a few particles within the model," Dr. Durgadoo explains. "We have traced back almost two million 'virtual' particles over a period of 16 months," Durgadoo continues. "For each month back, we subsequently calculated the probable region of the particles positions."

The Geomar model shows that the most likely origin of the flaperon is far to the North of the current search area, and compatible with the Christmas Island region terminal hypothesis.  The Geomar probability map is shown below. Other models such as the CSIRO model are suspect, IMO. There is virtually no probability (see map) that the flaperon originated from the current search area.

There is still no word from the French relative to the flaperon damage forensics or the pathology of the barnacles.

Mid-flight Speed - MH370

Mid-flight Speed - MH370

A mid-flight speed estimate for MH370 was made by Brian Anderson using a time of flight methodology. Brian's method can be found at the link below.

http://www.duncansteel.com/archives/1330

Since it is a slow week, and I can't stand to look at any more Maldives posts without getting upset, I decided to revisit this calculation using a different method than Brian. My method refers to the Doppler diagram below.

At 19:40 the satellite is virtually standing still at the extreme end of the "dither". V_s is essentially zero, and the Doppler shift associated with satellite movement relative to both Perth and the aircraft is negligible. All of the residual Doppler is the result of aircraft motion, D2 - D1 above.

Setting V_s = 0, V_p = V_pt, and | R_s - R_p | ~ | R_e - R_p | in the expression for D2 and subtracting D1 results in the following:

D2 - D1 ~ V_pt dot (R_s - R_e) x FL / | R_e - R_p | x C

R_s - R_e  is a vector in the North - South direction of magnitude 1206 km. 

| R_e - R_p | is 36, 738 km (by calculation from satellite to 19:40 range ring)

Using the values above and the values for FL and C we have:

D2 - D1 = (V_pt dot z) x 0.18 (approximation valid near equator)

where V_pt dot z is the scalar component of aircraft tangential (to the earth) velocity in the North - South direction.  

Using a residual Doppler value at 19:40 of 37.8 Hz (D2 - D1), and solving for V_pt dot z yields:

V_pt dot z = 210 m/s = 756 km/hr = 408 knots.

408 knots is the slowest speed that the aircraft could have been traveling to produce a residual Doppler of 37.8 Hz at the 19:40 range ring. That speed would be at a heading of 180 degrees. Headings on either side of 180 degrees would require a higher speed to produce the 37.8 Hz residual Doppler, limited by the aircraft performance envelop.

This value is in general agreement with Brian's calculation. 

Update: 08/05/2016

The graphic below shows the mid-flight speed probability distribution calculated by the DSTG and published in their book "Bayesian Methods..." Figure 5.7. The highest probability speed calculated by the DSTG corresponds exactly with the 408 knot value calculated above.

It should not be inferred that the DSTG result and my result somehow reinforce  each other - although the most probable DSTG speed corresponding to zero BFO error at 180 degrees is identical to my result. The DSTG result is a probability distribution. My result is simply a determination of the minimum speed necessary to produce the 19:40 BFO observation. In fact, I would regard speeds below 400 knots in the above graphic not only to be unlikely, but physically impossible if the 19:40 BFO value is valid. It would require a significant bias drift to account for speeds less than 400 knots at 19:40. A vertical line at 180 degrees in the above figure would depict the probability function the DSTG used for BFO error.

Update: 9/11/16

Given the calculation of speed and heading required at 19:40 it is reasonable to ask what combination of speed and heading, if maintained, would produce an arrival time at 20:40 consistent with the distance to that range ring. The answer is about 425 knots at a heading of about 168 degrees. This answer assumes a late FMT with a latitude at 19:40 of about 8N.

It turns out this heading is directly at the Cocos waypoint. 168 degrees is also consistent with the recent work of Iannello and Godrey which postulates a terminus on the 7th arc around 27S using McMurdo Station, Antartica as a programmed destination.

Update: 9/12/16 

If one carefully extracts the likelihood relative magnitude vs speed in DSTG Figure 5.7 above it is possible to derive the BFO probability density function used by the DSTG in their modeling. This information is displayed graphically in the figure below. The data extraction yields a standard deviation (sigma) of approximately 5Hz. Track maintained at 180 degrees true. Speed is ground speed.

BFO Errors (again)

BFO Errors (again)

Note: The original post contained errors in the M matrix (NEU to ECEF conversion). The matrix has been corrected as of 25 September 2015)

There are many ways for errors to creep into the BFO data:

- Drift in the AES, satellite, and ground station oscillator chain (my estimate +/- 5Hz).

- Unknowable rates of climb/descent which are not included in the AES Doppler compensation algorithm (~  +/- 4Hz/m/sec).

- Heading errors (the subject of this post) - caused by sensor lag and sensor drift.

Sensor lag refers to the fact that the heading sensors, GPS or gyro, have a fixed update rate. The heading values which the AES uses to pre-compensate Doppler are never current. They will always lag the current heading by some amount depending on rate of turn (if any) and the sensor update rate.

The BFO graphic below serves as a useful starting point.

The BFO graphic includes a number of vectors associated with the locations and velocities of the aircraft, the satellite, and the ground station. These vectors are directly coupled to the problem physics. Some are known, such as the location of the LES, and some are unknown, such as the location and velocity of the aircraft. One vector, however, is estimated from sensors on the aircraft - V_pt, the velocity of the aircraft in the local tangent plane. The AES pre-compensates the L-band signal sent to the satellite by subtracting the Doppler associated with V_pt relative to the nominal position of the satellite over the equator. Vertical aircraft velocity is not included in V_pt.  Clearly, any errors associated with the estimate of V_pt will find their way into the Doppler compensation, D1. The purpose of this post is to estimate D1 errors associated with an incorrect heading.

Parameters and relationships are defined below:

The error vector contained in V_pt is quantified above. Matrix M is simply a conversion matrix used to generate velocities in the ECEF coordinate system when the NEU velocities are known. Note that most people use ENU rather than NEU. The matrix above is anticipating a velocity vector in the NEU format. 

Continuing using the D1 equation from the BFO graphic:

Note: A heading error of +1 degree corresponds to an actual heading of 179 degrees. 

The values used in the example should be familiar. The location, speed, and heading were selected to be in approximate conformity to a location and speed in the SIO favored by the ATSB and IG.

The D1 compensation is clearly quite sensitive to heading. More sensitive than I would have guessed before attempting to quantify it.  So, in addition to ROC, the BFO values will be corrupted by rate of turn (sensor lag) and sensor drift. In the ATSB and IG models the aircraft never turns (after the "Final Major Turn"), so drift is the primary consideration. 

It might be that heading errors (along with oscillator chain drift) are the major sources of error in the tracking examples in the ATSB report. 

Edit 3 February 2017

Graphic below added at request of a collaborator. It allows the effect of tracking and speed errors in the vicinity of an assumed FMT to be estimated.

Added graphic below is the "Doppler Residual", the sum of D1 + D2 in the BFO graphic above at the time 19:41 UTC. The Doppler Residual is the total Doppler due to aircraft motion relative to the satellite and the Doppler Compensation applied by the AES. It includes motion of the aircraft relative to the actual satellite position (aircraft and satellite motion), D2, and the Doppler Compensation applied by the AES for aircraft motion relative to the nominal satellite position over the equator, D1. Note longitude change to 93E.

The dashed line in the graphic above is the Doppler Residual measured by the ground station in Perth after all other sources of frequency variations have been  accounted for (AES offset, satellite motion relative to Perth, and GES correction term). The residual shown by the dashed line was derived at 19:41 UTC.