Bayesian Methods in the Search for MH370 pp 2334  Cite as
Measurement Model, Satellite Communications
Abstract
The Bayesian filter discussed in Chap. 3 relies on knowledge of three probability density functions: the state prior distribution, the state stochastic model, and the measurement conditional probability density.
Keywords
Frequency Slot Satellite Communication System Round Trip Delay Frequency Compensation Aircraft PositionThe Bayesian filter discussed in Chap. 3 relies on knowledge of three probability density functions: the state prior distribution, the state stochastic model, and the measurement conditional probability density. The prior used for the analysis in this book was discussed in Chap. 4. This chapter addresses the measurement probability density and Chaps. 6 and 7 discuss the state dynamics model.
This chapter gives a brief overview of the satellite communication system and then describes the nonlinear measurement functions and empirical noise models for the timing and frequency measurements. The majority of the communications messages available were automated signalling messages, but there were also two telephone calls made to the plane that remained unanswered. The first of these is particularly important because of when it occurred. The chapter concludes with a description of the measurement model for telephony. Further details may be found in [2].
5.1 Satellite Communications System
An AES is equipped with a satellite data unit that comprises a satellite modem and auxiliary hardware and software. Transmission of data over the satellite is via bursts which are scheduled to arrive at the GES at a specified time and frequency. Communications from multiple users are coordinated by the allocation of different time and frequency slots to each user. Return channel (AES to GES) time slot boundaries are referenced to the forward channel (GES to AES) [2]. The duration of each time slot is sufficient to account for all possible positions of the aircraft with respect to the satellite. The width of each frequency slot is determined by the data rate and a guard width that accounts for possible variations in the satellite oscillator frequency and other possible frequency offsets. Frequency compensations applied onboard the aircraft (aircraft induced Doppler precompensation) and at the ground station serve to reduce the possible difference between the expected and actual frequency of the messages received from the aircraft. The onground compensation makes use of a second ground station located in Burum, Netherlands that transmits a reference signal to the Inmarsat3F1 satellite which is relayed to the Perth GES. Its purpose is to enable the receive modem in the GES to compensate for the Doppler frequency shift from the satellite to the Perth GES. This compensation process is referred to as Enhanced Automatic Frequency Correction.
After the Enhanced Automatic Frequency Correction process, the expected time of arrival of each communications burst is compared with the actual time of arrival and the difference between the two is referred to as the Burst Timing Offset (BTO). The BTO is minimised when the elevation angle to the satellite is 90\(^\circ \) and increases as the aircraft moves away from the subsatellite position. Hence, the BTO is a measure of how far the aircraft is from the subsatellite position. Similarly, the difference between the expected frequency of each communications burst and the actual received frequency is referred to as the Burst Frequency Offset (BFO). The BFO is a function of the relative speed between the aircraft and the satellite. Given that the satellite position and speed are known, the BFO provides information about the aircraft velocity vector. The BTO and BFO are logged by the ground station for every communications burst. This logging was a relatively recent addition to the ground station following the Air France 447 accident [2, 45] and was intended to assist in locating an aircraft. Statistical models for these two measurement functions are now developed.
5.2 Burst Timing Offset

\(T\left( \mathbf {x}_k, \mathbf {s}_k\right) \) is the round trip propagation delay from the ground station to the aircraft via the satellite;

\(\mathbf {s}_k\) is the state of the satellite at time \(t_k\), that is its position and velocity in three dimensions, along with the satellite oscillator’s state;

\(T^{\mathsf {nom}}\) is the nominal round trip delay;

\(T^{\mathsf {channel}}\) is a channel dependent bias term due to processing in the satellite data unit;

\(T^{\mathsf {anomaly}}_k\) is an anomaly correction term discussed below;

\(w_k^{\mathsf {BTO}}\) is a zero mean scalar noise process with statistics to be determined from measurement data logs.
Combining the nominal locations of the satellite and aircraft with the known location of the Perth ground station gives a value of \(T^{\mathsf {nom}} = {499{,}962}\,{\upmu {\mathrm {s}}}\).
There are a number of different channel types used that carry different traffic types and have different baud rates. Communications from the aircraft to the ground are typically over the R and Tdata channels with Cchannel used for voice telephone calls. Communications from the ground to the aircraft are over the Pchannel. The channel dependent calibration term \(T^{\mathsf {channel}}\) is assumed to be constant over a single flight but can vary between flights. A fixed value for each flight assessed was empirically derived by comparing the communications logs with known aircraft positions: the calculated value of \(T^{\mathsf {channel}}\) was the mean difference between the measured BTO and the expected BTO calculated using the known aircraft location. For the accident flight this calibration is only available for the time when the plane was at the tarmac and for the first half hour of flight. As such, values from the previous flight were also used in the calculation of \(T^{\mathsf {channel}}\). The majority of the messages available from the accident flight are R1200 messages for which \(T^{\mathsf {channel}} = {4{,}283}\,{\upmu {\mathrm {s}}}\).
The anomaly correction term \(T^{\mathsf {anomaly}}_k\) was empirically derived through analysis of a collection of communications logs. For some communication messages, typically during initial logon, there was a very large difference between the measured BTO and the nominal delay. Analysis showed that rather than simple outliers, these anomalous BTO measurements could be corrected by a factor of \(N\times {7{,}820}\,{\upmu {\mathrm {s}}}\) where N is a positive integer. The origin of these anomalous BTO measurements has not been fully determined, but the empirical correction time is quite close to the transmission interrupt clock period of \({7{,}812.5}\,{\upmu {\mathrm {s}}}\) and the BTO collection process contains quantisation.
The histogram in Fig. 5.2 has an underlying mean of \({10}\,{\upmu {\mathrm {s}}}\). This is due to the channel dependent calibration term \(T^{\mathsf {channel}}\) not being stationary. Over the span of a day it appears constant but in the context of the 20 flights represented in Fig. 5.2 there is a slow variation. As discussed above, different values were fitted for each flight. Figure 5.3 shows a scatter plot of the BTO errors against time. The channel dependent calibration term \(T^{\mathsf {channel}}\) was matched to the final flight before the accident flight, MH371 (and the beginning of the accident flight) and the BTO errors from MH371 on 7 March 2014 are marked as red crosses. The variation in bias is sufficiently slow that assuming it is the same for the accident flight as the previous flight is satisfactory.
5.3 Burst Frequency Offset
The Burst Frequency Offset is a function of the Doppler shifts imparted on the communication signal due to the motion of the satellite and the aircraft. The relationship is more complicated than a direct Doppler calculation because the aircraft software contains Doppler compensation that offsets the Doppler shift due to the aircraft motion. Although the aircraft attempts to compensate for its own motion, it does this under the assumption that the communications satellite is in motionless geostationary orbit and it does not include the vertical component of the aircraft velocity (which is nonzero when it is ascending or descending) [2]. Since Inmarsat3F1 is not exactly geostationary, the compensation is unable to completely remove Doppler effects. Empirical analysis of the BFO was conducted for the 20 flights of 9MMRO prior to the accident flight. This analysis used the same Doppler correction software as the 9MMRO satellite data unit to determine the expected BFO given a known reported aircraft position and velocity and compared this with the observed measurements.

\(\varDelta F_k^{\mathsf {up}}(\mathbf {x}_k,\mathbf {s}_k)\) is the uplink (aircraft to satellite) Doppler shift;

\(\varDelta F_k^{\mathsf {down}}(\mathbf {s}_k)\) is the downlink (satellite to ground station) Doppler shift;

\(\delta f_k^\mathsf {comp}(\mathbf {x}_k)\) is the frequency compensation applied by the aircraft;

\(\delta f_k^\mathsf {sat}(\mathbf {s}_k) \) is the variation in satellite translation frequency: the satellite uses a local oscillator to translate the carrier frequency of the message;

\(\delta f_k^\mathsf {AFC}(\mathbf {s}_k) \) is the frequency compensation applied by the ground station receive chain;

\(\delta f_k^\mathsf {bias}(\mathbf {x}_k,\mathbf {s}_k)\) is a slowly varying bias due to errors in the aircraft and satellite oscillators and processing in the satellite data unit;

\(w_k^{\mathsf {BFO}}\) is a zero mean scalar noise process with statistics to be determined from measurement data logs.
This function was described in detail in [2], we review it briefly and elaborate where the analysis herein makes different modeling assumptions to those in [2]. Again, the function \(h_k^{\mathsf {BFO}}\left( \mathbf {x}_k, \mathbf {s}_k\right) \) is assumed to be deterministic, so the measurement variance is the variance of the noise term \(w_k^{\mathsf {BFO}}\). This is a less reliable assumption than for BTO because the bias term \(\delta f_k^\mathsf {bias}(\mathbf {x}_k,\mathbf {s}_k)\) changes. To compensate for this, the measurement variance was inflated from the empirically derived \(w_k^{\mathsf {BFO}}\) variance.
The satellite translates the frequency of messages using a local oscillator that is maintained in a temperaturecontrolled enclosure to improve its stability. During eclipse periods, when the satellite passes through the Earth’s shadow, the satellite temperature drops, resulting in a small variation in translation frequency [2]. An eclipse period occurred during the accident flight and some of the validation flights were also affected by eclipses. The oscillator temperature also varies with time of day as the satellite orientation to the sun changes and as the temperature control system applies its controls. All of these thermal effects are included in the term \(\delta f_k^\mathsf {sat}(\mathbf {s}_k)\). The specific details of the functions that define \(\delta f_k^\mathsf {sat}(\mathbf {s}_k)\) and \(\delta f_k^\mathsf {AFC}(\mathbf {s}_k) \) are proprietary of Inmarsat.
The variations in bias shown in Fig. 5.4 happen over a timescale of minutes rather than hours. In the accident flight the available BFO values are generally at least an hour apart. This is a relatively long time compared with the correlation structure of the error, so the model does not use a coloured noise model for the BFO. However, the drift of the BFO bias means that it is not sufficient to assume that \(\delta f_k^\mathsf {bias}(\mathbf {x}_k,\mathbf {s}_k)\) will be the same in flight as on the tarmac before takeoff. The potential variations were incorporated by modeling the BFO bias as an unknown constant with a prior mean given by the tarmac value and a standard deviation of 25 Hz. Since the BFO measurement Eq. (5.6) is linear in the bias its distribution conditioned on the other states can be estimated with a Kalman filter. This is the RaoBlackwellised particle filter described in Sect. 3.3.
Statistics of BFO errors for 20 flights of aircraft 9MMRO prior to MH370
Mean BFO error (outliers included) (Hz)  Standard deviation of BFO (outliers included) (Hz)  Mean BFO error (outliers excluded) (Hz)  Standard deviation of BFO (outliers excluded) (Hz)  

Including tarmac  0.2246  4.9592  0.2745  4.0192 
Inflight only  0.1079  5.4840  0.1755  4.3177 
5.4 CChannel Telephone Calls
There were two unanswered telephone calls from the ground to MH370 after the loss of radar data. These communications use the Cchannel and result in measurements of BFO but not BTO. Initially the Cchannel data was not included in the flight prediction but analysis from DST Group highlighted that the first of these calls provides critical information. The first call occurred from 18:39:53 to 18:40:56 and is important because the measured BFO is significantly different from the BFO on the R1200 measurement preceding it at 18:28:15. The R1200 BFO value is consistent with the speed and direction of the aircraft while under radar coverage whereas the later Cchannel BFO value is not. Assuming that the change in BFO implies a turn, the difference between the BFO predicted by using a MATLAB model of the SDU software^{1} and the measured BFO on the Cchannel was analysed as a function of postturn direction and for a range of aircraft speeds and turn times between 18:28:15 and 18:39:53. Figure 5.6 shows the residual error and it clearly demonstrates that only Southerly track angles are consistent with the Cchannel measurements. The model predicted BFO values of Northerly paths are more than 10 standard deviations away from the measured BFO.
5.5 Information Content of Measurements
The diagrams show that the BTO provides reasonable localisation along a circle of a given range from the satellite. The information provided by the BFO is less precise, providing information on speed, with standard deviation on the order of 50 kn, and direction on the order of 40\(^\circ \).
Footnotes
 1.
Note The difference between the MATLAB model output and the SDU software output was found to be inconsequential to this analysis for determining \(\delta f_k^\mathsf {comp}(\mathbf {x}_k)\) in the SATCOM system model.
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